Finite Group DatabasebetaFinite Group DatabasebetaOrder 20864.
1492 conjugacy classes of subgroups; only the 553 normal subgroups are shown.
This group has 1492 conjugacy classes of subgroups — too many to draw. Only its 553 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D163 |
| commutator, socle, cosocle | C22×C163 | C25 |
| Fitting | #519×C163 | C2 |
| hypercenter | #519 | D163 |
| hypocenter | C163 | #3486 |
| LCS | G ▷ C22×C163 ▷ C163 | |
| UCS | C1 ◁ C22 ◁ #519 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#19104783 | C2 | semidirect |
| 2. C22×Q8×D163 | C2 | semidirect |
| 3. Q8×D652 | C2 | semidirect |
| 4. #23782327 | C2 | semidirect |
| 5. #23782241 | C2 | semidirect |
| 6. #23782177 | C2 | semidirect |
| 7. #519×C163 | C2 | semidirect |
| 9. C2×#14579700 | C22 | semidirect |
| 14. C2×Q8×D163 | C22 | semidirect |
| 16. #19104735 | C22 | semidirect |
| 17. #19104723 | C22 | semidirect |
| 20. #19104715 | C22 | semidirect |
| 25. C4×Q8×C163 | C22 | semidirect |
| 45. C2×Q8×C163 | D4 | semidirect |
| 78. C2×D163 | 2−1+4 | semidirect |
| 88. C163 | #3486 | semidirect |
| 96. #519 | D163 | semidirect |
| 113. C4×Q8 | C2×D163 | semidirect |
| 124. C2×Q8 | D652 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#19104783 | 2 |
| C22×Q8×D163 | 2 |
| Q8×D652 | 2 |
| #23782327 | 2 |
| #23782241 | 2 |
| #23782177 | 2 |
| #519×C163 | 2 |
| #3486 | 163 |