Finite Group DatabasebetaFinite Group DatabasebetaOrder 84.
10 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22×C7 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | ΓL(1,𝔽8) | — | 4 | 1 | no | maximal, Hall |
| 3 | C2×C7 | — | 3 | 1 | no | — |
| 4 | A4 | — | 7 | 1 | no | maximal, Hall |
| 5 | C7 | A4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 6 | C22 | ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 7 | C3 | — | 28 | 1 | no | Sylow |
| 8 | C2 | — | 3 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C22×C7 | C3 |
| LCS | G ▷ C22×C7 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C7 | C3 | semidirect |
| 5. C7 | A4 | semidirect |
| 6. C22 | ΓL(1,𝔽8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C7 | 3 |
| ΓL(1,𝔽8) | 4 |
| A4 | 7 |