Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
1712 conjugacy classes of subgroups; only the 315 normal subgroups are shown.
This group has 1712 conjugacy classes of subgroups — too many to draw. Only its 315 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 | C23×C5⋊D5 |
| commutator, socle, cosocle | C22×C52 | C24 |
| Fitting | C22×C52×D4 | C2 |
| hypercenter | C22×D4 | C5⋊D5 |
| hypocenter | C52 | D42 |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ C22×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C52×D4 | C2 | semidirect |
| 2. C22×C22⋊(C5⋊D5) | C2 | semidirect |
| 3. C2×D4×C5⋊D5 | C2 | semidirect |
| 4. #11717273 | C2 | semidirect |
| 5. #11717260 | C2 | semidirect |
| 6. #11717259 | C2 | semidirect |
| 7. #11717258 | C2 | semidirect |
| 8. #11717257 | C2 | semidirect |
| 9. D4×#1039 | C2 | semidirect |
| 10. #11717250 | C2 | semidirect |
| 11. C4×C22⋊(C5⋊D5) | C2 | semidirect |
| 12. C24×C52 | C22 | semidirect |
| 13. C23×C5⋊D5 | C22 | semidirect |
| 16. C2×C52×D4 | C22 | semidirect |
| 17. C2×C52×D4 | C22 | semidirect |
| 18. C22×C4×C52 | C22 | semidirect |
| 21. C2×C22⋊(C5⋊D5) | C22 | semidirect |
| 22. C2×C22⋊(C5⋊D5) | C22 | semidirect |
| 23. C2×C22⋊(C5⋊D5) | C22 | semidirect |
| 24. C2×C22⋊(C5⋊D5) | C22 | semidirect |
| 25. C2×C22⋊(C5⋊D5) | C22 | semidirect |
| 26. C22×#1039 | C22 | semidirect |
| 28. C2×C4⋊(C5⋊D5) | C22 | semidirect |
| 29. C2×C4×C5⋊D5 | C22 | semidirect |
| 30. C22⋊#1039 | C22 | semidirect |
| 31. C22⋊#1039 | C22 | semidirect |
| 32. C22⋊#1039 | C22 | semidirect |
| 33. #87808 | C22 | semidirect |
| 34. #87808 | C22 | semidirect |
| 35. C4⋊#1039 | C22 | semidirect |
| 36. #87806 | C22 | semidirect |
| 37. C4×#1039 | C22 | semidirect |
| 52. C52×D4 | D4 | semidirect |
| 61. C22⋊(C5⋊D5) | D4 | direct (N × H) |
| 69. C22×C5×D4 | D5 | semidirect |
| 88. C22×C52 | C2×D4 | semidirect |
| 89. C22×C52 | C2×D4 | semidirect |
| 97. C2×C5⋊D5 | C2×D4 | semidirect |
| 99. C4×C52 | C2×D4 | semidirect |
| 103. #1039 | C2×D4 | semidirect |
| 104. C24×C5 | C2×D5 | semidirect |
| 108. C2×C5×D4 | C2×D5 | semidirect |
| 109. C2×C5×D4 | C2×D5 | semidirect |
| 110. C22×C4×C5 | C2×D5 | semidirect |
| 152. C5×D4 | C22⋊D5 | semidirect |
| 171. C22×D4 | C5⋊D5 | semidirect |
| 180. C52 | D42 | semidirect |
| 189. C22×C5 | D4×D5 | semidirect |
| 190. C22×C5 | C2×C22⋊D5 | semidirect |
| 200. C4×C5 | C2×C22⋊D5 | semidirect |
| 206. C24 | C2×C5⋊D5 | semidirect |
| 218. C2×D4 | C2×C5⋊D5 | semidirect |
| 219. C2×D4 | C2×C5⋊D5 | semidirect |
| 222. C22×C4 | C2×C5⋊D5 | semidirect |
| 259. D4 | C22⋊(C5⋊D5) | direct (N × H) |
| 269. C5 | D4×C22⋊D5 | semidirect |
| 285. C22 | D4×C5⋊D5 | semidirect |
| 286. C22 | C2×C22⋊(C5⋊D5) | semidirect |
| 293. C4 | C2×C22⋊(C5⋊D5) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C52×D4 | 2 |
| C22×C22⋊(C5⋊D5) | 2 |
| C2×D4×C5⋊D5 | 2 |
| #11717273 | 2 |
| #11717260 | 2 |
| #11717259 | 2 |
| #11717258 | 2 |
| #11717257 | 2 |
| D4×#1039 | 2 |
| #11717250 | 2 |
| C4×C22⋊(C5⋊D5) | 2 |
| D4×C22⋊D5 | 5 |