Subgroups — C₇⋊C₉ #316

Order 63.

6 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇	C₃	1	1	yes	maximal, characteristic
2	C₉	—	7	1	no	maximal, Sylow
3	C₇	C₉	1	1	yes	characteristic, complemented, Sylow
4	C₃	ΓL(1,𝔽₈)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₃	ΓL(1,𝔽₈)
commutator, hypocenter	C₇	C₉
Fitting, socle, prime radical, cosocle	C₃×C₇	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₃

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₇	C₉	semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×C₇	3
C₉	7