Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
596 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 596 conjugacy classes of subgroups — too many to draw. Only its 137 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×#495 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #495 | D53 |
| hypocenter | C53 | #3155 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #495 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#13133244 | C2 | semidirect |
| 2. C2×#13133240 | C2 | semidirect |
| 3. C2×D8×D53 | C2 | semidirect |
| 4. #16811191 | C2 | semidirect |
| 5. #16811013 | C2 | semidirect |
| 6. #16811009 | C2 | semidirect |
| 7. #16811008 | C2 | semidirect |
| 8. #16811000 | C2 | semidirect |
| 9. #16810998 | C2 | semidirect |
| 10. #16810577 | C2 | semidirect |
| 11. C53×#495 | C2 | semidirect |
| 16. C2×#11719924 | C22 | semidirect |
| 17. C2×#11719924 | C22 | semidirect |
| 19. C2×D424 | C22 | semidirect |
| 20. C2×D424 | C22 | semidirect |
| 22. C2×#11719922 | C22 | semidirect |
| 26. #13133214 | C22 | semidirect |
| 27. #13133213 | C22 | semidirect |
| 28. #13133206 | C22 | semidirect |
| 32. #13133128 | C22 | semidirect |
| 33. #13133127 | C22 | semidirect |
| 34. C8×Dic53 | C22 | semidirect |
| 35. C2×Q16×C53 | C22 | semidirect |
| 36. C2×SD16×C53 | C22 | semidirect |
| 38. #124×C53 | C22 | semidirect |
| 39. C4×C8×C53 | C22 | semidirect |
| 75. #11719912 | D4 | semidirect |
| 76. C8×D53 | D4 | semidirect |
| 137. #495 | D53 | semidirect |
| 141. C53 | #3155 | semidirect |
| 146. C2×Q16 | C2×D53 | semidirect |
| 149. C2×SD16 | C2×D53 | semidirect |
| 162. #124 | C2×D53 | semidirect |
| 169. C4×C8 | C2×D53 | semidirect |