Element-by-element listings of modular groups are easy to produce for small moduli, but they conceal the arithmetic mechanisms that determine element orders, cyclicity, and subgroup multiplicity. This article asks whether the complete order distribution of the additive group $\mathbb{Z}_n$ and the unit group $U(n)=(\mathbb{Z}/n\mathbb{Z})^{\times}$ can be recovered from the prime-power factorization of $n$ without tracing every power orbit. A factorization-aware procedure is developed in which additive orders are obtained from the divisor lattice, local unit groups are represented by cyclic factors, and their order distributions are combined through a least-common-multiple convolution. Invariant-factor normalization supplies a compact structural certificate, while an independent residuewise order reducer verifies every count. The computational corpus contains all 299 moduli from 2 through 300. Two detailed arithmetic cases, $\mathbb{Z}_{209}$ and $U(210)$, are used to establish correspondence with independently tabulated records. The method recovers the additive spectrum of $\mathbb{Z}_{209}$ as $1^1,11^{10},19^{18},209^{180}$ and the unit spectrum of $U(210)$ as $1^1,2^7,3^2,4^8,6^{14},12^{16}$. Across the full corpus, 114 unit groups are cyclic, the richest unit-order distribution occurs at $n=241$ with 20 distinct orders, and the largest compression is observed at $n=263$, where 262 residues are represented by four spectral entries. The invariant-factor ranks occur with frequencies $1,113,133,47,$ and $5$ from rank zero through rank four, and the corresponding numbers of involutions are $0,1,3,7,$ and $15$. The 299 unit groups occupy 142 invariant-factor classes, of which 85 recur at more than one modulus. Least-common-multiple convolution uses 4,425 divisor-pair combinations, compared with 1,587,655 modular multiplication steps required by direct orbit traversal. Repeated whole-corpus timings give a median reduction from 75.58 to 4.25 ms. The results show that the order spectrum is both a mathematically interpretable group signature and an efficient computational object for modular-group analysis.

finite abelian group, modular arithmetic, element order, Chinese remainder theorem, Carmichael function, invariant factors, cyclic subgroup