Finite Group Databasebeta

← C₅×C₇×C₁₅₁⋊C₅ #40996170

Subgroups — C₅×C₇×C₁₅₁⋊C₅ #40996170

Order 26425.

32 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₇×C₁₅₁	C₅	1	1	yes	maximal, characteristic, complemented
2	C₇×C₁₅₁⋊C₅	C₅	1	5	yes	maximal, direct product factor
3	C₅×C₁₅₁⋊C₅	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₇×C₁₅₁	C₅²	1	1	yes	characteristic, complemented, Hall
5	C₅×C₁₅₁	C₅×C₇	1	1	yes	characteristic, complemented
6	C₁₅₁⋊C₅	C₅×C₇	1	5	yes	direct product factor
7	C₅²×C₇	—	151	1	no	maximal, Hall
8	C₁₅₁	C₅²×C₇	1	1	yes	characteristic, complemented, Sylow
9	C₅×C₇	—	151	5	no	—
10	C₅×C₇	C₁₅₁⋊C₅	1	1	yes	characteristic, direct product factor
11	C₅²	—	151	1	no	Sylow
12	C₇	C₅×C₁₅₁⋊C₅	1	1	yes	characteristic, direct product factor, Sylow
13	C₅	—	151	5	no	—
14	C₅	C₇×C₁₅₁⋊C₅	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₅×C₇	C₁₅₁⋊C₅
commutator, cosocle, hypocenter	C₁₅₁	C₅²×C₇
Frattini	C₁	G
Fitting, socle	C₅×C₇×C₁₅₁	C₅
LCS	G ▷ C₁₅₁
UCS	C₁ ◁ C₅×C₇

Complements

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

N	H ≅ G/N	type
1. C₅×C₇×C₁₅₁	C₅	semidirect
2. C₇×C₁₅₁⋊C₅	C₅	direct (N × H)
3. C₅×C₁₅₁⋊C₅	C₇	direct (N × H)
4. C₇×C₁₅₁	C₅²	semidirect
5. C₅×C₁₅₁	C₅×C₇	semidirect
6. C₁₅₁⋊C₅	C₅×C₇	direct (N × H)
8. C₁₅₁	C₅²×C₇	semidirect
10. C₅×C₇	C₁₅₁⋊C₅	direct (N × H)
12. C₇	C₅×C₁₅₁⋊C₅	direct (N × H)
14. C₅	C₇×C₁₅₁⋊C₅	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×C₇×C₁₅₁	5
C₇×C₁₅₁⋊C₅	5
C₅×C₁₅₁⋊C₅	7
C₅²×C₇	151