1. Direct method: This is the most direct method. By observing and analyzing the properties of groups, subgroups can be found directly. For example, for integer set z and addition operation, it is obvious that Z itself and any non-empty subset are subgroups of Z.
2. Method of generating by subgroups: If a subset H of a group G can generate all the elements of G (that is, any element of G can be written as a finite power of the elements of H), then H is a subgroup of G ... This method needs to solve some problems about generators.
3. Using the quotient group method: If a group G has a subgroup H, then G/H (that is, the set of all pairs (G, H) shaped like gh, in which both G and H are in H) is also a group. This method needs to solve some problems about enterprise groups.
4. Through homomorphism: If there is a homomorphism F from group G to another group H, and F is injective and the kernel of F is H, then H is a subgroup of G ... This method needs to solve some problems about homomorphism.
5. According to Lagrange theorem, if the order of a group G is divisible by the order of any subgroup, then G is a commutative group. This method needs to solve some problems about Lagrange theorem.
The above are some calculation methods of mathematical subgroups. Different methods are suitable for different situations, and it is necessary to choose the appropriate method according to specific problems.