Subgroup structure of symmetric group:S6

This article discusses the subgroup structure of symmetric group:S6, which is the symmetric group on the set $$\{ 1, 2,3,4,5,6\}$$. The group has order 720.

Table classifying subgroups up to conjugacy
The below lists subgroups up to conjugacy, i.e., up to automorphisms arising from conjugation in symmetric group:S6. This is not the same as the classification up to automorphisms because of the presence of other automorphisms, a phenomenon unique to degree six (see symmetric groups on finite sets are complete).

The table needs to be completed.