Non-solvable groups
Groups whose derived series never reaches the trivial subgroup — equivalently, with no subnormal series of abelian factors. Each is shown with the first step of that series.
| order | group | G′ | G/G′ |
|---|
| 60 | A5 #304 | perfect |
| 120 | SL(2,𝔽5) #1195 | perfect |
| 120 | S5 #1224 | A5 | C2 |
| 120 | C2×A5 #1225 | A5 | C2 |
| 168 | GL(3,𝔽2) #4272 | perfect |
| 180 | C3×A5 #4374 | A5 | C3 |
| 240 | #6728 | SL(2,𝔽5) | C2 |
| 240 | #6729 | SL(2,𝔽5) | C2 |
| 240 | A5⋊C4 #6730 | A5 | C4 |
| 240 | C4×A5 #6731 | A5 | C4 |
| 240 | #6732 | SL(2,𝔽5) | C2 |
| 240 | C2×SL(2,𝔽5) #6733 | SL(2,𝔽5) | C2 |
| 240 | C2×S5 #6828 | A5 | C22 |
| 240 | C22×A5 #6829 | A5 | C22 |
| 300 | C5×A5 #64479 | A5 | C5 |
| 336 | SL(2,𝔽7) #66645 | perfect |
| 336 | SO(3,𝔽7) #66739 | GL(3,𝔽2) | C2 |
| 336 | C2×GL(3,𝔽2) #66740 | GL(3,𝔽2) | C2 |
| 360 | C3×SL(2,𝔽5) #67121 | SL(2,𝔽5) | C3 |
| 360 | A6 #67188 | perfect |
| 360 | C3×S5 #67189 | A5 | C2×C3 |
| 360 | ΓL(2,𝔽4) #67190 | C3×A5 | C2 |
| 360 | D3×A5 #67191 | C3×A5 | C2 |
| 360 | C2×C3×A5 #67192 | A5 | C2×C3 |
| 420 | C7×A5 #88288 | A5 | C7 |
| 480 | A5⋊C8 #91141 | A5 | C8 |
| 480 | GL(2,𝔽5) #91142 | SL(2,𝔽5) | C4 |
| 480 | #91143 | SL(2,𝔽5) | C4 |
| 480 | C8×A5 #91144 | A5 | C8 |
| 480 | #91145 | SL(2,𝔽5) | C4 |
| 480 | C4×SL(2,𝔽5) #91146 | SL(2,𝔽5) | C4 |
| 480 | C4×S5 #91867 | A5 | C2×C4 |
| 480 | C4⋊S5 #91868 | C2×A5 | C22 |
| 480 | A5⋊Q8 #91869 | C2×A5 | C22 |
| 480 | #91870 | SL(2,𝔽5) | C22 |
| 480 | #91871 | SL(2,𝔽5) | C22 |
| 480 | #91872 | SL(2,𝔽5) | C22 |
| 480 | C2×#6728 #91873 | SL(2,𝔽5) | C22 |
| 480 | C2×#6729 #91874 | SL(2,𝔽5) | C22 |
| 480 | C22⋊S5 #91875 | C2×A5 | C22 |
| 480 | C2×A5⋊C4 #91876 | A5 | C2×C4 |
| 480 | #91877 | SL(2,𝔽5) | C22 |
| 480 | C2×C4×A5 #91878 | A5 | C2×C4 |
| 480 | C2×#6732 #91879 | SL(2,𝔽5) | C22 |
| 480 | D4×A5 #91880 | C2×A5 | C22 |
| 480 | #91881 | SL(2,𝔽5) | C22 |
| 480 | Q8×A5 #91882 | C2×A5 | C22 |
| 480 | #91883 | SL(2,𝔽5) | C22 |
| 480 | C22×SL(2,𝔽5) #91884 | SL(2,𝔽5) | C22 |
| 480 | C22×S5 #92110 | A5 | C23 |
| 480 | C23×A5 #92111 | A5 | C23 |
| 504 | SL(2,𝔽8) #92730 | perfect |
