Subgroups — C₂×C₅×D₁₂ #6806

Order 240.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×D₃	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₄×C₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×D₁₂	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅×D₄	—	3	1	no	maximal, Hall
5	C₂²×C₃×C₅	C₂²	1	1	yes	characteristic
6	C₂×C₅×D₃	—	2	4	no	—
7	C₂×C₅×D₃	C₂²	1	4	yes	complemented
8	C₃×C₄×C₅	C₂²	1	2	yes	complemented
9	C₂×D₁₂	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂³×C₅	—	3	2	no	—
11	C₅×D₄	—	3	4	no	—
12	C₂×C₄×C₅	D₃	1	1	yes	characteristic, complemented
13	C₂×C₃×C₅	D₄	1	2	yes	complemented
14	C₂×C₃×C₅	C₂³	1	1	yes	characteristic
15	C₅×D₃	—	2	4	no	—
16	C₂²×D₃	C₂×C₅	1	2	yes	complemented
17	C₂×C₃×C₄	C₂×C₅	1	1	yes	characteristic, complemented
18	D₁₂	C₂×C₅	1	4	yes	direct product factor
19	C₂²×C₅	—	3	4	no	—
20	C₂²×C₅	—	6	4	no	—
21	C₂²×C₅	C₂×D₃	1	1	yes	characteristic
22	C₄×C₅	C₂×D₃	1	2	yes	complemented
23	C₂×D₄	—	3	1	no	Sylow
24	C₃×C₅	C₂×D₄	1	1	yes	characteristic, complemented, Hall
25	C₂²×C₃	C₂²×C₅	1	1	yes	characteristic
26	C₂×D₃	—	2	4	no	—
27	C₂×D₃	C₂²×C₅	1	4	yes	complemented
28	C₃×C₄	C₂²×C₅	1	2	yes	complemented
29	C₂×C₅	—	6	4	no	—
30	C₂×C₅	D₁₂	1	2	yes	direct product factor
31	C₂×C₅	C₂²×D₃	1	1	yes	characteristic
32	C₂³	—	3	2	no	—
33	D₄	—	3	4	no	—
34	C₂×C₄	C₅×D₃	1	1	yes	characteristic, complemented
35	C₂×C₃	C₅×D₄	1	2	yes	complemented
36	C₂×C₃	C₂³×C₅	1	1	yes	characteristic
37	D₃	—	2	4	no	—
38	C₅	C₂×D₁₂	1	1	yes	characteristic, direct product factor, Sylow
39	C₂²	—	3	4	no	—
40	C₂²	—	6	4	no	—
41	C₂²	C₂×C₅×D₃	1	1	yes	characteristic
42	C₄	C₂×C₅×D₃	1	2	yes	complemented
43	C₃	C₂×C₅×D₄	1	1	yes	characteristic, complemented, Sylow
44	C₂	—	6	4	no	—
45	C₂	C₅×D₁₂	1	2	yes	direct product factor
46	C₂	C₂²×C₅×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×D₃
commutator, cosocle	C₂×C₃	C₂³×C₅
Frattini	C₂	C₂²×C₅×D₃
Fitting	C₂×C₃×C₄×C₅	C₂
socle	C₂²×C₃×C₅	C₂²
hypercenter	C₂×C₄×C₅	D₃
hypocenter	C₃	C₂×C₅×D₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×D₃	C₂	semidirect
2. C₂×C₃×C₄×C₅	C₂	semidirect
3. C₅×D₁₂	C₂	direct (N × H)
7. C₂×C₅×D₃	C₂²	semidirect
8. C₃×C₄×C₅	C₂²	semidirect
9. C₂×D₁₂	C₅	direct (N × H)
12. C₂×C₄×C₅	D₃	semidirect
13. C₂×C₃×C₅	D₄	semidirect
16. C₂²×D₃	C₂×C₅	semidirect
17. C₂×C₃×C₄	C₂×C₅	semidirect
18. D₁₂	C₂×C₅	direct (N × H)
22. C₄×C₅	C₂×D₃	semidirect
24. C₃×C₅	C₂×D₄	semidirect
27. C₂×D₃	C₂²×C₅	semidirect
28. C₃×C₄	C₂²×C₅	semidirect
30. C₂×C₅	D₁₂	direct (N × H)
34. C₂×C₄	C₅×D₃	semidirect
35. C₂×C₃	C₅×D₄	semidirect
38. C₅	C₂×D₁₂	direct (N × H)
42. C₄	C₂×C₅×D₃	semidirect
43. C₃	C₂×C₅×D₄	semidirect
45. C₂	C₅×D₁₂	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×C₅×D₃	2
C₂×C₃×C₄×C₅	2
C₅×D₁₂	2
C₂×C₅×D₄	3
C₂×D₁₂	5

Subgroups — C2×C5×D12 #6806

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×D₁₂ #6806