Subgroups — C₂²×C₇×Dic₁₅ #13132888

Order 1680.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₅×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×Dic₁₅	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₇×Dic₅	—	3	1	no	maximal, Hall
4	C₂²×C₃×C₅×C₇	C₄	1	4	yes	complemented
5	C₂²×C₃×C₅×C₇	C₂²	1	3	yes	—
6	C₇×Dic₁₅	C₂²	1	4	yes	direct product factor
7	C₂²×C₇×Dic₃	—	5	1	no	maximal, Hall
8	C₂³×C₅×C₇	D₃	1	1	yes	characteristic
9	C₂×C₇×Dic₅	—	3	6	no	—
10	C₂²×Dic₁₅	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂×C₃×C₅×C₇	C₂×C₄	1	6	yes	complemented
12	C₂×C₃×C₅×C₇	C₂³	1	1	yes	characteristic
13	C₂³×C₃×C₇	D₅	1	1	yes	characteristic
14	C₂×C₇×Dic₃	—	5	6	no	—
15	C₂²×C₅×C₇	Dic₃	1	4	yes	complemented
16	C₂²×C₅×C₇	C₂×D₃	1	3	yes	—
17	C₇×Dic₅	—	3	4	no	—
18	C₂³×C₃×C₅	C₂×C₇	1	1	yes	characteristic
19	C₂×Dic₁₅	C₂×C₇	1	6	yes	direct product factor
20	C₂²×C₄×C₇	—	15	1	no	Hall
21	C₃×C₅×C₇	C₂²×C₄	1	1	yes	characteristic, complemented, Hall
22	C₂²×C₃×C₇	Dic₅	1	4	yes	complemented
23	C₂²×C₃×C₇	C₂×D₅	1	3	yes	—
24	C₇×Dic₃	—	5	4	no	—
25	C₂²×Dic₅	—	3	1	no	Hall
26	C₂×C₅×C₇	C₂×Dic₃	1	6	yes	complemented
27	C₂×C₅×C₇	C₂²×D₃	1	1	yes	characteristic
28	C₂²×C₃×C₅	C₄×C₇	1	4	yes	complemented
29	C₂²×C₃×C₅	C₂²×C₇	1	3	yes	—
30	Dic₁₅	C₂²×C₇	1	4	yes	direct product factor
31	C₂³×C₇	D₁₅	1	1	yes	characteristic
32	C₂×C₄×C₇	—	15	6	no	—
33	C₂²×Dic₃	—	5	1	no	Hall
34	C₂×C₃×C₇	C₂×Dic₅	1	6	yes	complemented
35	C₂×C₃×C₇	C₂²×D₅	1	1	yes	characteristic
36	C₂³×C₅	D₃×C₇	1	1	yes	characteristic
37	C₂×Dic₅	—	3	6	no	—
38	C₅×C₇	C₂²×Dic₃	1	1	yes	characteristic, complemented, Hall
39	C₂×C₃×C₅	C₂×C₄×C₇	1	6	yes	complemented
40	C₂×C₃×C₅	C₂³×C₇	1	1	yes	characteristic
41	C₂²×C₇	Dic₁₅	1	4	yes	direct product factor
42	C₂²×C₇	C₂×D₁₅	1	3	yes	—
43	C₄×C₇	—	15	4	no	—
44	C₂³×C₃	C₇×D₅	1	1	yes	characteristic
45	C₂×Dic₃	—	5	6	no	—
46	C₃×C₇	C₂²×Dic₅	1	1	yes	characteristic, complemented, Hall
47	C₂²×C₅	C₇×Dic₃	1	4	yes	complemented
48	C₂²×C₅	C₂×D₃×C₇	1	3	yes	—
49	Dic₅	—	3	4	no	—
50	C₂²×C₄	—	15	1	no	Sylow
51	C₃×C₅	C₂²×C₄×C₇	1	1	yes	characteristic, complemented, Hall
52	C₂×C₇	C₂×Dic₁₅	1	6	yes	direct product factor
53	C₂×C₇	C₂²×D₁₅	1	1	yes	characteristic
54	C₂²×C₃	C₇×Dic₅	1	4	yes	complemented
55	C₂²×C₃	C₂×C₇×D₅	1	3	yes	—
56	Dic₃	—	5	4	no	—
57	C₂×C₅	C₂×C₇×Dic₃	1	6	yes	complemented
58	C₂×C₅	C₂²×D₃×C₇	1	1	yes	characteristic
59	C₂³	C₇×D₁₅	1	1	yes	characteristic
60	C₂×C₄	—	15	6	no	—
61	C₇	C₂²×Dic₁₅	1	1	yes	characteristic, direct product factor, Sylow
62	C₂×C₃	C₂×C₇×Dic₅	1	6	yes	complemented
63	C₂×C₃	C₂²×C₇×D₅	1	1	yes	characteristic
64	C₅	C₂²×C₇×Dic₃	1	1	yes	characteristic, complemented, Sylow
65	C₂²	C₇×Dic₁₅	1	4	yes	direct product factor
66	C₂²	C₂×C₇×D₁₅	1	3	yes	—
67	C₄	—	15	4	no	—
68	C₃	C₂²×C₇×Dic₅	1	1	yes	characteristic, complemented, Sylow
69	C₂	C₂×C₇×Dic₁₅	1	6	yes	direct product factor
70	C₂	C₂²×C₇×D₁₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₇	D₁₅
commutator, hypocenter	C₃×C₅	C₂²×C₄×C₇
Frattini	C₂	C₂²×C₇×D₁₅
Fitting, socle, prime radical	C₂³×C₃×C₅×C₇	C₂
cosocle	C₂×C₃×C₅	C₂³×C₇
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂³×C₇

