Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
402 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 402 conjugacy classes of subgroups — too many to draw. Only its 183 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D5 |
| commutator, socle | C22×C5 | C23×C4 |
| Fitting | C5×C8×Q8 | C2 |
| prime radical | C2×#713 | C4 |
| cosocle | C2×C4×C5 | C24 |
| hypercenter | C8×Q8 | D5 |
| hypocenter | C5 | #2971 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C4 ◁ C8×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C5×C8×Q8 | C2 | semidirect |
| 3. Q8×#168 | C2 | semidirect |
| 4. #65127 | C2 | semidirect |
| 5. #65124 | C2 | semidirect |
| 6. #64977 | C2 | semidirect |
| 7. C4×#676 | C2 | semidirect |
| 46. C8×Q8 | D5 | semidirect |
| 52. C5×Q8 | M16 | semidirect |
| 65. C5×C8 | #41 | semidirect |
| 67. #168 | #41 | semidirect |
| 119. Q8 | #676 | semidirect |
| 134. C8 | #713 | semidirect |
| 135. C5 | #2971 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×#713 | 2 |
| C5×C8×Q8 | 2 |
| Q8×#168 | 2 |
| #65127 | 2 |
| #65124 | 2 |
| #64977 | 2 |
| C4×#676 | 2 |
| #2971 | 5 |