Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
162 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 60 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 | #99 |
| commutator | C2×C4 | C42×C251 |
| Frattini, cosocle | C2×#41 | C22×C251 |
| socle | C2×C251 | #342 |
| prime radical | C2×#41×C251 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C251 ◁ C2×C4×C251 ◁ C2×#41×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. C2×M16×C251 | C4 | semidirect |
| 7. C22×C8×C251 | C4 | semidirect |
| 9. #101×C251 | C4 | semidirect |
| 73. C251 | #1395 | direct (N × H) |
| 74. #1395 | C251 | direct (N × H) |
| 80. C2×M16 | C4×C251 | semidirect |
| 81. C22×C8 | C4×C251 | semidirect |
| 83. #101 | C4×C251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #413×C251 | 2 |
| #410×C251 | 2 |
| #408×C251 | 2 |
| #1395 | 251 |