Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
456 conjugacy classes of subgroups; only the 217 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 217 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×D4×D53 |
| commutator | C4×C53 | C23×C4 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#575 | C2 |
| socle | C2×C53 | C2×C4×D4 |
| hypercenter | #575 | D53 |
| hypocenter | C53 | #2955 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #575 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#13133224 | C2 | semidirect |
| 2. #16810745 | C2 | semidirect |
| 3. #16810740 | C2 | semidirect |
| 4. #16810672 | C2 | semidirect |
| 5. #16810640 | C2 | semidirect |
| 6. #16810581 | C2 | semidirect |
| 10. #13133224 | C22 | semidirect |
| 11. #13133224 | C22 | semidirect |
| 17. #13133153 | C22 | semidirect |
| 18. #13133144 | C22 | semidirect |
| 19. #13133127 | C22 | semidirect |
| 20. C8×Dic53 | C22 | semidirect |
| 22. #137×C53 | C4 | semidirect |
| 23. AGL(1,ℤ/8)×C53 | C4 | semidirect |
| 97. C53 | #2955 | semidirect |
| 99. #137 | Dic53 | semidirect |
| 100. AGL(1,ℤ/8) | Dic53 | semidirect |