Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
644 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 644 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23×C7 | C22×D3 |
| commutator, cosocle | C23×C3 | C23×C7 |
| Frattini | C23 | C22×D3×C7 |
| Fitting | C3×C7×#95 | C2 |
| socle | C23×C3×C7 | C23 |
| hypercenter | C7×#95 | D3 |
| hypocenter | C3 | C7×#392 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23×C7 ◁ C7×#95 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C7×D12 | C2 | semidirect |
| 2. C2×C7×(C2×D3)⋊C4 | C2 | semidirect |
| 3. C3×C7×#95 | C2 | semidirect |
| 13. C7×#95 | D3 | semidirect |
| 14. #4671 | C7 | direct (N × H) |
| 20. C22×D3×C7 | D4 | semidirect |
| 22. C2×C3×C4×C7 | D4 | semidirect |
| 31. C22×D12 | C2×C7 | semidirect |
| 32. C2×(C2×D3)⋊C4 | C2×C7 | semidirect |
| 33. C3×#95 | C2×C7 | semidirect |
| 54. C2×C4×C7 | D12 | semidirect |
| 67. #95 | D3×C7 | semidirect |
| 83. C22×D3 | C7×D4 | semidirect |
| 85. C2×C3×C4 | C7×D4 | semidirect |
| 88. C3×C7 | #392 | semidirect |
| 118. C2×C4 | C7×D12 | semidirect |
| 119. C7 | #4671 | direct (N × H) |
| 134. C3 | C7×#392 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C7×D12 | 2 |
| C2×C7×(C2×D3)⋊C4 | 2 |
| C3×C7×#95 | 2 |
| C7×#392 | 3 |
| #4671 | 7 |