Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
1224 conjugacy classes of subgroups; only the 237 normal subgroups are shown.
This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 237 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D251 |
| commutator, socle, cosocle | C23×C251 | C24 |
| Fitting | C24×C4×C251 | C2 |
| hypercenter | C24×C4 | D251 |
| hypocenter | C251 | #2847 |
| LCS | G ▷ C23×C251 ▷ C251 | |
| UCS | C1 ◁ C23 ◁ C24×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#22548689 | C2 | semidirect |
| 2. C2×#22548684 | C2 | semidirect |
| 3. #30631566 | C2 | semidirect |
| 4. #30631547 | C2 | semidirect |
| 5. C24×C4×C251 | C2 | semidirect |
| 6. C22×D1004 | C22 | semidirect |
| 10. C2×C4⋊Dic251 | C22 | semidirect |
| 12. C2×(C2×D251)⋊C4 | C22 | semidirect |
| 13. C251⋊#95 | C22 | semidirect |
| 29. C24×C251 | D4 | semidirect |
| 35. C22×C4×C251 | D4 | semidirect |
| 65. C22×C251 | #121 | semidirect |
| 73. C4×C251 | #120 | semidirect |
| 80. C251 | #2847 | semidirect |
| 82. C24×C4 | D251 | semidirect |
| 102. C24 | D1004 | semidirect |
| 113. C22×C4 | C22⋊D251 | semidirect |
| 149. C22 | #22548689 | semidirect |
| 155. C4 | #22548684 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#22548689 | 2 |
| C2×#22548684 | 2 |
| #30631566 | 2 |
| #30631547 | 2 |
| C24×C4×C251 | 2 |
| #2847 | 251 |