Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
548 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 125 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 | #399×C251 | C2 |
| prime radical | C24×D251 | C22 |
| hypercenter | #399 | D251 |
| hypocenter | C251 | #2842 |
| LCS | G ▷ C23×C251 ▷ C251 | |
| UCS | C1 ◁ C23 ◁ #399 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #30631623 | C2 | semidirect |
| 2. #30631611 | C2 | semidirect |
| 3. #30631609 | C2 | semidirect |
| 5. #399×C251 | C2 | semidirect |
| 7. C2×C4⋊Dic251 | C22 | semidirect |
| 8. C2×Dic251⋊C4 | C22 | semidirect |
| 12. C251⋊#95 | C22 | semidirect |
| 13. C2×C4⋊C4×C251 | C22 | semidirect |
| 15. #95×C251 | C22 | semidirect |
| 66. C251 | #2842 | semidirect |
| 69. #399 | D251 | semidirect |
| 76. C2×C4⋊C4 | C2×D251 | semidirect |
| 82. #95 | C2×D251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #30631623 | 2 |
| #30631611 | 2 |
| #30631609 | 2 |
| #30631542 | 2 |
| #399×C251 | 2 |
| #2842 | 251 |