Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
212 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 68 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 | #120 |
| commutator, Frattini, cosocle | C42 | C23×C251 |
| prime radical | #127×C251 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C251 ◁ C42×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #475×C251 | C2 | semidirect |
| 4. #462×C251 | C2 | semidirect |
| 6. Q8⋊C8×C251 | C2 | semidirect |
| 7. Q8⋊C8×C251 | C2 | semidirect |
| 90. C251 | #1640 | direct (N × H) |
| 91. #1640 | C251 | direct (N × H) |
| 94. #475 | C2×C251 | semidirect |
| 95. #462 | C2×C251 | semidirect |
| 97. Q8⋊C8 | C2×C251 | semidirect |
| 98. Q8⋊C8 | C2×C251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #559×C251 | 2 |
| #486×C251 | 2 |
| #475×C251 | 2 |
| #462×C251 | 2 |
| #331×C251 | 2 |
| Q8⋊C8×C251 | 2 |
| #1640 | 251 |