What is the relationship between the Galilean group and the Poincaré group? What is the relationship between the Galilean group and the Poincaré group?
Are they siblings within the Lie group? Or does the Poincaré group contain the Galilean group as a subgroup?
I'm not so much interested in the Galilean group being the limit of the Poincaré group for c -> inf.
 A: It is assumed you have appreciated Inönü, E.; Wigner, E. P. (1953), "On the Contraction of Groups and Their Representations" Proc. Natl. Acad. Sci. 39 (6): 510–24, and the super-helpful Gilmore text in Group contraction.
Very crudely, the Poincaré Lie algebra,
$$
[J_m,P_n] = i \epsilon_{mnk} P_k ~, \qquad  [J_i, P_0] =  0 ~, \\
 [K_i,P_k] = i \delta_{ik}  P_0 ~, \\
[P_0,P_i]=0 \qquad [P_i,P_j]=0 \qquad [K_i, P_0] = -i P_i ~, \\
 [J_m,J_n] = i \epsilon_{mnk} J_k ~, \qquad 
 [J_m,K_n] = i \epsilon_{mnk} K_k ~, \\
 [K_m,K_n] = -i \epsilon_{mnk} J_k  ~,
$$
given relabelings $E=-cP_0$ and $K_i=cC_i$  contracts upon $c\to \infty$ to  the Galilean algebra,
$$
[J_m,P_n] = i \epsilon_{mnk} P_k ~, \qquad 
 [J_i,E]=0 \\
 [C_i,P_j]= 0,~\\
[E,P_i]=0, \qquad 
 [P_i,P_j]=0, \qquad 
 [C_i,E]=i  P_i \\  
[J_m,J_n] = i \epsilon_{mnk} J_k ~,  \qquad 
 [J_m,C_n] = i \epsilon_{mnk} C_k ~, \\
 [C_i,C_j]=0 .
  $$
There are a few subtleties and wrinkles, extensions, to be sure, which I gather you are not focussing on, but,  crudely, the third and the last commutation relations trivialized/collapsed. (There is more, but I am oversimplifying...).
This collapse/amputation is the Lie algebraic manifestation of a group contraction.
