Finite Group DatabasebetaFinite Group DatabasebetaOrder 31155.
16 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C67×ΓL(1,𝔽32) | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | C67×C31⋊C3 | C5 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 3 | C31×C67 | C3×C5 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | C3×C5×C67 | — | 31 | 1 | no | maximal, Hall |
| 5 | (C31⋊C3)⋊C5 | C67 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 6 | C5×C67 | — | 31 | 1 | no | Hall |
| 7 | C3×C67 | — | 31 | 1 | no | Hall |
| 8 | ΓL(1,𝔽32) | C3×C67 | 1 | 1 | yes | characteristic, complemented, Hall |
| 9 | C31⋊C3 | C5×C67 | 1 | 1 | yes | characteristic, complemented, Hall |
| 10 | C67 | (C31⋊C3)⋊C5 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 11 | C31 | C3×C5×C67 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 12 | C3×C5 | — | 31 | 1 | no | Hall |
| 13 | C5 | — | 31 | 1 | no | Sylow |
| 14 | C3 | — | 31 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C67 | (C31⋊C3)⋊C5 |
| commutator, cosocle, hypocenter | C31 | C3×C5×C67 |
| Frattini | C1 | G |
| Fitting, socle | C31×C67 | C3×C5 |
| LCS | G ▷ C31 | |
| UCS | C1 ◁ C67 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C67×ΓL(1,𝔽32) | C3 | semidirect |
| 2. C67×C31⋊C3 | C5 | semidirect |
| 3. C31×C67 | C3×C5 | semidirect |
| 5. (C31⋊C3)⋊C5 | C67 | direct (N × H) |
| 8. ΓL(1,𝔽32) | C3×C67 | semidirect |
| 9. C31⋊C3 | C5×C67 | semidirect |
| 10. C67 | (C31⋊C3)⋊C5 | direct (N × H) |
| 11. C31 | C3×C5×C67 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C67×ΓL(1,𝔽32) | 3 |
| C67×C31⋊C3 | 5 |
| C3×C5×C67 | 31 |
| (C31⋊C3)⋊C5 | 67 |