Finite Group Databasebeta

← C₂×D₇×C₁₂₃₁ #46828086

Subgroups — C₂×D₇×C₁₂₃₁ #46828086

Order 34468.

20 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₁₂₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	D₇×C₁₂₃₁	C₂	1	2	yes	maximal, direct product factor
3	C₇×C₁₂₃₁	C₂²	1	1	yes	characteristic, complemented, Hall
4	C₂²×C₁₂₃₁	—	7	1	no	maximal, Hall
5	C₂×C₁₂₃₁	—	7	2	no	—
6	C₂×C₁₂₃₁	D₇	1	1	yes	characteristic, direct product factor
7	C₁₂₃₁	C₂×D₇	1	1	yes	characteristic, direct product factor, Sylow
8	C₂×D₇	C₁₂₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₇	C₂×C₁₂₃₁	1	1	yes	characteristic, complemented
10	D₇	C₂×C₁₂₃₁	1	2	yes	direct product factor
11	C₇	C₂²×C₁₂₃₁	1	1	yes	characteristic, complemented, Sylow
12	C₂²	—	7	1	no	Sylow
13	C₂	—	7	2	no	—
14	C₂	D₇×C₁₂₃₁	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₁₂₃₁	D₇
commutator, cosocle, hypocenter	C₇	C₂²×C₁₂₃₁
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₁₂₃₁	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₁₂₃₁

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×C₁₂₃₁	C₂	semidirect
2. D₇×C₁₂₃₁	C₂	direct (N × H)
3. C₇×C₁₂₃₁	C₂²	semidirect
6. C₂×C₁₂₃₁	D₇	direct (N × H)
7. C₁₂₃₁	C₂×D₇	direct (N × H)
8. C₂×D₇	C₁₂₃₁	direct (N × H)
9. C₂×C₇	C₂×C₁₂₃₁	semidirect
10. D₇	C₂×C₁₂₃₁	direct (N × H)
11. C₇	C₂²×C₁₂₃₁	semidirect
14. C₂	D₇×C₁₂₃₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×C₁₂₃₁	2
D₇×C₁₂₃₁	2
C₂²×C₁₂₃₁	7
C₂×D₇	1231