Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
384 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 136 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 | C2×C4 | C22×C4×C251 |
| Frattini, cosocle | C22×C4 | C23×C251 |
| prime radical | C22×D8×C251 | C2 |
| 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 |
|---|---|---|
| 2. C2×#107×C251 | C2 | semidirect |
| 3. C2×#102×C251 | C2 | semidirect |
| 4. C2×C4×C8×C251 | C2 | semidirect |
| 5. #390×C251 | C2 | semidirect |
| 9. C2×D8×C251 | C4 | semidirect |
| 54. C2×C4×C251 | D8 | semidirect |
| 62. C8×C251 | C22⋊C4 | semidirect |
| 85. C251 | #1970 | direct (N × H) |
| 86. #1970 | C251 | direct (N × H) |
| 88. C2×#107 | C2×C251 | semidirect |
| 89. C2×#102 | C2×C251 | semidirect |
| 90. C2×C4×C8 | C2×C251 | semidirect |
| 91. #390 | C2×C251 | semidirect |
| 95. C2×D8 | C4×C251 | semidirect |
| 140. C2×C4 | D8×C251 | semidirect |
| 148. C8 | C22⋊C4×C251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D8×C251 | 2 |
| C2×#107×C251 | 2 |
| C2×#102×C251 | 2 |
| C2×C4×C8×C251 | 2 |
| #390×C251 | 2 |
| #1970 | 251 |