Discussion
The subgroup lattice of a finite cyclic group is :
Answer: A
The subgroup lattice is a finite direct product of chains.
A cyclic group of order \(n\) has exactly one subgroup of each order \(d\) dividing \(n\), and \(H_{d} \subseteq H_{e}\) exactly when \(d \mid e\). So its subgroup lattice is the lattice of divisors of \(n\). If \(n = p_{1}^{a_{1}} \cdots p_{k}^{a_{k}}\), a divisor is chosen by picking an exponent \(0, \ldots, a_{i}\) for each prime independently, which makes the lattice the product of the chains \(\{0, 1, \ldots, a_{i}\}\).
A product of chains is distributive, and every distributive lattice is modular, so 'modular but not distributive', 'not modular' and 'not distributive' are all false.
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