Finite Group DatabasebetaFinite Group DatabasebetaOrder 63.
12 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×C7 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | ΓL(1,𝔽8) | C3 | 1 | 3 | yes | maximal, direct product factor |
| 3 | C32 | — | 7 | 1 | no | maximal, Sylow |
| 4 | C7 | C32 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 5 | C3 | — | 7 | 3 | no | — |
| 6 | C3 | ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C7 | C32 |
| Frattini | C1 | G |
| Fitting, socle | C3×C7 | C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C7 | C3 | semidirect |
| 2. ΓL(1,𝔽8) | C3 | direct (N × H) |
| 4. C7 | C32 | semidirect |
| 6. C3 | ΓL(1,𝔽8) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C7 | 3 |
| ΓL(1,𝔽8) | 3 |
| C32 | 7 |