Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
158 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 158 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C7 | C4⋊S4 |
| commutator, cosocle | C4×A4 | C22×C7 |
| Frattini | C4 | C2×C7×S4 |
| Fitting | C22×C7×C8 | D3 |
| socle | C23×C7 | D12 |
| hypercenter | C7×C8 | S4 |
| hypocenter | A4 | C7×D8 |
| LCS | G ▷ C4×A4 ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2×C7 ◁ C4×C7 ◁ C7×C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×C4⋊S4 | C2 | semidirect |
| 2. C7×C8×A4 | C2 | semidirect |
| 8. C22×C7×C8 | D3 | semidirect |
| 12. #5402 | C7 | direct (N × H) |
| 26. C4⋊S4 | C2×C7 | semidirect |
| 27. C8×A4 | C2×C7 | semidirect |
| 29. C7×A4 | D8 | semidirect |
| 42. C7×C8 | S4 | semidirect |
| 49. C22×C8 | D3×C7 | semidirect |
| 58. C22×C7 | D24 | semidirect |
| 81. A4 | C7×D8 | semidirect |
| 93. C8 | C7×S4 | semidirect |
| 94. C7 | #5402 | direct (N × H) |
| 102. C22 | C7×D24 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×C4⋊S4 | 2 |
| C7×C8×A4 | 2 |
| C7×#466 | 3 |
| C7×D24 | 4 |
| #5402 | 7 |