Finite Group DatabasebetaFinite Group DatabasebetaOrder 168.
32 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 | C2×AGL(1,𝔽7) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | ΓL(1,𝔽8)⋊C4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C22⋊D7 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 5 | C2×ΓL(1,𝔽8) | — | 2 | 1 | no | — |
| 6 | C2×ΓL(1,𝔽8) | C22 | 1 | 1 | yes | characteristic |
| 7 | AGL(1,𝔽7) | — | 2 | 1 | no | — |
| 8 | C22×C7 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 9 | C2×D7 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 10 | Dic7 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 11 | C3×D4 | — | 7 | 1 | no | maximal, Hall |
| 12 | ΓL(1,𝔽8) | D4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 13 | C2×C7 | — | 2 | 1 | no | — |
| 14 | C2×C7 | C22×C3 | 1 | 1 | yes | characteristic |
| 15 | D7 | — | 2 | 1 | no | — |
| 16 | C22×C3 | — | 7 | 1 | no | — |
| 17 | C22×C3 | — | 7 | 1 | no | — |
| 18 | C3×C4 | — | 7 | 1 | no | — |
| 19 | D4 | — | 7 | 1 | no | Sylow |
| 20 | C7 | C3×D4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 21 | C2×C3 | — | 7 | 1 | no | — |
| 22 | C2×C3 | — | 14 | 1 | no | — |
| 23 | C2×C3 | — | 14 | 1 | no | — |
| 24 | C22 | — | 7 | 1 | no | — |
| 25 | C22 | AGL(1,𝔽7) | 1 | 1 | yes | characteristic, complemented |
| 26 | C4 | — | 7 | 1 | no | — |
| 27 | C3 | — | 7 | 1 | no | Sylow |
| 28 | C2 | — | 2 | 1 | no | — |
| 29 | C2 | — | 14 | 1 | no | — |
| 30 | C2 | C2×AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×AGL(1,𝔽7) |
| commutator, socle, cosocle | C2×C7 | C22×C3 |
| Fitting | C22×C7 | C2×C3 |
| hypercenter | C22 | AGL(1,𝔽7) |
| hypocenter | C7 | C3×D4 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×ΓL(1,𝔽8) | C2 | semidirect |
| 2. C2×AGL(1,𝔽7) | C2 | semidirect |
| 3. ΓL(1,𝔽8)⋊C4 | C2 | semidirect |
| 4. C22⋊D7 | C3 | semidirect |
| 8. C22×C7 | C2×C3 | semidirect |
| 9. C2×D7 | C2×C3 | semidirect |
| 10. Dic7 | C2×C3 | semidirect |
| 12. ΓL(1,𝔽8) | D4 | semidirect |
| 20. C7 | C3×D4 | semidirect |
| 25. 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 |
| C2×AGL(1,𝔽7) | 2 |
| ΓL(1,𝔽8)⋊C4 | 2 |
| C22⋊D7 | 3 |
| C3×D4 | 7 |