Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
526 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 526 conjugacy classes of subgroups — too many to draw. Only its 149 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×D5 |
| commutator, socle, cosocle | C23×C5 | C24 |
| Fitting | C22×C5×C4⋊C4 | C2 |
| prime radical | C22×C22⋊D5 | C22 |
| hypercenter | C22×C4⋊C4 | D5 |
| hypocenter | C5 | #2623 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ C22×C4⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C5×C4⋊C4 | C2 | semidirect |
| 4. #65500 | C2 | semidirect |
| 5. #65499 | C2 | semidirect |
| 6. #65282 | C2 | semidirect |
| 7. #65281 | C2 | semidirect |
| 8. #65271 | C2 | semidirect |
| 9. #65269 | C2 | semidirect |
| 56. C22×C4⋊C4 | D5 | semidirect |
| 189. C5 | #2623 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C5×C4⋊C4 | 2 |
| C2×#4075 | 2 |
| #65502 | 2 |
| #65500 | 2 |
| #65499 | 2 |
| #65282 | 2 |
| #65281 | 2 |
| #65271 | 2 |
| #65269 | 2 |
| #2623 | 5 |