Finite Group DatabasebetaFinite Group DatabasebetaOrder 20864.
1348 conjugacy classes of subgroups; only the 213 normal subgroups are shown.
This group has 1348 conjugacy classes of subgroups — too many to draw. Only its 213 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D163 |
| commutator, socle, cosocle | C23×C163 | C24 |
| Fitting | C23×D4×C163 | C2 |
| hypercenter | C23×D4 | D163 |
| hypocenter | C163 | #2433 |
| LCS | G ▷ C23×C163 ▷ C163 | |
| UCS | C1 ◁ C23 ◁ C23×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#19104731 | C2 | semidirect |
| 2. C2×#19104709 | C2 | semidirect |
| 3. C2×#19104704 | C2 | semidirect |
| 4. #23781573 | C2 | semidirect |
| 5. #23781568 | C2 | semidirect |
| 6. C4×C22⋊Dic163 | C2 | semidirect |
| 7. C23×D4×C163 | C2 | semidirect |
| 8. C22×D652 | C22 | semidirect |
| 10. C22×C22⋊D163 | C22 | semidirect |
| 14. C2×C4⋊Dic163 | C22 | semidirect |
| 16. C2×C22⋊Dic163 | C22 | semidirect |
| 17. C2×(C2×D163)⋊C4 | C22 | semidirect |
| 18. C2×C4×Dic163 | C22 | semidirect |
| 19. C163⋊#95 | C22 | semidirect |
| 38. C22⋊Dic163 | D4 | semidirect |
| 52. C2×D4×C163 | D4 | semidirect |
| 54. C22×C4×C163 | D4 | semidirect |
| 68. C2×Dic163 | C2×D4 | semidirect |
| 99. C22×C163 | #127 | semidirect |
| 109. C4×C163 | #120 | semidirect |
| 119. C163 | #2433 | semidirect |
| 121. C23×D4 | D163 | semidirect |
| 165. C2×D4 | C22⋊D163 | semidirect |
| 169. C22×C4 | C22⋊D163 | semidirect |
| 223. C22 | #19104731 | semidirect |
| 231. C4 | #19104704 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#19104731 | 2 |
| C2×#19104709 | 2 |
| C2×#19104704 | 2 |
| #23781573 | 2 |
| #23781568 | 2 |
| C4×C22⋊Dic163 | 2 |
| C23×D4×C163 | 2 |
| #2433 | 163 |