Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
272 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C251 | D8 |
| commutator | C8 | C22×C4×C251 |
| Frattini, cosocle | C2×C8 | C23×C251 |
| prime radical | C22×D8×C251 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C251 ◁ C22×C4×C251 ◁ C22×C8×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C16×C251 | C2 | semidirect |
| 3. C2×#107×C251 | C2 | semidirect |
| 4. #357×C251 | C2 | direct (N × H) |
| 8. C2×D8×C251 | C4 | semidirect |
| 13. C2×C16×C251 | C22 | semidirect |
| 15. #107×C251 | C22 | semidirect |
| 25. D8×C251 | C2×C4 | semidirect |
| 63. C2×C251 | #357 | direct (N × H) |
| 67. C251 | C2×#357 | direct (N × H) |
| 68. C2×#357 | C251 | direct (N × H) |
| 70. C22×C16 | C2×C251 | semidirect |
| 71. C2×#107 | C2×C251 | semidirect |
| 72. #357 | C2×C251 | direct (N × H) |
| 76. C2×D8 | C4×C251 | semidirect |
| 81. C2×C16 | C22×C251 | semidirect |
| 83. #107 | C22×C251 | semidirect |
| 93. D8 | C2×C4×C251 | semidirect |
| 131. C2 | #357×C251 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D8×C251 | 2 |
| C22×C16×C251 | 2 |
| C2×#107×C251 | 2 |
| #357×C251 | 2 |
| C2×#357 | 251 |