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×Dic₁₅	C₂	direct (N × H)
4. C₂²×C₃×C₅×C₇	C₄	semidirect
6. C₇×Dic₁₅	C₂²	direct (N × H)
10. C₂²×Dic₁₅	C₇	direct (N × H)
11. C₂×C₃×C₅×C₇	C₂×C₄	semidirect
15. C₂²×C₅×C₇	Dic₃	semidirect
19. C₂×Dic₁₅	C₂×C₇	direct (N × H)
21. C₃×C₅×C₇	C₂²×C₄	semidirect
22. C₂²×C₃×C₇	Dic₅	semidirect
26. C₂×C₅×C₇	C₂×Dic₃	semidirect
28. C₂²×C₃×C₅	C₄×C₇	semidirect
30. Dic₁₅	C₂²×C₇	direct (N × H)
34. C₂×C₃×C₇	C₂×Dic₅	semidirect
38. C₅×C₇	C₂²×Dic₃	semidirect
39. C₂×C₃×C₅	C₂×C₄×C₇	semidirect
41. C₂²×C₇	Dic₁₅	direct (N × H)
46. C₃×C₇	C₂²×Dic₅	semidirect
47. C₂²×C₅	C₇×Dic₃	semidirect
51. C₃×C₅	C₂²×C₄×C₇	semidirect
52. C₂×C₇	C₂×Dic₁₅	direct (N × H)
54. C₂²×C₃	C₇×Dic₅	semidirect
57. C₂×C₅	C₂×C₇×Dic₃	semidirect
61. C₇	C₂²×Dic₁₅	direct (N × H)
62. C₂×C₃	C₂×C₇×Dic₅	semidirect
64. C₅	C₂²×C₇×Dic₃	semidirect
65. C₂²	C₇×Dic₁₅	direct (N × H)
68. C₃	C₂²×C₇×Dic₅	semidirect
69. C₂	C₂×C₇×Dic₁₅	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₅×C₇	2
C₂×C₇×Dic₁₅	2
C₂²×C₇×Dic₅	3
C₂²×C₇×Dic₃	5
C₂²×Dic₁₅	7

Subgroups — C22×C7×Dic15 #13132888

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×Dic₁₅ #13132888