Finite Group DatabasebetaFinite Group DatabasebetaOrder 20864.
1056 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 159 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 | #392×C163 | C2 |
| hypercenter | #392 | D163 |
| hypocenter | C163 | #2404 |
| LCS | G ▷ C23×C163 ▷ C163 | |
| UCS | C1 ◁ C23 ◁ #392 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#14579698 | C2 | semidirect |
| 2. C2×#19104703 | C2 | semidirect |
| 3. C2×C22⋊C4×D163 | C2 | semidirect |
| 4. #23781578 | C2 | semidirect |
| 5. #23781577 | C2 | semidirect |
| 6. #23781573 | C2 | semidirect |
| 7. #23781545 | C2 | semidirect |
| 8. #23781542 | C2 | semidirect |
| 9. #23781541 | C2 | semidirect |
| 10. #23781527 | C2 | semidirect |
| 11. #392×C163 | C2 | semidirect |
| 15. C22×Dic326 | C22 | semidirect |
| 18. C23×Dic163 | C22 | semidirect |
| 24. C2×C22⋊Dic163 | C22 | semidirect |
| 30. C163⋊#95 | C22 | semidirect |
| 31. C163⋊#95 | C22 | semidirect |
| 32. C163⋊#95 | C22 | semidirect |
| 33. C163⋊#95 | C22 | semidirect |
| 35. C22×D4×C163 | C22 | semidirect |
| 37. C2×C22⋊C4×C163 | C22 | semidirect |
| 38. #95×C163 | C22 | semidirect |
| 54. C2×C22⋊D163 | D4 | semidirect |
| 65. C2×C4×D163 | D4 | semidirect |
| 99. C22×D163 | #41 | semidirect |
| 127. C23×C163 | #41 | semidirect |
| 182. C163 | #2404 | semidirect |
| 190. #392 | D163 | semidirect |
| 202. C22×D4 | C2×D163 | semidirect |
| 216. C2×C22⋊C4 | C2×D163 | semidirect |
| 221. #95 | C2×D163 | semidirect |
| 294. C23 | #14579698 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#14579698 | 2 |
| C2×#19104703 | 2 |
| C2×C22⋊C4×D163 | 2 |
| #23781578 | 2 |
| #23781577 | 2 |
| #23781573 | 2 |
| #23781545 | 2 |
| #23781542 | 2 |
| #23781541 | 2 |
| #23781527 | 2 |
| #392×C163 | 2 |
| #2404 | 163 |