Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
13310 conjugacy classes of subgroups; only the 11166 normal subgroups are shown.
This group has 13310 conjugacy classes of subgroups — too many to draw. Only its 11166 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C24×C4×C53 | C22 |
| commutator | C2 | C25×C4×C53 |
| Frattini, cosocle | C4 | C26×C53 |
| socle | C25×C53 | C2×C4 |
| prime radical | C26×C53 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C23×M16×C53 | C2 | direct (N × H) |
| 3. C24×C8×C53 | C2 | semidirect |
| 9. C22×M16×C53 | C22 | direct (N × H) |
| 10. C23×C8×C53 | C22 | semidirect |
| 19. C2×M16×C53 | C23 | direct (N × H) |
| 20. C22×C8×C53 | C23 | semidirect |
| 22. C24×C53 | M16 | direct (N × H) |
| 30. M16×C53 | C24 | direct (N × H) |
| 31. C2×C8×C53 | C24 | semidirect |
| 33. C23×C53 | C2×M16 | direct (N × H) |
| 41. C8×C53 | C25 | semidirect |
| 42. C24×M16 | C53 | direct (N × H) |
| 44. C22×C53 | C22×M16 | direct (N × H) |
| 51. C23×M16 | C2×C53 | direct (N × H) |
| 52. C24×C8 | C2×C53 | semidirect |
| 54. C2×C53 | C23×M16 | direct (N × H) |
| 61. C22×M16 | C22×C53 | direct (N × H) |
| 62. C23×C8 | C22×C53 | semidirect |
| 63. C53 | C24×M16 | direct (N × H) |
| 72. C2×M16 | C23×C53 | direct (N × H) |
| 73. C22×C8 | C23×C53 | semidirect |
| 75. C24 | M16×C53 | direct (N × H) |
| 83. M16 | C24×C53 | direct (N × H) |
| 84. C2×C8 | C24×C53 | semidirect |
| 86. C23 | C2×M16×C53 | direct (N × H) |
| 94. C8 | C25×C53 | semidirect |
| 96. C22 | C22×M16×C53 | direct (N × H) |
| 103. C2 | C23×M16×C53 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C25×C4×C53 | 2 |
| C23×M16×C53 | 2 |
| C24×C8×C53 | 2 |
| C24×M16 | 53 |