Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
412 conjugacy classes of subgroups; only the 200 normal subgroups are shown.
This group has 412 conjugacy classes of subgroups — too many to draw. Only its 200 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 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C251 |
| prime radical | C22×D4×C251 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C251 ◁ C2×C42×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#125×C251 | C2 | semidirect |
| 3. #488×C251 | C2 | direct (N × H) |
| 4. C2×#103×C251 | C2 | semidirect |
| 6. C2×C4×C8×C251 | C2 | semidirect |
| 11. #125×C251 | C22 | semidirect |
| 18. #103×C251 | C22 | semidirect |
| 20. C4×C8×C251 | C22 | semidirect |
| 70. C2×C251 | #488 | direct (N × H) |
| 73. C251 | C2×#488 | direct (N × H) |
| 74. C2×#488 | C251 | direct (N × H) |
| 75. C2×#125 | C2×C251 | semidirect |
| 77. #488 | C2×C251 | direct (N × H) |
| 78. C2×#103 | C2×C251 | semidirect |
| 80. C2×C4×C8 | C2×C251 | semidirect |
| 85. #125 | C22×C251 | semidirect |
| 92. #103 | C22×C251 | semidirect |
| 94. C4×C8 | C22×C251 | semidirect |
| 144. C2 | #488×C251 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#125×C251 | 2 |
| C2×#124×C251 | 2 |
| #488×C251 | 2 |
| C2×#103×C251 | 2 |
| C2×#102×C251 | 2 |
| C2×C4×C8×C251 | 2 |
| C2×#488 | 251 |