Subgroups — C₂×C₂²⋊ΓL(1,𝔽₈) #4283

Order 168.

24 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, direct product factor
2	C₂³×C₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₂×ΓL(1,𝔽₈)	—	4	1	no	maximal
4	C₂²×C₇	—	3	1	no	—
5	C₂²×C₇	—	3	1	no	—
6	C₂²×C₇	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂×A₄	—	7	1	no	maximal, Hall
8	ΓL(1,𝔽₈)	—	4	1	no	Hall
9	C₂×C₇	—	3	1	no	—
10	C₂×C₇	—	3	1	no	—
11	C₂×C₇	A₄	1	1	yes	characteristic, complemented
12	A₄	—	7	1	no	—
13	C₂³	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
14	C₇	C₂×A₄	1	1	yes	characteristic, complemented, Sylow
15	C₂×C₃	—	28	1	no	—
16	C₂²	—	3	1	no	—
17	C₂²	—	3	1	no	—
18	C₂²	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
19	C₃	—	28	1	no	Sylow
20	C₂	—	3	1	no	—
21	C₂	—	3	1	no	—
22	C₂	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	C₂²⋊ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂²×C₇	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₂³×C₇	C₃
LCS	G ▷ C₂²×C₇
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.