Subgroups — C₂²×C₃×ΓL(1,𝔽₈) #6999

Order 252.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×ΓL(1,𝔽₈)	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₃×C₇	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂²×ΓL(1,𝔽₈)	C₃	1	3	yes	maximal, direct product factor
4	C₃×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, direct product factor, Hall
5	C₂×C₃×C₇	C₂×C₃	1	3	yes	complemented
6	C₂×ΓL(1,𝔽₈)	C₂×C₃	1	9	yes	direct product factor
7	C₂²×C₃²	—	7	1	no	maximal, Hall
8	C₂²×C₇	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃×C₇	C₂²×C₃	1	1	yes	characteristic, complemented
10	ΓL(1,𝔽₈)	C₂²×C₃	1	3	yes	direct product factor
11	C₂×C₃²	—	7	3	no	—
12	C₂×C₇	C₂×C₃²	1	3	yes	complemented
13	C₂²×C₃	—	7	3	no	—
14	C₂²×C₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
15	C₃²	—	7	1	no	Sylow
16	C₇	C₂²×C₃²	1	1	yes	characteristic, complemented, Sylow
17	C₂×C₃	—	7	9	no	—
18	C₂×C₃	C₂×ΓL(1,𝔽₈)	1	3	yes	direct product factor
19	C₂²	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
20	C₃	—	7	3	no	—
21	C₃	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
22	C₂	C₂×C₃×ΓL(1,𝔽₈)	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₃	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₂²×C₃²
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₇	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂²×C₃

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.