Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
1786 conjugacy classes of subgroups; only the 886 normal subgroups are shown.
This group has 1786 conjugacy classes of subgroups — too many to draw. Only its 886 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×H(5,𝔽2) | C2 |
| socle | C2×C52 | C25 |
| hypercenter | C5×H(5,𝔽2) | D5 |
| hypocenter | C5 | C5×#585 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C5 ◁ C5×H(5,𝔽2) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C52×H(5,𝔽2) | C2 | semidirect |
| 2. C5×#4150 | C2 | semidirect |
| 3. C5×D5×#41 | C2 | semidirect |
| 4. C5×#4144 | C2 | semidirect |
| 5. C2×C5×#711 | C2 | semidirect |
| 6. C52×#41 | C22 | semidirect |
| 7. C2×C52×D4 | C22 | semidirect |
| 8. C2×C5×C22⋊D5 | C22 | semidirect |
| 9. C22×C5×Dic5 | C22 | semidirect |
| 10. C5×#713 | C22 | semidirect |
| 11. C5×Q8×D5 | C22 | semidirect |
| 12. C5×#711 | C22 | semidirect |
| 13. C5×#711 | C22 | semidirect |
| 14. C5×D4×D5 | C22 | semidirect |
| 15. C5×#709 | C22 | semidirect |
| 16. C2×C4×C5×D5 | C22 | semidirect |
| 17. C2×C5×Dic10 | C22 | semidirect |
| 19. #66285 | C5 | direct (N × H) |
| 35. C5×H(5,𝔽2) | D5 | semidirect |
| 36. C5×H(5,𝔽2) | C2×C5 | semidirect |
| 39. #4150 | C2×C5 | semidirect |
| 40. D5×#41 | C2×C5 | semidirect |
| 41. #4144 | C2×C5 | semidirect |
| 42. C2×#711 | C2×C5 | semidirect |
| 55. C5×#41 | C2×D5 | semidirect |
| 56. C5×#41 | C22×C5 | semidirect |
| 62. C2×C5×D4 | C2×D5 | semidirect |
| 63. C2×C5×D4 | C22×C5 | semidirect |
| 66. C2×C22⋊D5 | C22×C5 | semidirect |
| 67. C22×Dic5 | C22×C5 | semidirect |
| 68. #713 | C22×C5 | semidirect |
| 69. Q8×D5 | C22×C5 | semidirect |
| 70. #711 | C22×C5 | semidirect |
| 71. #711 | C22×C5 | semidirect |
| 72. D4×D5 | C22×C5 | semidirect |
| 73. #709 | C22×C5 | semidirect |
| 74. C2×C4×D5 | C22×C5 | semidirect |
| 75. C2×Dic10 | C22×C5 | semidirect |
| 110. H(5,𝔽2) | C5×D5 | semidirect |
| 113. C52 | #585 | semidirect |
| 135. #41 | C2×C5×D5 | semidirect |
| 140. C2×D4 | C2×C5×D5 | semidirect |
| 164. C5 | #66285 | direct (N × H) |
| 165. C5 | C5×#585 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C52×H(5,𝔽2) | 2 |
| C5×#4150 | 2 |
| C5×D5×#41 | 2 |
| C5×#4144 | 2 |
| C2×C5×#711 | 2 |
| C5×#585 | 5 |
| #66285 | 5 |