Finite Group Databasebeta

← C₃×C₂²⋊(C₁₀₂₁⋊C₃) #49145244

Subgroups — C₃×C₂²⋊(C₁₀₂₁⋊C₃) #49145244

Order 36756.

28 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊(C₁₀₂₁⋊C₃)	C₃	1	3	yes	maximal, direct product factor
2	C₂²×C₃×C₁₀₂₁	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₁₀₂₁⋊C₃	—	4	1	no	maximal, Hall
4	C₂×C₃×C₁₀₂₁	—	3	1	no	—
5	C₂²×C₁₀₂₁	C₃²	1	1	yes	characteristic, complemented, Hall
6	C₃×C₁₀₂₁	A₄	1	1	yes	characteristic, complemented
7	C₁₀₂₁⋊C₃	—	4	3	no	—
8	C₂×C₁₀₂₁	—	3	1	no	—
9	C₁₀₂₁	C₃×A₄	1	1	yes	characteristic, complemented, Sylow
10	C₃×A₄	—	1021	1	no	maximal, Hall
11	C₂²×C₃	C₁₀₂₁⋊C₃	1	1	yes	characteristic, complemented
12	A₄	—	1021	3	no	—
13	C₃²	—	4084	1	no	Sylow
14	C₂×C₃	—	3	1	no	—
15	C₂²	C₃×C₁₀₂₁⋊C₃	1	1	yes	characteristic, complemented, Sylow
16	C₃	—	4084	3	no	—
17	C₃	C₂²⋊(C₁₀₂₁⋊C₃)	1	1	yes	characteristic, direct product factor
18	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₃	C₂²⋊(C₁₀₂₁⋊C₃)
commutator, cosocle, hypocenter	C₂²×C₁₀₂₁	C₃²
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₁₀₂₁	C₃
LCS	G ▷ C₂²×C₁₀₂₁
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. C₂²⋊(C₁₀₂₁⋊C₃)	C₃	direct (N × H)
2. C₂²×C₃×C₁₀₂₁	C₃	semidirect
5. C₂²×C₁₀₂₁	C₃²	semidirect
6. C₃×C₁₀₂₁	A₄	semidirect
9. C₁₀₂₁	C₃×A₄	semidirect
11. C₂²×C₃	C₁₀₂₁⋊C₃	semidirect
15. C₂²	C₃×C₁₀₂₁⋊C₃	semidirect
17. C₃	C₂²⋊(C₁₀₂₁⋊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
C₂²⋊(C₁₀₂₁⋊C₃)	3
C₂²×C₃×C₁₀₂₁	3
C₃×C₁₀₂₁⋊C₃	4
C₃×A₄	1021