Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
336 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | #13133185 |
| commutator | C42×C53 | C23 |
| Frattini | C42 | C22×D53 |
| Fitting | C53×#459 | C2 |
| socle | C22×C53 | #120 |
| hypercenter | #459 | D53 |
| hypocenter | C53 | #1619 |
| LCS | G ▷ C42×C53 ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #459 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D4×D212 | C2 | semidirect |
| 2. #16810999 | C2 | semidirect |
| 3. #16810815 | C2 | semidirect |
| 4. #16810366 | C2 | semidirect |
| 5. D4⋊#88322 | C2 | semidirect |
| 6. #16810350 | C2 | semidirect |
| 7. C53×#459 | C2 | semidirect |
| 17. C4⋊#88322 | C22 | semidirect |
| 19. C4×#88322 | C22 | semidirect |
| 21. #127×C53 | C22 | semidirect |
| 23. C4⋊C8×C53 | C22 | semidirect |
| 67. D4×C53 | D8 | semidirect |
| 83. D212 | D8 | semidirect |
| 121. #459 | D53 | semidirect |
| 125. C53 | #1619 | semidirect |
| 132. #127 | C2×D53 | semidirect |
| 139. C4⋊C8 | C2×D53 | semidirect |
| 194. D4 | #11719922 | semidirect |