Finite Group DatabasebetaFinite Group DatabasebetaOrder 34432.
5956 conjugacy classes of subgroups; only the 3203 normal subgroups are shown.
This group has 5956 conjugacy classes of subgroups — too many to draw. Only its 3203 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C24×D269 |
| commutator, socle, cosocle | C2×C269 | C26 |
| Frattini | C2 | C25×D269 |
| Fitting | #585×C269 | C2 |
| hypercenter | #585 | D269 |
| hypocenter | C269 | C2×#585 |
| LCS | G ▷ C2×C269 ▷ C269 | |
| UCS | C1 ◁ C4 ◁ #585 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #31771442 | C2 | semidirect |
| 2. #31771441 | C2 | semidirect |
| 3. #31771440 | C2 | semidirect |
| 4. 2−1+4×D269 | C2 | semidirect |
| 5. H(5,𝔽2)×D269 | C2 | semidirect |
| 6. C2×#41×D269 | C2 | semidirect |
| 7. #585×C269 | C2 | semidirect |
| 8. #22575501 | C22 | semidirect |
| 9. #22575500 | C22 | semidirect |
| 10. #22575499 | C22 | semidirect |
| 11. #22575498 | C22 | semidirect |
| 12. C2×#16864403 | C22 | semidirect |
| 13. C2×#16864402 | C22 | semidirect |
| 14. C2×#16864401 | C22 | semidirect |
| 15. #41×D269 | C22 | semidirect |
| 16. #41×D269 | C22 | semidirect |
| 17. #41×D269 | C22 | semidirect |
| 18. #41×D269 | C22 | semidirect |
| 19. C2×Q8×D269 | C22 | semidirect |
| 20. C2×D4×D269 | C22 | semidirect |
| 21. C22×C4×D269 | C22 | semidirect |
| 22. 2−1+4×C269 | C22 | semidirect |
| 23. H(5,𝔽2)×C269 | C22 | semidirect |
| 24. C2×#41×C269 | C22 | semidirect |
| 26. #16864403 | C23 | semidirect |
| 27. #16864403 | C23 | semidirect |
| 28. #16864402 | C23 | semidirect |
| 29. #16864402 | C23 | semidirect |
| 30. #16864401 | C23 | semidirect |
| 31. #16864401 | C23 | semidirect |
| 32. C2×Dic538 | C23 | semidirect |
| 34. Q8×D269 | C23 | semidirect |
| 35. C2×D1076 | C23 | semidirect |
| 36. C2×C22⋊D269 | C23 | semidirect |
| 38. D4×D269 | C23 | semidirect |
| 39. C22×Dic269 | C23 | semidirect |
| 42. C2×C4×D269 | C23 | semidirect |
| 43. C2×C4×D269 | C23 | semidirect |
| 44. #41×C269 | C23 | semidirect |
| 45. #41×C269 | C23 | semidirect |
| 46. C2×Q8×C269 | C23 | semidirect |
| 47. C2×D4×C269 | C23 | semidirect |
| 48. C22×C4×C269 | C23 | semidirect |
| 80. D269 | #585 | direct (N × H) |
| 81. C269 | C2×#585 | semidirect |
| 86. #585 | D269 | direct (N × H) |
| 92. 2−1+4 | C2×D269 | semidirect |
| 95. H(5,𝔽2) | C2×D269 | semidirect |
| 103. C2×#41 | C2×D269 | semidirect |
| 114. #41 | C22×D269 | semidirect |
| 115. #41 | C22×D269 | semidirect |
| 119. C2×Q8 | C22×D269 | semidirect |
| 124. C2×D4 | C22×D269 | semidirect |
| 130. C22×C4 | C22×D269 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #31771442 | 2 |
| #31771441 | 2 |
| #31771440 | 2 |
| 2−1+4×D269 | 2 |
| H(5,𝔽2)×D269 | 2 |
| C2×#41×D269 | 2 |
| #585×C269 | 2 |
| C2×#585 | 269 |