Finite Group DatabasebetaFinite Group DatabasebetaOrder 34688.
852 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 852 conjugacy classes of subgroups — too many to draw. Only its 173 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D271 |
| commutator, socle, cosocle | C23×C271 | C24 |
| Fitting | #391×C271 | C2 |
| prime radical | C2×#22576135 | C2 |
| hypercenter | #391 | D271 |
| hypocenter | C271 | #2436 |
| LCS | G ▷ C23×C271 ▷ C271 | |
| UCS | C1 ◁ C23 ◁ #391 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×Q8×D271 | C2 | semidirect |
| 2. C2×#22576149 | C2 | semidirect |
| 3. C2×#22576148 | C2 | semidirect |
| 5. #31773810 | C2 | semidirect |
| 6. #31773802 | C2 | semidirect |
| 7. #31773775 | C2 | semidirect |
| 8. C4×(C2×D271)⋊C4 | C2 | semidirect |
| 9. #391×C271 | C2 | semidirect |
| 40. C2×Dic542 | D4 | semidirect |
| 63. C2×Q8×C271 | D4 | semidirect |
| 122. C2×D271 | #124 | semidirect |
| 137. C271 | #2436 | semidirect |
| 145. #391 | D271 | semidirect |
| 182. C2×Q8 | D1084 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q8×D271 | 2 |
| C2×#22576149 | 2 |
| C2×#22576148 | 2 |
| C2×#22576135 | 2 |
| #31773810 | 2 |
| #31773802 | 2 |
| #31773775 | 2 |
| C4×(C2×D271)⋊C4 | 2 |
| #391×C271 | 2 |
| #2436 | 271 |