Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
244 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 244 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #13133204 |
| commutator | C22×C4×C53 | C23 |
| Frattini | C22×C4 | C22×D53 |
| Fitting | C53×#421 | C2 |
| socle | C2×C53 | #393 |
| hypercenter | #421 | D53 |
| hypocenter | C53 | #2031 |
| LCS | G ▷ C22×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #421 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811176 | C2 | semidirect |
| 4. #16810394 | C2 | semidirect |
| 5. #16810384 | C2 | semidirect |
| 6. #16810379 | C2 | semidirect |
| 27. C2×Q8×C53 | Q8 | semidirect |
| 29. C2×D4×C53 | Q8 | semidirect |
| 110. C53 | #2031 | semidirect |
| 131. C2×Q8 | Dic106 | semidirect |
| 133. C2×D4 | Dic106 | semidirect |