Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
398 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 398 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D212 |
| commutator | C2×C4×C53 | C24 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#424 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #424 | D53 |
| hypocenter | C53 | #3081 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #424 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811251 | C2 | semidirect |
| 3. #16810924 | C2 | semidirect |
| 4. #16810924 | C2 | semidirect |
| 5. #16810914 | C2 | semidirect |
| 6. #16810904 | C2 | semidirect |
| 7. #16810665 | C2 | semidirect |
| 8. #16810661 | C2 | semidirect |
| 11. C53×#424 | C2 | semidirect |
| 13. #13133217 | C22 | semidirect |
| 15. #13133209 | C22 | semidirect |
| 26. #13133151 | C22 | semidirect |
| 27. #13133151 | C22 | semidirect |
| 28. #13133151 | C22 | semidirect |
| 29. #13133149 | C22 | semidirect |
| 30. #13133149 | C22 | semidirect |
| 31. #13133149 | C22 | semidirect |
| 39. C4⋊C8×C53 | C22 | semidirect |
| 40. C4⋊C8×C53 | C22 | semidirect |
| 46. C2×D212 | Q8 | semidirect |
| 133. #424 | D53 | semidirect |
| 136. C53 | #3081 | semidirect |
| 160. C4⋊C8 | C2×D53 | semidirect |
| 161. C4⋊C8 | C2×D53 | semidirect |