Subgroup structure of symmetric group:S5

The symmetric group of degree five has many subgroups. We'll take the five letters as $$\{ 1,2,3,4,5\}$$. The group has order 120.

Note that since $$S_5$$ is a complete group, every automorphism of it is inner, so the classification of subgroups upto conjugacy is the same as the classification of subgroups upto automorphism. In other words, every subgroup is an automorph-conjugate subgroup.

Table classifying subgroups up to automorphisms
Note that the only normal subgroups are the trivial subgroup, the whole group, and A5 in S5, so we do not waste a column on specifying whether the subgroup is normal and on the quotient group.

Table listing number of subgroups by order
Note that these orders satisfy the congruence condition on number of subgroups of given prime power order: the number of subgroups of order $$p^r$$ is congruent to $$1$$ modulo $$p$$.