Finite Group DatabasebetaFinite Group DatabasebetaOrder 20864.
852 conjugacy classes of subgroups; only the 161 normal subgroups are shown.
This group has 852 conjugacy classes of subgroups — too many to draw. Only its 161 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 | #393×C163 | C2 |
| prime radical | C2×#19104708 | C2 |
| hypercenter | #393 | D163 |
| hypocenter | C163 | #2436 |
| LCS | G ▷ C23×C163 ▷ C163 | |
| UCS | C1 ◁ C23 ◁ #393 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×Q8×D163 | C2 | semidirect |
| 2. C2×#19104721 | C2 | semidirect |
| 4. #23781578 | C2 | semidirect |
| 5. #23781577 | C2 | semidirect |
| 6. #23781576 | C2 | semidirect |
| 7. #23781569 | C2 | semidirect |
| 8. #23781543 | C2 | semidirect |
| 9. #23781541 | C2 | semidirect |
| 10. #23781512 | C2 | semidirect |
| 11. #393×C163 | C2 | semidirect |
| 45. C2×Dic326 | D4 | semidirect |
| 141. C2×D163 | #124 | semidirect |
| 158. C163 | #2436 | semidirect |
| 166. #393 | D163 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q8×D163 | 2 |
| C2×#19104721 | 2 |
| C2×#19104708 | 2 |
| #23781578 | 2 |
| #23781577 | 2 |
| #23781576 | 2 |
| #23781569 | 2 |
| #23781543 | 2 |
| #23781541 | 2 |
| #23781512 | 2 |
| #393×C163 | 2 |
| #2436 | 163 |