Finite Group DatabasebetaFinite Group DatabasebetaOrder 34432.
5796 conjugacy classes of subgroups; only the 3201 normal subgroups are shown.
This group has 5796 conjugacy classes of subgroups — too many to draw. Only its 3201 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C25×D269 |
| commutator, socle, cosocle | C2×C269 | C26 |
| Fitting | C2×2−1+4×C269 | C2 |
| hypercenter | C2×2−1+4 | D269 |
| hypocenter | C269 | 2−1+6 |
| LCS | G ▷ C2×C269 ▷ C269 | |
| UCS | C1 ◁ C2 ◁ C2×2−1+4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #31771442 | C2 | semidirect |
| 2. #31771441 | C2 | semidirect |
| 3. C2×#22575501 | C2 | semidirect |
| 4. 2−1+4×D269 | C2 | semidirect |
| 5. C2×2−1+4×C269 | C2 | semidirect |
| 6. #22575501 | C22 | semidirect |
| 7. #22575501 | C22 | semidirect |
| 8. #22575500 | C22 | semidirect |
| 9. #22575499 | C22 | semidirect |
| 10. #22575498 | C22 | semidirect |
| 11. C2×#16864403 | C22 | semidirect |
| 12. C2×#16864402 | C22 | semidirect |
| 13. #41×D269 | C22 | semidirect |
| 14. C2×Q8×D269 | C22 | semidirect |
| 15. 2−1+4×C269 | C22 | semidirect |
| 16. C2×#41×C269 | C22 | semidirect |
| 17. C22×Q8×C269 | C22 | semidirect |
| 56. C269 | 2−1+6 | semidirect |
| 62. C2×2−1+4 | D269 | semidirect |
| 66. 2−1+4 | C2×D269 | semidirect |
| 72. C2×#41 | C2×D269 | semidirect |
| 74. C22×Q8 | C2×D269 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #31771442 | 2 |
| #31771441 | 2 |
| C2×#22575501 | 2 |
| 2−1+4×D269 | 2 |
| C2×2−1+4×C269 | 2 |
| 2−1+6 | 269 |