Finite Group Databasebeta

← C₇×C₃₇×D₆₁ #44539138

Subgroups — C₇×C₃₇×D₆₁ #44539138

Order 31598.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₃₇×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃₇×D₆₁	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₃₇×C₆₁	C₂×C₇	1	1	yes	characteristic, complemented, Hall
4	C₇×D₆₁	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₇×C₃₇	—	61	1	no	maximal, Hall
6	C₇×C₆₁	C₂×C₃₇	1	1	yes	characteristic, complemented, Hall
7	C₇×C₃₇	D₆₁	1	1	yes	characteristic, direct product factor, Hall
8	D₆₁	C₇×C₃₇	1	1	yes	characteristic, direct product factor, Hall
9	C₂×C₃₇	—	61	1	no	Hall
10	C₆₁	C₂×C₇×C₃₇	1	1	yes	characteristic, complemented, Sylow
11	C₃₇	C₇×D₆₁	1	1	yes	characteristic, direct product factor, Sylow
12	C₂×C₇	—	61	1	no	Hall
13	C₇	C₃₇×D₆₁	1	1	yes	characteristic, direct product factor, Sylow
14	C₂	—	61	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₇×C₃₇	D₆₁
commutator, cosocle, hypocenter	C₆₁	C₂×C₇×C₃₇
Frattini	C₁	G
Fitting, socle	C₇×C₃₇×C₆₁	C₂
LCS	G ▷ C₆₁
UCS	C₁ ◁ C₇×C₃₇

Complements

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

N	H ≅ G/N	type
1. C₇×C₃₇×C₆₁	C₂	semidirect
2. C₃₇×D₆₁	C₇	direct (N × H)
3. C₃₇×C₆₁	C₂×C₇	semidirect
4. C₇×D₆₁	C₃₇	direct (N × H)
6. C₇×C₆₁	C₂×C₃₇	semidirect
7. C₇×C₃₇	D₆₁	direct (N × H)
8. D₆₁	C₇×C₃₇	direct (N × H)
10. C₆₁	C₂×C₇×C₃₇	semidirect
11. C₃₇	C₇×D₆₁	direct (N × H)
13. C₇	C₃₇×D₆₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×C₃₇×C₆₁	2
C₃₇×D₆₁	7
C₇×D₆₁	37
C₂×C₇×C₃₇	61