Finite Group DatabasebetaFinite Group DatabasebetaOrder 31510.
16 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C5×C23×C137 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | D23×C137 | — | 5 | 1 | no | maximal, Hall |
| 3 | C23×C137 | D5 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | D5×C137 | — | 23 | 1 | no | maximal, Hall |
| 5 | C5×C137 | D23 | 1 | 1 | yes | characteristic, complemented, Hall |
| 6 | C2×C137 | — | 115 | 1 | no | Hall |
| 7 | D115 | C137 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 8 | C137 | D115 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 9 | C5×C23 | C2×C137 | 1 | 1 | yes | characteristic, complemented, Hall |
| 10 | D23 | — | 5 | 1 | no | Hall |
| 11 | C23 | D5×C137 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 12 | D5 | — | 23 | 1 | no | Hall |
| 13 | C5 | D23×C137 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 14 | C2 | — | 115 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C137 | D115 |
| commutator, cosocle, hypocenter | C5×C23 | C2×C137 |
| Frattini | C1 | G |
| Fitting, socle | C5×C23×C137 | C2 |
| LCS | G ▷ C5×C23 | |
| UCS | C1 ◁ C137 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×C23×C137 | C2 | semidirect |
| 3. C23×C137 | D5 | semidirect |
| 5. C5×C137 | D23 | semidirect |
| 7. D115 | C137 | direct (N × H) |
| 8. C137 | D115 | direct (N × H) |
| 9. C5×C23 | C2×C137 | semidirect |
| 11. C23 | D5×C137 | semidirect |
| 13. C5 | D23×C137 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×C23×C137 | 2 |
| D23×C137 | 5 |
| D5×C137 | 23 |
| D115 | 137 |