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Let M be the class of binary matroids without a Fano plane as a submatroid. We show that every supersolvable matroid in M is graphic, corresponding to a chordal graph. Then we characterize the case that the "modular join" of two matroids is supersolvable. This is used to study modular flats and modular joins of binary supersolvable matroids. We decompose supersolvable matroids in M as modular joins with respect to hyperplanes. For such matroids every modular flat is contained in a maximal chaindoi:10.1090/s0002-9939-1991-1068134-3 fatcat:d6etfxmvn5cafkurahiv75hbnu