Subgroups — C₂×C₃×AGL(1,𝔽₇) #6989

Order 252.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×AGL(1,𝔽₇)	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×D₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×AGL(1,𝔽₇)	C₃	1	3	yes	maximal, direct product factor
5	C₃×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₂×C₃×C₇	C₂×C₃	1	1	yes	characteristic, complemented
7	C₃×D₇	C₂×C₃	1	2	yes	complemented
8	C₂×ΓL(1,𝔽₈)	C₂×C₃	1	3	yes	complemented
9	AGL(1,𝔽₇)	C₂×C₃	1	6	yes	direct product factor
10	C₂²×C₃²	—	7	1	no	maximal, Hall
11	C₂×D₇	C₃²	1	1	yes	characteristic, complemented, Hall
12	C₃×C₇	C₂²×C₃	1	1	yes	characteristic, complemented
13	ΓL(1,𝔽₈)	C₂²×C₃	1	3	yes	complemented
14	C₂×C₃²	—	7	2	no	—
15	C₂×C₃²	—	7	1	no	—
16	C₂×C₇	C₂×C₃²	1	1	yes	characteristic, complemented
17	D₇	C₂×C₃²	1	2	yes	complemented
18	C₂²×C₃	—	7	3	no	—
19	C₂²×C₃	—	7	1	no	—
20	C₃²	—	7	1	no	Sylow
21	C₇	C₂²×C₃²	1	1	yes	characteristic, complemented, Sylow
22	C₂×C₃	—	7	6	no	—
23	C₂×C₃	—	7	2	no	—
24	C₂×C₃	—	7	3	no	—
25	C₂×C₃	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
26	C₂²	—	7	1	no	Sylow
27	C₃	—	7	3	no	—
28	C₃	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
29	C₂	—	7	2	no	—
30	C₂	C₃×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃	AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₇	C₂²×C₃²
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₇	C₂×C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₃

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.