So basically you know how to find the unbroken Lie algebra but do not know how to find the associated Lie group.

To a given Lie algebra $\mathfrak g$ there exists a unique group $\tilde G$, called the universal covering group, with the property of being [simply connected][1]. For example, the covering group of the algebra $\mathfrak{su}(2)$ is $SU(2)$.

The other groups, $\{G\}$, associated to the same algebra can be obtained from the covering group in the following way

$$G=\frac{\tilde G}{Ker(\rho)},$$

where $Ker(\rho)$ is the kernel of the group homomorphism $\rho:\tilde G\rightarrow G$. Once you have defined a particular representation by choosing a particular highest weight, you are able to compute this kernel. For example, you start with an $\mathfrak{su}(2)$ algebra. Then if you choose the adjoint representation (the highest weight being the highest root) you can show that $Ker(\rho)=\mathbb Z_2$ and the group will be $G=SU(2)/\mathbb Z_2=SO(3)$. On the other hand, if you choose the defining representation you get $Ker(\rho)=\mathbb 1$ and $G=SU(2)/\mathbb 1=SU(2)$.

There are some technical details needed to compute those kernel but in general,

$$Ker(\rho)\subset\mathcal Z(\tilde G),$$

where $\mathcal Z(\tilde G)$ is the center of $\tilde G$, and this center is a finite group which can be obtained from the extended Dynkin diagram.

Same references:

Cornwell, group theory in physics, 1984;

Olive, Turok, Nucl Phys B215, 1983, p470;