We present the pressure-temperature phase diagram La$_5$Co$_2$Ge$_3$ up to $sim$ 5,GPa, which was constructed from magnetization, resistivity and specific heat measurements. At ambient pressure, La$_5$Co$_2$Ge$_3$ is an itinerant ferromagnet with a Curie temperature $T_textrm Csim$ 4,K. Upon increasing pressure up to $sim$ 1.7,GPa, $T_textrm C$ is suppressed down to $sim$ 3,K. Upon further increasing pressure, our results suggest that La$_5$Co$_2$Ge$_3$ enters a different low-temperature ground state. The corresponding transition temperature, $T^*$, has a nonmonotonic pressure dependence up to $sim$ 5,GPa. Our results demonstrate that the ferromagnetic quantum critical point in La$_5$Co$_2$Ge$_3$ is avoided by the appearance of a different, likely magnetically ordered state that has an antiferromagnetic component.
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