Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
186 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C251 | C2×D4 |
| commutator | C2×C4 | C22×C4×C251 |
| Frattini, cosocle | C22×C4 | C23×C251 |
| socle | C22×C251 | C4×D4 |
| prime radical | C23×C251 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C251 ◁ C22×C8×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 15. #107×C251 | C4 | semidirect |
| 16. #107×C251 | C4 | semidirect |
| 17. #106×C251 | C4 | semidirect |
| 18. #106×C251 | C4 | semidirect |
| 88. C251 | #1922 | direct (N × H) |
| 89. #1922 | C251 | direct (N × H) |
| 104. #107 | C4×C251 | semidirect |
| 105. #107 | C4×C251 | semidirect |
| 106. #106 | C4×C251 | semidirect |
| 107. #106 | C4×C251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #427×C251 | 2 |
| #424×C251 | 2 |
| #405×C251 | 2 |
| #341×C251 | 2 |
| #339×C251 | 2 |
| #336×C251 | 2 |
| #1922 | 251 |