Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
480 conjugacy classes of subgroups; only the 360 normal subgroups are shown.
This group has 480 conjugacy classes of subgroups — too many to draw. Only its 360 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C251 | D4 |
| commutator | C4 | C23×C4×C251 |
| Frattini, cosocle | C2×C4 | C24×C251 |
| socle | C23×C251 | C4⋊C4 |
| prime radical | C24×C251 | Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C251 ◁ C23×C4×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#108×C251 | C2 | direct (N × H) |
| 11. #108×C251 | C22 | direct (N × H) |
| 42. C22×C251 | #108 | direct (N × H) |
| 52. C2×C251 | C2×#108 | direct (N × H) |
| 54. C251 | C22×#108 | direct (N × H) |
| 55. C22×#108 | C251 | direct (N × H) |
| 58. C2×#108 | C2×C251 | direct (N × H) |
| 66. #108 | C22×C251 | direct (N × H) |
| 97. C22 | #108×C251 | direct (N × H) |
| 107. C2 | C2×#108×C251 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×M16×C251 | 2 |
| C23×C8×C251 | 2 |
| C2×#108×C251 | 2 |
| C22×#108 | 251 |