Finite Group DatabasebetaFinite Group DatabasebetaOrder 168.
64 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22×ΓL(1,𝔽8) | C2 | 1 | 7 | yes | maximal, direct product factor |
| 2 | C23×C7 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 3 | C2×ΓL(1,𝔽8) | C22 | 1 | 7 | yes | direct product factor |
| 4 | C22×C7 | C2×C3 | 1 | 7 | yes | complemented |
| 5 | C23×C3 | — | 7 | 1 | no | maximal, Hall |
| 6 | ΓL(1,𝔽8) | C23 | 1 | 1 | yes | characteristic, direct product factor, Hall |
| 7 | C2×C7 | C22×C3 | 1 | 7 | yes | complemented |
| 8 | C22×C3 | — | 7 | 7 | no | — |
| 9 | C23 | ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 10 | C7 | C23×C3 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 11 | C2×C3 | — | 7 | 7 | no | — |
| 12 | C22 | C2×ΓL(1,𝔽8) | 1 | 7 | yes | direct product factor |
| 13 | C3 | — | 7 | 1 | no | Sylow |
| 14 | C2 | C22×ΓL(1,𝔽8) | 1 | 7 | yes | direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C7 | C23×C3 |
| Frattini | C1 | G |
| Fitting, socle | C23×C7 | C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×ΓL(1,𝔽8) | C2 | direct (N × H) |
| 2. C23×C7 | C3 | semidirect |
| 3. C2×ΓL(1,𝔽8) | C22 | direct (N × H) |
| 4. C22×C7 | C2×C3 | semidirect |
| 6. ΓL(1,𝔽8) | C23 | direct (N × H) |
| 7. C2×C7 | C22×C3 | semidirect |
| 9. C23 | ΓL(1,𝔽8) | direct (N × H) |
| 10. C7 | C23×C3 | semidirect |
| 12. C22 | C2×ΓL(1,𝔽8) | direct (N × H) |
| 14. C2 | C22×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 |
|---|---|
| C22×ΓL(1,𝔽8) | 2 |
| C23×C7 | 3 |
| C23×C3 | 7 |