Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
134 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 32 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 | #560⋊C3 |
| commutator, cosocle, hypocenter | C42 | C22×C3×C7 |
| Frattini | C22 | C22×C7×A4 |
| Fitting | C7×#560 | C3 |
| socle | C22×C7 | C22×A4 |
| LCS | G ▷ C42 | |
| 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×#126⋊C3 | C2 | semidirect |
| 3. C7×#560 | C3 | semidirect |
| 5. C7×C42⋊C3 | C22 | semidirect |
| 6. C7×#127 | C2×C3 | semidirect |
| 7. C7×#126 | C2×C3 | semidirect |
| 11. #560⋊C3 | C7 | direct (N × H) |
| 23. C42×C7 | C22×C3 | semidirect |
| 24. #127⋊C3 | C2×C7 | semidirect |
| 25. #126⋊C3 | C2×C7 | semidirect |
| 28. #560 | C3×C7 | semidirect |
| 40. C42⋊C3 | C22×C7 | semidirect |
| 43. #127 | C2×C3×C7 | semidirect |
| 44. #126 | C2×C3×C7 | semidirect |
| 69. C42 | C22×C3×C7 | semidirect |
| 86. C7 | #560⋊C3 | 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×#126⋊C3 | 2 |
| C7×#560 | 3 |
| C22×C7×A4 | 4 |
| #560⋊C3 | 7 |