Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
182 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 182 conjugacy classes of subgroups — too many to draw. Only its 68 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 | #121 |
| commutator, Frattini, cosocle | C42 | C23×C251 |
| prime radical | #459×C251 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C251 ◁ C42×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #486×C251 | C2 | semidirect |
| 2. #461×C251 | C2 | semidirect |
| 4. C8×D4×C251 | C2 | semidirect |
| 5. #335×C251 | C2 | semidirect |
| 6. #332×C251 | C2 | semidirect |
| 7. #331×C251 | C2 | semidirect |
| 84. C251 | #1681 | direct (N × H) |
| 85. #1681 | C251 | direct (N × H) |
| 86. #486 | C2×C251 | semidirect |
| 87. #461 | C2×C251 | semidirect |
| 89. C8×D4 | C2×C251 | semidirect |
| 90. #335 | C2×C251 | semidirect |
| 91. #332 | C2×C251 | semidirect |
| 92. #331 | C2×C251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #486×C251 | 2 |
| #461×C251 | 2 |
| #459×C251 | 2 |
| C8×D4×C251 | 2 |
| #335×C251 | 2 |
| #332×C251 | 2 |
| #331×C251 | 2 |
| #1681 | 251 |