Subgroups — C₂×C₅×AGL(1,𝔽₇) #88292

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	C₅×AGL(1,𝔽₇)	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₅×D₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₅×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×AGL(1,𝔽₇)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₅×C₇	C₂×C₃	1	1	yes	characteristic, complemented
7	C₅×D₇	C₂×C₃	1	2	yes	complemented
8	C₂²×C₃×C₅	—	7	1	no	maximal, Hall
9	C₂×ΓL(1,𝔽₈)	C₂×C₅	1	1	yes	characteristic, complemented
10	AGL(1,𝔽₇)	C₂×C₅	1	2	yes	direct product factor
11	C₅×C₇	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
12	C₂×C₃×C₅	—	7	2	no	—
13	C₂×C₃×C₅	—	7	1	no	—
14	C₂×D₇	C₃×C₅	1	1	yes	characteristic, complemented, Hall
15	ΓL(1,𝔽₈)	C₂²×C₅	1	1	yes	characteristic, complemented, Hall
16	C₂²×C₅	—	7	1	no	Hall
17	C₃×C₅	—	7	1	no	Hall
18	C₂×C₇	C₂×C₃×C₅	1	1	yes	characteristic, complemented
19	D₇	C₂×C₃×C₅	1	2	yes	complemented
20	C₂²×C₃	—	7	1	no	Hall
21	C₂×C₅	—	7	2	no	—
22	C₂×C₅	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
23	C₇	C₂²×C₃×C₅	1	1	yes	characteristic, complemented, Sylow
24	C₂×C₃	—	7	2	no	—
25	C₂×C₃	—	7	1	no	—
26	C₅	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor, Sylow
27	C₂²	—	7	1	no	Sylow
28	C₃	—	7	1	no	Sylow
29	C₂	—	7	2	no	—
30	C₂	C₅×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₅	AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₇	C₂²×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₅×C₇	C₂×C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×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.