Finite Group Databasebeta

← C₂×D₃₇×C₁₆₇ #37612786

Subgroups — C₂×D₃₇×C₁₆₇ #37612786

Order 24716.

20 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	D₃₇×C₁₆₇	C₂	1	2	yes	maximal, direct product factor
3	C₃₇×C₁₆₇	C₂²	1	1	yes	characteristic, complemented, Hall
4	C₂²×C₁₆₇	—	37	1	no	maximal, Hall
5	C₂×C₁₆₇	—	37	2	no	—
6	C₂×C₁₆₇	D₃₇	1	1	yes	characteristic, direct product factor
7	C₁₆₇	C₂×D₃₇	1	1	yes	characteristic, direct product factor, Sylow
8	C₂×D₃₇	C₁₆₇	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃₇	C₂×C₁₆₇	1	1	yes	characteristic, complemented
10	D₃₇	C₂×C₁₆₇	1	2	yes	direct product factor
11	C₃₇	C₂²×C₁₆₇	1	1	yes	characteristic, complemented, Sylow
12	C₂²	—	37	1	no	Sylow
13	C₂	—	37	2	no	—
14	C₂	D₃₇×C₁₆₇	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₁₆₇	D₃₇
commutator, cosocle, hypocenter	C₃₇	C₂²×C₁₆₇
Frattini	C₁	G
Fitting, socle	C₂×C₃₇×C₁₆₇	C₂
LCS	G ▷ C₃₇
UCS	C₁ ◁ C₂×C₁₆₇

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₇×C₁₆₇	C₂	semidirect
2. D₃₇×C₁₆₇	C₂	direct (N × H)
3. C₃₇×C₁₆₇	C₂²	semidirect
6. C₂×C₁₆₇	D₃₇	direct (N × H)
7. C₁₆₇	C₂×D₃₇	direct (N × H)
8. C₂×D₃₇	C₁₆₇	direct (N × H)
9. C₂×C₃₇	C₂×C₁₆₇	semidirect
10. D₃₇	C₂×C₁₆₇	direct (N × H)
11. C₃₇	C₂²×C₁₆₇	semidirect
14. C₂	D₃₇×C₁₆₇	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×C₃₇×C₁₆₇	2
D₃₇×C₁₆₇	2
C₂²×C₁₆₇	37
C₂×D₃₇	167