Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
480 conjugacy classes of subgroups; only the 296 normal subgroups are shown.
This group has 480 conjugacy classes of subgroups — too many to draw. Only its 296 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C251 | C2×D4 |
| commutator | C4 | C23×C4×C251 |
| Frattini, cosocle | C2×C4 | C24×C251 |
| LCS | G ▷ 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 |
|---|---|---|
| 3. #441×C251 | C2 | direct (N × H) |
| 8. C2×Q16×C251 | C4 | semidirect |
| 29. Q16×C251 | C2×C4 | semidirect |
| 65. C2×C251 | #441 | direct (N × H) |
| 68. C251 | C2×#441 | direct (N × H) |
| 69. C2×#441 | C251 | direct (N × H) |
| 72. #441 | C2×C251 | direct (N × H) |
| 77. C2×Q16 | C4×C251 | semidirect |
| 98. Q16 | C2×C4×C251 | semidirect |
| 134. C2 | #441×C251 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q16×C251 | 2 |
| C2×C4×Q8×C251 | 2 |
| #441×C251 | 2 |
| C2×#106×C251 | 2 |
| C2×#103×C251 | 2 |
| C2×#97×C251 | 2 |
| C2×#441 | 251 |