Subgroup structure of symmetric groups

The symmetric group on a set is the group, under multiplication, of permutations of that set. The symmetric group of degree $$n$$ is the symmetric group on a set of size $$n$$. For convenience, we consider the set to be $$\{ 1,2, \dots, n \}$$.

This article discusses the element structure of the symmetric group of degree $$n$$.

Here are links to more detailed information for small values of degree $$n$$.

Transitive subgroups
We first list key statistics on the transitive subgroups of $$S_n$$ for small values of $$n$$. Note that $$n$$ must divide the order of any transitive subgroup of $$S_n$$.

All subgroups in terms of partition of orbit sizes
For each partition into orbit sizes, the subgroups giving rise to such a partition are subdirect products of the transitive subgroups corresponding to the orbit sizes. The table below needs to be completed.