# Subgroup structure of symmetric group:S5

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

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.

1. The trivial subgroup. Isomorphic to trivial group. (1)
2. The two-element subgroup generated by a transposition, such as $(1,2)$. Isomorphic to cyclic group of order two. (10).
3. The two-element subgroup generated by a double transposition, such as $(1,2)(3,4)$. Isomorphic to cyclic group of order two. (15)
4. The four-element subgroup spanned by two disjoint transpositions, such as $\langle (1,2) , (3,4) \rangle$. Isomorphic to Klein-four group. (15)
5. The four-element subgroup containing the identity and three double transpositions on a subset of size four. Isomorphic to Klein-four group. (5)
6. The four-element subgroup spanned by a 4-cycle. Isomorphic to cyclic group of order four. (15)
7. The eight-element subgroup spanned by a 4-cycle and a transposition that conjugates this cycle to its inverse. Isomorphic to dihedral group of order eight. (5)
8. The three-element subgroup spanned by a three-cycle. Isomorphic to cyclic group of order three. (10)
9. The six-element spanned by a 3-cycle and a transposition disjoint from it. Isomorphic to cyclic group of order six. (10)
10. The six-element subgroup spanned by all permutations on a subset of size three. Isomorphic to symmetric group on three elements. (10)
11. The six-element subgroup obtained by taking permutations on a particular subset of size three, and multiplying this by a transposition on the remaining two elements if the permutation is odd. Isomorphic to symmetric group on three letters.
12. The twelve-element subgroup generated by the symmetric group on three letters and the symmetric group on the remaining two letters. (10)
13. The twelve-element subgroup obtained as the alternating group on four letters. Isomorphic to alternating group on four letters. (5)
14. The 24-element subgroup obtained as the symmetric group on four letters. Isomorphic to symmetric group on four letters. (5)
15. The five-element subgroup generated by a five-cycle. Isomorphic to cyclic group on five leters. (6)
16. The ten-element subgroup generated by a five-cycle and a double transposition that conjugate it to its inverse. Isomorphic to dihedral group of order ten. (6)
17. The twenty-element subgroup generated by a five-cycle and a four-cycle that conjugates it to its square. (6)
18. The alternating group on all five letters. Isomorphic to alternating group on five letters. (1)
19. The whole group. (1)