Exhibit 10.35

INDEMNIFICATION PRIORITY AND INFORMATION SHARING AGREEMENT

     This INDEMNIFICATION PRIORITY AND INFORMATION SHARING AGREEMENT, dated as of November 1, 2009 (this “Agreement”), between the management companies and other entities set forth on Annex II (the “Management Companies”), and HCA, Inc., a Delaware corporation (the “Company”).

     WHEREAS, the Company has entered into one or more monitoring, stockholder, indemnification and other agreements (any such agreement or agreements, collectively, the “Company Indemnification Agreements”) providing for, among other things, the indemnification of and advancement of expenses incurred by the funds set forth on Annex I (the “Funds”), the Management Companies, and their respective directors, members, managers, partners, affiliates and controlling persons for certain matters described therein (the Funds, the Management Companies and their respective directors, members, managers, partners, affiliates and controlling persons, collectively, the “Sponsor Indemnified Parties”);

     WHEREAS, one or more executives of the Management Companies or their respective affiliates may serve as a directors, officers or consultants of the Company and one or more other persons (who are not executives of the Management Companies or their respective affiliates) may serve as a directors, officers or consultants of the Company as an appointee or designee of the Funds or the Management Companies (any such person or persons, the “Designated Directors”);

     WHEREAS, the Designated Directors may have entered into indemnification agreements with the Company providing for indemnification and advancement of expenses for the Designated Directors in connection with their service as a director of the Company and the Designated Directors may, in their capacities as directors of the Company, be indemnified and/or entitled to advancement of expenses under the Company’s certificate or articles of incorporation, by-laws, limited liability company operating agreement, limited partnership agreement or other organizational documents (in each case, a “Company Director Indemnity”);

     WHEREAS, the Funds, the Management Companies and/or their respective affiliates and controlling persons (in this capacity, collectively, the “Sponsor Indemnitors”) have (i) entered into one or more limited partnership agreements, limited liability company operating agreements and other agreements or arrangements, (ii) certificates and articles of incorporation, by-laws, and other organizational documents and (iii) obtained insurance (any such agreements, documents or insurance, collectively, the “Sponsor Indemnification Agreements”), in each case, providing for, among other things, indemnification of and advancement of expenses for the Designated Directors designated by such Sponsor Indemnitor for, among other things, the same matters that are subject to indemnification and advancement of expenses under the Company Indemnification Agreements and the Company Director Indemnity; and

     WHEREAS, the Company, the Funds and the Management Companies wish to clarify certain matters regarding the indemnification and advancement of expenses provided under the Company Indemnification Agreements and the Company Director Indemnity as it relates to the indemnification and advancement of expenses provided for under the Sponsor Indemnification Agreements.

     NOW, THEREFORE, in consideration of the foregoing recitals and the premises hereinafter set forth, the Company, the Funds and the Management Companies hereby agree as follows.

     1. The Company hereby acknowledges and agrees that the obligation of the Company under either any Company Indemnification Agreements or the Company Director Indemnity to indemnify or advance expenses to any Designated Director for the matters covered thereby shall be the primary source of indemnification and advancement of such Designated Director in connection therewith and any obligation on the part of any Sponsor Indemnitor under any Sponsor Indemnification Agreement to indemnify or advance expenses to such Designated Director shall be secondary to the Company’s obligation and shall be reduced by any amount that the Designated Director may collect as indemnification or advancement from the Company. In the event that the Company fails to indemnify or advance expenses to a Designated Director as required or contemplated by any Company Indemnification Agreement or Company Director Indemnity (such amounts, the “Unpaid Director Indemnity Amounts”) and any Sponsor Indemnitor makes any payment to such Designated Director in respect of indemnification or advancement of expenses under any Sponsor Indemnification Agreement on account of such Unpaid Director Indemnity Amounts, such Sponsor Indemnitor shall be subrogated to the rights of such Designated Director under any Company Indemnification Agreement or Company Director Indemnity, as the case may be, in respect of such Unpaid Director Indemnity Amounts.

     2. The Company hereby agrees that, to the fullest extent permitted by applicable law, its obligation to indemnify Sponsor Indemnified Parties under the Company Indemnification Agreements shall include any amounts expended by any Sponsor Indemnitor under the Sponsor Indemnification Agreements in respect of indemnification or advancement of expenses to any Designated Director in connection with litigation or other proceedings involving his or her service as a director of the Company to the extent such amounts expended by such Sponsor Indemnitor are on account of any Unpaid Director Indemnity Amounts.

     3. The Company hereby agrees that it will not amend any Company Director Indemnity as in effect on the date hereof to alter the rights of any Designated Director in any manner that would alter any Designated Director’s rights with respect to conduct pre-dating the date of any such amendment without the consent of each of the Management Companies.

     4. The Company hereby consents to (i) the Designated Directors sharing any information such Designated Directors receive in their capacity as directors of the Company and (ii) representatives of the Management Companies sharing any information sent to such representatives by or on behalf the Company, in either case, with officers, directors, members, employees and representatives of such Management Company and its affiliates (other than portfolio companies) provided that such Management Company maintain internal procedures with respect to the use and safekeeping of such information.

     5. Except as otherwise provided herein, this Agreement may be amended or modified only by a writing executed by each of the parties hereto.

     6. The provisions of this Agreement shall inure to the benefit and be binding upon the parties hereto and (i) the provisions of Section 3 shall inure to the benefit of the Designated Directors and (ii) all of the provisions of this Agreement shall inure to the benefit of the Funds, all of whom are intended to be third party beneficiaries hereof.

     7. This Agreement shall be governed by and construed in accordance with the laws of the state of incorporation of the Company regardless of the law that might be applied under principles of conflict of laws to the extent such principles would require or permit the application of the laws of another jurisdiction. No suit, action or proceeding with respect to this Agreement may be brought in any court or before any similar authority other than in a court of competent jurisdiction in the state of incorporation of the Company, and the parties hereto hereby submit to the exclusive jurisdiction of such courts for the purpose of such suit, proceeding or judgment. Each party irrevocably waives trial by jury in any legal action or proceeding in relation to this Agreement and for any counterclaim therein.

     8. Wherever possible, each provision of this Agreement shall be interpreted in such manner as to be effective and valid under applicable law, but if any provision of this Agreement is held to be invalid, illegal or unenforceable in any respect under applicable law or rule in any jurisdiction, such invalidity, illegality or unenforceability shall not affect any other provision or any other jurisdiction, but this Agreement shall be reformed, construed and enforced in such jurisdiction as if such invalid, illegal or unenforceable provision had never been contained herein.

     9. This Agreement may be executed in two or more counterparts, each of which shall be deemed an original, and all of which together shall constitute one and the same instrument. A signature of a party transmitted by facsimile or other electronic means shall constitute an original for all purposes.

     IN WITNESS WHEREOF, the parties hereto have executed this Agreement as of the date set forth in the first paragraph hereof.

ANNEX II — MANAGEMENT COMPANIES AND OTHER ENTITIES

Kohlberg Kravis Roberts & Co. L.P.

Bain Capital Partners, LLC

Merrill Lynch Global Private Equity, Inc.

Frisco, Inc.

Frisco Partners