Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
2528 conjugacy classes of subgroups; only the 685 normal subgroups are shown.
This group has 2528 conjugacy classes of subgroups — too many to draw. Only its 685 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C24 | C22×D53 |
| commutator | C22×C53 | C25 |
| Frattini | C22 | C24×D53 |
| Fitting | C23×D4×C53 | C2 |
| socle | C24×C53 | C23 |
| hypercenter | C23×D4 | D53 |
| hypocenter | C53 | C22×#121 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C24 ◁ C23×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C22⋊D53 | C2 | semidirect |
| 2. C23×C22⋊D53 | C2 | semidirect |
| 3. C24×Dic53 | C2 | semidirect |
| 4. C2×#13133189 | C2 | direct (N × H) |
| 5. C22×#11719919 | C2 | semidirect |
| 6. C22×C22⋊Dic53 | C2 | semidirect |
| 7. C22×#11719921 | C2 | semidirect |
| 8. C23×D4×C53 | C2 | semidirect |
| 12. C22×C22⋊D53 | C22 | semidirect |
| 13. C22×C22⋊D53 | C22 | semidirect |
| 18. C23×Dic53 | C22 | semidirect |
| 19. #13133189 | C22 | direct (N × H) |
| 20. C2×#11719919 | C22 | semidirect |
| 21. C2×C22⋊Dic53 | C22 | semidirect |
| 22. C2×#11719921 | C22 | semidirect |
| 26. C22×D4×C53 | C22 | semidirect |
| 33. C24×C53 | D4 | semidirect |
| 42. C2×D4×C53 | C23 | semidirect |
| 47. C2×C22⋊D53 | C23 | semidirect |
| 48. C2×C22⋊D53 | C23 | semidirect |
| 52. C22×Dic53 | D4 | semidirect |
| 56. C22×Dic53 | C23 | semidirect |
| 57. C22⋊Dic53 | C23 | semidirect |
| 58. #11719921 | C23 | semidirect |
| 59. #11719919 | C23 | semidirect |
| 69. C23×C53 | C2×D4 | semidirect |
| 87. C2×Dic53 | C2×D4 | semidirect |
| 98. C22×C53 | #121 | semidirect |
| 101. C22×C53 | C22×D4 | semidirect |
| 111. Dic53 | C22×D4 | semidirect |
| 115. C2×C53 | C2×#121 | semidirect |
| 122. C23×D4 | D53 | semidirect |
| 128. C53 | C22×#121 | semidirect |
| 137. C22×D4 | C2×D53 | semidirect |
| 157. C24 | C22⋊D53 | semidirect |
| 166. C2×D4 | C22×D53 | semidirect |
| 195. C23 | C2×C22⋊D53 | semidirect |
| 222. C22 | #13133189 | direct (N × H) |
| 225. C22 | C22×C22⋊D53 | semidirect |
| 235. C2 | C2×#13133189 | direct (N × H) |