Subgroups — C₇×#17987083 #44545209

Order 32046.

34 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#14556233	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×D₇×C₁₀₉	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇²×C₁₀₉	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	#17987083	C₇	1	1	yes	maximal, characteristic, direct product factor
5	C₂×C₇×C₁₀₉⋊C₃	—	7	1	no	maximal
6	#14556233	C₂×C₇	1	1	yes	characteristic, complemented
7	C₇×C₁₀₉⋊C₃	—	7	1	no	—
8	C₂×C₇×C₁₀₉	—	7	1	no	—
9	D₇×C₁₀₉	C₃×C₇	1	1	yes	characteristic, complemented
10	C₇×C₁₀₉	—	6	1	no	—
11	C₇×C₁₀₉	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
12	C₇×C₁₀₉	C₂×C₃×C₇	1	1	yes	characteristic, complemented
13	C₂×C₁₀₉⋊C₃	—	7	1	no	Hall
14	C₁₀₉⋊C₃	—	7	1	no	Hall
15	C₇×AGL(1,𝔽₇)	—	109	1	no	maximal, Hall
16	C₂×C₁₀₉	—	7	1	no	Hall
17	C₇×ΓL(1,𝔽₈)	—	109	1	no	Hall
18	C₁₀₉	C₇×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
19	C₇×D₇	C₁₀₉⋊C₃	1	1	yes	characteristic, complemented, Hall
20	C₇²	C₂×C₁₀₉⋊C₃	1	1	yes	characteristic, complemented, Sylow
21	C₂×C₃×C₇	—	763	1	no	—
22	AGL(1,𝔽₇)	—	109	1	no	—
23	C₃×C₇	—	763	1	no	—
24	ΓL(1,𝔽₈)	—	109	1	no	—
25	C₂×C₇	—	7	1	no	—
26	D₇	C₇×C₁₀₉⋊C₃	1	1	yes	characteristic, complemented
27	C₇	—	6	1	no	—
28	C₇	C₂×C₇×C₁₀₉⋊C₃	1	1	yes	characteristic, complemented
29	C₇	#17987083	1	1	yes	characteristic, direct product factor
30	C₂×C₃	—	763	1	no	Hall
31	C₃	—	763	1	no	Sylow
32	C₂	—	7	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₇	#17987083
commutator, cosocle, hypocenter	C₇×C₁₀₉	C₂×C₃×C₇
Frattini	C₁	G
Fitting, socle	C₇²×C₁₀₉	C₂×C₃
LCS	G ▷ C₇×C₁₀₉
UCS	C₁ ◁ C₇

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.