Large reflector antennas, such as the European Space Agency deep space antennas (DSAs), practically always require struts to support the sub-reflector. While inevitable, they deteriorate the antenna performance. To minimize this deterioration, it is pivotal to understand the role played by different features, including struts diameter and shape. This paper proposes a detailed numerical investigation on the impact of these features on antenna efficiency and side lobes, for a test case comprising both DSA3 and DSA4. It is demonstrated, for the first time in a comprehensive and quantitative way that includes different permutations for the strut design, that both features are significant to define the deterioration, thus providing a significant feedback for struts design.