Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
564 conjugacy classes of subgroups; only the 177 normal subgroups are shown.
This group has 564 conjugacy classes of subgroups — too many to draw. Only its 177 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 | #380×C251 | C2 |
| prime radical | C24×C251 | C23 |
| hypercenter | #380 | D251 |
| hypocenter | C251 | #2855 |
| LCS | G ▷ C23×C251 ▷ C251 | |
| UCS | C1 ◁ C23 ◁ #380 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #30631592 | C2 | semidirect |
| 4. #30631586 | C2 | semidirect |
| 5. #30631572 | C2 | semidirect |
| 33. C22×Dic251 | Q8 | semidirect |
| 102. C251 | #2855 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#22548694 | 2 |
| C22×C4×Dic251 | 2 |
| #30631592 | 2 |
| #30631586 | 2 |
| #30631572 | 2 |
| #30631562 | 2 |
| #30631548 | 2 |
| #380×C251 | 2 |
| #2855 | 251 |