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×C4×D53 | C2 | semidirect |
| 3. C2×#13133188 | C2 | direct (N × H) |
| 4. C22×C4⋊Dic53 | C2 | semidirect |
| 5. C22×C22⋊Dic53 | C2 | semidirect |
| 6. C23×D4×C53 | C2 | semidirect |
| 9. C22×C22⋊D53 | C22 | semidirect |
| 13. C22×C4×D53 | C22 | semidirect |
| 14. #13133188 | C22 | direct (N × H) |
| 15. C2×C4⋊Dic53 | C22 | semidirect |
| 16. C2×C22⋊Dic53 | C22 | semidirect |
| 19. C22×D4×C53 | C22 | semidirect |
| 28. C23×D53 | D4 | semidirect |
| 31. C2×D4×C53 | C23 | semidirect |
| 32. C22×C4×C53 | D4 | semidirect |
| 35. C2×C22⋊D53 | C23 | semidirect |
| 41. C2×C4×D53 | C23 | semidirect |
| 42. C22⋊Dic53 | C23 | semidirect |
| 43. C4⋊Dic53 | C23 | semidirect |
| 56. C22×D53 | C2×D4 | semidirect |
| 59. C2×C4×C53 | C2×D4 | semidirect |
| 71. C22×C53 | #121 | semidirect |
| 79. C2×D53 | C22×D4 | semidirect |
| 80. C4×C53 | C22×D4 | semidirect |
| 85. C2×C53 | C2×#121 | semidirect |
| 91. C23×D4 | D53 | semidirect |
| 96. C53 | C22×#121 | semidirect |
| 102. C22×D4 | C2×D53 | semidirect |
| 123. C2×D4 | C22×D53 | semidirect |
| 130. C22×C4 | C22⋊D53 | semidirect |
| 155. C2×C4 | C2×C22⋊D53 | semidirect |
| 165. C22 | #13133188 | direct (N × H) |
| 172. C4 | C22×C22⋊D53 | semidirect |
| 175. C2 | C2×#13133188 | direct (N × H) |