Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
436 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 436 conjugacy classes of subgroups — too many to draw. Only its 125 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×#489 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #489 | D53 |
| hypocenter | C53 | C2×#489 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #489 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C4⋊Q8×D53 | C2 | semidirect |
| 3. #16810985 | C2 | semidirect |
| 4. #16810984 | C2 | semidirect |
| 5. #16810983 | C2 | semidirect |
| 6. #16810982 | C2 | semidirect |
| 7. #16810981 | C2 | semidirect |
| 8. #103×D53 | C2 | semidirect |
| 10. #97×D53 | C2 | semidirect |
| 11. C53×#489 | C2 | semidirect |
| 16. #13133217 | C22 | semidirect |
| 18. #13133216 | C22 | semidirect |
| 30. #13133142 | C22 | semidirect |
| 31. #13133141 | C22 | semidirect |
| 32. #13133140 | C22 | semidirect |
| 36. #13133128 | C22 | semidirect |
| 37. #13133127 | C22 | semidirect |
| 38. C4⋊Q8×C53 | C22 | semidirect |
| 40. #103×C53 | C22 | semidirect |
| 42. #97×C53 | C22 | semidirect |
| 141. D53 | #489 | direct (N × H) |
| 149. #489 | D53 | direct (N × H) |
| 153. C53 | C2×#489 | semidirect |
| 162. C4⋊Q8 | C2×D53 | semidirect |
| 177. #103 | C2×D53 | semidirect |
| 184. #97 | C2×D53 | semidirect |