Deriving the largest expected number of elementary particles in the standard model from the maximal compact subgroup H of the exceptional Lie group E7(-5)

The maximal number of elementary particles which could be expected to be found within a modestly extended energy scale of the standard model was found using various methods to be N = 69. In particular using E-infinity theory the present Author found the exact transfinite expectation value to be 〈N〉=α¯o/2≅69 where α¯o=137.082039325 is the exact inverse fine structure constant. In the present work we show among other things how to derive the exact integer value 69 from the exceptional Lie symmetry groups hierarchy. It is found that the relevant number is given by dim H = 69 where H is the maximal compact subspace of E7(-5) so that N = dim H = 69 while dim E7 = 133.