Finite Group DatabasebetaFinite Group DatabasebetaOrder 31375.
20 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C5×C251⋊C5 | C5 | 1 | 1 | yes | maximal, characteristic |
| 2 | C25×C251 | C5 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #20262472 | C5 | 1 | 4 | yes | maximal, complemented |
| 4 | C5×C251 | C52 | 1 | 1 | yes | characteristic |
| 5 | C251⋊C5 | — | 5 | 1 | no | — |
| 6 | C251 | 5−1+2 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 7 | 5−1+2 | — | 251 | 1 | no | maximal, Sylow |
| 8 | C52 | — | 251 | 1 | no | — |
| 9 | C25 | — | 251 | 4 | no | — |
| 10 | C25 | C251⋊C5 | 1 | 1 | yes | characteristic, complemented |
| 11 | C5 | — | 1255 | 1 | no | — |
| 12 | C5 | C5×C251⋊C5 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C5 | C5×C251⋊C5 |
| commutator, socle, cosocle | C5×C251 | C52 |
| Fitting | C25×C251 | C5 |
| prime radical | C5×C251⋊C5 | C5 |
| hypercenter | C25 | C251⋊C5 |
| hypocenter | C251 | 5−1+2 |
| LCS | G ▷ C5×C251 ▷ C251 | |
| UCS | C1 ◁ C5 ◁ C25 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C25×C251 | C5 | semidirect |
| 3. #20262472 | C5 | semidirect |
| 6. C251 | 5−1+2 | semidirect |
| 10. C25 | C251⋊C5 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×C251⋊C5 | 5 |
| C25×C251 | 5 |
| #20262472 | 5 |
| 5−1+2 | 251 |