Finite Group Databasebeta

← C₃×C₃₇×D₁₃₇ #44477343

Subgroups — C₃×C₃₇×D₁₃₇ #44477343

Order 30414.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃₇×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃₇×D₁₃₇	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₃₇×C₁₃₇	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	C₃×D₁₃₇	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₃×C₁₃₇	C₂×C₃₇	1	1	yes	characteristic, complemented, Hall
6	D₁₃₇	C₃×C₃₇	1	1	yes	characteristic, direct product factor, Hall
7	C₂×C₃×C₃₇	—	137	1	no	maximal, Hall
8	C₁₃₇	C₂×C₃×C₃₇	1	1	yes	characteristic, complemented, Sylow
9	C₃×C₃₇	D₁₃₇	1	1	yes	characteristic, direct product factor, Hall
10	C₂×C₃₇	—	137	1	no	Hall
11	C₃₇	C₃×D₁₃₇	1	1	yes	characteristic, direct product factor, Sylow
12	C₂×C₃	—	137	1	no	Hall
13	C₃	C₃₇×D₁₃₇	1	1	yes	characteristic, direct product factor, Sylow
14	C₂	—	137	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃×C₃₇	D₁₃₇
commutator, cosocle, hypocenter	C₁₃₇	C₂×C₃×C₃₇
Frattini	C₁	G
Fitting, socle	C₃×C₃₇×C₁₃₇	C₂
LCS	G ▷ C₁₃₇
UCS	C₁ ◁ C₃×C₃₇

Complements

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

N	H ≅ G/N	type
1. C₃×C₃₇×C₁₃₇	C₂	semidirect
2. C₃₇×D₁₃₇	C₃	direct (N × H)
3. C₃₇×C₁₃₇	C₂×C₃	semidirect
4. C₃×D₁₃₇	C₃₇	direct (N × H)
5. C₃×C₁₃₇	C₂×C₃₇	semidirect
6. D₁₃₇	C₃×C₃₇	direct (N × H)
8. C₁₃₇	C₂×C₃×C₃₇	semidirect
9. C₃×C₃₇	D₁₃₇	direct (N × H)
11. C₃₇	C₃×D₁₃₇	direct (N × H)
13. C₃	C₃₇×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
C₃×C₃₇×C₁₃₇	2
C₃₇×D₁₃₇	3
C₃×D₁₃₇	37
C₂×C₃×C₃₇	137