Finite Group DatabasebetaFinite Group DatabasebetaOrder 20864.
836 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 836 conjugacy classes of subgroups — too many to draw. Only its 173 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×#14579698 | C2 |
| hypercenter | #390 | D163 |
| hypocenter | C163 | #2432 |
| LCS | G ▷ C23×C163 ▷ C163 | |
| UCS | C1 ◁ C23 ◁ #390 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#19104735 | C2 | semidirect |
| 3. C2×#19104722 | C2 | semidirect |
| 4. C2×#19104715 | C2 | semidirect |
| 5. #23781581 | C2 | semidirect |
| 6. #23781570 | C2 | semidirect |
| 7. #23781540 | C2 | semidirect |
| 8. C4×(C2×D163)⋊C4 | C2 | semidirect |
| 9. #390×C163 | C2 | semidirect |
| 106. C2×C4×C163 | #41 | semidirect |
| 136. C163 | #2432 | semidirect |
| 144. #390 | D163 | semidirect |
| 243. C2×C4 | #14579698 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#14579698 | 2 |
| C2×#19104735 | 2 |
| C2×#19104722 | 2 |
| C2×#19104715 | 2 |
| #23781581 | 2 |
| #23781570 | 2 |
| #23781540 | 2 |
| C4×(C2×D163)⋊C4 | 2 |
| #390×C163 | 2 |
| #2432 | 163 |