Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
688 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 688 conjugacy classes of subgroups — too many to draw. Only its 189 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×C42×C5 | C2 |
| prime radical | C2×#4076 | C2 |
| hypercenter | C22×C42 | D5 |
| hypocenter | C5 | #2854 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ C22×C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C42×C5 | C2 | semidirect |
| 3. C2×#4075 | C2 | semidirect |
| 4. #65502 | C2 | semidirect |
| 5. #65501 | C2 | semidirect |
| 6. #65229 | C2 | semidirect |
| 7. #65228 | C2 | semidirect |
| 8. #65217 | C2 | semidirect |
| 37. C22×C4×C5 | D4 | semidirect |
| 60. C22×C42 | D5 | semidirect |
| 135. C4×C5 | #123 | semidirect |
| 152. C22×C4 | D20 | semidirect |
| 206. C5 | #2854 | semidirect |
| 227. C4 | #4075 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C42×C5 | 2 |
| C2×#4076 | 2 |
| C2×#4075 | 2 |
| #65502 | 2 |
| #65501 | 2 |
| #65229 | 2 |
| #65228 | 2 |
| #65217 | 2 |
| #2854 | 5 |