Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
596 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 596 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 | #3155 |
| 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×#13133245 | C2 | semidirect |
| 2. C2×#13133239 | C2 | semidirect |
| 3. C22×D424 | C2 | semidirect |
| 4. #16811192 | C2 | semidirect |
| 5. #16811012 | C2 | semidirect |
| 6. #16811007 | C2 | semidirect |
| 7. #16811000 | C2 | semidirect |
| 8. C53×#405 | C2 | semidirect |
| 13. C2×Dic212 | C22 | semidirect |
| 14. C2×#11719913 | C22 | semidirect |
| 15. C2×#11719913 | C22 | semidirect |
| 19. C2×D424 | C22 | semidirect |
| 20. C2×D424 | C22 | semidirect |
| 21. #13133214 | C22 | semidirect |
| 22. #13133205 | C22 | semidirect |
| 28. #97×C53 | C22 | semidirect |
| 29. C4×C8×C53 | C22 | semidirect |
| 43. M16×C53 | D4 | semidirect |
| 44. C2×C8×C53 | D4 | semidirect |
| 109. #405 | D53 | semidirect |
| 110. C53 | #3155 | semidirect |
| 130. #97 | C2×D53 | semidirect |
| 131. C4×C8 | C2×D53 | semidirect |
| 152. M16 | D212 | semidirect |
| 153. C2×C8 | D212 | semidirect |