Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
3000 conjugacy classes of subgroups; only the 661 normal subgroups are shown.
This group has 3000 conjugacy classes of subgroups — too many to draw. Only its 661 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D53 |
| commutator | C22×C53 | C25 |
| Frattini | C22 | C24×D53 |
| Fitting | C2×#127×C53 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | C2×#127 | D53 |
| hypocenter | C53 | C22×#127 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×#127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×D4×D53 | C2 | semidirect |
| 2. C2×#13133214 | C2 | semidirect |
| 3. C2×#13133213 | C2 | semidirect |
| 4. #127×D53 | C2 | direct (N × H) |
| 5. C2×C42×D53 | C2 | semidirect |
| 6. C2×#127×C53 | C2 | semidirect |
| 8. C22×D212 | C22 | semidirect |
| 9. C22×C22⋊D53 | C22 | semidirect |
| 11. C2×D4×D53 | C22 | semidirect |
| 13. #13133214 | C22 | semidirect |
| 14. #13133213 | C22 | semidirect |
| 15. C2×C4×Dic53 | C22 | semidirect |
| 16. C42×D53 | C22 | semidirect |
| 17. C22×D4×C53 | C22 | semidirect |
| 18. #127×C53 | C22 | semidirect |
| 19. C2×C42×C53 | C22 | semidirect |
| 26. C2×D4×C53 | C23 | semidirect |
| 29. C2×C22⋊D53 | C23 | semidirect |
| 33. C2×D212 | C23 | semidirect |
| 34. C2×C4×D53 | D4 | semidirect |
| 36. C42×C53 | C23 | semidirect |
| 37. C4×Dic53 | C23 | semidirect |
| 48. C2×C4×C53 | C2×D4 | semidirect |
| 51. C2×Dic53 | C2×D4 | semidirect |
| 54. C4×D53 | C2×D4 | semidirect |
| 62. C2×D53 | #127 | direct (N × H) |
| 64. C4×C53 | C22×D4 | semidirect |
| 65. Dic53 | C22×D4 | semidirect |
| 68. C2×C53 | C2×#127 | semidirect |
| 71. D53 | C2×#127 | direct (N × H) |
| 76. C2×#127 | D53 | direct (N × H) |
| 78. C53 | C22×#127 | semidirect |
| 84. C22×D4 | C2×D53 | semidirect |
| 88. #127 | C2×D53 | direct (N × H) |
| 91. C2×C42 | C2×D53 | semidirect |
| 103. C2×D4 | C22×D53 | semidirect |
| 109. C42 | C22×D53 | semidirect |
| 125. C2×C4 | D4×D53 | semidirect |
| 137. C4 | C2×D4×D53 | semidirect |
| 141. C2 | #127×D53 | direct (N × H) |