Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
166 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7 | #857⋊C2 |
| commutator, cosocle, hypocenter | C42⋊C3 | C22×C7 |
| Frattini | C22 | C2×C7×S4 |
| Fitting | C7×#127 | D3 |
| socle | C22×C7 | C2×S4 |
| LCS | G ▷ C42⋊C3 | |
| UCS | C1 ◁ C7 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#127⋊C3 | C2 | semidirect |
| 2. C7×#857 | C2 | semidirect |
| 5. C7×C42⋊C3 | C22 | semidirect |
| 8. C7×#127 | D3 | semidirect |
| 11. #857⋊C2 | C7 | direct (N × H) |
| 24. C42×C7 | C2×D3 | semidirect |
| 25. #127⋊C3 | C2×C7 | semidirect |
| 26. #857 | C2×C7 | semidirect |
| 47. C42⋊C3 | C22×C7 | semidirect |
| 52. #127 | D3×C7 | semidirect |
| 77. C42 | C2×D3×C7 | semidirect |
| 100. C7 | #857⋊C2 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#127⋊C3 | 2 |
| C7×#857 | 2 |
| C7×#453 | 3 |
| C2×C7×S4 | 4 |
| #857⋊C2 | 7 |