During the Civil War, both the Union and Confederate drafts had certain exemptions, but these varied between the two sides and were subject to change.
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Immigrant men who had filed for citizenship - Generally, immigrants who were not yet citizens could be drafted in both the Union and Confederate armies, but fully naturalized citizens would not be exempt.
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Wealthy men - Wealth could provide a way out of the draft, as wealthier individuals could pay for substitutes or buy their way out of service in both the Union and Confederate armies.
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Married men - Marriages provided some exemption from the draft, especially early in the war. However, policies varied and more married men were called as the war progressed.
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African American men - African Americans were not eligible for the draft in the Confederacy, and while the Union initially did not draft African Americans either, they eventually enlisted as soldiers.
Therefore, while the exemption could apply to wealthy men or married men based on specific provisions, the broadest exemption from both drafts would apply most clearly to wealthy men, as they had more financial means to avoid combat duty. However, it's important to note that the situation was complex, and the specifics of the draft laws could change throughout the war.