<?xml version="1.0"?>
<ownershipDocument>

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2018-07-02</periodOfReport>

    <notSubjectToSection16>1</notSubjectToSection16>

    <issuer>
        <issuerCik>0001195933</issuerCik>
        <issuerName>INFINITY PROPERTY &amp; CASUALTY CORP</issuerName>
        <issuerTradingSymbol>IPCC</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001712813</rptOwnerCik>
            <rptOwnerName>Jordan Amy</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>2201 4TH AVENUE NORTH</rptOwnerStreet1>
            <rptOwnerStreet2></rptOwnerStreet2>
            <rptOwnerCity>BIRMINGHAM</rptOwnerCity>
            <rptOwnerState>AL</rptOwnerState>
            <rptOwnerZipCode>35203</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>0</isDirector>
            <isOfficer>1</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle>Sr VP &amp; Controller</officerTitle>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <transactionDate>
                <value>2018-07-02</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>D</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionTimeliness>
                <value></value>
            </transactionTimeliness>
            <transactionAmounts>
                <transactionShares>
                    <value>300.24</value>
                    <footnoteId id="F1"/>
                    <footnoteId id="F2"/>
                </transactionShares>
                <transactionPricePerShare>
                    <footnoteId id="F2"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>0</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
    </nonDerivativeTable>

    <footnotes>
        <footnote id="F1">Includes 300.24 shares acquired through Infinity Property and Casualty Corporation's (the &quot;Company&quot;) employee stock purchase plan, including 14.88 shares acquired subsequent to August 1, 2017.</footnote>
        <footnote id="F2">Pursuant to the Agreement and Plan of Merger, dated as February 13, 2018 (the &quot;Merger Agreement&quot;), by and among Kemper Corporation (&quot;Kemper&quot;), Vulcan Sub, Inc., a wholly owned subsidiary of Kemper (&quot;Sub&quot;), and the Company, effective as of the effective time of the merger of the Company and Sub, these shares of Company common stock, no par value per share, were converted into the right to receive, at the election of the stockholder, (i) $51.60 in cash and 1.2019 shares of common stock of Kemper, par value $0.01 per share (&quot;Kemper Common Stock&quot;), without interest and less any applicable withholding for taxes, (ii) $1.29.00 in cash, without interest and less any applicable withholding for taxes, or (iii) $49.58 in cash and 1.2332 shares of Kemper Common Stock, without interest and less any applicable withholding for taxes.</footnote>
    </footnotes>

    <remarks></remarks>

    <ownerSignature>
        <signatureName>/s/ James H. Romaker, by Power of Attorney</signatureName>
        <signatureDate>2018-07-02</signatureDate>
    </ownerSignature>
</ownershipDocument>
