Finite Group DatabasebetaFinite Group DatabasebetaOrder 15625.
43 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C625⋊C5 | C5 | 1 | 5 | yes | maximal |
| 2 | #15691073 | C5 | 1 | 1 | yes | maximal, characteristic |
| 3 | C5×C125 | C25 | 1 | 3 | yes | — |
| 4 | C5×C125 | C25 | 1 | 1 | yes | characteristic |
| 5 | C5×C125 | C52 | 1 | 1 | yes | characteristic |
| 6 | C252 | C25 | 1 | 1 | yes | characteristic |
| 7 | C625 | — | 5 | 5 | no | — |
| 8 | C5×C25 | — | 5 | 1 | no | — |
| 9 | C5×C25 | C5×C25 | 1 | 1 | yes | characteristic |
| 10 | C125 | — | 5 | 3 | no | — |
| 11 | C125 | C125 | 1 | 5 | yes | — |
| 12 | C125 | 5−1+2 | 1 | 5 | yes | — |
| 13 | C52 | C125⋊C5 | 1 | 1 | yes | characteristic |
| 14 | C25 | — | 25 | 1 | no | — |
| 15 | C25 | C25⋊C25 | 1 | 1 | yes | characteristic |
| 16 | C25 | C5×C125 | 1 | 1 | yes | characteristic |
| 17 | C25 | C125⋊C5 | 1 | 3 | yes | — |
| 18 | C5 | — | 5 | 1 | no | — |
| 19 | C5 | C25⋊C125 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C25 | C25⋊C25 |
| commutator | C25 | C5×C125 |
| Frattini, cosocle | C5×C125 | C52 |
| socle | C5 | C25⋊C125 |
| prime radical | C52 | C125⋊C5 |
| LCS | G ▷ C25 ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C5×C125 ◁ G | |
| LECS | G ▷ C5×C125 ▷ C25 ▷ C5 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C625⋊C5 | 5 |
| #15691073 | 5 |