Finite Group Databasebeta

← C₂×D₅×C₁₁₇₁ #37552802

Subgroups — C₂×D₅×C₁₁₇₁ #37552802

Order 23420.

20 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₁₁₇₁	C₂	1	1	yes	maximal, characteristic, complemented
2	D₅×C₁₁₇₁	C₂	1	2	yes	maximal, direct product factor
3	C₅×C₁₁₇₁	C₂²	1	1	yes	characteristic, complemented, Hall
4	C₂²×C₁₁₇₁	—	5	1	no	maximal, Hall
5	C₂×C₁₁₇₁	—	5	2	no	—
6	C₂×C₁₁₇₁	D₅	1	1	yes	characteristic, direct product factor
7	C₁₁₇₁	C₂×D₅	1	1	yes	characteristic, direct product factor, Sylow
8	C₂×D₅	C₁₁₇₁	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₅	C₂×C₁₁₇₁	1	1	yes	characteristic, complemented
10	D₅	C₂×C₁₁₇₁	1	2	yes	direct product factor
11	C₅	C₂²×C₁₁₇₁	1	1	yes	characteristic, complemented, Sylow
12	C₂²	—	5	1	no	Sylow
13	C₂	—	5	2	no	—
14	C₂	D₅×C₁₁₇₁	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₁₁₇₁	D₅
commutator, cosocle, hypocenter	C₅	C₂²×C₁₁₇₁
Frattini	C₁	G
Fitting, socle	C₂×C₅×C₁₁₇₁	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₂×C₁₁₇₁

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×C₁₁₇₁	C₂	semidirect
2. D₅×C₁₁₇₁	C₂	direct (N × H)
3. C₅×C₁₁₇₁	C₂²	semidirect
6. C₂×C₁₁₇₁	D₅	direct (N × H)
7. C₁₁₇₁	C₂×D₅	direct (N × H)
8. C₂×D₅	C₁₁₇₁	direct (N × H)
9. C₂×C₅	C₂×C₁₁₇₁	semidirect
10. D₅	C₂×C₁₁₇₁	direct (N × H)
11. C₅	C₂²×C₁₁₇₁	semidirect
14. C₂	D₅×C₁₁₇₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×C₁₁₇₁	2
D₅×C₁₁₇₁	2
C₂²×C₁₁₇₁	5
C₂×D₅	1171