Finite Group Databasebeta

← C₂×C₇×D₁₁₂₃ #44537745

Subgroups — C₂×C₇×D₁₁₂₃ #44537745

Order 31444.

20 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₁₁₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×D₁₁₂₃	C₂	1	2	yes	maximal, direct product factor
3	C₇×C₁₁₂₃	C₂²	1	1	yes	characteristic, complemented, Hall
4	C₂×D₁₁₂₃	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₁₁₂₃	C₂×C₇	1	1	yes	characteristic, complemented
6	D₁₁₂₃	C₂×C₇	1	2	yes	direct product factor
7	C₁₁₂₃	C₂²×C₇	1	1	yes	characteristic, complemented, Sylow
8	C₂²×C₇	—	1123	1	no	maximal, Hall
9	C₂×C₇	—	1123	2	no	—
10	C₂×C₇	D₁₁₂₃	1	1	yes	characteristic, direct product factor
11	C₇	C₂×D₁₁₂₃	1	1	yes	characteristic, direct product factor, Sylow
12	C₂²	—	1123	1	no	Sylow
13	C₂	—	1123	2	no	—
14	C₂	C₇×D₁₁₂₃	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₇	D₁₁₂₃
commutator, cosocle, hypocenter	C₁₁₂₃	C₂²×C₇
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₁₁₂₃	C₂
LCS	G ▷ C₁₁₂₃
UCS	C₁ ◁ C₂×C₇

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×C₁₁₂₃	C₂	semidirect
2. C₇×D₁₁₂₃	C₂	direct (N × H)
3. C₇×C₁₁₂₃	C₂²	semidirect
4. C₂×D₁₁₂₃	C₇	direct (N × H)
5. C₂×C₁₁₂₃	C₂×C₇	semidirect
6. D₁₁₂₃	C₂×C₇	direct (N × H)
7. C₁₁₂₃	C₂²×C₇	semidirect
10. C₂×C₇	D₁₁₂₃	direct (N × H)
11. C₇	C₂×D₁₁₂₃	direct (N × H)
14. C₂	C₇×D₁₁₂₃	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×C₁₁₂₃	2
C₇×D₁₁₂₃	2
C₂×D₁₁₂₃	7
C₂²×C₇	1123