Finite Group DatabasebetaFinite Group DatabasebetaOrder 31115.
12 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C127⋊C49 | C5 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 2 | C5×C7×C127 | C7 | 1 | 1 | yes | maximal, characteristic |
| 3 | C7×C127 | C5×C7 | 1 | 1 | yes | characteristic |
| 4 | C5×C127 | C49 | 1 | 1 | yes | characteristic, complemented, Hall |
| 5 | C5×C49 | — | 127 | 1 | no | maximal, Hall |
| 6 | C127 | C5×C49 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 7 | C49 | — | 127 | 1 | no | Sylow |
| 8 | C5×C7 | C127⋊C7 | 1 | 1 | yes | characteristic |
| 9 | C7 | C5×C127⋊C7 | 1 | 1 | yes | characteristic |
| 10 | C5 | C127⋊C49 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5×C7 | C127⋊C7 |
| commutator, hypocenter | C127 | C5×C49 |
| Frattini | C7 | C5×C127⋊C7 |
| Fitting, socle, prime radical | C5×C7×C127 | C7 |
| cosocle | C7×C127 | C5×C7 |
| LCS | G ▷ C127 | |
| UCS | C1 ◁ C5×C7 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C127⋊C49 | C5 | direct (N × H) |
| 4. C5×C127 | C49 | semidirect |
| 6. C127 | C5×C49 | semidirect |
| 10. C5 | C127⋊C49 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C127⋊C49 | 5 |
| C5×C7×C127 | 7 |
| C5×C49 | 127 |