Finite Group DatabasebetaFinite Group DatabasebetaOrder 1736.
12 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C23×C31 | C7 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | C7×C31 | — | 8 | 1 | no | maximal, Hall |
| 3 | C22×C31 | — | 7 | 1 | no | — |
| 4 | C2×C31 | — | 7 | 1 | no | — |
| 5 | AGL(1,𝔽8) | C31 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 6 | C31 | AGL(1,𝔽8) | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 7 | C23 | C7×C31 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 8 | C7 | — | 8 | 1 | no | Sylow |
| 9 | C22 | — | 7 | 1 | no | — |
| 10 | C2 | — | 7 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C31 | AGL(1,𝔽8) |
| commutator, cosocle, hypocenter | C23 | C7×C31 |
| Frattini | C1 | G |
| Fitting, socle | C23×C31 | C7 |
| LCS | G ▷ C23 | |
| UCS | C1 ◁ C31 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C31 | C7 | semidirect |
| 5. AGL(1,𝔽8) | C31 | direct (N × H) |
| 6. C31 | AGL(1,𝔽8) | direct (N × H) |
| 7. C23 | C7×C31 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C31 | 7 |
| C7×C31 | 8 |
| AGL(1,𝔽8) | 31 |