Finite Group Databasebeta

Subgroups — #10587722

Order 546.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#63270	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	D₉₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇×C₁₃	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	D₁₃⋊C₃	—	7	1	no	maximal, Hall
5	AGL(1,𝔽₇)	—	13	1	no	maximal, Hall
6	C₁₃⋊C₃	—	7	1	no	Hall
7	D₁₃	—	7	1	no	Hall
8	ΓL(1,𝔽₈)	—	13	1	no	Hall
9	D₇	—	13	1	no	Hall
10	C₁₃	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
11	C₇	D₁₃⋊C₃	1	1	yes	characteristic, complemented, Sylow
12	C₂×C₃	—	91	1	no	Hall
13	C₃	—	91	1	no	Sylow
14	C₂	—	91	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₃	C₂×C₃
LCS	G ▷ C₇×C₁₃
UCS	C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #63270	C₂	semidirect
2. D₉₁	C₃	semidirect
3. C₇×C₁₃	C₂×C₃	semidirect
10. C₁₃	AGL(1,𝔽₇)	semidirect
11. C₇	D₁₃⋊C₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#63270	2
D₉₁	3
D₁₃⋊C₃	7
AGL(1,𝔽₇)	13