For any (finite) group $G$ its length $l(G)$ is the length of maximal chain of proper subgroups (it's known and pretty widely used invariant). But we can also define width function $w_G(k)$ in such fashion: $w_G(k) := \#\{H<G: l(H) = k\}$. Then we can do some adjustments — make it a function $W_G:[0, 1] \to \Bbb R$ by setting $W_G(k/l(G)) = w_G(k)$, interpolating linearly and then maybe normalizing by setting integral over $[0, 1]$ to $1$. For example, $W_{\Bbb Z/n}$ is constant and $W_{\Bbb (Z/p)^n}$ is $p$-binomial distribution.
So, my question is
What is limit of $W_{S_n}$ for large $n$ — is there some "central limit theorem"? Is it dominated by $W_{Syl_2(S_n)}$?
(Exact length of $S_n$ is known (Cameron-Solomon-Turull, 1989) and asymptotically equal to length of 2-Sylow.)