Finite Group DatabasebetaFinite Group DatabasebetaOrder 20864.
688 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 688 conjugacy classes of subgroups — too many to draw. Only its 165 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 | #2854 |
| 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. C2×#19104719 | C2 | semidirect |
| 2. C2×#19104716 | C2 | semidirect |
| 4. C22×C4×Dic163 | C2 | semidirect |
| 5. #23781565 | C2 | semidirect |
| 6. #23781562 | C2 | semidirect |
| 7. #23781558 | C2 | semidirect |
| 8. #23781554 | C2 | semidirect |
| 9. #23781544 | C2 | semidirect |
| 10. #23781523 | C2 | semidirect |
| 11. #386×C163 | C2 | semidirect |
| 49. C22×Dic163 | D4 | semidirect |
| 150. Dic163 | #123 | semidirect |
| 159. C163 | #2854 | semidirect |
| 170. #386 | D163 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#19104719 | 2 |
| C2×#19104716 | 2 |
| C2×#19104707 | 2 |
| C22×C4×Dic163 | 2 |
| #23781565 | 2 |
| #23781562 | 2 |
| #23781558 | 2 |
| #23781554 | 2 |
| #23781544 | 2 |
| #23781523 | 2 |
| #386×C163 | 2 |
| #2854 | 163 |