Finite Group DatabasebetaFinite Group DatabasebetaOrder 34432.
1536 conjugacy classes of subgroups; only the 549 normal subgroups are shown.
This group has 1536 conjugacy classes of subgroups — too many to draw. Only its 549 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 | C24×D269 |
| commutator, socle, cosocle | C22×C269 | C25 |
| Fitting | #519×C269 | C2 |
| hypercenter | #519 | D269 |
| hypocenter | C269 | #3487 |
| LCS | G ▷ C22×C269 ▷ C269 | |
| UCS | C1 ◁ C22 ◁ #519 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#22575501 | C2 | semidirect |
| 2. C22×#16864403 | C2 | semidirect |
| 3. #31771277 | C2 | semidirect |
| 4. #31771267 | C2 | semidirect |
| 5. #31771174 | C2 | semidirect |
| 6. #31771117 | C2 | semidirect |
| 7. #519×C269 | C2 | semidirect |
| 11. C2×#16864403 | C22 | semidirect |
| 13. C2×Q8×D269 | C22 | semidirect |
| 16. #22575450 | C22 | semidirect |
| 17. #22575441 | C22 | semidirect |
| 19. #22575433 | C22 | semidirect |
| 20. #22575428 | C22 | semidirect |
| 21. #22575428 | C22 | semidirect |
| 23. C4×D1076 | C22 | semidirect |
| 25. C4×Q8×C269 | C22 | semidirect |
| 45. C2×Q8×C269 | D4 | semidirect |
| 87. C269 | #3487 | semidirect |
| 95. #519 | D269 | semidirect |
| 112. C4×Q8 | C2×D269 | semidirect |
| 121. C2×Q8 | D1076 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#22575501 | 2 |
| C22×#16864403 | 2 |
| #31771277 | 2 |
| #31771267 | 2 |
| #31771174 | 2 |
| #31771117 | 2 |
| #519×C269 | 2 |
| #3487 | 269 |