Finite Group DatabasebetaFinite Group DatabasebetaOrder 2000.
188 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C5 | #1042 |
| commutator, hypocenter | C52 | C42×C5 |
| Frattini | C2 | C2×C5×#1042 |
| Fitting | C4×C53 | C4 |
| socle | C2×C53 | C2×C4 |
| prime radical | C2×C52×D5 | C22 |
| cosocle | C2×C5×D5 | C22×C5 |
| LCS | G ▷ C52 | |
| UCS | C1 ◁ C4×C5 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C5×#1042 | C4 | direct (N × H) |
| 5. C4×C53 | C4 | semidirect |
| 6. C52×Dic5 | C4 | semidirect |
| 7. C4×#1042 | C5 | direct (N × H) |
| 19. C53 | C42 | semidirect |
| 24. #1042 | C4×C5 | direct (N × H) |
| 31. C4×C52 | Dic5 | semidirect |
| 32. C4×C52 | C4×C5 | semidirect |
| 33. C4×C52 | AGL(1,𝔽5) | semidirect |
| 36. C5×Dic5 | Dic5 | semidirect |
| 37. C5×Dic5 | C4×C5 | semidirect |
| 63. C52 | C4×Dic5 | semidirect |
| 64. C52 | C42×C5 | semidirect |
| 65. C52 | C4×AGL(1,𝔽5) | semidirect |
| 79. C4×C5 | C5×Dic5 | semidirect |
| 80. C4×C5 | C5×AGL(1,𝔽5) | semidirect |
| 81. C4×C5 | #1042 | direct (N × H) |
| 83. Dic5 | C5×Dic5 | semidirect |
| 102. C5 | C4×C5×Dic5 | semidirect |
| 103. C5 | C4×C5×AGL(1,𝔽5) | semidirect |
| 104. C5 | C4×#1042 | direct (N × H) |
| 108. C4 | C5×#1042 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C5×#1042 | 2 |
| C4×C52×D5 | 2 |
| C4×#1042 | 5 |
| C4×C5×AGL(1,𝔽5) | 5 |
| C4×C5×Dic5 | 5 |