Finite Group Databasebeta

← C₁₃×C₂²⋊(C₁₅₇⋊C₃) #37587653

Subgroups — C₁₃×C₂²⋊(C₁₅₇⋊C₃) #37587653

Order 24492.

20 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×C₁₅₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₁₃×C₁₅₇⋊C₃	—	4	1	no	maximal, Hall
3	C₂×C₁₃×C₁₅₇	—	3	1	no	—
4	C₁₃×C₁₅₇	A₄	1	1	yes	characteristic, complemented, Hall
5	C₂²⋊(C₁₅₇⋊C₃)	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₁₅₇	C₃×C₁₃	1	1	yes	characteristic, complemented, Hall
7	C₁₅₇⋊C₃	—	4	1	no	Hall
8	C₂×C₁₅₇	—	3	1	no	—
9	C₁₅₇	A₄×C₁₃	1	1	yes	characteristic, complemented, Sylow
10	A₄×C₁₃	—	157	1	no	maximal, Hall
11	C₂²×C₁₃	C₁₅₇⋊C₃	1	1	yes	characteristic, complemented, Hall
12	C₃×C₁₃	—	628	1	no	Hall
13	C₂×C₁₃	—	3	1	no	—
14	C₁₃	C₂²⋊(C₁₅₇⋊C₃)	1	1	yes	characteristic, direct product factor, Sylow
15	A₄	—	157	1	no	Hall
16	C₂²	C₁₃×C₁₅₇⋊C₃	1	1	yes	characteristic, complemented, Sylow
17	C₃	—	628	1	no	Sylow
18	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₁₃	C₂²⋊(C₁₅₇⋊C₃)
commutator, cosocle, hypocenter	C₂²×C₁₅₇	C₃×C₁₃
Frattini	C₁	G
Fitting, socle	C₂²×C₁₃×C₁₅₇	C₃
LCS	G ▷ C₂²×C₁₅₇
UCS	C₁ ◁ C₁₃

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₃×C₁₅₇	C₃	semidirect
4. C₁₃×C₁₅₇	A₄	semidirect
5. C₂²⋊(C₁₅₇⋊C₃)	C₁₃	direct (N × H)
6. C₂²×C₁₅₇	C₃×C₁₃	semidirect
9. C₁₅₇	A₄×C₁₃	semidirect
11. C₂²×C₁₃	C₁₅₇⋊C₃	semidirect
14. C₁₃	C₂²⋊(C₁₅₇⋊C₃)	direct (N × H)
16. C₂²	C₁₃×C₁₅₇⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×C₁₅₇	3
C₁₃×C₁₅₇⋊C₃	4
C₂²⋊(C₁₅₇⋊C₃)	13
A₄×C₁₃	157