Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
548 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 131 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×#461 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #461 | D53 |
| hypocenter | C53 | #3036 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #461 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#13133244 | C2 | semidirect |
| 2. #16811174 | C2 | semidirect |
| 3. C4×D4×D53 | C2 | semidirect |
| 4. #16810841 | C2 | semidirect |
| 5. #16810840 | C2 | semidirect |
| 6. #16810825 | C2 | semidirect |
| 7. #16810819 | C2 | semidirect |
| 8. #16810818 | C2 | semidirect |
| 9. #16810645 | C2 | semidirect |
| 10. #16810623 | C2 | semidirect |
| 11. C53×#461 | C2 | semidirect |
| 20. #13133214 | C22 | semidirect |
| 21. #13133206 | C22 | semidirect |
| 34. C4⋊#88322 | C22 | semidirect |
| 35. #13133146 | C22 | semidirect |
| 36. #13133141 | C22 | semidirect |
| 37. #13133139 | C22 | semidirect |
| 38. #13133137 | C22 | semidirect |
| 40. C4⋊Q8×C53 | C22 | semidirect |
| 42. C4⋊C8×C53 | C22 | semidirect |
| 43. #103×C53 | C22 | semidirect |
| 53. D4×D53 | D4 | semidirect |
| 146. C4×C53 | AGL(1,ℤ/8) | semidirect |
| 167. #461 | D53 | semidirect |
| 172. C53 | #3036 | semidirect |
| 182. C4⋊Q8 | C2×D53 | semidirect |
| 200. C4⋊C8 | C2×D53 | semidirect |
| 202. #103 | C2×D53 | semidirect |
| 311. C4 | #13133244 | semidirect |