Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
708 conjugacy classes of subgroups; only the 169 normal subgroups are shown.
This group has 708 conjugacy classes of subgroups — too many to draw. Only its 169 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator, socle | C23×C53 | C24 |
| Fitting | C53×#384 | C2 |
| hypercenter | #384 | D53 |
| hypocenter | C53 | #2501 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #384 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#13133202 | C2 | semidirect |
| 3. C2×C4⋊C4×D53 | C2 | semidirect |
| 4. #16810558 | C2 | semidirect |
| 5. #16810525 | C2 | semidirect |
| 6. #16810524 | C2 | semidirect |
| 7. #16810492 | C2 | semidirect |
| 8. C4×#11719921 | C2 | semidirect |
| 9. C53×#384 | C2 | semidirect |
| 56. #11719921 | D4 | semidirect |
| 112. C4×C53 | #123 | semidirect |
| 131. #384 | D53 | semidirect |
| 133. C53 | #2501 | semidirect |
| 239. C4 | #13133202 | semidirect |