Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
24030 conjugacy classes of subgroups; only the 4939 normal subgroups are shown.
This group has 24030 conjugacy classes of subgroups — too many to draw. Only its 4939 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C25 | C22×D53 |
| commutator, cosocle | C22×C53 | C26 |
| Frattini | C22 | C25×D53 |
| Fitting | C24×D4×C53 | C2 |
| socle | C25×C53 | C23 |
| hypercenter | C24×D4 | D53 |
| hypocenter | C53 | C23×#127 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C25 ◁ C24×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×D212 | C2 | semidirect |
| 2. C24×C22⋊D53 | C2 | semidirect |
| 3. C22×#13133213 | C2 | direct (N × H) |
| 4. C23×C4×Dic53 | C2 | semidirect |
| 5. C24×D4×C53 | C2 | semidirect |
| 8. C23×D212 | C22 | semidirect |
| 10. C23×C22⋊D53 | C22 | semidirect |
| 12. C2×#13133213 | C22 | direct (N × H) |
| 13. C22×C4×Dic53 | C22 | semidirect |
| 16. C23×D4×C53 | C22 | semidirect |
| 22. C22×D212 | C23 | semidirect |
| 24. C22×C22⋊D53 | C23 | semidirect |
| 25. C23×Dic53 | D4 | semidirect |
| 27. #13133213 | C23 | direct (N × H) |
| 28. C2×C4×Dic53 | C23 | semidirect |
| 34. C22×D4×C53 | C23 | semidirect |
| 35. C23×C4×C53 | D4 | semidirect |
| 49. C2×D4×C53 | C24 | semidirect |
| 50. C22×C4×C53 | C2×D4 | semidirect |
| 53. C2×C22⋊D53 | C24 | semidirect |
| 54. C22×Dic53 | C2×D4 | semidirect |
| 57. C2×D212 | C24 | semidirect |
| 58. C4×Dic53 | C24 | semidirect |
| 62. C23×C53 | #127 | semidirect |
| 72. C2×C4×C53 | C22×D4 | semidirect |
| 75. C2×Dic53 | C22×D4 | semidirect |
| 82. C22×C53 | C2×#127 | semidirect |
| 89. C4×C53 | C23×D4 | semidirect |
| 90. Dic53 | C23×D4 | semidirect |
| 93. C24×D4 | D53 | semidirect |
| 97. C2×C53 | C22×#127 | semidirect |
| 108. C23×D4 | C2×D53 | semidirect |
| 113. C53 | C23×#127 | semidirect |
| 126. C22×D4 | C22×D53 | semidirect |
| 129. C23×C4 | C22⋊D53 | semidirect |
| 149. C2×D4 | C23×D53 | semidirect |
| 152. C22×C4 | C2×C22⋊D53 | semidirect |
| 162. C23 | #13133213 | direct (N × H) |
| 172. C2×C4 | C22×C22⋊D53 | semidirect |
| 180. C22 | C2×#13133213 | direct (N × H) |
| 185. C4 | C23×C22⋊D53 | semidirect |
| 188. C2 | C22×#13133213 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×D212 | 2 |
| C24×C22⋊D53 | 2 |
| C22×#13133213 | 2 |
| C23×C4×Dic53 | 2 |
| C24×D4×C53 | 2 |
| C23×#127 | 53 |