Finite Group DatabasebetaFinite Group DatabasebetaOrder 34688.
840 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 840 conjugacy classes of subgroups — too many to draw. Only its 171 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 | #393×C271 | C2 |
| prime radical | C22×C22⋊D271 | C22 |
| hypercenter | #393 | D271 |
| hypocenter | C271 | #2409 |
| LCS | G ▷ C23×C271 ▷ C271 | |
| UCS | C1 ◁ C23 ◁ #393 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×Q8×D271 | C2 | semidirect |
| 2. C2×#22576137 | C2 | semidirect |
| 4. #31773818 | C2 | semidirect |
| 5. #31773816 | C2 | semidirect |
| 6. #31773809 | C2 | semidirect |
| 7. #31773801 | C2 | semidirect |
| 9. #31773771 | C2 | semidirect |
| 10. #31773756 | C2 | semidirect |
| 11. #393×C271 | C2 | semidirect |
| 44. C2×Dic542 | D4 | semidirect |
| 48. C2×C22⋊D271 | Q8 | semidirect |
| 99. C2×Dic271 | C2×Q8 | semidirect |
| 106. C23×C271 | C2×Q8 | semidirect |
| 133. C2×D271 | C22⋊Q8 | semidirect |
| 150. C271 | #2409 | semidirect |
| 158. #393 | D271 | semidirect |
| 238. C23 | Q8×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 |
| C2×#22576137 | 2 |
| C2×(C22⋊D271)⋊C4 | 2 |
| #31773818 | 2 |
| #31773816 | 2 |
| #31773809 | 2 |
| #31773801 | 2 |
| #31773773 | 2 |
| #31773771 | 2 |
| #31773756 | 2 |
| #393×C271 | 2 |
| #2409 | 271 |