Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
668 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 668 conjugacy classes of subgroups — too many to draw. Only its 149 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×#389 | C2 |
| hypercenter | #389 | D53 |
| hypocenter | C53 | #2682 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #389 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#13133202 | C2 | semidirect |
| 2. C2×#13133192 | C2 | semidirect |
| 3. #16810545 | C2 | semidirect |
| 4. #16810540 | C2 | semidirect |
| 5. #16810533 | C2 | semidirect |
| 6. #16810525 | C2 | semidirect |
| 7. #16810525 | C2 | semidirect |
| 8. #16810497 | C2 | semidirect |
| 9. C53×#389 | C2 | semidirect |
| 53. C2×D212 | D4 | semidirect |
| 59. C4⋊C4×C53 | D4 | semidirect |
| 157. #389 | D53 | semidirect |
| 159. C53 | #2682 | semidirect |
| 212. C4⋊C4 | D212 | semidirect |