Subgroups — C₂²×C₅×ΓL(1,𝔽₈) #88306

Order 420.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×ΓL(1,𝔽₈)	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₅×C₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, direct product factor, Hall
4	C₂²×ΓL(1,𝔽₈)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₅×C₇	C₂×C₃	1	3	yes	complemented
6	C₂²×C₃×C₅	—	7	1	no	maximal, Hall
7	C₂×ΓL(1,𝔽₈)	C₂×C₅	1	3	yes	direct product factor
8	C₅×C₇	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
9	C₂×C₃×C₅	—	7	3	no	—
10	C₂²×C₇	C₃×C₅	1	1	yes	characteristic, complemented, Hall
11	ΓL(1,𝔽₈)	C₂²×C₅	1	1	yes	characteristic, direct product factor, Hall
12	C₂²×C₅	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
13	C₃×C₅	—	7	1	no	Hall
14	C₂×C₇	C₂×C₃×C₅	1	3	yes	complemented
15	C₂²×C₃	—	7	1	no	Hall
16	C₂×C₅	C₂×ΓL(1,𝔽₈)	1	3	yes	direct product factor
17	C₇	C₂²×C₃×C₅	1	1	yes	characteristic, complemented, Sylow
18	C₂×C₃	—	7	3	no	—
19	C₅	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
20	C₂²	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
21	C₃	—	7	1	no	Sylow
22	C₂	C₂×C₅×ΓL(1,𝔽₈)	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₅	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₂²×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₂²×C₅×C₇	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂²×C₅

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.