Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
8094 conjugacy classes of subgroups; only the 4296 normal subgroups are shown.
This group has 8094 conjugacy classes of subgroups — too many to draw. Only its 4296 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C42 | #6157 |
| commutator, hypocenter | C53 | C22×C43 |
| Frattini | C22 | C24×#6157 |
| Fitting | C22×C42×C53 | C4 |
| socle | C24×C53 | C22×C4 |
| prime radical | C24×D53 | C23 |
| cosocle | C22×D53 | C25 |
| LCS | G ▷ C53 | |
| UCS | C1 ◁ C22×C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C42×#6157 | C2 | direct (N × H) |
| 5. C22×C4×#6157 | C4 | direct (N × H) |
| 7. C42×#6157 | C22 | direct (N × H) |
| 9. C22×C4×Dic53 | C4 | semidirect |
| 10. C2×C42×D53 | C4 | semidirect |
| 12. C22×C42×C53 | C4 | semidirect |
| 15. C2×C4×#6157 | C2×C4 | direct (N × H) |
| 22. C2×C4×Dic53 | C2×C4 | semidirect |
| 23. C42×D53 | C2×C4 | semidirect |
| 26. C2×C42×C53 | C2×C4 | semidirect |
| 31. C22×#6157 | C42 | direct (N × H) |
| 34. C22×C4×C53 | C42 | semidirect |
| 36. C22×Dic53 | C42 | semidirect |
| 38. C2×C4×D53 | C42 | semidirect |
| 42. C4×#6157 | C22×C4 | direct (N × H) |
| 43. C42×C53 | C22×C4 | semidirect |
| 44. C4×Dic53 | C22×C4 | semidirect |
| 52. C2×#6157 | C2×C42 | direct (N × H) |
| 54. C2×C4×C53 | C2×C42 | semidirect |
| 56. C2×Dic53 | C2×C42 | semidirect |
| 58. C4×D53 | C2×C42 | semidirect |
| 61. C22×C53 | C43 | semidirect |
| 64. C2×D53 | C43 | semidirect |
| 68. #6157 | C22×C42 | direct (N × H) |
| 69. C4×C53 | C22×C42 | semidirect |
| 70. Dic53 | C22×C42 | semidirect |
| 74. C2×C53 | C2×C43 | semidirect |
| 76. D53 | C2×C43 | semidirect |
| 85. C22×C42 | #6157 | direct (N × H) |
| 87. C53 | C22×C43 | semidirect |
| 101. C2×C42 | C2×#6157 | direct (N × H) |
| 115. C22×C4 | C4×#6157 | direct (N × H) |
| 119. C42 | C22×#6157 | direct (N × H) |
| 133. C2×C4 | C2×C4×#6157 | direct (N × H) |
| 139. C22 | C42×#6157 | direct (N × H) |
| 144. C4 | C22×C4×#6157 | direct (N × H) |
| 147. C2 | C2×C42×#6157 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C4×#6157 | 2 |
| C2×C42×#6157 | 2 |
| C22×C42×D53 | 2 |
| C22×C43 | 53 |