Iteration limit reached - too few observations in a category

Hey!

In order to identify important variables in a dataset, I am starting out with making a univariate logistic regression model for each of the variables. However, for some categorical variables, there are only a few observations in some categories (the total number of observations is 1155; there are no missing values). Running the code

 modelspec1='Y ~ X';
 B1 = fitglm(table,modelspec1,'Distribution','binomial');

returns the warning

 Warning: Iteration limit reached. 
 In glmfit (line 324)
 In GeneralizedLinearModel/fitter (line 575)
 In classreg.regr.FitObject/doFit (line 94)
 In GeneralizedLinearModel.fit (line 882)
 In fitglm (line 142)

and the estimate of beta becomes around 100, with a huge standard error and a p-value close to 1. I have tried to increase the number of iterations using

 opts = statset('glmfit');
 opts.MaxIter = 10000; % default value for glmfit is 100.

but it did not help.

I cannot share my data, but I have 1155 observations, where for example I have a dummy variable where 0 occurs 1132 times, and 1 occurs only 23 times. Where the dummy is 1, the response variable is always 0, while for dummy 0, the responses are both 0 and 1.

I really need to get some kind of an estimate (just dropping a category is sadly not an option). How can I fix the problem? I read ( here ) that I could only use a part of the data where the dummy is zero, but I would like to use it all (as only using a part of it becomes a problem when I include more variables in the model), is it possible?

Thank you!