Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
208 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C42 | C22×D5 |
| commutator, socle | C22×C5 | C2×C42 |
| Fitting | C4×C5×M16 | C2 |
| prime radical | C23×C5 | C42 |
| cosocle | C42×C5 | C23 |
| hypercenter | C4×M16 | D5 |
| hypocenter | C5 | #1451 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C42 ◁ C4×M16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #64674 | C2 | semidirect |
| 5. #64674 | C2 | semidirect |
| 49. #168 | M16 | semidirect |
| 102. C5 | #1451 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×C5×M16 | 2 |
| #65214 | 2 |
| C2×C4×#168 | 2 |
| #64674 | 2 |
| #1451 | 5 |