Finite Group DatabasebetaFinite Group DatabasebetaOrder 34688.
836 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 836 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 | #390×C271 | C2 |
| prime radical | C22×#16864653 | C2 |
| hypercenter | #390 | D271 |
| hypocenter | C271 | #2432 |
| LCS | G ▷ C23×C271 ▷ C271 | |
| UCS | C1 ◁ C23 ◁ #390 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#22576160 | C2 | semidirect |
| 3. C2×#22576147 | C2 | semidirect |
| 4. C2×#22576140 | C2 | semidirect |
| 5. #31773813 | C2 | semidirect |
| 6. #31773802 | C2 | semidirect |
| 7. #31773772 | C2 | semidirect |
| 8. C4×(C2×D271)⋊C4 | C2 | semidirect |
| 9. #390×C271 | C2 | semidirect |
| 106. C2×C4×C271 | #41 | semidirect |
| 136. C271 | #2432 | semidirect |
| 144. #390 | D271 | semidirect |
| 243. C2×C4 | #16864653 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#16864653 | 2 |
| C2×#22576160 | 2 |
| C2×#22576147 | 2 |
| C2×#22576140 | 2 |
| #31773813 | 2 |
| #31773802 | 2 |
| #31773772 | 2 |
| C4×(C2×D271)⋊C4 | 2 |
| #390×C271 | 2 |
| #2432 | 271 |