Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
556 conjugacy classes of subgroups; only the 504 normal subgroups are shown.
This group has 556 conjugacy classes of subgroups — too many to draw. Only its 504 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 | C24 |
| commutator | C2 | C22×C42×C251 |
| Frattini, cosocle | C2×C4 | C24×C251 |
| socle | C22×C251 | C23×C4 |
| prime radical | C2×#41×C251 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #405×C251 | C2 | semidirect |
| 4. C4×M16×C251 | C2 | semidirect |
| 5. C2×#97×C251 | C2 | semidirect |
| 7. #131×C251 | C4 | semidirect |
| 23. M16×C251 | C2×C4 | semidirect |
| 24. C2×C8×C251 | C2×C4 | semidirect |
| 53. C251 | #2875 | direct (N × H) |
| 54. #2875 | C251 | direct (N × H) |
| 57. #405 | C2×C251 | semidirect |
| 58. C4×M16 | C2×C251 | semidirect |
| 59. C2×#97 | C2×C251 | semidirect |
| 61. #131 | C4×C251 | semidirect |
| 77. M16 | C2×C4×C251 | semidirect |
| 78. C2×C8 | C2×C4×C251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#131×C251 | 2 |
| C4×#41×C251 | 2 |
| #405×C251 | 2 |
| C4×M16×C251 | 2 |
| C2×#97×C251 | 2 |
| #2875 | 251 |