Subgroups — C₂²×AGL(1,𝔽₇) #4277

Order 168.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×AGL(1,𝔽₇)	C₂	1	6	yes	maximal, direct product factor
3	C₂²×D₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂×ΓL(1,𝔽₈)	C₂²	1	3	yes	complemented
5	AGL(1,𝔽₇)	C₂²	1	4	yes	direct product factor
6	C₂²×C₇	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂×D₇	C₂×C₃	1	6	yes	complemented
8	C₂³×C₃	—	7	1	no	maximal, Hall
9	ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic, complemented, Hall
10	C₂×C₇	C₂²×C₃	1	3	yes	complemented
11	D₇	C₂²×C₃	1	4	yes	complemented
12	C₂²×C₃	—	7	6	no	—
13	C₂²×C₃	—	7	1	no	—
14	C₂³	—	7	1	no	Sylow
15	C₇	C₂³×C₃	1	1	yes	characteristic, complemented, Sylow
16	C₂×C₃	—	7	4	no	—
17	C₂×C₃	—	7	3	no	—
18	C₂²	—	7	6	no	—
19	C₂²	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
20	C₃	—	7	1	no	Sylow
21	C₂	—	7	4	no	—
22	C₂	C₂×AGL(1,𝔽₇)	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²	AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₇	C₂³×C₃
Frattini	C₁	G
Fitting, socle	C₂²×C₇	C₂×C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂²

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.