Finite Group DatabasebetaFinite Group DatabasebetaOrder 20864.
908 conjugacy classes of subgroups; only the 203 normal subgroups are shown.
This group has 908 conjugacy classes of subgroups — too many to draw. Only its 203 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 | #399×C163 | C2 |
| prime radical | C24×D163 | C22 |
| hypercenter | #399 | D163 |
| hypocenter | C163 | C2×#399 |
| LCS | G ▷ C23×C163 ▷ C163 | |
| UCS | C1 ◁ C23 ◁ #399 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C4⋊C4×D163 | C2 | semidirect |
| 3. #23781602 | C2 | semidirect |
| 4. #23781601 | C2 | semidirect |
| 5. #23781600 | C2 | semidirect |
| 6. #95×D163 | C2 | semidirect |
| 7. #399×C163 | C2 | semidirect |
| 11. C2×C4⋊Dic163 | C22 | semidirect |
| 12. C2×Dic163⋊C4 | C22 | semidirect |
| 18. C163⋊#95 | C22 | semidirect |
| 19. C2×C4⋊C4×C163 | C22 | semidirect |
| 21. #95×C163 | C22 | semidirect |
| 88. D163 | #399 | direct (N × H) |
| 89. C163 | C2×#399 | semidirect |
| 96. #399 | D163 | direct (N × H) |
| 104. C2×C4⋊C4 | C2×D163 | semidirect |
| 111. #95 | C2×D163 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4⋊C4×D163 | 2 |
| C2×C22⋊C4×D163 | 2 |
| #23781602 | 2 |
| #23781601 | 2 |
| #23781600 | 2 |
| #95×D163 | 2 |
| #399×C163 | 2 |
| C2×#399 | 163 |