Finite Group DatabasebetaFinite Group DatabasebetaOrder 31270.
16 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C5×C53×C59 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | C59×D53 | — | 5 | 1 | no | maximal, Hall |
| 3 | C53×C59 | D5 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | D5×C59 | — | 53 | 1 | no | maximal, Hall |
| 5 | D265 | C59 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 6 | C5×C59 | D53 | 1 | 1 | yes | characteristic, complemented, Hall |
| 7 | C5×C53 | C2×C59 | 1 | 1 | yes | characteristic, complemented, Hall |
| 8 | C2×C59 | — | 265 | 1 | no | Hall |
| 9 | D53 | — | 5 | 1 | no | Hall |
| 10 | C59 | D265 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 11 | C53 | D5×C59 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 12 | D5 | — | 53 | 1 | no | Hall |
| 13 | C5 | C59×D53 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 14 | C2 | — | 265 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C59 | D265 |
| commutator, cosocle, hypocenter | C5×C53 | C2×C59 |
| Frattini | C1 | G |
| Fitting, socle | C5×C53×C59 | C2 |
| LCS | G ▷ C5×C53 | |
| UCS | C1 ◁ C59 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×C53×C59 | C2 | semidirect |
| 3. C53×C59 | D5 | semidirect |
| 5. D265 | C59 | direct (N × H) |
| 6. C5×C59 | D53 | semidirect |
| 7. C5×C53 | C2×C59 | semidirect |
| 10. C59 | D265 | direct (N × H) |
| 11. C53 | D5×C59 | semidirect |
| 13. C5 | C59×D53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×C53×C59 | 2 |
| C59×D53 | 5 |
| D5×C59 | 53 |
| D265 | 59 |