Finite Group DatabasebetaFinite Group DatabasebetaOrder 800.
222 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 222 conjugacy classes of subgroups — too many to draw. Only its 62 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 | C2×D52 |
| commutator | C2×C52 | C22×C4 |
| Fitting | C2×C4×C52 | C22 |
| socle, cosocle | C22×C52 | C23 |
| prime radical | C2×#6038 | C2 |
| hypercenter | C2×C4 | D52 |
| hypocenter | C52 | C4×D4 |
| LCS | G ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C4×C5×D5 | C2 | semidirect |
| 3. C2×#6036 | C2 | semidirect |
| 4. #87808 | C2 | semidirect |
| 5. C5×#683 | C2 | semidirect |
| 6. #87778 | C2 | semidirect |
| 7. Dic52 | C2 | semidirect |
| 16. #6038 | C4 | semidirect |
| 23. C2×C5⋊D5 | C2×C4 | semidirect |
| 26. C2×C5×D5 | C2×C4 | semidirect |
| 30. C5×Dic5 | C2×C4 | semidirect |
| 31. C5×Dic5 | D4 | semidirect |
| 37. C2×C4×D5 | D5 | semidirect |
| 44. #683 | D5 | semidirect |
| 76. C52 | C4×D4 | semidirect |
| 94. C2×D5 | C4×D5 | semidirect |
| 105. Dic5 | C4×D5 | semidirect |
| 106. Dic5 | C22⋊D5 | semidirect |
| 139. C5 | #4039 | semidirect |
| 140. C5 | C4×C22⋊D5 | 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×#6038 | 2 |
| C2×#6036 | 2 |
| #87808 | 2 |
| C5×#683 | 2 |
| #87778 | 2 |
| Dic52 | 2 |
| C4×C22⋊D5 | 5 |
| #4039 | 5 |