Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
492 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 492 conjugacy classes of subgroups — too many to draw. Only its 189 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 | C4×C53 | C23×C4 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#405 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #405 | D53 |
| hypocenter | C53 | #2949 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #405 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #16811100 | C2 | semidirect |
| 4. #16810725 | C2 | semidirect |
| 5. C4×#11719913 | C2 | semidirect |
| 6. #16810661 | C2 | semidirect |
| 7. #16810633 | C2 | semidirect |
| 9. C53×#405 | C2 | semidirect |
| 12. C2×#11719913 | C4 | semidirect |
| 15. C4×Dic106 | C22 | semidirect |
| 23. #13133151 | C22 | semidirect |
| 24. #13133149 | C22 | semidirect |
| 25. #13133142 | C22 | semidirect |
| 30. #97×C53 | C22 | semidirect |
| 31. C4×C8×C53 | C22 | semidirect |
| 42. C2×Dic106 | C2×C4 | semidirect |
| 45. C2×C8×C53 | C2×C4 | semidirect |
| 61. #11719913 | C2×C4 | semidirect |
| 62. #11719913 | C2×C4 | semidirect |
| 112. #405 | D53 | semidirect |
| 113. C53 | #2949 | semidirect |
| 134. #97 | C2×D53 | semidirect |
| 135. C4×C8 | C2×D53 | semidirect |
| 151. C2×C8 | C4×D53 | semidirect |