Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
352 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 352 conjugacy classes of subgroups — too many to draw. Only its 123 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×#499 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #499 | D53 |
| hypocenter | C53 | #3380 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #499 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811329 | C2 | semidirect |
| 2. #16811027 | C2 | semidirect |
| 3. #16811024 | C2 | semidirect |
| 4. #16810922 | C2 | semidirect |
| 5. #16810920 | C2 | semidirect |
| 6. #16810912 | C2 | semidirect |
| 7. C8×Dic106 | C2 | semidirect |
| 11. C53×#499 | C2 | semidirect |
| 66. D424 | Q8 | semidirect |
| 112. #499 | D53 | semidirect |
| 120. C53 | #3380 | semidirect |