Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
404 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 404 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 | C23×C251 | C2×D4 |
| commutator | C2×C4 | C22×C4×C251 |
| Frattini, cosocle | C22×C4 | C23×C251 |
| prime radical | C24×C251 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C251 ◁ C23×C4×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#103×C251 | C2 | semidirect |
| 5. #340×C251 | C2 | semidirect |
| 11. C22⋊Q8×C251 | C4 | semidirect |
| 24. C2×Q8×C251 | C2×C4 | semidirect |
| 44. C4⋊C4×C251 | C2×C4 | semidirect |
| 83. C22×C251 | #103 | semidirect |
| 105. C251 | #1872 | direct (N × H) |
| 106. #1872 | C251 | direct (N × H) |
| 109. C2×#103 | C2×C251 | semidirect |
| 111. #340 | C2×C251 | semidirect |
| 117. C22⋊Q8 | C4×C251 | semidirect |
| 130. C2×Q8 | C2×C4×C251 | semidirect |
| 150. C4⋊C4 | C2×C4×C251 | semidirect |
| 189. C22 | #103×C251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C22⋊Q8×C251 | 2 |
| C22×C4⋊C4×C251 | 2 |
| C2×#103×C251 | 2 |
| C2×C22⋊C8×C251 | 2 |
| #340×C251 | 2 |
| #1872 | 251 |