Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
492 conjugacy classes of subgroups; only the 213 normal subgroups are shown.
This group has 492 conjugacy classes of subgroups — too many to draw. Only its 213 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D4×D53 |
| commutator | C4×C53 | C23×C4 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#427 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #427 | D53 |
| hypocenter | C53 | #2911 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ #427 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #16810664 | C2 | semidirect |
| 4. #16810662 | C2 | semidirect |
| 5. #16810661 | C2 | semidirect |
| 6. #16810659 | C2 | semidirect |
| 7. C53×#427 | C2 | semidirect |
| 9. C2×#11719912 | C4 | semidirect |
| 20. #13133151 | C22 | semidirect |
| 21. #13133150 | C22 | semidirect |
| 22. #13133149 | C22 | semidirect |
| 23. #13133148 | C22 | semidirect |
| 26. #107×C53 | C22 | semidirect |
| 27. #106×C53 | C22 | semidirect |
| 36. C2×C8×C53 | C2×C4 | semidirect |
| 48. C2×#88322 | C2×C4 | semidirect |
| 50. #11719912 | C2×C4 | semidirect |
| 51. #11719912 | C2×C4 | semidirect |
| 94. #427 | D53 | semidirect |
| 95. C53 | #2911 | semidirect |
| 112. #107 | C2×D53 | semidirect |
| 115. #106 | C2×D53 | semidirect |
| 128. C2×C8 | C4×D53 | semidirect |