Finite Group Databasebeta

← C₄₁×C₂²⋊(C₆₇⋊C₃) #45682102

Subgroups — C₄₁×C₂²⋊(C₆₇⋊C₃) #45682102

Order 32964.

20 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄₁×C₆₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₄₁×C₆₇⋊C₃	—	4	1	no	maximal, Hall
3	C₂×C₄₁×C₆₇	—	3	1	no	—
4	C₄₁×C₆₇	A₄	1	1	yes	characteristic, complemented, Hall
5	C₂²⋊(C₆₇⋊C₃)	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Hall
6	A₄×C₄₁	—	67	1	no	maximal, Hall
7	C₂²×C₆₇	C₃×C₄₁	1	1	yes	characteristic, complemented, Hall
8	C₆₇⋊C₃	—	4	1	no	Hall
9	C₂²×C₄₁	C₆₇⋊C₃	1	1	yes	characteristic, complemented, Hall
10	C₂×C₆₇	—	3	1	no	—
11	C₃×C₄₁	—	268	1	no	Hall
12	C₂×C₄₁	—	3	1	no	—
13	C₆₇	A₄×C₄₁	1	1	yes	characteristic, complemented, Sylow
14	C₄₁	C₂²⋊(C₆₇⋊C₃)	1	1	yes	characteristic, direct product factor, Sylow
15	A₄	—	67	1	no	Hall
16	C₂²	C₄₁×C₆₇⋊C₃	1	1	yes	characteristic, complemented, Sylow
17	C₃	—	268	1	no	Sylow
18	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₄₁	C₂²⋊(C₆₇⋊C₃)
commutator, cosocle, hypocenter	C₂²×C₆₇	C₃×C₄₁
Frattini	C₁	G
Fitting, socle	C₂²×C₄₁×C₆₇	C₃
LCS	G ▷ C₂²×C₆₇
UCS	C₁ ◁ C₄₁

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄₁×C₆₇	C₃	semidirect
4. C₄₁×C₆₇	A₄	semidirect
5. C₂²⋊(C₆₇⋊C₃)	C₄₁	direct (N × H)
7. C₂²×C₆₇	C₃×C₄₁	semidirect
9. C₂²×C₄₁	C₆₇⋊C₃	semidirect
13. C₆₇	A₄×C₄₁	semidirect
14. C₄₁	C₂²⋊(C₆₇⋊C₃)	direct (N × H)
16. C₂²	C₄₁×C₆₇⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄₁×C₆₇	3
C₄₁×C₆₇⋊C₃	4
C₂²⋊(C₆₇⋊C₃)	41
A₄×C₄₁	67