Finite Group DatabasebetaFinite Group DatabasebetaOrder 34688.
1492 conjugacy classes of subgroups; only the 553 normal subgroups are shown.
This group has 1492 conjugacy classes of subgroups — too many to draw. Only its 553 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×D271 |
| commutator, socle, cosocle | C22×C271 | C25 |
| Fitting | C2×2−1+4×C271 | C2 |
| hypercenter | C2×2−1+4 | D271 |
| hypocenter | C271 | #3486 |
| LCS | G ▷ C22×C271 ▷ C271 | |
| UCS | C1 ◁ C22 ◁ C2×2−1+4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×Q8×D271 | C2 | semidirect |
| 2. Q8×C22⋊D271 | C2 | semidirect |
| 3. #31774554 | C2 | semidirect |
| 4. #31774468 | C2 | semidirect |
| 5. #31774404 | C2 | semidirect |
| 6. #31774375 | C2 | semidirect |
| 7. C2×2−1+4×C271 | C2 | semidirect |
| 12. #22576158 | C22 | semidirect |
| 13. #22576147 | C22 | semidirect |
| 15. #22576139 | C22 | semidirect |
| 19. Q8×Dic271 | C22 | semidirect |
| 24. C2×#41×C271 | C22 | semidirect |
| 26. C22×Q8×C271 | C22 | semidirect |
| 44. C2×Q8×C271 | D4 | semidirect |
| 78. C2×D271 | 2−1+4 | semidirect |
| 88. C271 | #3486 | semidirect |
| 90. C2×2−1+4 | D271 | semidirect |
| 99. C2×#41 | C2×D271 | semidirect |
| 104. C22×Q8 | C2×D271 | semidirect |
| 123. C2×Q8 | C22⋊D271 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q8×D271 | 2 |
| Q8×C22⋊D271 | 2 |
| #31774554 | 2 |
| #31774468 | 2 |
| #31774404 | 2 |
| #31774375 | 2 |
| C2×2−1+4×C271 | 2 |
| #3486 | 271 |