Subgroups — Dic₅×ΓL(1,𝔽₈) #88277

Order 420.

24 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	C₇×Dic₅	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	C₂×C₅×C₇	C₂×C₃	1	1	yes	characteristic
6	C₃×Dic₅	—	7	1	no	maximal, Hall
7	C₂×ΓL(1,𝔽₈)	D₅	1	1	yes	characteristic
8	C₅×C₇	C₃×C₄	1	1	yes	characteristic, complemented, Hall
9	C₂×C₃×C₅	—	7	1	no	—
10	C₄×C₇	—	5	1	no	Hall
11	ΓL(1,𝔽₈)	Dic₅	1	1	yes	characteristic, direct product factor, Hall
12	Dic₅	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
13	C₃×C₅	—	7	1	no	Hall
14	C₂×C₇	C₃×D₅	1	1	yes	characteristic
15	C₃×C₄	—	35	1	no	Hall
16	C₂×C₅	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
17	C₇	C₃×Dic₅	1	1	yes	characteristic, complemented, Sylow
18	C₂×C₃	—	7	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	D₅×ΓL(1,𝔽₈)	1	1	yes	characteristic

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