Finite Group DatabasebetaFinite Group DatabasebetaOrder 126.
12 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×C9 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | D21 | — | 3 | 1 | no | maximal |
| 3 | C3×C7 | D3 | 1 | 1 | yes | characteristic |
| 4 | D9 | — | 7 | 1 | no | maximal, Hall |
| 5 | D7 | — | 9 | 1 | no | Hall |
| 6 | C9 | D7 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 7 | C7 | D9 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 8 | D3 | — | 21 | 1 | no | — |
| 9 | C3 | D21 | 1 | 1 | yes | characteristic |
| 10 | C2 | — | 63 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C7×C9 | C2 |
| Frattini | C3 | D21 |
| socle | C3×C7 | D3 |
| LCS | G ▷ C7×C9 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×C9 | C2 | semidirect |
| 6. C9 | D7 | semidirect |
| 7. C7 | D9 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×C9 | 2 |
| D21 | 3 |
| D9 | 7 |