Finite Group DatabasebetaFinite Group DatabasebetaOrder 20864.
3422 conjugacy classes of subgroups; only the 245 normal subgroups are shown.
This group has 3422 conjugacy classes of subgroups — too many to draw. Only its 245 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 | C2×#120×C163 | C2 |
| hypercenter | C2×#120 | D163 |
| hypocenter | C163 | #2846 |
| LCS | G ▷ C23×C163 ▷ C163 | |
| UCS | C1 ◁ C23 ◁ C2×#120 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C25×D163 | C2 | semidirect |
| 2. C2×#19104703 | C2 | semidirect |
| 3. #23781521 | C2 | semidirect |
| 4. #23781520 | C2 | semidirect |
| 5. C2×#120×C163 | C2 | semidirect |
| 9. C22×C22⋊D163 | C22 | semidirect |
| 11. C2×C22⋊Dic163 | C22 | semidirect |
| 12. C2×(C2×D163)⋊C4 | C22 | semidirect |
| 14. C22×D4×C163 | C22 | semidirect |
| 16. C2×C22⋊C4×C163 | C22 | semidirect |
| 22. C23×D163 | D4 | semidirect |
| 25. C22×Dic163 | C23 | semidirect |
| 29. C24×C163 | D4 | semidirect |
| 31. C24×C163 | C23 | semidirect |
| 34. C22×C4×C163 | C23 | semidirect |
| 62. C22×C163 | #120 | semidirect |
| 68. C2×D163 | #120 | semidirect |
| 76. C163 | #2846 | semidirect |
| 80. C2×#120 | D163 | semidirect |
| 88. C22×D4 | C2×D163 | semidirect |
| 93. C2×C22⋊C4 | C2×D163 | semidirect |
| 102. C24 | C22⋊D163 | semidirect |
| 104. C24 | C22×D163 | semidirect |
| 109. C22×C4 | C22×D163 | semidirect |
| 143. C22 | #19104703 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C25×D163 | 2 |
| C2×#19104703 | 2 |
| #23781521 | 2 |
| #23781520 | 2 |
| C2×#120×C163 | 2 |
| #2846 | 163 |