Finite Group DatabasebetaFinite Group DatabasebetaOrder 20864.
924 conjugacy classes of subgroups; only the 237 normal subgroups are shown.
This group has 924 conjugacy classes of subgroups — too many to draw. Only its 237 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 | C22×C42×C163 | C2 |
| hypercenter | C22×C42 | D163 |
| hypocenter | C163 | #2856 |
| LCS | G ▷ C23×C163 ▷ C163 | |
| UCS | C1 ◁ C23 ◁ C22×C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#19104732 | C2 | semidirect |
| 2. C2×#19104709 | C2 | semidirect |
| 3. #23781567 | C2 | semidirect |
| 4. #23781535 | C2 | semidirect |
| 5. C22×C42×C163 | C2 | semidirect |
| 27. C22×C4×C163 | D4 | semidirect |
| 31. C42×C163 | D4 | semidirect |
| 51. C22×C163 | #127 | semidirect |
| 59. C4×C163 | #121 | semidirect |
| 66. C163 | #2856 | semidirect |
| 70. C22×C42 | D163 | semidirect |
| 92. C22×C4 | D652 | semidirect |
| 98. C42 | C22⋊D163 | semidirect |
| 121. C22 | #19104732 | semidirect |
| 127. C4 | #19104709 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#19104732 | 2 |
| C2×#19104709 | 2 |
| #23781567 | 2 |
| #23781535 | 2 |
| C22×C42×C163 | 2 |
| #2856 | 163 |