TITLE: How many non-isomorphic abelian subgroups of the permutation group $S_n$?
QUESTION [11 upvotes]: I am interested in how many (pairwise non-isomorphic) subgroups of the permutation group $S_n$ are abelian. ($n \in \mathbb{N}$ arbitrary and possibly big)
Are you aware of any references which treat this problem?

REPLY [2 votes]: This is a (cw) post to illustrate Claim 2 from the answer by Sean Eberhard. Below is the plot of the list $(\frac1n,\frac{\log f(n)}{\pi\sqrt{2/3}(n/\log n)^{1/2}})$ for $n$ from $1000$ to $10000$.

I used the list of values from OEIS A023893.
As you see after $n$ about 3000 the sequence sort of changes behavior; at about $n=5000$ (to be precise, at $5226$) it achieves a (local ?) maximum and then starts to drop with increasing speed.
It looks so strange that I suspected some error in the computation of values. But then I recomputed it myself in Mathematica using power series expansion and obtained precise coincidence.
Update
I now extended computations to $n\leqslant100000$ and it seems that the tendency continues, so that at least in the range from $10000$ to $100000$ $\log f(n)$ grows qualitatively slower than $\sqrt{\frac n{\log n}}$...

Update 2
With the constant $2\pi/\sqrt3$ instead of $\pi\sqrt{2/3}$ it is