Subgroups — C₇⋊SL(2,𝔽₃) #4253

Order 168.

14 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×Q₈	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₂×ΓL(1,𝔽₈)	—	4	1	no	maximal
3	C₄×C₇	—	3	1	no	—
4	SL(2,𝔽₃)	—	7	1	no	maximal, Hall
5	ΓL(1,𝔽₈)	—	4	1	no	Hall
6	C₂×C₇	A₄	1	1	yes	characteristic
7	Q₈	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
8	C₇	SL(2,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
9	C₂×C₃	—	28	1	no	—
10	C₄	—	3	1	no	—
11	C₃	—	28	1	no	Sylow
12	C₂	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₂²⋊ΓL(1,𝔽₈)
commutator, Fitting, cosocle, hypocenter	C₇×Q₈	C₃
socle	C₂×C₇	A₄
LCS	G ▷ C₇×Q₈
UCS	C₁ ◁ C₂

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

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