Finite Group DatabasebetaFinite Group DatabasebetaOrder 414.
12 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C9×C23 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | C3×D23 | C3 | 1 | 1 | yes | maximal, characteristic |
| 3 | C3×C23 | C2×C3 | 1 | 1 | yes | characteristic |
| 4 | D23 | C9 | 1 | 1 | yes | characteristic, direct product factor, Hall |
| 5 | C23 | C2×C9 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 6 | C2×C9 | — | 23 | 1 | no | maximal, Hall |
| 7 | C9 | D23 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 8 | C2×C3 | — | 23 | 1 | no | — |
| 9 | C3 | C3×D23 | 1 | 1 | yes | characteristic |
| 10 | C2 | — | 23 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | D23 |
| commutator, hypocenter | C23 | C2×C9 |
| Frattini | C3 | C3×D23 |
| Fitting | C9×C23 | C2 |
| socle, cosocle | C3×C23 | C2×C3 |
| prime radical | C3×D23 | C3 |
| LCS | G ▷ C23 | |
| UCS | C1 ◁ C9 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C9×C23 | C2 | semidirect |
| 4. D23 | C9 | direct (N × H) |
| 5. C23 | C2×C9 | semidirect |
| 7. C9 | D23 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9×C23 | 2 |
| C3×D23 | 3 |
| C2×C9 | 23 |