Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
236 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | #13133187 |
| commutator | C42×C53 | C23 |
| Frattini | C42 | C22×D53 |
| Fitting | C53×D4⋊C8 | C2 |
| socle | C22×C53 | #120 |
| hypercenter | D4⋊C8 | D53 |
| hypocenter | C53 | #1622 |
| LCS | G ▷ C42×C53 ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ D4⋊C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811330 | C2 | semidirect |
| 2. #16810996 | C2 | semidirect |
| 3. #16810824 | C2 | semidirect |
| 5. #16810371 | C2 | semidirect |
| 6. #16810355 | C2 | semidirect |
| 7. C53×D4⋊C8 | C2 | semidirect |
| 53. D4×C53 | SD16 | semidirect |
| 103. D4⋊C8 | D53 | semidirect |
| 104. C53 | #1622 | semidirect |
| 158. D4 | #11719913 | semidirect |