| 504 | C3×GL(3,𝔽2) #92731 | GL(3,𝔽2) | C3 |
| 540 | C9×A5 #10587371 | A5 | C9 |
| 540 | C32×A5 #10587428 | A5 | C32 |
| 600 | C5×SL(2,𝔽5) #10597073 | SL(2,𝔽5) | C5 |
| 600 | C5×S5 #10597163 | A5 | C2×C5 |
| 600 | C5⋊S5 #10597164 | C5×A5 | C2 |
| 600 | D5×A5 #10597165 | C5×A5 | C2 |
| 600 | C2×C5×A5 #10597166 | A5 | C2×C5 |
| 660 | PSL(2,𝔽11) #10620343 | perfect |
| 660 | C11×A5 #10620344 | A5 | C11 |
| 672 | GL(3,𝔽2)⋊C4 #10621461 | GL(3,𝔽2) | C4 |
| 672 | #10621462 | SL(2,𝔽7) | C2 |
| 672 | #10621463 | SL(2,𝔽7) | C2 |
| 672 | C4×GL(3,𝔽2) #10621464 | GL(3,𝔽2) | C4 |
| 672 | #10621465 | SL(2,𝔽7) | C2 |
| 672 | C2×SL(2,𝔽7) #10621466 | SL(2,𝔽7) | C2 |
| 672 | C2×SO(3,𝔽7) #10621672 | GL(3,𝔽2) | C22 |
| 672 | C22×GL(3,𝔽2) #10621673 | GL(3,𝔽2) | C22 |
| 720 | SL(2,𝔽9) #10623940 | perfect |
| 720 | C3×#6728 #10623941 | SL(2,𝔽5) | C2×C3 |
| 720 | C3×#6729 #10623942 | SL(2,𝔽5) | C2×C3 |
| 720 | C3×A5⋊C4 #10623943 | A5 | C3×C4 |
| 720 | C3⋊(A5⋊C4) #10623944 | C3×A5 | C4 |
| 720 | C3⋊#6728 #10623945 | C3×SL(2,𝔽5) | C2 |
| 720 | C3⋊#6729 #10623946 | C3×SL(2,𝔽5) | C2 |
| 720 | Dic3×A5 #10623947 | C3×A5 | C4 |
| 720 | D3×SL(2,𝔽5) #10623948 | C3×SL(2,𝔽5) | C2 |
| 720 | C3⋊#6732 #10623949 | C3×SL(2,𝔽5) | C2 |
| 720 | C3×C4×A5 #10623950 | A5 | C3×C4 |
| 720 | C3×#6732 #10623951 | SL(2,𝔽5) | C2×C3 |
| 720 | C2×C3×SL(2,𝔽5) #10623952 | SL(2,𝔽5) | C2×C3 |
| 720 | S6 #10624294 | A6 | C2 |
| 720 | PGL(2,𝔽9) #10624295 | A6 | C2 |
| 720 | #10624296 | A6 | C2 |
| 720 | C2×A6 #10624297 | A6 | C2 |
| 720 | D3×S5 #10624298 | C3×A5 | C22 |
| 720 | A4×A5 #10624299 | C22×A5 | C3 |
| 720 | C2×C3×S5 #10624300 | A5 | C22×C3 |
| 720 | C2×ΓL(2,𝔽4) #10624301 | C3×A5 | C22 |
| 720 | C2×D3×A5 #10624302 | C3×A5 | C22 |
| 720 | C22×C3×A5 #10624303 | A5 | C22×C3 |
| 780 | C13×A5 #11715930 | A5 | C13 |
| 840 | C7×SL(2,𝔽5) #11719711 | SL(2,𝔽5) | C7 |
| 840 | C5×GL(3,𝔽2) #11719832 | GL(3,𝔽2) | C5 |
| 840 | C7×S5 #11719833 | A5 | C2×C7 |
| 840 | C7⋊S5 #11719834 | C7×A5 | C2 |
| 840 | D7×A5 #11719835 | C7×A5 | C2 |
| 840 | C2×C7×A5 #11719836 | A5 | C2×C7 |
About non-solvable groups →
Finite Group Databasebeta