Finite Group DatabasebetaFinite Group DatabasebetaOrder 168.
24 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22⋊ΓL(1,𝔽8) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C22×D7 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 3 | AGL(1,𝔽7) | — | 4 | 1 | no | maximal |
| 4 | C22×C7 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 5 | C2×D7 | — | 3 | 1 | no | — |
| 6 | C2×D7 | — | 3 | 1 | no | — |
| 7 | C2×A4 | — | 7 | 1 | no | maximal, Hall |
| 8 | ΓL(1,𝔽8) | — | 4 | 1 | no | Hall |
| 9 | C2×C7 | — | 3 | 1 | no | — |
| 10 | D7 | — | 3 | 1 | no | — |
| 11 | D7 | A4 | 1 | 1 | yes | characteristic, complemented |
| 12 | A4 | — | 7 | 1 | no | — |
| 13 | C23 | — | 7 | 1 | no | Sylow |
| 14 | C7 | C2×A4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 15 | C2×C3 | — | 28 | 1 | no | — |
| 16 | C22 | — | 21 | 1 | no | — |
| 17 | C22 | — | 21 | 1 | no | — |
| 18 | C22 | AGL(1,𝔽7) | 1 | 1 | yes | characteristic, complemented |
| 19 | C3 | — | 28 | 1 | no | Sylow |
| 20 | C2 | — | 3 | 1 | no | — |
| 21 | C2 | — | 7 | 1 | no | — |
| 22 | C2 | — | 21 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C22×C7 | C2×C3 |
| LCS | G ▷ C22×C7 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22⋊ΓL(1,𝔽8) | C2 | semidirect |
| 2. C22×D7 | C3 | semidirect |
| 4. C22×C7 | C2×C3 | semidirect |
| 11. D7 | A4 | semidirect |
| 14. C7 | C2×A4 | semidirect |
| 18. C22 | AGL(1,𝔽7) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22⋊ΓL(1,𝔽8) | 2 |
| C22×D7 | 3 |
| AGL(1,𝔽7) | 4 |
| C2×A4 | 7 |