Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
494 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 494 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D53 |
| commutator | C2×C4×C53 | C24 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#459 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #459 | D53 |
| hypocenter | C53 | #3284 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #459 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#13133236 | C2 | semidirect |
| 2. #16811293 | C2 | semidirect |
| 3. #16811277 | C2 | semidirect |
| 4. #16810995 | C2 | semidirect |
| 5. #16810851 | C2 | semidirect |
| 6. #16810834 | C2 | semidirect |
| 7. #16810786 | C2 | semidirect |
| 8. #16810765 | C2 | semidirect |
| 9. C4×#11719923 | C2 | semidirect |
| 10. #16810688 | C2 | semidirect |
| 11. C53×#459 | C2 | semidirect |
| 68. #11719923 | D4 | semidirect |
| 127. C4×C53 | #135 | semidirect |
| 147. #459 | D53 | semidirect |
| 152. C53 | #3284 | semidirect |
| 272. C4 | #13133236 | semidirect |