Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
760 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 760 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 | C2×D4×D53 |
| commutator | C2×C4×C53 | C24 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#493 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #493 | D53 |
| hypocenter | C53 | C2×#493 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #493 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×D8×D53 | C2 | semidirect |
| 2. #127×D53 | C2 | semidirect |
| 3. #16811000 | C2 | semidirect |
| 4. #16810999 | C2 | semidirect |
| 5. #16810998 | C2 | semidirect |
| 6. C4×C8×D53 | C2 | semidirect |
| 7. C53×#493 | C2 | semidirect |
| 10. C2×D424 | C22 | semidirect |
| 11. C2×#11719922 | C22 | semidirect |
| 14. #13133214 | C22 | semidirect |
| 16. #13133213 | C22 | semidirect |
| 18. C4×#88322 | C22 | semidirect |
| 19. C8×Dic53 | C22 | semidirect |
| 20. C2×D8×C53 | C22 | semidirect |
| 21. #127×C53 | C22 | semidirect |
| 22. C4×C8×C53 | C22 | semidirect |
| 42. C8×D53 | D4 | semidirect |
| 58. C4×D53 | D8 | semidirect |
| 60. C8×C53 | C2×D4 | semidirect |
| 61. #88322 | C2×D4 | semidirect |
| 70. C4×C53 | C2×D8 | semidirect |
| 73. Dic53 | C2×D8 | semidirect |
| 79. D53 | #493 | direct (N × H) |
| 85. #493 | D53 | direct (N × H) |
| 87. C53 | C2×#493 | semidirect |
| 93. C2×D8 | C2×D53 | semidirect |
| 98. #127 | C2×D53 | semidirect |
| 102. C4×C8 | C2×D53 | semidirect |
| 141. C8 | D4×D53 | semidirect |
| 152. C4 | D8×D53 | semidirect |