Finite Group DatabasebetaFinite Group DatabasebetaOrder 20864.
852 conjugacy classes of subgroups; only the 169 normal subgroups are shown.
This group has 852 conjugacy classes of subgroups — too many to draw. Only its 169 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 | #386×C163 | C2 |
| prime radical | C2×#19104707 | C2 |
| hypercenter | #386 | D163 |
| hypocenter | C163 | #2436 |
| LCS | G ▷ C23×C163 ▷ C163 | |
| UCS | C1 ◁ C23 ◁ #386 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×Dic326 | C2 | semidirect |
| 2. C2×#19104721 | C2 | semidirect |
| 4. #23781578 | C2 | semidirect |
| 5. #23781577 | C2 | semidirect |
| 6. #23781569 | C2 | semidirect |
| 7. #23781545 | C2 | semidirect |
| 8. #23781544 | C2 | semidirect |
| 9. #23781542 | C2 | semidirect |
| 10. C22⋊C4×Dic163 | C2 | semidirect |
| 11. #386×C163 | C2 | semidirect |
| 46. C2×Dic326 | D4 | semidirect |
| 136. C22×C163 | #124 | semidirect |
| 163. C163 | #2436 | semidirect |
| 174. #386 | D163 | semidirect |
| 305. C22 | #19104721 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Dic326 | 2 |
| C2×#19104721 | 2 |
| C2×#19104707 | 2 |
| #23781578 | 2 |
| #23781577 | 2 |
| #23781569 | 2 |
| #23781545 | 2 |
| #23781544 | 2 |
| #23781542 | 2 |
| C22⋊C4×Dic163 | 2 |
| #386×C163 | 2 |
| #2436 | 163 |