Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
402 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 402 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D20 |
| commutator, cosocle | C2×C4×C5 | C24 |
| Frattini | C2×C4 | C23×D5 |
| Fitting | C5×#408 | C2 |
| socle | C22×C5 | C22×D4 |
| hypercenter | #408 | D5 |
| hypocenter | C5 | #3203 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C2×#41 ◁ #408 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #66167 | C2 | semidirect |
| 2. C5×#408 | C2 | semidirect |
| 3. #65423 | C2 | semidirect |
| 4. #65405 | C2 | semidirect |
| 5. #65404 | C2 | semidirect |
| 6. #65025 | C2 | semidirect |
| 7. #65023 | C2 | semidirect |
| 8. #65022 | C2 | semidirect |
| 9. #65011 | C2 | semidirect |
| 77. #408 | D5 | semidirect |
| 215. C5 | #3203 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #66167 | 2 |
| C5×#408 | 2 |
| #65423 | 2 |
| #65405 | 2 |
| #65404 | 2 |
| #65025 | 2 |
| #65023 | 2 |
| #65022 | 2 |
| #65011 | 2 |
| #3203 | 5 |