Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
210 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 72 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 |
|---|---|---|
| 1. #557×C251 | C2 | semidirect |
| 5. #332×C251 | C2 | semidirect |
| 6. Q8⋊C8×C251 | C2 | semidirect |
| 7. Q8⋊C8×C251 | C2 | semidirect |
| 55. Q8×C251 | SD16 | semidirect |
| 90. C251 | #1626 | direct (N × H) |
| 91. #1626 | C251 | direct (N × H) |
| 92. #557 | C2×C251 | semidirect |
| 96. #332 | C2×C251 | semidirect |
| 97. Q8⋊C8 | C2×C251 | semidirect |
| 98. Q8⋊C8 | C2×C251 | semidirect |
| 146. Q8 | SD16×C251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #557×C251 | 2 |
| #486×C251 | 2 |
| #460×C251 | 2 |
| #332×C251 | 2 |
| Q8⋊C8×C251 | 2 |
| #1626 | 251 |