Finite Group DatabasebetaFinite Group DatabasebetaOrder 20864.
432 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 432 conjugacy classes of subgroups — too many to draw. Only its 131 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 | #401×C163 | C2 |
| prime radical | C23×D163 | C23 |
| hypercenter | #401 | D163 |
| hypocenter | C163 | #2401 |
| LCS | G ▷ C23×C163 ▷ C163 | |
| UCS | C1 ◁ C23 ◁ #401 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #23781606 | C2 | semidirect |
| 4. #23781516 | C2 | semidirect |
| 6. #23781510 | C2 | semidirect |
| 7. #401×C163 | C2 | semidirect |
| 70. C163 | #2401 | semidirect |
| 72. #401 | D163 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23781606 | 2 |
| #23781594 | 2 |
| #23781589 | 2 |
| #23781516 | 2 |
| #23781512 | 2 |
| #23781510 | 2 |
| #401×C163 | 2 |
| #2401 | 163 |