The subgroup structure of pseudo-reductive groups

Let k be a field. We investigate the relationship between subgroups of a pseudo-reductive k-group G and its maximal reductive quotient $G'$, with applications to the subgroup structure of G. Let $k'/k$ be the minimal field of definition for the geometric unipotent radical of G, and let π':Gk' → G' be the quotient map. We first characterise those smooth subgroups H of G for which π'(Hk')=G'. We next consider the following questions: given a subgroup $H'$ of $G'$, does there exist a subgroup H of G such that π'(Hk')=H', and if $H'$ is smooth can we find such a H that is smooth? We find sufficient conditions for a positive answer to these questions. In general there are various obstructions to the existence of such a subgroup H, which we illustrate with several examples. Finally, we apply these results to relate the maximal smooth subgroups of G with those of $G'$.