Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
1734 conjugacy classes of subgroups; only the 890 normal subgroups are shown.
This group has 1734 conjugacy classes of subgroups — too many to draw. Only its 890 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C5 | C24×D5 |
| commutator, cosocle | C2×C5 | C25×C5 |
| Frattini | C2 | C24×C5×D5 |
| Fitting | C52×2−1+4 | C2 |
| socle | C2×C52 | C25 |
| hypercenter | C5×2−1+4 | D5 |
| hypocenter | C5 | C2×C5×2−1+4 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C5 ◁ C5×2−1+4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C52×2−1+4 | C2 | semidirect |
| 2. C5×#4150 | C2 | semidirect |
| 3. C5×D5×#41 | C2 | semidirect |
| 4. C5×#4147 | C2 | semidirect |
| 5. C2×C5×Q8×D5 | C2 | semidirect |
| 6. C52×#41 | C22 | semidirect |
| 7. C2×C52×Q8 | C22 | semidirect |
| 8. C5×#713 | C22 | semidirect |
| 10. C5×Q8×D5 | C22 | semidirect |
| 11. C5×#711 | C22 | semidirect |
| 13. C5×#709 | C22 | semidirect |
| 15. C2×C5×Dic10 | C22 | semidirect |
| 17. D5×2−1+4 | C5 | direct (N × H) |
| 31. C5×2−1+4 | D5 | direct (N × H) |
| 32. C5×2−1+4 | C2×C5 | semidirect |
| 35. #4150 | C2×C5 | semidirect |
| 36. D5×#41 | C2×C5 | semidirect |
| 37. #4147 | C2×C5 | semidirect |
| 38. C2×Q8×D5 | C2×C5 | semidirect |
| 49. C5×#41 | C2×D5 | semidirect |
| 50. C5×#41 | C22×C5 | semidirect |
| 55. C2×C5×Q8 | C2×D5 | semidirect |
| 56. C2×C5×Q8 | C22×C5 | semidirect |
| 59. #713 | C22×C5 | semidirect |
| 61. Q8×D5 | C22×C5 | semidirect |
| 62. #711 | C22×C5 | semidirect |
| 64. #709 | C22×C5 | semidirect |
| 66. C2×Dic10 | C22×C5 | semidirect |
| 71. C5×D5 | 2−1+4 | direct (N × H) |
| 97. 2−1+4 | C5×D5 | direct (N × H) |
| 100. C52 | C2×2−1+4 | semidirect |
| 118. #41 | C2×C5×D5 | semidirect |
| 122. C2×Q8 | C2×C5×D5 | semidirect |
| 134. D5 | C5×2−1+4 | direct (N × H) |
| 146. C5 | D5×2−1+4 | direct (N × H) |
| 147. C5 | C2×C5×2−1+4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C52×2−1+4 | 2 |
| C5×#4150 | 2 |
| C5×D5×#41 | 2 |
| C5×#4147 | 2 |
| C2×C5×Q8×D5 | 2 |
| C2×C5×2−1+4 | 5 |
| D5×2−1+4 | 5 |