Finite Group Databasebeta

← C₄₇×C₂²⋊(C₄₃⋊C₃) #37564656

Subgroups — C₄₇×C₂²⋊(C₄₃⋊C₃) #37564656

Order 24252.

20 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄₃×C₄₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₄₇×C₄₃⋊C₃	—	4	1	no	maximal, Hall
3	C₂×C₄₃×C₄₇	—	3	1	no	—
4	C₄₃×C₄₇	A₄	1	1	yes	characteristic, complemented, Hall
5	A₄×C₄₇	—	43	1	no	maximal, Hall
6	C₂²⋊(C₄₃⋊C₃)	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₄₇	C₄₃⋊C₃	1	1	yes	characteristic, complemented, Hall
8	C₂²×C₄₃	C₃×C₄₇	1	1	yes	characteristic, complemented, Hall
9	C₃×C₄₇	—	172	1	no	Hall
10	C₄₃⋊C₃	—	4	1	no	Hall
11	C₂×C₄₇	—	3	1	no	—
12	C₂×C₄₃	—	3	1	no	—
13	C₄₇	C₂²⋊(C₄₃⋊C₃)	1	1	yes	characteristic, direct product factor, Sylow
14	C₄₃	A₄×C₄₇	1	1	yes	characteristic, complemented, Sylow
15	A₄	—	43	1	no	Hall
16	C₂²	C₄₇×C₄₃⋊C₃	1	1	yes	characteristic, complemented, Sylow
17	C₃	—	172	1	no	Sylow
18	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₄₇	C₂²⋊(C₄₃⋊C₃)
commutator, cosocle, hypocenter	C₂²×C₄₃	C₃×C₄₇
Frattini	C₁	G
Fitting, socle	C₂²×C₄₃×C₄₇	C₃
LCS	G ▷ C₂²×C₄₃
UCS	C₁ ◁ C₄₇

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄₃×C₄₇	C₃	semidirect
4. C₄₃×C₄₇	A₄	semidirect
6. C₂²⋊(C₄₃⋊C₃)	C₄₇	direct (N × H)
7. C₂²×C₄₇	C₄₃⋊C₃	semidirect
8. C₂²×C₄₃	C₃×C₄₇	semidirect
13. C₄₇	C₂²⋊(C₄₃⋊C₃)	direct (N × H)
14. C₄₃	A₄×C₄₇	semidirect
16. C₂²	C₄₇×C₄₃⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄₃×C₄₇	3
C₄₇×C₄₃⋊C₃	4
A₄×C₄₇	43
C₂²⋊(C₄₃⋊C₃)	47