Finite Group DatabasebetaFinite Group DatabasebetaOrder 800.
526 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 526 conjugacy classes of subgroups — too many to draw. Only its 124 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D52 |
| commutator, cosocle | C2×C52 | C24 |
| Frattini | C2 | C22×D52 |
| Fitting | C2×C4×C52 | C22 |
| socle | C22×C52 | C23 |
| hypercenter | C2×C4 | D52 |
| hypocenter | C52 | C22×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. C22×D52 | C2 | semidirect |
| 2. C2×C4×C5⋊D5 | C2 | semidirect |
| 3. C2×C5×D20 | C2 | semidirect |
| 4. C2×#6037 | C2 | semidirect |
| 5. #87877 | C2 | direct (N × H) |
| 7. C22×C5×D5 | C22 | semidirect |
| 9. C2×D52 | C22 | semidirect |
| 10. C2×C4×C52 | C22 | semidirect |
| 11. C2×#1039 | C22 | semidirect |
| 12. C4×C5⋊D5 | C22 | semidirect |
| 13. C5×D20 | C22 | semidirect |
| 14. #6037 | C22 | semidirect |
| 17. C2×C5⋊D5 | D4 | semidirect |
| 20. C2×C5×D5 | C23 | semidirect |
| 22. C4×C52 | C23 | semidirect |
| 23. #1039 | C23 | semidirect |
| 28. C2×D20 | D5 | semidirect |
| 31. C2×C52 | C2×D4 | semidirect |
| 33. C5⋊D5 | C2×D4 | semidirect |
| 40. C22×D5 | C2×D5 | semidirect |
| 43. C2×C4×C5 | C2×D5 | semidirect |
| 47. D20 | C2×D5 | semidirect |
| 51. C52 | C22×D4 | semidirect |
| 62. C2×D5 | C22×D5 | semidirect |
| 64. C4×C5 | C22×D5 | semidirect |
| 75. C2×C5 | D4×D5 | semidirect |
| 88. C2×C4 | D52 | semidirect |
| 90. C5 | C2×D4×D5 | semidirect |
| 98. C4 | C2×D52 | semidirect |
| 101. C2 | #87877 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D52 | 2 |
| C2×C4×C5⋊D5 | 2 |
| C2×C5×D20 | 2 |
| C2×#6037 | 2 |
| #87877 | 2 |
| C2×D4×D5 | 5 |