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 | C4×C22⋊C4×C163 | C2 |
| hypercenter | C4×C22⋊C4 | D163 |
| hypocenter | C163 | #2433 |
| LCS | G ▷ C23×C163 ▷ C163 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×D652 | C2 | semidirect |
| 2. C2×#19104732 | C2 | semidirect |
| 3. C2×#19104709 | C2 | semidirect |
| 4. C2×#19104705 | C2 | semidirect |
| 5. #23781575 | C2 | semidirect |
| 6. #23781567 | C2 | semidirect |
| 7. C4×C22⋊C4×C163 | C2 | semidirect |
| 13. C22×D652 | C22 | semidirect |
| 14. C22×D652 | C22 | semidirect |
| 19. C2×C4⋊Dic163 | C22 | semidirect |
| 20. C2×(C2×D163)⋊C4 | C22 | semidirect |
| 21. C2×(C2×D163)⋊C4 | C22 | semidirect |
| 24. C2×C42×C163 | C22 | semidirect |
| 25. #95×C163 | C22 | semidirect |
| 42. C2×D652 | D4 | semidirect |
| 51. C22×C4×C163 | D4 | semidirect |
| 55. C22⋊C4×C163 | D4 | semidirect |
| 89. C2×C4×C163 | C2×D4 | semidirect |
| 93. C22×C163 | #127 | semidirect |
| 110. C4×C163 | #120 | semidirect |
| 119. C163 | #2433 | semidirect |
| 127. C4×C22⋊C4 | D163 | semidirect |
| 144. C2×C42 | C2×D163 | semidirect |
| 145. #95 | C2×D163 | semidirect |
| 168. C22×C4 | D652 | semidirect |
| 175. C22⋊C4 | D652 | semidirect |
| 209. C2×C4 | C2×D652 | semidirect |
| 223. C22 | #19104732 | semidirect |
| 231. C4 | #19104705 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×D652 | 2 |
| C2×#19104732 | 2 |
| C2×#19104709 | 2 |
| C2×#19104705 | 2 |
| #23781575 | 2 |
| #23781567 | 2 |
| C4×C22⋊C4×C163 | 2 |
| #2433 | 163 |