Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
704 conjugacy classes of subgroups; only the 340 normal subgroups are shown.
This group has 704 conjugacy classes of subgroups — too many to draw. Only its 340 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C251 | C22×C22⋊C4 |
| commutator | C22 | C23×C4×C251 |
| Frattini, cosocle | C2×C4 | C24×C251 |
| prime radical | C2×2−1+4×C251 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C251 ◁ C2×#41×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #568×C251 | C2 | semidirect |
| 3. #413×C251 | C2 | semidirect |
| 4. C2×#101×C251 | C2 | semidirect |
| 59. C251 | #2889 | direct (N × H) |
| 60. #2889 | C251 | direct (N × H) |
| 62. #568 | C2×C251 | semidirect |
| 63. #413 | C2×C251 | semidirect |
| 64. C2×#101 | C2×C251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×2−1+4×C251 | 2 |
| #568×C251 | 2 |
| #413×C251 | 2 |
| C2×#101×C251 | 2 |
| #2889 | 251 |