Finite Group DatabasebetaFinite Group DatabasebetaOrder 20864.
516 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 165 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 | C43×C163 | C2 |
| prime radical | C22×D652 | C22 |
| hypercenter | C43 | D163 |
| hypocenter | C163 | #2860 |
| LCS | G ▷ C23×C163 ▷ C163 | |
| UCS | C1 ◁ C23 ◁ C43 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#19104729 | C2 | semidirect |
| 3. #23781563 | C2 | semidirect |
| 4. #23781556 | C2 | semidirect |
| 5. C43×C163 | C2 | semidirect |
| 28. C42×C163 | D4 | semidirect |
| 52. C4×C163 | #126 | semidirect |
| 58. C163 | #2860 | semidirect |
| 64. C43 | D163 | semidirect |
| 87. C42 | D652 | semidirect |
| 112. C4 | #19104729 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#19104729 | 2 |
| #23781567 | 2 |
| #23781563 | 2 |
| #23781556 | 2 |
| C43×C163 | 2 |
| #2860 | 163 |