Finite Group Databasebeta

← C₅×C₆₁×C₃₇⋊C₃ #46799653

Subgroups — C₅×C₆₁×C₃₇⋊C₃ #46799653

Order 33855.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₃₇×C₆₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₆₁×C₃₇⋊C₃	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₃₇×C₆₁	C₃×C₅	1	1	yes	characteristic, complemented, Hall
4	C₃×C₅×C₆₁	—	37	1	no	maximal, Hall
5	C₅×C₃₇⋊C₃	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₅×C₆₁	C₃₇⋊C₃	1	1	yes	characteristic, direct product factor, Hall
7	C₅×C₃₇	C₃×C₆₁	1	1	yes	characteristic, complemented, Hall
8	C₃×C₆₁	—	37	1	no	Hall
9	C₃₇⋊C₃	C₅×C₆₁	1	1	yes	characteristic, direct product factor, Hall
10	C₆₁	C₅×C₃₇⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
11	C₃₇	C₃×C₅×C₆₁	1	1	yes	characteristic, complemented, Sylow
12	C₃×C₅	—	37	1	no	Hall
13	C₅	C₆₁×C₃₇⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
14	C₃	—	37	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₅×C₆₁	C₃₇⋊C₃
commutator, cosocle, hypocenter	C₃₇	C₃×C₅×C₆₁
Frattini	C₁	G
Fitting, socle	C₅×C₃₇×C₆₁	C₃
LCS	G ▷ C₃₇
UCS	C₁ ◁ C₅×C₆₁

Complements

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

N	H ≅ G/N	type
1. C₅×C₃₇×C₆₁	C₃	semidirect
2. C₆₁×C₃₇⋊C₃	C₅	direct (N × H)
3. C₃₇×C₆₁	C₃×C₅	semidirect
5. C₅×C₃₇⋊C₃	C₆₁	direct (N × H)
6. C₅×C₆₁	C₃₇⋊C₃	direct (N × H)
7. C₅×C₃₇	C₃×C₆₁	semidirect
9. C₃₇⋊C₃	C₅×C₆₁	direct (N × H)
10. C₆₁	C₅×C₃₇⋊C₃	direct (N × H)
11. C₃₇	C₃×C₅×C₆₁	semidirect
13. C₅	C₆₁×C₃₇⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×C₃₇×C₆₁	3
C₆₁×C₃₇⋊C₃	5
C₃×C₅×C₆₁	37
C₅×C₃₇⋊C₃	61