Number of conjugacy classes of subgroups in symmetric group is logarithmically bounded by square of degree

History
This result is due to Laci Pyber (reference needs to be added).

Statement
There exist positive constants $$A$$ and $$c$$ such that the following holds.

Consider the fact about::symmetric group (specifically, fact about::symmetric group on finite set) $$S_n$$ of degree $$n$$ (i.e., it acts on a set of size $$n$$). Then, the number of conjugacy classes of subgroups in $$S_n$$ is at most $$\! Ac^{n^2}$$.

Compare with numerical information available at subgroup structure of symmetric groups.