Finite Group DatabasebetaFinite Group DatabasebetaOrder 20864.
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×D163 |
| commutator, socle, cosocle | C2×C163 | C26 |
| Frattini | C2 | C25×D163 |
| Fitting | #585×C163 | C2 |
| hypercenter | #585 | D163 |
| hypocenter | C163 | C2×#585 |
| LCS | G ▷ C2×C163 ▷ C163 | |
| UCS | C1 ◁ C4 ◁ #585 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #23782502 | C2 | semidirect |
| 2. #23782501 | C2 | semidirect |
| 3. #23782500 | C2 | semidirect |
| 4. 2−1+4×D163 | C2 | semidirect |
| 5. H(5,𝔽2)×D163 | C2 | semidirect |
| 6. C2×#41×D163 | C2 | semidirect |
| 7. #585×C163 | C2 | semidirect |
| 8. #19104783 | C22 | semidirect |
| 9. #19104782 | C22 | semidirect |
| 10. #19104781 | C22 | semidirect |
| 11. #19104780 | C22 | semidirect |
| 12. C2×#14579700 | C22 | semidirect |
| 13. C2×#14579699 | C22 | semidirect |
| 14. C2×#14579698 | C22 | semidirect |
| 15. #41×D163 | C22 | semidirect |
| 16. #41×D163 | C22 | semidirect |
| 17. #41×D163 | C22 | semidirect |
| 18. #41×D163 | C22 | semidirect |
| 19. C2×Q8×D163 | C22 | semidirect |
| 20. C2×D4×D163 | C22 | semidirect |
| 21. C22×C4×D163 | C22 | semidirect |
| 22. 2−1+4×C163 | C22 | semidirect |
| 23. H(5,𝔽2)×C163 | C22 | semidirect |
| 24. C2×#41×C163 | C22 | semidirect |
| 26. #14579700 | C23 | semidirect |
| 27. #14579700 | C23 | semidirect |
| 28. #14579699 | C23 | semidirect |
| 29. #14579699 | C23 | semidirect |
| 30. #14579698 | C23 | semidirect |
| 31. #14579698 | C23 | semidirect |
| 32. C2×Dic326 | C23 | semidirect |
| 34. Q8×D163 | C23 | semidirect |
| 35. C2×D652 | C23 | semidirect |
| 36. C2×C22⋊D163 | C23 | semidirect |
| 38. D4×D163 | C23 | semidirect |
| 39. C22×Dic163 | C23 | semidirect |
| 42. C2×C4×D163 | C23 | semidirect |
| 43. C2×C4×D163 | C23 | semidirect |
| 44. #41×C163 | C23 | semidirect |
| 45. #41×C163 | C23 | semidirect |
| 46. C2×Q8×C163 | C23 | semidirect |
| 47. C2×D4×C163 | C23 | semidirect |
| 48. C22×C4×C163 | C23 | semidirect |
| 80. D163 | #585 | direct (N × H) |
| 81. C163 | C2×#585 | semidirect |
| 86. #585 | D163 | direct (N × H) |
| 92. 2−1+4 | C2×D163 | semidirect |
| 95. H(5,𝔽2) | C2×D163 | semidirect |
| 103. C2×#41 | C2×D163 | semidirect |
| 114. #41 | C22×D163 | semidirect |
| 115. #41 | C22×D163 | semidirect |
| 119. C2×Q8 | C22×D163 | semidirect |
| 124. C2×D4 | C22×D163 | semidirect |
| 130. C22×C4 | C22×D163 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23782502 | 2 |
| #23782501 | 2 |
| #23782500 | 2 |
| 2−1+4×D163 | 2 |
| H(5,𝔽2)×D163 | 2 |
| C2×#41×D163 | 2 |
| #585×C163 | 2 |
| C2×#585 | 163 |