Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
362 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 362 conjugacy classes of subgroups — too many to draw. Only its 148 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×AGL(1,𝔽5) |
| commutator, socle | C22×C5 | C23×C4 |
| Fitting, prime radical | C5×#121 | C4 |
| cosocle | C2×Dic5 | C24 |
| hypercenter | #121 | AGL(1,𝔽5) |
| hypocenter | C5 | #2967 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #121 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #65774 | C2 | semidirect |
| 3. #65773 | C2 | semidirect |
| 4. #65773 | C2 | semidirect |
| 5. #65757 | C2 | semidirect |
| 6. C4×#704 | C2 | semidirect |
| 7. #65707 | C2 | semidirect |
| 8. #65697 | C2 | semidirect |
| 9. #65691 | C2 | semidirect |
| 201. C5 | #2967 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #65934 | 2 |
| #65774 | 2 |
| #65773 | 2 |
| #65757 | 2 |
| C4×#704 | 2 |
| #65707 | 2 |
| #65697 | 2 |
| #65691 | 2 |
| #2967 | 5 |