Finite Group DatabasebetaFinite Group DatabasebetaOrder 34432.
1212 conjugacy classes of subgroups; only the 509 normal subgroups are shown.
This group has 1212 conjugacy classes of subgroups — too many to draw. Only its 509 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22×D269 |
| commutator, socle, cosocle | C22×C269 | C25 |
| Frattini | C22 | C24×D269 |
| Fitting | C22×C42×C269 | C2 |
| hypercenter | C22×C42 | D269 |
| hypocenter | C269 | #3444 |
| LCS | G ▷ C22×C269 ▷ C269 | |
| UCS | C1 ◁ C42 ◁ C22×C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #31771162 | C2 | semidirect |
| 2. #31771155 | C2 | semidirect |
| 3. #31771139 | C2 | semidirect |
| 4. #31771116 | C2 | semidirect |
| 5. C4×#16864402 | C2 | semidirect |
| 6. C22×C42×C269 | C2 | semidirect |
| 8. #22575453 | C22 | semidirect |
| 9. #22575450 | C22 | semidirect |
| 10. #22575447 | C22 | semidirect |
| 11. #22575445 | C22 | semidirect |
| 12. #22575441 | C22 | semidirect |
| 17. C4×Dic538 | C22 | semidirect |
| 18. C4×D1076 | C22 | semidirect |
| 21. #22575410 | C22 | semidirect |
| 22. C42×D269 | C22 | semidirect |
| 60. C2×C4×C269 | #41 | semidirect |
| 76. C269 | #3444 | semidirect |
| 83. C22×C42 | D269 | semidirect |
| 137. C2×C4 | #16864402 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #31771162 | 2 |
| #31771155 | 2 |
| #31771139 | 2 |
| #31771116 | 2 |
| C4×#16864402 | 2 |
| C22×C42×C269 | 2 |
| #3444 | 269 |