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I would like to have a sense of the number of distinct subgroups (that is, isomorphic subgroups are counted multiple times) the symmetric group $S_n$ has as $n$ goes to infinity. I suspect nailing down the exact number is harder than understanding the general size. Can someone help me understand this?

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marked as duplicate by Community Apr 12 '18 at 23:33

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    $\begingroup$ Pyber has solved this, see this question and its answers. $\endgroup$ – Dietrich Burde Apr 12 '18 at 15:01
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    $\begingroup$ There's a general heuristicts that if you want to do some asymptotic enumeration in "uniform among primes" class of finite groups, then 2-primary part quickly become dominating and you can restrict to it. It's almost always true, and this case is not an exception. $\endgroup$ – xsnl Apr 12 '18 at 15:07