Subgroups — #89204

Order 441.

30 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇²	C₃	1	1	yes	maximal, characteristic
2	C₇⋊C₉	—	7	8	no	maximal
3	C₇²	C₉	1	1	yes	characteristic, complemented, Sylow
4	C₃×C₇	ΓL(1,𝔽₈)	1	8	yes	—
5	C₉	—	49	1	no	Sylow
6	C₇	C₇⋊C₉	1	8	yes	complemented
7	C₃	#3861	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₃	#3861
commutator, hypocenter	C₇²	C₉
Fitting, socle, prime radical, cosocle	C₃×C₇²	C₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₃

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

N	H ≅ G/N	type
3. C₇²	C₉	semidirect
6. C₇	C₇⋊C₉	semidirect

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

subgroup	index
C₃×C₇²	3
C₇⋊C₉	7