Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
864 conjugacy classes of subgroups; only the 185 normal subgroups are shown.
This group has 864 conjugacy classes of subgroups — too many to draw. Only its 185 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, prime radical | C22×C5×#41 | C2 |
| hypercenter | C22×#41 | D5 |
| hypocenter | C5 | #2406 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ C22×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#4081 | C2 | semidirect |
| 3. C22×#683 | C2 | semidirect |
| 4. #65518 | C2 | semidirect |
| 5. #65511 | C2 | semidirect |
| 6. #65509 | C2 | semidirect |
| 7. #65500 | C2 | semidirect |
| 8. #65499 | C2 | semidirect |
| 100. C2×Dic5 | #41 | semidirect |
| 209. C5 | #2406 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C5×#41 | 2 |
| C2×#4081 | 2 |
| C22×#683 | 2 |
| #65518 | 2 |
| #65511 | 2 |
| #65509 | 2 |
| #65500 | 2 |
| #65499 | 2 |
| #2406 | 5 |