Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
59186 conjugacy classes of subgroups; only the 32041 normal subgroups are shown.
This group has 59186 conjugacy classes of subgroups — too many to draw. Only its 32041 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C26×D53 |
| commutator, socle, cosocle | C2×C53 | C27 |
| Fitting | C53×2−1+6 | C2 |
| hypercenter | 2−1+6 | D53 |
| hypocenter | C53 | C2×2−1+6 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ 2−1+6 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20289528 | C2 | semidirect |
| 2. #20289527 | C2 | semidirect |
| 3. #585×D53 | C2 | semidirect |
| 4. C2×2−1+4×D53 | C2 | semidirect |
| 5. C53×2−1+6 | C2 | semidirect |
| 6. #16811460 | C22 | semidirect |
| 7. #16811459 | C22 | semidirect |
| 8. #16811458 | C22 | semidirect |
| 9. C2×#13133265 | C22 | semidirect |
| 10. C2×#13133264 | C22 | semidirect |
| 11. 2−1+4×D53 | C22 | semidirect |
| 12. 2−1+4×D53 | C22 | semidirect |
| 13. H(5,𝔽2)×D53 | C22 | semidirect |
| 14. C2×#41×D53 | C22 | semidirect |
| 15. C22×Q8×D53 | C22 | semidirect |
| 16. C53×#585 | C22 | semidirect |
| 17. C2×2−1+4×C53 | C22 | semidirect |
| 18. #13133265 | C23 | semidirect |
| 19. #13133264 | C23 | semidirect |
| 20. #13133263 | C23 | semidirect |
| 21. #13133262 | C23 | semidirect |
| 22. C2×#11719949 | C23 | semidirect |
| 23. C2×#11719945 | C23 | semidirect |
| 24. C2×#11719947 | C23 | semidirect |
| 26. #41×D53 | C23 | semidirect |
| 27. C22×Dic106 | C23 | semidirect |
| 29. C2×Q8×D53 | C23 | semidirect |
| 32. 2−1+4×C53 | C23 | semidirect |
| 33. H(5,𝔽2)×C53 | C23 | semidirect |
| 34. C2×#41×C53 | C23 | semidirect |
| 35. C22×Q8×C53 | C23 | semidirect |
| 78. 2−1+6 | D53 | direct (N × H) |
| 84. D53 | 2−1+6 | direct (N × H) |
| 88. #585 | C2×D53 | semidirect |
| 93. C2×2−1+4 | C2×D53 | semidirect |
| 97. C53 | C2×2−1+6 | semidirect |
| 100. 2−1+4 | C22×D53 | semidirect |
| 103. H(5,𝔽2) | C22×D53 | semidirect |
| 109. C2×#41 | C22×D53 | semidirect |
| 113. C22×Q8 | C22×D53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20289528 | 2 |
| #20289527 | 2 |
| #585×D53 | 2 |
| C2×2−1+4×D53 | 2 |
| C53×2−1+6 | 2 |
| C2×2−1+6 | 53 |