Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
1492 conjugacy classes of subgroups; only the 553 normal subgroups are shown.
This group has 1492 conjugacy classes of subgroups — too many to draw. Only its 553 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 | C24×D5 |
| commutator, socle, cosocle | C22×C5 | C25 |
| Fitting | C2×C4×C5×Q8 | C2 |
| hypercenter | C2×C4×Q8 | D5 |
| hypocenter | C5 | #3497 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#4147 | C2 | semidirect |
| 2. C2×C4×C5×Q8 | C2 | semidirect |
| 3. #65911 | C2 | semidirect |
| 4. #65910 | C2 | semidirect |
| 5. Q8×D20 | C2 | semidirect |
| 6. #65844 | C2 | semidirect |
| 7. #65811 | C2 | semidirect |
| 27. C2×C5×Q8 | D4 | semidirect |
| 54. C2×C4×Q8 | D5 | semidirect |
| 100. C4×C5 | 2−1+4 | semidirect |
| 106. C2×Q8 | D20 | semidirect |
| 145. C5 | #3497 | semidirect |
| 157. C4 | #4147 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#4147 | 2 |
| C2×C4×C5×Q8 | 2 |
| #65911 | 2 |
| #65910 | 2 |
| Q8×D20 | 2 |
| #65844 | 2 |
| #65811 | 2 |
| #3497 | 5 |