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How is normality reflected in Todd-Coxeter Algorithm tables?

I can think of the generic cases like if there are only 2 indices then the subgroup has index 2 and thus is normal. Same case for p indices if $G$ is a p-group. But the general case eludes me.

I thought about using left/ right cosets, but T-C deals with only right cosets so that's no use; same goes for the normalizer approach. Unless there is some trick to manipulate the table into that of left cosets, but I really doubt it.

Any help would be appreciated. Thanks.

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If you are doing Todd-Coxeter coset enumeration of a group $G$ over a subgroup $H$, then $H \unlhd G$ if and only if, in the final completed table, we have $cy = c$ for all cosets $c$ and all generators $y$ of $H$. If you know in advance that $H$ is normal, then you can assume that condition during the enumeration, but usually you will not be able to verify it until the table is complete.

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    As you have seen through my previous questions, I am Master to do this algorithm. +12012-11-21