Cellular immune control of HIV is mediated, in part, by induction of single amino acid mutations that reduce viral fitness, but compensatory mutations limit this effect. Here, we sought to determine if higher order constraints on viral evolution exist, because some coordinately linked combinations of mutations may hurt viability. Immune targeting of multiple sites in such a multidimensionally conserved region might render the virus particularly vulnerable, because viable escape pathways would be greatly restricted. We analyzed available HIV sequences using a method from physics to reveal distinct groups of amino acids whose mutations are collectively coordinated ("HIV sectors"). From the standpoint of mutations at individual sites, one such group in Gag is as conserved as other collectively coevolving groups of sites in Gag. However, it exhibits higher order conservation indicating constraints on the viability of viral strains with multiple mutations. Mapping amino acids from this group onto protein structures shows that combined mutations likely destabilize multiprotein structural interactions critical for viral function. Persons who durably control HIV without medications preferentially target the sector in Gag predicted to be most vulnerable. By sequencing circulating viruses from these individuals, we find that individual mutations occur with similar frequency in this sector as in other targeted Gag sectors. However, multiple mutations within this sector are very rare, indicating previously unrecognized multidimensional constraints on HIV evolution. Targeting such regions with higher order evolutionary constraints provides a novel approach to immunogen design for a vaccine against HIV and other rapidly mutating viruses.