Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
106 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 106 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | D16×D53 |
| commutator | C16×C53 | C23 |
| Frattini | C16 | C22×D53 |
| Fitting | C53×SD64 | C2 |
| socle | C2×C53 | C2×D16 |
| hypercenter | SD64 | D53 |
| hypocenter | C53 | #2262 |
| LCS | G ▷ C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ SD64 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811056 | C2 | semidirect |
| 2. #16811052 | C2 | semidirect |
| 3. #16810463 | C2 | semidirect |
| 4. #16810462 | C2 | semidirect |
| 5. #16810458 | C2 | semidirect |
| 6. C32×D53 | C2 | semidirect |
| 7. C53×SD64 | C2 | semidirect |
| 57. SD64 | D53 | semidirect |
| 60. C53 | #2262 | semidirect |