Finite Group DatabasebetaFinite Group DatabasebetaOrder 20864.
524 conjugacy classes of subgroups; only the 153 normal subgroups are shown.
This group has 524 conjugacy classes of subgroups — too many to draw. Only its 153 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 | #390×C163 | C2 |
| prime radical | C22×D4×C163 | C22 |
| hypercenter | #390 | D163 |
| hypocenter | C163 | #2861 |
| LCS | G ▷ C23×C163 ▷ C163 | |
| UCS | C1 ◁ C23 ◁ #390 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#19104726 | C2 | semidirect |
| 3. #23781571 | C2 | semidirect |
| 4. #23781570 | C2 | semidirect |
| 6. #23781558 | C2 | semidirect |
| 7. #23781553 | C2 | semidirect |
| 8. C42×Dic163 | C2 | semidirect |
| 109. C163 | #2861 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#19104726 | 2 |
| C2×#19104722 | 2 |
| #23781571 | 2 |
| #23781570 | 2 |
| #23781568 | 2 |
| #23781558 | 2 |
| #23781553 | 2 |
| C42×Dic163 | 2 |
| #390×C163 | 2 |
| #2861 | 163 |