Finite Group DatabasebetaFinite Group DatabasebetaOrder 20864.
568 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 568 conjugacy classes of subgroups — too many to draw. Only its 143 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 | #397×C163 | C2 |
| prime radical | C22×C22⋊D163 | C22 |
| hypercenter | #397 | D163 |
| hypocenter | C163 | #2398 |
| LCS | G ▷ C23×C163 ▷ C163 | |
| UCS | C1 ◁ C23 ◁ #397 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#19104700 | C2 | semidirect |
| 3. #23781606 | C2 | semidirect |
| 4. #23781591 | C2 | semidirect |
| 6. #23781578 | C2 | semidirect |
| 7. #23781577 | C2 | semidirect |
| 8. C22⋊C4×Dic163 | C2 | semidirect |
| 9. #23781512 | C2 | semidirect |
| 10. #23781510 | C2 | semidirect |
| 11. #397×C163 | C2 | semidirect |
| 138. C163 | #2398 | semidirect |
| 144. #397 | D163 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#19104700 | 2 |
| C2×(C22⋊D163)⋊C4 | 2 |
| #23781606 | 2 |
| #23781591 | 2 |
| #23781580 | 2 |
| #23781578 | 2 |
| #23781577 | 2 |
| C22⋊C4×Dic163 | 2 |
| #23781512 | 2 |
| #23781510 | 2 |
| #397×C163 | 2 |
| #2398 | 163 |