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    $\begingroup$ Let $G = C_{q_1} \times C_{q_2} \times \cdots \times C_{q_k}$, where $q_1, \dots, q_k$ are prime powers. Then $G$ is isomorphic to a subgroup of $S_n$ if and only if $q_1 + q_2 + \cdots + q_k \leq n$. $\endgroup$ Commented Dec 1, 2021 at 14:11
  • 8
    $\begingroup$ The number is tabulated (up to $n=10000$) at oeis.org/A023893 – no lliterature is noted. $\endgroup$ Commented Dec 1, 2021 at 22:31