Subgroups — AGL(1,𝔽₅)×ΓL(1,𝔽₈) #88289

Order 420.

24 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	C₇×AGL(1,𝔽₅)	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic, complemented, Hall
4	C₄×ΓL(1,𝔽₈)	—	5	1	no	maximal, Hall
5	C₇×D₅	C₂×C₃	1	1	yes	characteristic
6	C₃×AGL(1,𝔽₅)	—	7	1	no	maximal, Hall
7	C₂×ΓL(1,𝔽₈)	—	5	1	no	—
8	C₅×C₇	C₃×C₄	1	1	yes	characteristic, complemented, Hall
9	C₃×D₅	—	7	1	no	—
10	C₄×C₇	—	5	1	no	Hall
11	ΓL(1,𝔽₈)	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor, Hall
12	AGL(1,𝔽₅)	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
13	C₃×C₅	—	7	1	no	Hall
14	C₂×C₇	—	5	1	no	—
15	C₃×C₄	—	35	1	no	Hall
16	D₅	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
17	C₇	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
18	C₂×C₃	—	35	1	no	—
19	C₅	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
20	C₄	—	5	1	no	Sylow
21	C₃	—	7	1	no	Sylow
22	C₂	—	5	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅×C₇	C₃×C₄
prime radical	D₅×ΓL(1,𝔽₈)	C₂
cosocle	C₇×D₅	C₂×C₃
LCS	G ▷ C₅×C₇
UCS	C₁

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.