Finite Group DatabasebetaFinite Group DatabasebetaOrder 34624.
240 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 93 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×D541 |
| commutator, socle, cosocle | C22×C541 | C24 |
| Fitting | #123×C541 | C2 |
| hypercenter | #123 | D541 |
| hypocenter | C541 | #540 |
| LCS | G ▷ C22×C541 ▷ C541 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #31772624 | C2 | semidirect |
| 2. #31772618 | C2 | semidirect |
| 3. #31772613 | C2 | semidirect |
| 4. #31772612 | C2 | semidirect |
| 5. #31772610 | C2 | semidirect |
| 6. #31772603 | C2 | semidirect |
| 7. #31772602 | C2 | semidirect |
| 8. #31772601 | C2 | semidirect |
| 9. #31772601 | C2 | semidirect |
| 10. #31772601 | C2 | semidirect |
| 11. #31772600 | C2 | semidirect |
| 12. C4×C22⋊D541 | C2 | semidirect |
| 13. #31772588 | C2 | semidirect |
| 14. #31772586 | C2 | semidirect |
| 15. #123×C541 | C2 | semidirect |
| 115. C541 | #540 | semidirect |
| 122. #123 | D541 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #31772624 | 2 |
| #31772618 | 2 |
| #31772613 | 2 |
| #31772612 | 2 |
| #31772610 | 2 |
| #31772603 | 2 |
| #31772602 | 2 |
| #31772601 | 2 |
| #31772600 | 2 |
| C4×C22⋊D541 | 2 |
| #31772588 | 2 |
| #31772586 | 2 |
| #123×C541 | 2 |
| #540 | 541 |