Finite Group DatabasebetaFinite Group DatabasebetaOrder 26176.
228 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 105 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×D409 |
| commutator, socle, cosocle | C22×C409 | C24 |
| Fitting | #125×C409 | C2 |
| prime radical | C2×D1636 | C22 |
| hypercenter | #125 | D409 |
| hypocenter | C409 | #554 |
| LCS | G ▷ C22×C409 ▷ C409 | |
| UCS | C1 ◁ C22 ◁ #125 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #27189716 | C2 | semidirect |
| 2. #27189708 | C2 | semidirect |
| 3. #27189697 | C2 | semidirect |
| 4. #27189695 | C2 | semidirect |
| 5. #27189695 | C2 | semidirect |
| 6. C4×Dic818 | C2 | semidirect |
| 10. C4⋊C4×D409 | C2 | semidirect |
| 11. #125×C409 | C2 | semidirect |
| 38. D1636 | Q8 | semidirect |
| 70. C409 | #554 | semidirect |
| 74. #125 | D409 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #27189716 | 2 |
| #27189708 | 2 |
| #27189697 | 2 |
| #27189695 | 2 |
| C4×Dic818 | 2 |
| C4×D1636 | 2 |
| #27189679 | 2 |
| C4⋊C4×D409 | 2 |
| #125×C409 | 2 |
| #554 | 409 |