Subgroups — C₃×#686 #6683

Order 240.

44 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	#686	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×C₅	—	2	1	no	—
6	C₂×C₃×D₅	—	2	1	no	—
7	C₃×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₃×D₈	—	5	1	no	maximal, Hall
9	C₅×D₄	C₂×C₃	1	1	yes	characteristic, complemented
10	D₂₀	C₂×C₃	1	1	yes	characteristic, complemented
11	#168	C₂×C₃	1	1	yes	characteristic, complemented
12	C₂×C₃×C₅	—	4	1	no	—
13	C₂×C₃×C₅	D₄	1	1	yes	characteristic
14	C₃×D₅	—	4	1	no	—
15	C₃×D₄	—	5	1	no	—
16	C₃×D₄	D₅	1	1	yes	characteristic, complemented
17	C₃×C₈	—	5	1	no	—
18	C₂²×C₅	—	2	1	no	—
19	C₂×D₅	—	2	1	no	—
20	C₄×C₅	C₂²×C₃	1	1	yes	characteristic
21	D₈	—	5	1	no	Sylow
22	C₃×C₅	D₈	1	1	yes	characteristic, complemented, Hall
23	C₂²×C₃	—	2	1	no	—
24	C₂²×C₃	—	10	1	no	—
25	C₃×C₄	C₂×D₅	1	1	yes	characteristic
26	C₂×C₅	—	4	1	no	—
27	C₂×C₅	C₃×D₄	1	1	yes	characteristic
28	D₅	—	4	1	no	—
29	D₄	—	5	1	no	—
30	D₄	C₃×D₅	1	1	yes	characteristic, complemented
31	C₈	—	5	1	no	—
32	C₂×C₃	—	4	1	no	—
33	C₂×C₃	—	20	1	no	—
34	C₂×C₃	C₂²⋊D₅	1	1	yes	characteristic
35	C₅	C₃×D₈	1	1	yes	characteristic, complemented, Sylow
36	C₂²	—	2	1	no	—
37	C₂²	—	10	1	no	—
38	C₄	C₂×C₃×D₅	1	1	yes	characteristic
39	C₃	#686	1	1	yes	characteristic, direct product factor, Sylow
40	C₂	—	4	1	no	—
41	C₂	—	20	1	no	—
42	C₂	C₃×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂²⋊D₅
commutator, cosocle	C₄×C₅	C₂²×C₃
Frattini	C₄	C₂×C₃×D₅
Fitting	C₃×C₅×D₄	C₂
socle	C₂×C₃×C₅	D₄
hypercenter	C₃×D₄	D₅
hypocenter	C₅	C₃×D₈
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×D₄

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×D₄	C₂	semidirect
2. C₃×D₂₀	C₂	semidirect
3. C₃×#168	C₂	semidirect
4. #686	C₃	direct (N × H)
9. C₅×D₄	C₂×C₃	semidirect
10. D₂₀	C₂×C₃	semidirect
11. #168	C₂×C₃	semidirect
16. C₃×D₄	D₅	semidirect
22. C₃×C₅	D₈	semidirect
30. D₄	C₃×D₅	semidirect
35. C₅	C₃×D₈	semidirect
39. C₃	#686	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×D₄	2
C₃×D₂₀	2
C₃×#168	2
#686	3
C₃×D₈	5

Subgroups — C3×#686 #6683

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#686 #6683