Finite Group Databasebeta

← C₂×C₄₃×D₁₃₇ #37555680

Subgroups — C₂×C₄₃×D₁₃₇ #37555680

Order 23564.

20 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₃×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₃×D₁₃₇	C₂	1	2	yes	maximal, direct product factor
3	C₄₃×C₁₃₇	C₂²	1	1	yes	characteristic, complemented, Hall
4	C₂×D₁₃₇	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₁₃₇	C₂×C₄₃	1	1	yes	characteristic, complemented
6	D₁₃₇	C₂×C₄₃	1	2	yes	direct product factor
7	C₂²×C₄₃	—	137	1	no	maximal, Hall
8	C₁₃₇	C₂²×C₄₃	1	1	yes	characteristic, complemented, Sylow
9	C₂×C₄₃	—	137	2	no	—
10	C₂×C₄₃	D₁₃₇	1	1	yes	characteristic, direct product factor
11	C₄₃	C₂×D₁₃₇	1	1	yes	characteristic, direct product factor, Sylow
12	C₂²	—	137	1	no	Sylow
13	C₂	—	137	2	no	—
14	C₂	C₄₃×D₁₃₇	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₄₃	D₁₃₇
commutator, cosocle, hypocenter	C₁₃₇	C₂²×C₄₃
Frattini	C₁	G
Fitting, socle	C₂×C₄₃×C₁₃₇	C₂
LCS	G ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄₃

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₃×C₁₃₇	C₂	semidirect
2. C₄₃×D₁₃₇	C₂	direct (N × H)
3. C₄₃×C₁₃₇	C₂²	semidirect
4. C₂×D₁₃₇	C₄₃	direct (N × H)
5. C₂×C₁₃₇	C₂×C₄₃	semidirect
6. D₁₃₇	C₂×C₄₃	direct (N × H)
8. C₁₃₇	C₂²×C₄₃	semidirect
10. C₂×C₄₃	D₁₃₇	direct (N × H)
11. C₄₃	C₂×D₁₃₇	direct (N × H)
14. C₂	C₄₃×D₁₃₇	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×C₄₃×C₁₃₇	2
C₄₃×D₁₃₇	2
C₂×D₁₃₇	43
C₂²×C₄₃	137