Subgroups — C₅×AΓL(1,𝔽₈) #11719838

Order 840.

24 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×AGL(1,𝔽₈)	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	AΓL(1,𝔽₈)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₂×C₅×A₄	—	7	1	no	maximal, Hall
4	C₅×ΓL(1,𝔽₈)	—	8	1	no	maximal, Hall
5	C₅×A₄	—	7	1	no	—
6	AGL(1,𝔽₈)	C₃×C₅	1	1	yes	characteristic, complemented, Hall
7	C₂³×C₅	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
8	C₅×C₇	—	8	1	no	Hall
9	C₂×C₃×C₅	—	28	1	no	—
10	C₂×A₄	—	7	1	no	Hall
11	ΓL(1,𝔽₈)	—	8	1	no	Hall
12	C₂²×C₅	—	7	1	no	—
13	C₃×C₅	—	28	1	no	Hall
14	A₄	—	7	1	no	—
15	C₂×C₅	—	7	1	no	—
16	C₂³	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
17	C₇	—	8	1	no	Sylow
18	C₂×C₃	—	28	1	no	—
19	C₅	AΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
20	C₂²	—	7	1	no	—
21	C₃	—	28	1	no	Sylow
22	C₂	—	7	1	no	—

	subgroup	G/H
center, hypercenter	C₅	AΓL(1,𝔽₈)
commutator, cosocle, hypocenter	AGL(1,𝔽₈)	C₃×C₅
Frattini	C₁	G
Fitting, socle	C₂³×C₅	ΓL(1,𝔽₈)
LCS	G ▷ AGL(1,𝔽₈)
UCS	C₁ ◁ 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.