Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
952 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 952 conjugacy classes of subgroups — too many to draw. Only its 173 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 | C4×C5×C22⋊C4 | C2 |
| hypercenter | C4×C22⋊C4 | D5 |
| hypocenter | C5 | #2431 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#709 | C2 | semidirect |
| 2. C2×#4076 | C2 | semidirect |
| 3. C2×#4021 | C2 | semidirect |
| 4. C2×#4020 | C2 | semidirect |
| 5. C4×C5×C22⋊C4 | C2 | semidirect |
| 6. #65249 | C2 | semidirect |
| 7. #65228 | C2 | semidirect |
| 8. #65218 | C2 | semidirect |
| 9. #64954 | C2 | semidirect |
| 39. C22×C4×C5 | D4 | semidirect |
| 60. C2×D20 | D4 | semidirect |
| 66. C2×Dic10 | D4 | semidirect |
| 82. C4×C22⋊C4 | D5 | semidirect |
| 111. C2×C4×C5 | #41 | semidirect |
| 201. C22×C4 | D20 | semidirect |
| 259. C2×C4 | #709 | semidirect |
| 262. C5 | #2431 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#709 | 2 |
| C2×#4076 | 2 |
| C2×#4021 | 2 |
| C2×#4020 | 2 |
| C4×C5×C22⋊C4 | 2 |
| #65249 | 2 |
| #65228 | 2 |
| #65218 | 2 |
| #64954 | 2 |
| #2431 | 5 |