Finite Group DatabasebetaFinite Group DatabasebetaOrder 34688.
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×D271 |
| commutator, socle, cosocle | C22×C271 | C25 |
| Frattini | C22 | C24×D271 |
| Fitting | C22×C42×C271 | C2 |
| hypercenter | C22×C42 | D271 |
| hypocenter | C271 | #3444 |
| LCS | G ▷ C22×C271 ▷ C271 | |
| UCS | C1 ◁ C42 ◁ C22×C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #31774454 | C2 | semidirect |
| 2. #31774447 | C2 | semidirect |
| 3. #31774431 | C2 | semidirect |
| 4. #31774408 | C2 | semidirect |
| 5. C4×#16864654 | C2 | semidirect |
| 6. C22×C42×C271 | C2 | semidirect |
| 8. #22576160 | C22 | semidirect |
| 9. #22576157 | C22 | semidirect |
| 10. #22576154 | C22 | semidirect |
| 11. #22576152 | C22 | semidirect |
| 12. #22576148 | C22 | semidirect |
| 17. C4×Dic542 | C22 | semidirect |
| 18. C4×D1084 | C22 | semidirect |
| 21. #22576117 | C22 | semidirect |
| 22. C42×D271 | C22 | semidirect |
| 60. C2×C4×C271 | #41 | semidirect |
| 76. C271 | #3444 | semidirect |
| 83. C22×C42 | D271 | semidirect |
| 137. C2×C4 | #16864654 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #31774454 | 2 |
| #31774447 | 2 |
| #31774431 | 2 |
| #31774408 | 2 |
| C4×#16864654 | 2 |
| C22×C42×C271 | 2 |
| #3444 | 271 |