Finite Group DatabasebetaFinite Group DatabasebetaOrder 32896.
286 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 286 conjugacy classes of subgroups — too many to draw. Only its 112 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×C257 | #120 |
| commutator | C2×C4 | C22×C4×C257 |
| Frattini, cosocle | C22×C4 | C23×C257 |
| prime radical | C2×#41×C257 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C257 ◁ C22×C4×C257 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #428×C257 | C2 | semidirect |
| 5. #382×C257 | C2 | semidirect |
| 6. #343×C257 | C2 | semidirect |
| 7. #339×C257 | C2 | semidirect |
| 22. #104×C257 | C4 | semidirect |
| 42. M16×C257 | C2×C4 | semidirect |
| 57. C42×C257 | C2×C4 | semidirect |
| 111. C257 | #1860 | direct (N × H) |
| 112. #1860 | C257 | direct (N × H) |
| 114. #428 | C2×C257 | semidirect |
| 117. #382 | C2×C257 | semidirect |
| 118. #343 | C2×C257 | semidirect |
| 119. #339 | C2×C257 | semidirect |
| 134. #104 | C4×C257 | semidirect |
| 154. M16 | C2×C4×C257 | semidirect |
| 169. C42 | C2×C4×C257 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #520×C257 | 2 |
| #428×C257 | 2 |
| C2×#104×C257 | 2 |
| #417×C257 | 2 |
| #382×C257 | 2 |
| #343×C257 | 2 |
| #339×C257 | 2 |
| #1860 | 257 |