Question

A group of order10 then find the order of its subgroup

Answer 1

No, your reasoning is not correct. For example, if there were two subgroups of order 55, they can only intersect at ee, the identity element or they are both the same subgroup.. This is because each subgroup has prime order and any intersection of subgroups is a subgroup. So, if there are two distinct subgroups of order 55, the original group would have to have at least 2525 elements.

In fact for a group of order 1010, it can only have one subgroup of order 22 and one of order 55. Let GG be the group of order 1010, and HH the subgroup of order 55. Note that [G:H]=2[G:H]=2. It's not hard to show that any subgroup of index 22 is normal.