Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
24030 conjugacy classes of subgroups; only the 12696 normal subgroups are shown.
This group has 24030 conjugacy classes of subgroups — too many to draw. Only its 12696 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24×C4 | #6157 |
| commutator, hypocenter | C53 | C24×C42 |
| Frattini | C2 | C25×#6157 |
| Fitting | C24×C4×C53 | C4 |
| socle | C25×C53 | C2×C4 |
| prime radical | C25×D53 | C22 |
| cosocle | C2×D53 | C26 |
| LCS | G ▷ C53 | |
| UCS | C1 ◁ C24×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C23×C4×#6157 | C2 | direct (N × H) |
| 4. C24×#6157 | C4 | direct (N × H) |
| 6. C22×C4×#6157 | C22 | direct (N × H) |
| 8. C24×Dic53 | C4 | semidirect |
| 9. C23×C4×D53 | C4 | semidirect |
| 11. C24×C4×C53 | C4 | semidirect |
| 12. C23×#6157 | C2×C4 | direct (N × H) |
| 14. C2×C4×#6157 | C23 | direct (N × H) |
| 18. C23×Dic53 | C2×C4 | semidirect |
| 19. C22×C4×D53 | C2×C4 | semidirect |
| 22. C23×C4×C53 | C2×C4 | semidirect |
| 23. C24×C53 | C42 | semidirect |
| 25. C23×D53 | C42 | semidirect |
| 29. C22×#6157 | C22×C4 | direct (N × H) |
| 31. C22×C4×C53 | C22×C4 | semidirect |
| 32. C22×Dic53 | C22×C4 | semidirect |
| 33. C2×C4×D53 | C22×C4 | semidirect |
| 35. C4×#6157 | C24 | direct (N × H) |
| 36. C23×C53 | C2×C42 | semidirect |
| 38. C22×D53 | C2×C42 | semidirect |
| 42. C2×#6157 | C23×C4 | direct (N × H) |
| 44. C2×C4×C53 | C23×C4 | semidirect |
| 45. C2×Dic53 | C23×C4 | semidirect |
| 46. C4×D53 | C23×C4 | semidirect |
| 49. C22×C53 | C22×C42 | semidirect |
| 51. C2×D53 | C22×C42 | semidirect |
| 55. #6157 | C24×C4 | direct (N × H) |
| 56. C4×C53 | C24×C4 | semidirect |
| 57. Dic53 | C24×C4 | semidirect |
| 61. C2×C53 | C23×C42 | semidirect |
| 63. D53 | C23×C42 | semidirect |
| 71. C24×C4 | #6157 | direct (N × H) |
| 73. C53 | C24×C42 | semidirect |
| 83. C23×C4 | C2×#6157 | direct (N × H) |
| 89. C24 | C4×#6157 | direct (N × H) |
| 96. C22×C4 | C22×#6157 | direct (N × H) |
| 102. C23 | C2×C4×#6157 | direct (N × H) |
| 109. C2×C4 | C23×#6157 | direct (N × H) |
| 114. C22 | C22×C4×#6157 | direct (N × H) |
| 118. C4 | C24×#6157 | direct (N × H) |
| 121. C2 | C23×C4×#6157 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C25×#6157 | 2 |
| C23×C4×#6157 | 2 |
| C24×C4×D53 | 2 |
| C24×C42 | 53 |