Finite Group DatabasebetaFinite Group DatabasebetaOrder 20864.
1058 conjugacy classes of subgroups; only the 451 normal subgroups are shown.
This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 451 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D163 |
| commutator, socle, cosocle | C22×C163 | C25 |
| Fitting | H(3,𝔽4)×C163 | C2 |
| hypercenter | H(3,𝔽4) | D163 |
| hypocenter | C163 | #3529 |
| LCS | G ▷ C22×C163 ▷ C163 | |
| UCS | C1 ◁ C22 ◁ H(3,𝔽4) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #23782415 | C2 | semidirect |
| 2. #23782409 | C2 | semidirect |
| 3. #23782404 | C2 | semidirect |
| 4. #23782214 | C2 | semidirect |
| 5. #23782194 | C2 | semidirect |
| 6. #23782171 | C2 | semidirect |
| 7. H(3,𝔽4)×C163 | C2 | semidirect |
| 8. C2×#14579700 | C22 | semidirect |
| 10. C23×Dic163 | C22 | semidirect |
| 11. #19104733 | C22 | semidirect |
| 13. #19104728 | C22 | semidirect |
| 14. #19104727 | C22 | semidirect |
| 15. #19104723 | C22 | semidirect |
| 16. #19104719 | C22 | semidirect |
| 17. #19104717 | C22 | semidirect |
| 19. #19104712 | C22 | semidirect |
| 20. #19104711 | C22 | semidirect |
| 21. #19104708 | C22 | semidirect |
| 24. #19104704 | C22 | semidirect |
| 25. Q8×Dic163 | C22 | semidirect |
| 27. (C22⋊D163)⋊C4 | C22 | semidirect |
| 28. #19104691 | C22 | semidirect |
| 29. C42×D163 | C22 | semidirect |
| 97. C163 | #3529 | semidirect |
| 101. H(3,𝔽4) | D163 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23782415 | 2 |
| #23782409 | 2 |
| #23782404 | 2 |
| #23782214 | 2 |
| #23782194 | 2 |
| #23782171 | 2 |
| H(3,𝔽4)×C163 | 2 |
| #3529 | 163 |