While it is well-recognized that chromatin loops play an important role in gene regulation, structural details regarding higher order chromatin loops are only emerging. Here we present a systematic study of restrained chromatin loops ranging from 25 to 427 nucleosomes (fibers of 580 Kb DNA in length), mimicking gene elements studied by 3C contact data. We find that hierarchical looping represents a stable configuration that can effectively bring distant regions of the GATA-4 gene together, satisfying connections reported by 3C experiments. Additionally, we find that restrained chromatin fibers larger than 100 nucleosomes (similar to 20Kb) form closed plectonemes, whereas fibers shorter than 100 nucleosomes form simple hairpin loops. By studying the dependence of loop structures on internal parameters, we show that loop features are sensitive to linker histone concentration, loop length, divalent ions, and DNA linker length. Specifically, increasing loop length, linker histone concentration, and divalent ion concentration are associated with increased persistence length (or decreased bending), while varying DNA linker length in a manner similar to experimentally observed "nucleosome free regions" (found near transcription start sites) disrupts intertwining and leads to loop opening and increased persistence length in linker histone depleted (-LH) fibers. Chromatin fiber structure sensitivity to these parameters, all of which vary throughout the cell cycle, tissue type, and species, suggests that caution is warranted when using uniform polymer models to fit chromatin conformation capture genome-wide data. Furthermore, the folding geometry we observe near the transcription initiation site of the GATA-4 gene suggests that hierarchical looping provides a structural mechanism for gene inhibition, and offers tunable parameters for design of gene regulation elements.