Subgroups — D₅×D₁₂ #6775

Order 240.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃×D₅	C₂	1	2	yes	maximal, complemented
2	D₆₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×D₅	C₂	1	1	yes	maximal, characteristic, complemented
5	#1203	C₂	1	2	yes	maximal, complemented
6	D₄×D₅	—	3	1	no	maximal, Hall
7	C₂×D₁₅	C₂²	1	2	yes	complemented
8	C₂×C₅×D₃	C₂²	1	2	yes	complemented
9	C₂×C₃×D₅	C₂²	1	1	yes	characteristic
10	D₃×D₅	—	2	4	no	—
11	C₃×C₄×C₅	C₂²	1	1	yes	characteristic, complemented
12	C₃×Dic₅	C₂²	1	1	yes	characteristic, complemented
13	C₂×D₁₂	—	5	1	no	maximal, Hall
14	C₂²×D₅	—	3	2	no	—
15	C₅×D₄	—	3	1	no	—
16	C₂²⋊D₅	—	3	2	no	—
17	D₂₀	—	3	1	no	—
18	C₄×D₅	D₃	1	1	yes	characteristic, complemented
19	C₂×C₃×C₅	C₂³	1	1	yes	characteristic
20	D₁₅	—	2	2	no	—
21	C₃×D₅	D₄	1	2	yes	complemented
22	C₅×D₃	—	2	2	no	—
23	C₂²×D₃	—	5	2	no	—
24	C₂×C₃×C₄	—	5	1	no	—
25	D₁₂	—	5	2	no	—
26	D₁₂	—	5	1	no	—
27	D₁₂	D₅	1	1	yes	characteristic, direct product factor
28	C₂²×C₅	—	3	2	no	—
29	C₂×D₅	—	3	2	no	—
30	C₂×D₅	—	6	4	no	—
31	C₂×D₅	C₂×D₃	1	1	yes	characteristic
32	C₄×C₅	C₂×D₃	1	1	yes	characteristic, complemented
33	Dic₅	C₂×D₃	1	1	yes	characteristic, complemented
34	C₂×D₄	—	15	1	no	Sylow
35	C₃×C₅	C₂×D₄	1	1	yes	characteristic, complemented, Hall
36	C₂²×C₃	—	5	1	no	—
37	C₂×D₃	—	5	2	no	—
38	C₂×D₃	—	10	4	no	—
39	C₂×D₃	C₂×D₅	1	2	yes	complemented
40	C₃×C₄	—	5	1	no	—
41	C₃×C₄	C₂×D₅	1	1	yes	characteristic, complemented
42	C₂×C₅	—	6	2	no	—
43	C₂×C₅	C₂²×D₃	1	1	yes	characteristic
44	D₅	—	6	2	no	—
45	D₅	D₁₂	1	2	yes	direct product factor
46	C₂³	—	15	2	no	—
47	D₄	—	3	1	no	—
48	D₄	—	15	2	no	—
49	D₄	—	15	1	no	—
50	C₂×C₄	—	5	1	no	—
51	C₂×C₃	—	5	2	no	—
52	C₂×C₃	C₂²×D₅	1	1	yes	characteristic
53	D₃	—	2	2	no	—
54	D₃	—	10	2	no	—
55	C₅	C₂×D₁₂	1	1	yes	characteristic, complemented, Sylow
56	C₂²	—	3	2	no	—
57	C₂²	—	5	1	no	—
58	C₂²	—	15	2	no	—
59	C₂²	—	30	4	no	—
60	C₄	—	5	1	no	—
61	C₄	D₃×D₅	1	1	yes	characteristic, complemented
62	C₃	D₄×D₅	1	1	yes	characteristic, complemented, Sylow
63	C₂	—	5	2	no	—
64	C₂	—	6	2	no	—
65	C₂	—	30	2	no	—
66	C₂	C₂×D₃×D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₃×D₅
commutator, socle, cosocle	C₂×C₃×C₅	C₂³
Fitting	C₃×C₄×C₅	C₂²
hypercenter	C₄	D₃×D₅
hypocenter	C₃×C₅	C₂×D₄
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. C₂×D₃×D₅	C₂	semidirect
2. D₆₀	C₂	semidirect
3. C₅×D₁₂	C₂	semidirect
4. C₃×C₄×D₅	C₂	semidirect
5. #1203	C₂	semidirect
7. C₂×D₁₅	C₂²	semidirect
8. C₂×C₅×D₃	C₂²	semidirect
11. C₃×C₄×C₅	C₂²	semidirect
12. C₃×Dic₅	C₂²	semidirect
18. C₄×D₅	D₃	semidirect
21. C₃×D₅	D₄	semidirect
27. D₁₂	D₅	direct (N × H)
32. C₄×C₅	C₂×D₃	semidirect
33. Dic₅	C₂×D₃	semidirect
35. C₃×C₅	C₂×D₄	semidirect
39. C₂×D₃	C₂×D₅	semidirect
41. C₃×C₄	C₂×D₅	semidirect
45. D₅	D₁₂	direct (N × H)
55. C₅	C₂×D₁₂	semidirect
61. C₄	D₃×D₅	semidirect
62. C₃	D₄×D₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃×D₅	2
D₆₀	2
C₅×D₁₂	2
C₃×C₄×D₅	2
#1203	2
D₄×D₅	3
C₂×D₁₂	5

Subgroups — D5×D12 #6775

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×D₁₂ #6775