Finite Group DatabasebetaFinite Group DatabasebetaOrder 26816.
252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 252 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×D419 |
| commutator, socle, cosocle | C22×C419 | C24 |
| Fitting | #123×C419 | C2 |
| hypercenter | #123 | D419 |
| hypocenter | C419 | #560 |
| 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. #27211714 | C2 | semidirect |
| 2. #27211713 | C2 | semidirect |
| 3. #27211700 | C2 | semidirect |
| 4. #27211693 | C2 | semidirect |
| 5. #27211692 | C2 | semidirect |
| 6. #27211691 | C2 | semidirect |
| 7. #27211688 | C2 | semidirect |
| 8. #27211686 | C2 | semidirect |
| 9. #123×C419 | C2 | semidirect |
| 13. C2×D1676 | C22 | semidirect |
| 16. C2×C22⋊D419 | C22 | semidirect |
| 17. C2×C4×D419 | C22 | semidirect |
| 18. C4⋊Dic419 | C22 | semidirect |
| 19. Dic419⋊C4 | C22 | semidirect |
| 24. (C2×D419)⋊C4 | 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 |
| 74. C419 | #560 | semidirect |
| 79. #123 | D419 | semidirect |
| 94. C22×C4 | C2×D419 | semidirect |
| 97. C4⋊C4 | C2×D419 | semidirect |
| 104. C22⋊C4 | C2×D419 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #27211714 | 2 |
| #27211713 | 2 |
| #27211700 | 2 |
| #27211693 | 2 |
| #27211692 | 2 |
| #27211691 | 2 |
| #27211688 | 2 |
| #27211686 | 2 |
| #123×C419 | 2 |
| #560 | 419 |