# Maximal normal $2$-subgroup of $D_{20}.$

I have to find maximal normal $$2$$-subgroup of the dihedral group $$D_{20}$$, of order $$40.$$ According to me as Sylow $$2$$-subgroups are isomorphic to $$D_4$$, are $$5$$ in numbers so not normal. Center of group is $$C_2$$ i.e. cyclic group of order $$2$$ is normal. There are also subgroups of order $$4$$, $$C_4$$ and $$C_2\times C_2$$ but i do not know that these are normal or not. Please suggest me. Thanks.

• Firstly, the maximum normal 2-subgroup has order 4, not 2. Secondly, the group $D_{20}$ has subgroups of order $2$ that are not normal. And thirdly, the maximal normal $2$-subgroup of any finite group is unique, so there can only be one answer, not several. – Derek Holt Jan 25 at 17:16
• @DerekHolt thanks ......can you please tell me which subgroup is maximal normal $2$-subgroup. – neelkanth Jan 25 at 17:18
• Here $D_{20}$, is of order $40.$ – neelkanth Jan 25 at 17:19
• @DerekHolt Can you tell me which subgroup is that one.... – neelkanth Jan 25 at 17:40
• I think $C_4$ is the required one ? – neelkanth Jan 25 at 17:55