Finite Group Databasebeta

← C₃×C₅²⋊C₃×C₁₅₇ #47977397

Subgroups — C₃×C₅²⋊C₃×C₁₅₇ #47977397

Order 35325.

32 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²⋊C₃×C₁₅₇	C₃	1	3	yes	maximal, direct product factor
2	C₃×C₅²×C₁₅₇	C₃	1	1	yes	maximal, characteristic, complemented
3	C₅²×C₁₅₇	C₃²	1	1	yes	characteristic, complemented, Hall
4	C₃×C₅×C₁₅₇	—	3	2	no	—
5	C₃²×C₁₅₇	—	25	1	no	maximal, Hall
6	C₅×C₁₅₇	—	3	2	no	—
7	C₃×C₁₅₇	—	25	3	no	—
8	C₃×C₁₅₇	C₅²⋊C₃	1	1	yes	characteristic, direct product factor
9	C₃×C₅²⋊C₃	C₁₅₇	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₁₅₇	C₃×C₅²⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
11	C₃×C₅²	C₃×C₁₅₇	1	1	yes	characteristic, complemented
12	C₅²⋊C₃	C₃×C₁₅₇	1	3	yes	direct product factor
13	C₅²	C₃²×C₁₅₇	1	1	yes	characteristic, complemented, Sylow
14	C₃×C₅	—	3	2	no	—
15	C₃²	—	25	1	no	Sylow
16	C₅	—	3	2	no	—
17	C₃	—	25	3	no	—
18	C₃	C₅²⋊C₃×C₁₅₇	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₃×C₁₅₇	C₅²⋊C₃
commutator, cosocle, hypocenter	C₅²	C₃²×C₁₅₇
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₁₅₇	C₃
LCS	G ▷ C₅²
UCS	C₁ ◁ C₃×C₁₅₇

Complements

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

N	H ≅ G/N	type
1. C₅²⋊C₃×C₁₅₇	C₃	direct (N × H)
2. C₃×C₅²×C₁₅₇	C₃	semidirect
3. C₅²×C₁₅₇	C₃²	semidirect
8. C₃×C₁₅₇	C₅²⋊C₃	direct (N × H)
9. C₃×C₅²⋊C₃	C₁₅₇	direct (N × H)
10. C₁₅₇	C₃×C₅²⋊C₃	direct (N × H)
11. C₃×C₅²	C₃×C₁₅₇	semidirect
12. C₅²⋊C₃	C₃×C₁₅₇	direct (N × H)
13. C₅²	C₃²×C₁₅₇	semidirect
18. C₃	C₅²⋊C₃×C₁₅₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅²⋊C₃×C₁₅₇	3
C₃×C₅²×C₁₅₇	3
C₃²×C₁₅₇	25
C₃×C₅²⋊C₃	157