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The layered ATMPn$_2$ (A = alkali earth or rare earth atom, TM = transition metal, Pn = Sb, Bi) compounds are widely studied for their rich magnetism and electronic structure topology. Here, we characterize the physical properties of LaMn$_x$Sb$_2$, an understudied member of the ATMPn$_2$ family. LaMn$_x$Sb$_2$ forms with intrinsic Mn vacancies, and we demonstrate synthetic control of the Mn occupancy to produce single crystals with x = 0.74-0.97. Magnetization and transport measurements indicate LaMn$_x$Sb$_2$ has a rich temperature-composition (T-x) magnetic phase diagram with physical properties strongly influenced by the Mn occupancy. LaMn$_x$Sb$_2$ orders antiferromagnetically at T$_{1}$ = 130--180 K, where T$_{1}$ increases with x. Below T$_{1}$, the T-x phase diagram is complicated. At high x, there is a second transition T$_2$ that decreases in temperature as x is lowered, vanishing below x $\leq$ 0.85. A third, first-order, transition T$_3$ is detected at x $\approx$ 0.92, and the transition temperature increases as x is lowered, crossing above T$_2$ near x $\approx$ 0.9. On moving below x $<$ 0.79, we find the crystal structure changes from the P4/nmm arrangement to a I$\bar{4}$2m structure with partially ordered Mn vacancies. The change in crystal structure results in the appearance of two new low temperature phases and a crossover between regimes of negative and positive magnetoresistance. Finally, we provide neutron diffraction for x = 0.93, and find that the high x compositions first adopt a G-type AFM structure with the Mn moments aligned within the ab-plane which is followed on cooling by a second transition to a different, non-collinear structure where the moments are rotated within the basal plane. Our results demonstrate that LaMn$_x$Sb$_2$ is a highly tunable material with six unique magnetically ordered phases, depending on T and x.