Finite Group DatabasebetaFinite Group DatabasebetaOrder 26816.
248 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 95 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×D419 |
| commutator, socle, cosocle | C22×C419 | C24 |
| Fitting | #123×C419 | C2 |
| prime radical | #27211686 | C2 |
| hypercenter | #123 | D419 |
| hypocenter | C419 | #551 |
| LCS | G ▷ C22×C419 ▷ C419 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #27211711 | C2 | semidirect |
| 2. #27211705 | C2 | semidirect |
| 3. #27211700 | C2 | semidirect |
| 4. #27211696 | C2 | semidirect |
| 6. C4×C22⋊D419 | C2 | semidirect |
| 7. #27211676 | C2 | semidirect |
| 8. C22⋊C4×D419 | C2 | semidirect |
| 9. #123×C419 | C2 | semidirect |
| 11. C2×Dic838 | C22 | semidirect |
| 15. C4⋊Dic419 | C22 | semidirect |
| 16. Dic419⋊C4 | C22 | semidirect |
| 17. Dic419⋊C4 | C22 | semidirect |
| 20. C22⋊Dic419 | C22 | semidirect |
| 24. C4×Dic419 | C22 | semidirect |
| 25. C4×Dic419 | C22 | semidirect |
| 27. C22×C4×C419 | C22 | semidirect |
| 28. C4⋊C4×C419 | C22 | semidirect |
| 30. C22⋊C4×C419 | C22 | semidirect |
| 68. C419 | #551 | semidirect |
| 73. #123 | D419 | semidirect |
| 85. C22×C4 | C2×D419 | semidirect |
| 89. C4⋊C4 | C2×D419 | semidirect |
| 97. C22⋊C4 | C2×D419 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #27211711 | 2 |
| #27211705 | 2 |
| #27211700 | 2 |
| #27211696 | 2 |
| #27211686 | 2 |
| C4×C22⋊D419 | 2 |
| #27211676 | 2 |
| C22⋊C4×D419 | 2 |
| #123×C419 | 2 |
| #551 | 419 |