Subgroups — C₂×D₅×ΓL(1,𝔽₈) #88293

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	1	yes	maximal, characteristic, complemented
2	D₅×ΓL(1,𝔽₈)	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₇×D₅	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₅×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂²×ΓL(1,𝔽₈)	—	5	1	no	maximal, Hall
6	C₂×C₅×C₇	C₂×C₃	1	1	yes	characteristic, complemented
7	C₇×D₅	C₂×C₃	1	2	yes	complemented
8	C₂×C₃×D₅	—	7	1	no	maximal, Hall
9	C₂×ΓL(1,𝔽₈)	—	5	2	no	—
10	C₂×ΓL(1,𝔽₈)	D₅	1	1	yes	characteristic, direct product factor
11	C₅×C₇	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
12	C₂×C₃×C₅	—	7	1	no	—
13	C₃×D₅	—	7	2	no	—
14	C₂²×C₇	—	5	1	no	Hall
15	ΓL(1,𝔽₈)	C₂×D₅	1	1	yes	characteristic, direct product factor, Hall
16	C₂×D₅	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
17	C₃×C₅	—	7	1	no	Hall
18	C₂×C₇	—	5	2	no	—
19	C₂×C₇	C₃×D₅	1	1	yes	characteristic, complemented
20	C₂²×C₃	—	35	1	no	Hall
21	C₂×C₅	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
22	D₅	C₂×ΓL(1,𝔽₈)	1	2	yes	direct product factor
23	C₇	C₂×C₃×D₅	1	1	yes	characteristic, complemented, Sylow
24	C₂×C₃	—	7	1	no	—
25	C₂×C₃	—	35	2	no	—
26	C₅	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
27	C₂²	—	5	1	no	Sylow
28	C₃	—	7	1	no	Sylow
29	C₂	—	5	2	no	—
30	C₂	D₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	D₅×ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₅×C₇	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₅×C₇	C₂×C₃
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.