Finite Group DatabasebetaFinite Group DatabasebetaOrder 800.
312 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 85 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 | C22×C5⋊D5 |
| commutator, socle, cosocle | C22×C52 | C23 |
| Fitting | C52×C4⋊C4 | C2 |
| prime radical | C2×C4⋊(C5⋊D5) | C2 |
| hypercenter | C4⋊C4 | C5⋊D5 |
| hypocenter | C52 | #123 |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C4×C5⋊D5 | C2 | semidirect |
| 3. C52×C4⋊C4 | C2 | semidirect |
| 4. #87808 | C2 | semidirect |
| 5. #87808 | C2 | semidirect |
| 6. #87808 | C2 | semidirect |
| 7. #87806 | C2 | semidirect |
| 32. C5×C4⋊C4 | D5 | semidirect |
| 52. C52 | #123 | semidirect |
| 68. C4⋊C4 | C5⋊D5 | semidirect |
| 86. C5 | #4040 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4⋊(C5⋊D5) | 2 |
| C2×C4×C5⋊D5 | 2 |
| C52×C4⋊C4 | 2 |
| #87808 | 2 |
| #87806 | 2 |
| #4040 | 5 |