Finite Group DatabasebetaFinite Group DatabasebetaOrder 34688.
852 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 852 conjugacy classes of subgroups — too many to draw. Only its 181 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, prime radical | C2×#121×C271 | C2 |
| hypercenter | C2×#121 | D271 |
| hypocenter | C271 | #2436 |
| LCS | G ▷ C23×C271 ▷ C271 | |
| UCS | C1 ◁ C23 ◁ C2×#121 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×Dic542 | C2 | semidirect |
| 2. C2×#22576147 | C2 | semidirect |
| 3. #31773809 | C2 | semidirect |
| 4. #31773802 | C2 | semidirect |
| 5. #31773777 | C2 | semidirect |
| 6. C4×C22⋊Dic271 | C2 | semidirect |
| 30. C2×Dic542 | D4 | semidirect |
| 85. C22×C271 | #124 | semidirect |
| 101. C271 | #2436 | semidirect |
| 187. C22 | #22576147 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Dic542 | 2 |
| C2×#22576147 | 2 |
| #31773809 | 2 |
| #31773802 | 2 |
| #31773777 | 2 |
| C4×C22⋊Dic271 | 2 |
| C2×#121×C271 | 2 |
| #2436 | 271 |