Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
220 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C251 | #102 |
| commutator | C8 | C22×C4×C251 |
| Frattini, cosocle | C2×C8 | C23×C251 |
| socle | C2×C251 | C2×#102 |
| prime radical | C2×#135×C251 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C251 ◁ C22×C4×C251 ◁ C22×C8×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×M32×C251 | C2 | semidirect |
| 3. C2×#108×C251 | C2 | semidirect |
| 4. #362×C251 | C2 | semidirect |
| 5. #361×C251 | C2 | semidirect |
| 84. C251 | #2148 | direct (N × H) |
| 85. #2148 | C251 | direct (N × H) |
| 87. C2×M32 | C2×C251 | semidirect |
| 88. C2×#108 | C2×C251 | semidirect |
| 89. #362 | C2×C251 | semidirect |
| 90. #361 | C2×C251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#135×C251 | 2 |
| C2×M32×C251 | 2 |
| C2×#108×C251 | 2 |
| #362×C251 | 2 |
| #361×C251 | 2 |
| #2148 | 251 |