Finite Group DatabasebetaFinite Group DatabasebetaOrder 84.
12 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×ΓL(1,𝔽8) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | Dic7 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 3 | ΓL(1,𝔽8) | C4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | C2×C7 | C2×C3 | 1 | 1 | yes | characteristic |
| 5 | C3×C4 | — | 7 | 1 | no | maximal, Hall |
| 6 | C7 | C3×C4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 7 | C2×C3 | — | 7 | 1 | no | — |
| 8 | C4 | — | 7 | 1 | no | Sylow |
| 9 | C3 | — | 7 | 1 | no | Sylow |
| 10 | C2 | AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | AGL(1,𝔽7) |
| commutator, hypocenter | C7 | C3×C4 |
| Fitting, socle, cosocle | C2×C7 | C2×C3 |
| prime radical | C2×ΓL(1,𝔽8) | C2 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. Dic7 | C3 | semidirect |
| 3. ΓL(1,𝔽8) | C4 | semidirect |
| 6. C7 | C3×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×ΓL(1,𝔽8) | 2 |
| Dic7 | 3 |
| C3×C4 | 7 |