Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
880 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 880 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C251 | C2×C22⋊C4 |
| commutator | C23 | C22×C4×C251 |
| Frattini, cosocle | C24 | C23×C251 |
| prime radical | C23×D4×C251 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C251 ◁ C25×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#99×C251 | C2 | semidirect |
| 3. #379×C251 | C2 | semidirect |
| 4. C25×C251 | C4 | semidirect |
| 8. C22×D4×C251 | C4 | semidirect |
| 10. C23×C4×C251 | C4 | semidirect |
| 12. C2×C22⋊C4×C251 | C22 | semidirect |
| 55. C23×C251 | C22⋊C4 | semidirect |
| 66. C2×C4×C251 | C22⋊C4 | semidirect |
| 82. C22×C251 | #99 | semidirect |
| 95. C251 | #1781 | direct (N × H) |
| 96. #1781 | C251 | direct (N × H) |
| 98. C2×#99 | C2×C251 | semidirect |
| 99. #379 | C2×C251 | semidirect |
| 100. C25 | C4×C251 | semidirect |
| 104. C22×D4 | C4×C251 | semidirect |
| 106. C23×C4 | C4×C251 | semidirect |
| 108. C2×C22⋊C4 | C22×C251 | semidirect |
| 151. C23 | C22⋊C4×C251 | semidirect |
| 162. C2×C4 | C22⋊C4×C251 | semidirect |
| 178. C22 | #99×C251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×D4×C251 | 2 |
| C2×#99×C251 | 2 |
| #379×C251 | 2 |
| #1781 | 251 |