Finite Group DatabasebetaFinite Group DatabasebetaOrder 168.
24 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×C4×C7 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C7×Dic3 | C2 | 1 | 2 | yes | maximal |
| 3 | C7×Q8 | — | 3 | 1 | no | maximal, Hall |
| 4 | C2×C3×C7 | C22 | 1 | 1 | yes | characteristic |
| 5 | C4×C7 | — | 3 | 2 | no | — |
| 6 | C4×C7 | D3 | 1 | 1 | yes | characteristic |
| 7 | Dic6 | C7 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 8 | C3×C7 | Q8 | 1 | 1 | yes | characteristic, complemented, Hall |
| 9 | C2×C7 | C2×D3 | 1 | 1 | yes | characteristic |
| 10 | C3×C4 | C2×C7 | 1 | 1 | yes | characteristic |
| 11 | Dic3 | C2×C7 | 1 | 2 | yes | — |
| 12 | Q8 | — | 3 | 1 | no | Sylow |
| 13 | C7 | Dic6 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 14 | C2×C3 | C22×C7 | 1 | 1 | yes | characteristic |
| 15 | C4 | — | 3 | 2 | no | — |
| 16 | C4 | D3×C7 | 1 | 1 | yes | characteristic |
| 17 | C3 | C7×Q8 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 18 | C2 | C2×D3×C7 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7 | C2×D3 |
| commutator, cosocle | C2×C3 | C22×C7 |
| Frattini | C2 | C2×D3×C7 |
| Fitting | C3×C4×C7 | C2 |
| socle, prime radical | C2×C3×C7 | C22 |
| hypercenter | C4×C7 | D3 |
| hypocenter | C3 | C7×Q8 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C7 ◁ C4×C7 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 7. Dic6 | C7 | direct (N × H) |
| 8. C3×C7 | Q8 | semidirect |
| 13. C7 | Dic6 | direct (N × H) |
| 17. C3 | C7×Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C4×C7 | 2 |
| C7×Dic3 | 2 |
| C7×Q8 | 3 |
| Dic6 | 7 |