Finite Group Databasebeta

← C₂²⋊(C₁₃⋊C₃)×C₁₅₇ #37587654

Subgroups — C₂²⋊(C₁₃⋊C₃)×C₁₅₇ #37587654

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⋊C₃×C₁₅₇	—	4	1	no	maximal, Hall
3	C₂×C₁₃×C₁₅₇	—	3	1	no	—
4	C₁₃×C₁₅₇	A₄	1	1	yes	characteristic, complemented, Hall
5	A₄×C₁₅₇	—	13	1	no	maximal, Hall
6	C₂²×C₁₅₇	C₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
7	C₃×C₁₅₇	—	52	1	no	Hall
8	C₂×C₁₅₇	—	3	1	no	—
9	C₁₅₇	C₂²⋊(C₁₃⋊C₃)	1	1	yes	characteristic, direct product factor, Sylow
10	C₂²⋊(C₁₃⋊C₃)	C₁₅₇	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂²×C₁₃	C₃×C₁₅₇	1	1	yes	characteristic, complemented, Hall
12	C₁₃⋊C₃	—	4	1	no	Hall
13	C₂×C₁₃	—	3	1	no	—
14	C₁₃	A₄×C₁₅₇	1	1	yes	characteristic, complemented, Sylow
15	A₄	—	13	1	no	Hall
16	C₂²	C₁₃⋊C₃×C₁₅₇	1	1	yes	characteristic, complemented, Sylow
17	C₃	—	52	1	no	Sylow
18	C₂	—	3	1	no	—

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

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
6. C₂²×C₁₅₇	C₁₃⋊C₃	semidirect
9. C₁₅₇	C₂²⋊(C₁₃⋊C₃)	direct (N × H)
10. C₂²⋊(C₁₃⋊C₃)	C₁₅₇	direct (N × H)
11. C₂²×C₁₃	C₃×C₁₅₇	semidirect
14. C₁₃	A₄×C₁₅₇	semidirect
16. C₂²	C₁₃⋊C₃×C₁₅₇	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⋊C₃×C₁₅₇	4
A₄×C₁₅₇	13
C₂²⋊(C₁₃⋊C₃)	157