Finite Group DatabasebetaFinite Group DatabasebetaOrder 168.
32 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×AGL(1,𝔽7) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C4×ΓL(1,𝔽8) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | ΓL(1,𝔽8)⋊C4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C4×D7 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 5 | C2×ΓL(1,𝔽8) | C22 | 1 | 1 | yes | characteristic |
| 6 | AGL(1,𝔽7) | C4 | 1 | 2 | yes | direct product factor |
| 7 | C2×D7 | C2×C3 | 1 | 1 | yes | characteristic |
| 8 | C4×C7 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 9 | Dic7 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 10 | C2×C3×C4 | — | 7 | 1 | no | maximal, Hall |
| 11 | ΓL(1,𝔽8) | C2×C4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 12 | C2×C7 | C22×C3 | 1 | 1 | yes | characteristic |
| 13 | D7 | C3×C4 | 1 | 2 | yes | complemented |
| 14 | C22×C3 | — | 7 | 1 | no | — |
| 15 | C3×C4 | — | 7 | 1 | no | — |
| 16 | C3×C4 | — | 7 | 1 | no | — |
| 17 | C2×C4 | — | 7 | 1 | no | Sylow |
| 18 | C7 | C2×C3×C4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 19 | C2×C3 | — | 7 | 2 | no | — |
| 20 | C2×C3 | — | 7 | 1 | no | — |
| 21 | C22 | — | 7 | 1 | no | — |
| 22 | C4 | — | 7 | 1 | no | — |
| 23 | C4 | AGL(1,𝔽7) | 1 | 1 | yes | characteristic, direct product factor |
| 24 | C3 | — | 7 | 1 | no | Sylow |
| 25 | C2 | — | 7 | 2 | no | — |
| 26 | C2 | C2×AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | AGL(1,𝔽7) |
| commutator, hypocenter | C7 | C2×C3×C4 |
| Frattini | C2 | C2×AGL(1,𝔽7) |
| Fitting | C4×C7 | C2×C3 |
| socle, cosocle | C2×C7 | C22×C3 |
| prime radical | C2×AGL(1,𝔽7) | C2 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C4×ΓL(1,𝔽8) | C2 | semidirect |
| 3. ΓL(1,𝔽8)⋊C4 | C2 | semidirect |
| 4. C4×D7 | C3 | semidirect |
| 6. AGL(1,𝔽7) | C4 | direct (N × H) |
| 8. C4×C7 | C2×C3 | semidirect |
| 9. Dic7 | C2×C3 | semidirect |
| 11. ΓL(1,𝔽8) | C2×C4 | semidirect |
| 13. D7 | C3×C4 | semidirect |
| 18. C7 | C2×C3×C4 | semidirect |
| 23. C4 | AGL(1,𝔽7) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×AGL(1,𝔽7) | 2 |
| C4×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)⋊C4 | 2 |
| C4×D7 | 3 |
| C2×C3×C4 | 7 |