Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
488 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 488 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D53 |
| commutator | C2×C4×C53 | C24 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#476 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #476 | D53 |
| hypocenter | C53 | #3109 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #476 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811174 | C2 | semidirect |
| 2. C4×D4×D53 | C2 | semidirect |
| 3. #16810943 | C2 | semidirect |
| 4. #16810910 | C2 | semidirect |
| 5. #16810819 | C2 | semidirect |
| 6. #16810818 | C2 | semidirect |
| 7. #16810758 | C2 | semidirect |
| 8. #16810734 | C2 | semidirect |
| 9. #16810732 | C2 | semidirect |
| 10. #16810679 | C2 | semidirect |
| 11. C53×#476 | C2 | semidirect |
| 17. #13133214 | C22 | semidirect |
| 25. #13133179 | C22 | semidirect |
| 31. #13133148 | C22 | semidirect |
| 32. C4⋊#88322 | C22 | semidirect |
| 33. #13133139 | C22 | semidirect |
| 35. #13133137 | C22 | semidirect |
| 37. #13133127 | C22 | semidirect |
| 38. C4⋊Q8×C53 | C22 | semidirect |
| 40. #106×C53 | C22 | semidirect |
| 41. C4⋊C8×C53 | C22 | semidirect |
| 93. D4×C53 | #41 | semidirect |
| 112. D212 | #41 | semidirect |
| 159. #476 | D53 | semidirect |
| 166. C53 | #3109 | semidirect |
| 172. C4⋊Q8 | C2×D53 | semidirect |
| 189. #106 | C2×D53 | semidirect |
| 191. C4⋊C8 | C2×D53 | semidirect |
| 256. D4 | #11719949 | semidirect |