Finite Group Databasebeta

← C₇×C₂²⋊(C₄₂₁⋊C₃) #47977950

Subgroups — C₇×C₂²⋊(C₄₂₁⋊C₃) #47977950

Order 35364.

20 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₄₂₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×C₄₂₁⋊C₃	—	4	1	no	maximal, Hall
3	C₂×C₇×C₄₂₁	—	3	1	no	—
4	C₂²⋊(C₄₂₁⋊C₃)	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₇×C₄₂₁	A₄	1	1	yes	characteristic, complemented, Hall
6	C₂²×C₄₂₁	C₃×C₇	1	1	yes	characteristic, complemented, Hall
7	C₄₂₁⋊C₃	—	4	1	no	Hall
8	C₂×C₄₂₁	—	3	1	no	—
9	C₄₂₁	C₇×A₄	1	1	yes	characteristic, complemented, Sylow
10	C₇×A₄	—	421	1	no	maximal, Hall
11	C₂²×C₇	C₄₂₁⋊C₃	1	1	yes	characteristic, complemented, Hall
12	C₃×C₇	—	1684	1	no	Hall
13	C₂×C₇	—	3	1	no	—
14	A₄	—	421	1	no	Hall
15	C₇	C₂²⋊(C₄₂₁⋊C₃)	1	1	yes	characteristic, direct product factor, Sylow
16	C₂²	C₇×C₄₂₁⋊C₃	1	1	yes	characteristic, complemented, Sylow
17	C₃	—	1684	1	no	Sylow
18	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₇	C₂²⋊(C₄₂₁⋊C₃)
commutator, cosocle, hypocenter	C₂²×C₄₂₁	C₃×C₇
Frattini	C₁	G
Fitting, socle	C₂²×C₇×C₄₂₁	C₃
LCS	G ▷ C₂²×C₄₂₁
UCS	C₁ ◁ C₇

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇×C₄₂₁	C₃	semidirect
4. C₂²⋊(C₄₂₁⋊C₃)	C₇	direct (N × H)
5. C₇×C₄₂₁	A₄	semidirect
6. C₂²×C₄₂₁	C₃×C₇	semidirect
9. C₄₂₁	C₇×A₄	semidirect
11. C₂²×C₇	C₄₂₁⋊C₃	semidirect
15. C₇	C₂²⋊(C₄₂₁⋊C₃)	direct (N × H)
16. C₂²	C₇×C₄₂₁⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇×C₄₂₁	3
C₇×C₄₂₁⋊C₃	4
C₂²⋊(C₄₂₁⋊C₃)	7
C₇×A₄	421