Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
420 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 420 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C251 | C22⋊C4 |
| commutator | C22 | C2×C42×C251 |
| Frattini, cosocle | C24 | C23×C251 |
| socle | C22×C251 | C2×C22⋊C4 |
| prime radical | C24×C251 | C23 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C251 ◁ C23×C4×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 8. #117×C251 | C4 | semidirect |
| 10. C2×C4⋊C4×C251 | C4 | semidirect |
| 14. C2×C42×C251 | C4 | semidirect |
| 75. C251 | #1740 | direct (N × H) |
| 76. #1740 | C251 | direct (N × H) |
| 84. #117 | C4×C251 | semidirect |
| 86. C2×C4⋊C4 | C4×C251 | semidirect |
| 90. C2×C42 | C4×C251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#117×C251 | 2 |
| #342×C251 | 2 |
| #1740 | 251 |