Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
1504 conjugacy classes of subgroups; only the 1384 normal subgroups are shown.
This group has 1504 conjugacy classes of subgroups — too many to draw. Only its 1384 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C251 | C22 |
| commutator | C2 | C24×C4×C251 |
| Frattini, cosocle | C4 | C25×C251 |
| socle | C23×C251 | C22×C4 |
| prime radical | C22×#41×C251 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#131×C251 | C2 | direct (N × H) |
| 3. C22×M16×C251 | C2 | semidirect |
| 4. C23×C8×C251 | C2 | semidirect |
| 10. #131×C251 | C22 | direct (N × H) |
| 11. C2×M16×C251 | C22 | semidirect |
| 14. C22×C8×C251 | C22 | semidirect |
| 24. M16×C251 | C23 | semidirect |
| 27. C2×C8×C251 | C23 | semidirect |
| 41. C22×C251 | #131 | direct (N × H) |
| 48. C2×C251 | C2×#131 | direct (N × H) |
| 50. C251 | C22×#131 | direct (N × H) |
| 51. C22×#131 | C251 | direct (N × H) |
| 53. C2×#131 | C2×C251 | direct (N × H) |
| 54. C22×M16 | C2×C251 | semidirect |
| 55. C23×C8 | C2×C251 | semidirect |
| 61. #131 | C22×C251 | direct (N × H) |
| 62. C2×M16 | C22×C251 | semidirect |
| 65. C22×C8 | C22×C251 | semidirect |
| 75. M16 | C23×C251 | semidirect |
| 78. C2×C8 | C23×C251 | semidirect |
| 92. C22 | #131×C251 | direct (N × H) |
| 99. C2 | C2×#131×C251 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#41×C251 | 2 |
| C2×#131×C251 | 2 |
| C22×M16×C251 | 2 |
| C23×C8×C251 | 2 |
| C22×#131 | 251 |