Finite Group DatabasebetaFinite Group DatabasebetaOrder 15616.
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×D61 |
| commutator, socle, cosocle | C2×C61 | C27 |
| Fitting | C61×2−1+6 | C2 |
| hypercenter | 2−1+6 | D61 |
| hypocenter | C61 | C2×2−1+6 |
| LCS | G ▷ C2×C61 ▷ C61 | |
| UCS | C1 ◁ C2 ◁ 2−1+6 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #21460632 | C2 | semidirect |
| 2. #21460631 | C2 | semidirect |
| 3. #585×D61 | C2 | semidirect |
| 4. C2×2−1+4×D61 | C2 | semidirect |
| 5. C61×2−1+6 | C2 | semidirect |
| 6. #16840599 | C22 | semidirect |
| 7. #16840598 | C22 | semidirect |
| 8. #16840597 | C22 | semidirect |
| 9. C2×#14519825 | C22 | semidirect |
| 10. C2×#14519824 | C22 | semidirect |
| 11. 2−1+4×D61 | C22 | semidirect |
| 12. 2−1+4×D61 | C22 | semidirect |
| 13. H(5,𝔽2)×D61 | C22 | semidirect |
| 14. C2×#41×D61 | C22 | semidirect |
| 15. C22×Q8×D61 | C22 | semidirect |
| 16. C61×#585 | C22 | semidirect |
| 17. C2×2−1+4×C61 | C22 | semidirect |
| 18. #14519825 | C23 | semidirect |
| 19. #14519824 | C23 | semidirect |
| 20. #14519823 | C23 | semidirect |
| 21. #14519822 | C23 | semidirect |
| 22. C2×#11758226 | C23 | semidirect |
| 23. C2×#11758222 | C23 | semidirect |
| 24. C2×#11758224 | C23 | semidirect |
| 26. #41×D61 | C23 | semidirect |
| 27. C22×Dic122 | C23 | semidirect |
| 29. C2×Q8×D61 | C23 | semidirect |
| 32. 2−1+4×C61 | C23 | semidirect |
| 33. H(5,𝔽2)×C61 | C23 | semidirect |
| 34. C2×#41×C61 | C23 | semidirect |
| 35. C22×Q8×C61 | C23 | semidirect |
| 78. 2−1+6 | D61 | direct (N × H) |
| 84. D61 | 2−1+6 | direct (N × H) |
| 88. #585 | C2×D61 | semidirect |
| 93. C2×2−1+4 | C2×D61 | semidirect |
| 97. C61 | C2×2−1+6 | semidirect |
| 100. 2−1+4 | C22×D61 | semidirect |
| 103. H(5,𝔽2) | C22×D61 | semidirect |
| 109. C2×#41 | C22×D61 | semidirect |
| 113. C22×Q8 | C22×D61 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21460632 | 2 |
| #21460631 | 2 |
| #585×D61 | 2 |
| C2×2−1+4×D61 | 2 |
| C61×2−1+6 | 2 |
| C2×2−1+6 | 61 |