Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
232 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C251 | C2×D8 |
| commutator | C8 | C22×C4×C251 |
| Frattini, cosocle | C2×C8 | C23×C251 |
| prime radical | C2×#135×C251 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×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×#107×C251 | C2 | semidirect |
| 4. #358×C251 | C2 | semidirect |
| 5. #357×C251 | C2 | semidirect |
| 8. #135×C251 | C4 | semidirect |
| 27. Q16×C251 | C2×C4 | semidirect |
| 30. D8×C251 | C2×C4 | semidirect |
| 73. C251 | #2139 | direct (N × H) |
| 74. #2139 | C251 | direct (N × H) |
| 76. C2×M32 | C2×C251 | semidirect |
| 77. C2×#107 | C2×C251 | semidirect |
| 78. #358 | C2×C251 | semidirect |
| 79. #357 | C2×C251 | semidirect |
| 82. #135 | C4×C251 | semidirect |
| 101. Q16 | C2×C4×C251 | semidirect |
| 104. D8 | C2×C4×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×#107×C251 | 2 |
| #358×C251 | 2 |
| #357×C251 | 2 |
| #2139 | 251 |