Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
59806 conjugacy classes of subgroups.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C53 | C24 |
| commutator, Frattini, cosocle | C2 | C27×C53 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×2−1+4×C53 | C2 | direct (N × H) |
| 2. C23×#41×C53 | C2 | semidirect |
| 3. C24×Q8×C53 | C2 | semidirect |
| 4. C2×2−1+4×C53 | C22 | direct (N × H) |
| 5. C22×#41×C53 | C22 | semidirect |
| 7. C23×Q8×C53 | C22 | semidirect |
| 11. 2−1+4×C53 | C23 | direct (N × H) |
| 12. C2×#41×C53 | C23 | semidirect |
| 14. C22×Q8×C53 | C23 | semidirect |
| 21. #41×C53 | C24 | semidirect |
| 23. C2×Q8×C53 | C24 | semidirect |
| 28. C23×C53 | 2−1+4 | direct (N × H) |
| 35. C23×2−1+4 | C53 | direct (N × H) |
| 37. C22×C53 | C2×2−1+4 | direct (N × H) |
| 41. C22×2−1+4 | C2×C53 | direct (N × H) |
| 42. C23×#41 | C2×C53 | semidirect |
| 43. C24×Q8 | C2×C53 | semidirect |
| 45. C2×C53 | C22×2−1+4 | direct (N × H) |
| 47. C2×2−1+4 | C22×C53 | direct (N × H) |
| 48. C22×#41 | C22×C53 | semidirect |
| 50. C23×Q8 | C22×C53 | semidirect |
| 53. C53 | C23×2−1+4 | direct (N × H) |
| 55. 2−1+4 | C23×C53 | direct (N × H) |
| 56. C2×#41 | C23×C53 | semidirect |
| 58. C22×Q8 | C23×C53 | semidirect |
| 65. #41 | C24×C53 | semidirect |
| 67. C2×Q8 | C24×C53 | semidirect |
| 72. C23 | 2−1+4×C53 | direct (N × H) |
| 80. C22 | C2×2−1+4×C53 | direct (N × H) |
| 85. C2 | C22×2−1+4×C53 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×2−1+4×C53 | 2 |
| C23×#41×C53 | 2 |
| C24×Q8×C53 | 2 |
| C23×2−1+4 | 53 |