Subgroups — #44545231

Order 32046.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30630924	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	#17987082	—	7	8	no	maximal
5	#14556234	—	7	8	no	—
6	D₇×C₁₀₉	—	7	8	no	—
7	C₇×C₁₀₉	AGL(1,𝔽₇)	1	8	yes	complemented
8	C₂×C₁₀₉⋊C₃	—	49	1	no	Hall
9	C₁₀₉⋊C₃	—	49	1	no	Hall
10	#64424	—	109	1	no	maximal, Hall
11	C₂×C₁₀₉	—	49	1	no	Hall
12	#3861	—	109	1	no	Hall
13	C₁₀₉	#64424	1	1	yes	characteristic, complemented, Sylow
14	C₇⋊D₇	C₁₀₉⋊C₃	1	1	yes	characteristic, complemented, Hall
15	C₇²	C₂×C₁₀₉⋊C₃	1	1	yes	characteristic, complemented, Sylow
16	AGL(1,𝔽₇)	—	763	8	no	—
17	ΓL(1,𝔽₈)	—	763	8	no	—
18	D₇	—	7	8	no	—
19	C₇	#17987082	1	8	yes	complemented
20	C₂×C₃	—	5341	1	no	Hall
21	C₃	—	5341	1	no	Sylow
22	C₂	—	49	1	no	Sylow

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

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.