Finite Group DatabasebetaFinite Group DatabasebetaOrder 20864.
560 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 560 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 | #2395 |
| LCS | G ▷ C23×C163 ▷ C163 | |
| UCS | C1 ◁ C23 ◁ #397 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#19104693 | C2 | semidirect |
| 3. #23781605 | C2 | semidirect |
| 4. #23781601 | C2 | semidirect |
| 5. #23781596 | C2 | semidirect |
| 6. #23781594 | C2 | semidirect |
| 8. #23781516 | C2 | semidirect |
| 9. C22⋊C4×Dic163 | C2 | semidirect |
| 10. #23781512 | C2 | semidirect |
| 11. #397×C163 | C2 | semidirect |
| 137. C163 | #2395 | semidirect |
| 145. #397 | D163 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×(C22⋊D163)⋊C4 | 2 |
| C2×#19104693 | 2 |
| #23781605 | 2 |
| #23781601 | 2 |
| #23781596 | 2 |
| #23781594 | 2 |
| #23781580 | 2 |
| #23781516 | 2 |
| C22⋊C4×Dic163 | 2 |
| #23781512 | 2 |
| #397×C163 | 2 |
| #2395 | 163 |