Finite Group DatabasebetaFinite Group DatabasebetaOrder 20864.
1056 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 173 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 | C22×C22⋊C4×C163 | C2 |
| hypercenter | C22×C22⋊C4 | D163 |
| hypocenter | C163 | #2404 |
| LCS | G ▷ C23×C163 ▷ C163 | |
| UCS | C1 ◁ C23 ◁ C22×C22⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#14579699 | C2 | semidirect |
| 2. C2×#19104704 | C2 | semidirect |
| 3. #23781577 | C2 | semidirect |
| 4. #23781574 | C2 | semidirect |
| 5. #23781544 | C2 | semidirect |
| 6. #23781541 | C2 | semidirect |
| 7. #23781528 | C2 | semidirect |
| 8. C22×C22⋊C4×C163 | C2 | semidirect |
| 11. C22×Dic326 | C22 | semidirect |
| 12. C22×D652 | C22 | semidirect |
| 15. C22×C4×D163 | C22 | semidirect |
| 19. C2×(C2×D163)⋊C4 | C22 | semidirect |
| 20. C163⋊#95 | C22 | semidirect |
| 21. C163⋊#95 | C22 | semidirect |
| 22. C163⋊#95 | C22 | semidirect |
| 40. C2×C22⋊D163 | D4 | semidirect |
| 58. C22×C4×C163 | D4 | semidirect |
| 92. C23×C163 | #41 | semidirect |
| 133. C163 | #2404 | semidirect |
| 137. C22×C22⋊C4 | D163 | semidirect |
| 189. C22×C4 | C22⋊D163 | semidirect |
| 215. C23 | #14579699 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#14579699 | 2 |
| C2×#19104704 | 2 |
| #23781577 | 2 |
| #23781574 | 2 |
| #23781544 | 2 |
| #23781541 | 2 |
| #23781528 | 2 |
| C22×C22⋊C4×C163 | 2 |
| #2404 | 163 |