Finite Group DatabasebetaFinite Group DatabasebetaOrder 2624.
196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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 | C23×D41 |
| commutator, socle, cosocle | C22×C41 | C24 |
| Fitting | #126×C41 | C2 |
| prime radical | C2×C22⋊D41 | C22 |
| hypercenter | #126 | D41 |
| hypocenter | C41 | #563 |
| LCS | G ▷ C22×C41 ▷ C41 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #13050164 | C2 | semidirect |
| 2. #13050163 | C2 | semidirect |
| 3. #13050159 | C2 | semidirect |
| 4. #13050158 | C2 | semidirect |
| 5. #13050157 | C2 | semidirect |
| 6. #13050155 | C2 | semidirect |
| 7. #13050154 | C2 | semidirect |
| 11. #13050144 | C2 | semidirect |
| 12. #13050142 | C2 | semidirect |
| 13. #13050141 | C2 | semidirect |
| 14. #13050141 | C2 | semidirect |
| 15. #126×C41 | C2 | semidirect |
| 98. C41 | #563 | semidirect |
| 104. #126 | D41 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13050164 | 2 |
| #13050163 | 2 |
| #13050159 | 2 |
| #13050158 | 2 |
| #13050157 | 2 |
| #13050155 | 2 |
| #13050154 | 2 |
| #13050150 | 2 |
| #13050148 | 2 |
| #13050146 | 2 |
| #13050144 | 2 |
| #13050142 | 2 |
| #13050141 | 2 |
| #126×C41 | 2 |
| #563 | 41 |