Subgroups — C₂×C₃×D₂₀ #6796

Order 240.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×D₅	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₄×C₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₂₀	C₂	1	4	yes	maximal, direct product factor
4	C₂×D₂₀	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×C₅	C₂²	1	1	yes	characteristic
6	C₂×C₃×D₅	—	2	4	no	—
7	C₂×C₃×D₅	C₂²	1	4	yes	complemented
8	C₃×C₄×C₅	C₂²	1	2	yes	complemented
9	C₂×C₃×D₄	—	5	1	no	maximal, Hall
10	C₂²×D₅	C₂×C₃	1	2	yes	complemented
11	C₂×C₄×C₅	C₂×C₃	1	1	yes	characteristic, complemented
12	D₂₀	C₂×C₃	1	4	yes	direct product factor
13	C₂×C₃×C₅	D₄	1	2	yes	complemented
14	C₂×C₃×C₅	C₂³	1	1	yes	characteristic
15	C₃×D₅	—	2	4	no	—
16	C₂³×C₃	—	5	2	no	—
17	C₃×D₄	—	5	4	no	—
18	C₂×C₃×C₄	D₅	1	1	yes	characteristic, complemented
19	C₂²×C₅	C₂²×C₃	1	1	yes	characteristic
20	C₂×D₅	—	2	4	no	—
21	C₂×D₅	C₂²×C₃	1	4	yes	complemented
22	C₄×C₅	C₂²×C₃	1	2	yes	complemented
23	C₂×D₄	—	5	1	no	Sylow
24	C₃×C₅	C₂×D₄	1	1	yes	characteristic, complemented, Hall
25	C₂²×C₃	—	5	4	no	—
26	C₂²×C₃	—	10	4	no	—
27	C₂²×C₃	C₂×D₅	1	1	yes	characteristic
28	C₃×C₄	C₂×D₅	1	2	yes	complemented
29	C₂×C₅	C₃×D₄	1	2	yes	complemented
30	C₂×C₅	C₂³×C₃	1	1	yes	characteristic
31	D₅	—	2	4	no	—
32	C₂³	—	5	2	no	—
33	D₄	—	5	4	no	—
34	C₂×C₄	C₃×D₅	1	1	yes	characteristic, complemented
35	C₂×C₃	—	10	4	no	—
36	C₂×C₃	D₂₀	1	2	yes	direct product factor
37	C₂×C₃	C₂²×D₅	1	1	yes	characteristic
38	C₅	C₂×C₃×D₄	1	1	yes	characteristic, complemented, Sylow
39	C₂²	—	5	4	no	—
40	C₂²	—	10	4	no	—
41	C₂²	C₂×C₃×D₅	1	1	yes	characteristic
42	C₄	C₂×C₃×D₅	1	2	yes	complemented
43	C₃	C₂×D₂₀	1	1	yes	characteristic, direct product factor, Sylow
44	C₂	—	10	4	no	—
45	C₂	C₃×D₂₀	1	2	yes	direct product factor
46	C₂	C₂²×C₃×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×D₅
commutator, cosocle	C₂×C₅	C₂³×C₃
Frattini	C₂	C₂²×C₃×D₅
Fitting	C₂×C₃×C₄×C₅	C₂
socle	C₂²×C₃×C₅	C₂²
hypercenter	C₂×C₃×C₄	D₅
hypocenter	C₅	C₂×C₃×D₄
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×D₅	C₂	semidirect
2. C₂×C₃×C₄×C₅	C₂	semidirect
3. C₃×D₂₀	C₂	direct (N × H)
4. C₂×D₂₀	C₃	direct (N × H)
7. C₂×C₃×D₅	C₂²	semidirect
8. C₃×C₄×C₅	C₂²	semidirect
10. C₂²×D₅	C₂×C₃	semidirect
11. C₂×C₄×C₅	C₂×C₃	semidirect
12. D₂₀	C₂×C₃	direct (N × H)
13. C₂×C₃×C₅	D₄	semidirect
18. C₂×C₃×C₄	D₅	semidirect
21. C₂×D₅	C₂²×C₃	semidirect
22. C₄×C₅	C₂²×C₃	semidirect
24. C₃×C₅	C₂×D₄	semidirect
28. C₃×C₄	C₂×D₅	semidirect
29. C₂×C₅	C₃×D₄	semidirect
34. C₂×C₄	C₃×D₅	semidirect
36. C₂×C₃	D₂₀	direct (N × H)
38. C₅	C₂×C₃×D₄	semidirect
42. C₄	C₂×C₃×D₅	semidirect
43. C₃	C₂×D₂₀	direct (N × H)
45. C₂	C₃×D₂₀	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×D₅	2
C₂×C₃×C₄×C₅	2
C₃×D₂₀	2
C₂×D₂₀	3
C₂×C₃×D₄	5

Subgroups — C2×C3×D20 #6796

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×D₂₀ #6796