Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
59186 conjugacy classes of subgroups.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C26 |
| commutator, Frattini, cosocle | C2 | C27×C53 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×2−1+6 | C2 | direct (N × H) |
| 2. C2×C53×#585 | C2 | semidirect |
| 3. C22×2−1+4×C53 | C2 | semidirect |
| 4. C53×#585 | C22 | semidirect |
| 5. C2×2−1+4×C53 | C22 | semidirect |
| 6. C2×2−1+4×C53 | C22 | semidirect |
| 7. C2×H(5,𝔽2)×C53 | C22 | semidirect |
| 8. C22×#41×C53 | C22 | semidirect |
| 9. C23×Q8×C53 | C22 | semidirect |
| 10. 2−1+4×C53 | C23 | semidirect |
| 11. H(5,𝔽2)×C53 | C23 | semidirect |
| 13. C2×#41×C53 | C23 | semidirect |
| 15. C22×Q8×C53 | C23 | semidirect |
| 33. C2×2−1+6 | C53 | direct (N × H) |
| 39. 2−1+6 | C2×C53 | direct (N × H) |
| 40. C2×#585 | C2×C53 | semidirect |
| 41. C22×2−1+4 | C2×C53 | semidirect |
| 43. C2×C53 | 2−1+6 | direct (N × H) |
| 45. #585 | C22×C53 | semidirect |
| 46. C2×2−1+4 | C22×C53 | semidirect |
| 47. C2×2−1+4 | C22×C53 | semidirect |
| 48. C2×H(5,𝔽2) | C22×C53 | semidirect |
| 49. C22×#41 | C22×C53 | semidirect |
| 50. C23×Q8 | C22×C53 | semidirect |
| 51. C53 | C2×2−1+6 | direct (N × H) |
| 52. 2−1+4 | C23×C53 | semidirect |
| 53. H(5,𝔽2) | C23×C53 | semidirect |
| 55. C2×#41 | C23×C53 | semidirect |
| 57. C22×Q8 | C23×C53 | semidirect |
| 81. C2 | C53×2−1+6 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×2−1+6 | 2 |
| C2×C53×#585 | 2 |
| C22×2−1+4×C53 | 2 |
| C2×2−1+6 | 53 |