The analysis of large-scale Boolean network dynamics is of great importance in understanding complex phenomena where systems are characterized by a large number of components. The computational cost to reveal the number of attractors and the period of each attractor increases exponentially as the number of nodes in the networks increases. This paper presents an efficient algorithm to find attractors for medium to large-scale networks. This is achieved by analyzing subnetworks within the network in a way that allows to reveal the attractors of the full network with little computational cost. In particular, for each subnetwork modeled as a Boolean control network, the input-state cycles are found and they are composed to reveal the attractors of the full network. The proposed algorithm reduces the computational cost significantly, especially in finding attractors of short period, or any periods if the aggregation network is acyclic. Also, this paper shows that finding the best acyclic aggregation is equivalent to finding the strongly connected components of the network graph. Finally, the efficiency of the algorithm is demonstrated on two biological systems, namely a T-cell receptor network and an early flower development network.