Suppose $Sub(G)$ is the set of all subgroups of a group $G$. What is the asymptotic of $|Sub(S_n)|$ with $n \to \infty$?
One can get the following bounds on $|Sub(S_n)|$:
$$2^{2^{\langle \frac{n}{2} \rangle}} - 1 \leq |Sub(S_n)| \leq 2^{n!}$$
Proof:
On one hand:
$|Sub(S_n)| \leq |P(S_n)| = 2^{n!}$
On the other hand:
Suppose $k = \langle \frac{n}{2} \rangle$. Then $S_n$ has a subgroup isomorphic to $Sym(\{1, … , k\} \times \{0, 1\})$. For all $A \subset \{1, … , k\} \times \{0, 1\})$ define $\sigma_A: (p, q) \mapsto (p, q + I_A(p) \mod 2)$. It is not hard to see, that $\{\sigma_A| A \subset \{1, … , k\}\} \cong C_2^{2^k}$. And it is known, that $C_2^m$ always has $2^m - 1$ subgroups isomorphic to $C_2$.
The only information on $|Sub(S_n)|$ I was able to find online was the list of exact values of $|Sym(S_n)|$ for $n \leq 18$, calculated by Derek Holt in «Enumerating subgroups of symmetric groups».
