Is it possible for a group (nonsimple and nonabelian) that solvability of all of its proper subgroups leads the whole group to be solvable?
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No. $SL(2,5)$ is a nonsimple nonsolvable group with the property that all its proper subgroups are solvable. 


An even simpler counter example is $A_5$. I believe that finite simple groups in which every proper subgroup is solvable are called minimal finite simple groups and as I recall they were classified by J. Thompson before the calssofication of all finite simple groups. This classification is useful, I think J. Wilson used them to study identities of solvable groups. 


The minimal nonsolvable group surely has the property that all proper subgroups are solvable. 

