Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×#405 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #405 | D53 |
| hypocenter | C53 | #3154 |
| LCS | G ▷ C2×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 |
|---|---|---|
| 1. C2×#13133250 | C2 | semidirect |
| 2. C2×#13133245 | C2 | semidirect |
| 3. C2×#13133239 | C2 | semidirect |
| 4. C22×#11719913 | C2 | semidirect |
| 5. #16811199 | C2 | semidirect |
| 6. #16811192 | C2 | semidirect |
| 7. #16811017 | C2 | semidirect |
| 8. #16811012 | C2 | semidirect |
| 9. #16811007 | C2 | semidirect |
| 10. #16810996 | C2 | semidirect |
| 11. C53×#405 | C2 | semidirect |
| 19. C2×Dic212 | C22 | semidirect |
| 20. C2×Dic212 | C22 | semidirect |
| 25. C2×#11719913 | C22 | semidirect |
| 26. C2×#11719913 | C22 | semidirect |
| 27. C2×#11719913 | C22 | semidirect |
| 29. C2×D424 | C22 | semidirect |
| 30. #13133217 | C22 | semidirect |
| 33. #13133205 | C22 | semidirect |
| 41. #97×C53 | C22 | semidirect |
| 42. C4×C8×C53 | C22 | semidirect |
| 67. M16×C53 | D4 | semidirect |
| 68. C2×C8×C53 | D4 | semidirect |
| 159. #405 | D53 | semidirect |
| 160. C53 | #3154 | semidirect |
| 190. #97 | C2×D53 | semidirect |
| 191. C4×C8 | C2×D53 | semidirect |
| 226. M16 | D212 | semidirect |
| 227. C2×C8 | D212 | semidirect |