Finite Group DatabasebetaFinite Group DatabasebetaOrder 374.
8 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C11×C17 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | C2×C17 | — | 11 | 1 | no | maximal, Hall |
| 3 | D11 | C17 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 4 | C17 | D11 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 5 | C11 | C2×C17 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 6 | C2 | — | 11 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C17 | D11 |
| commutator, cosocle, hypocenter | C11 | C2×C17 |
| Frattini | C1 | G |
| Fitting, socle | C11×C17 | C2 |
| LCS | G ▷ C11 | |
| UCS | C1 ◁ C17 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C11×C17 | C2 | semidirect |
| 3. D11 | C17 | direct (N × H) |
| 4. C17 | D11 | direct (N × H) |
| 5. C11 | C2×C17 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C11×C17 | 2 |
| C2×C17 | 11 |
| D11 | 17 |