Given the group \( \mathbb{Z}/7\mathbb{Z} \), we aim to determine its subgroups.

### Structure of \( \mathbb{Z}/7\mathbb{Z} \)

The group \( \mathbb{Z}/7\mathbb{Z} \) is the cyclic group of order 7, consisting of elements:
\[ \{0, 1, 2, 3, 4, 5, 6\} \]
under addition modulo 7.

### Subgroups of Cyclic Groups

A cyclic group \( \mathbb{Z}/n\mathbb{Z} \) of order \( n \) has exactly one subgroup of order \( d \) for each divisor \( d \) of \( n \). The subgroup of order \( d \) is generated by \( \frac{n}{d} \).

For \( \mathbb{Z}/7\mathbb{Z} \):
- The divisors of 7 are 1 and 7.

### Subgroups of \( \mathbb{Z}/7\mathbb{Z} \)

1. **Subgroup of order 1**:
   - The trivial subgroup: \(\{0\}\).

2. **Subgroup of order 7**:
   - The whole group itself: \(\mathbb{Z}/7\mathbb{Z}\).

Since 7 is prime, there are no other divisors, and therefore no other subgroups.

### Conclusion

The subgroups of \( \mathbb{Z}/7\mathbb{Z} \) are:
1. The trivial subgroup \(\{0\}\).
2. The whole group \(\mathbb{Z}/7\mathbb{Z}\).

Hence, there are exactly 2 subgroups of \( \mathbb{Z}/7\mathbb{Z} \).

**Summary**:
\[
\boxed{2}
\]

### Additional Queries and Expansions

1. What are the generators of the cyclic group \( \mathbb{Z}/7\mathbb{Z} \)?
2. How do you find the order of an element in a cyclic group?
3. What are the properties of cyclic groups of prime order?
4. How do you prove that a subgroup of a cyclic group is also cyclic?
5. What are the possible subgroups of \( \mathbb{Z}/n\mathbb{Z} \) for a non-prime \( n \)?
6. How do you identify the unique subgroup of each divisor in a cyclic group?
7. What is the significance of the trivial subgroup?
8. How does the structure of subgroups change with different prime orders?

**Tip:** In cyclic groups, understanding the divisors of the group's order helps to identify all possible subgroups quickly.

Explore the subgroups of the cyclic group Z/7Z, including the trivial subgroup {0} and the whole group Z/7Z itself. Learn about the structure of cyclic groups and the properties of subgroup formation in group theory.

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