Subgroups — C₂×C₅×C₇×Dic₆ #13132860

Order 1680.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₇×Dic₃	C₂	1	2	yes	maximal
3	C₅×C₇×Dic₆	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅×C₇×Q₈	—	3	1	no	maximal, Hall
5	C₂²×C₃×C₅×C₇	C₂²	1	1	yes	characteristic
6	C₃×C₄×C₅×C₇	C₂²	1	2	yes	—
7	C₅×C₇×Dic₃	C₂²	1	4	yes	—
8	C₂×C₇×Dic₆	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₅×C₇×Q₈	—	3	4	no	—
10	C₂×C₄×C₅×C₇	—	3	2	no	—
11	C₂×C₄×C₅×C₇	D₃	1	1	yes	characteristic
12	C₂×C₅×Dic₆	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₅×C₇	Q₈	1	2	yes	complemented
14	C₂×C₃×C₅×C₇	C₂³	1	1	yes	characteristic
15	C₂×C₃×C₄×C₇	C₂×C₅	1	1	yes	characteristic
16	C₂×C₇×Dic₃	C₂×C₅	1	2	yes	—
17	C₇×Dic₆	C₂×C₅	1	4	yes	direct product factor
18	C₂²×C₅×C₇	C₂×D₃	1	1	yes	characteristic
19	C₄×C₅×C₇	—	3	4	no	—
20	C₄×C₅×C₇	C₂×D₃	1	2	yes	—
21	C₂×C₃×C₄×C₅	C₂×C₇	1	1	yes	characteristic
22	C₂×C₅×Dic₃	C₂×C₇	1	2	yes	—
23	C₅×Dic₆	C₂×C₇	1	4	yes	direct product factor
24	C₂×C₇×Q₈	—	3	1	no	Hall
25	C₃×C₅×C₇	C₂×Q₈	1	1	yes	characteristic, complemented, Hall
26	C₂²×C₃×C₇	C₂²×C₅	1	1	yes	characteristic
27	C₃×C₄×C₇	C₂²×C₅	1	2	yes	—
28	C₇×Dic₃	C₂²×C₅	1	4	yes	—
29	C₂×C₅×Q₈	—	3	1	no	Hall
30	C₂×C₅×C₇	Dic₆	1	2	yes	direct product factor
31	C₂×C₅×C₇	C₂²×D₃	1	1	yes	characteristic
32	C₂²×C₃×C₅	C₂²×C₇	1	1	yes	characteristic
33	C₃×C₄×C₅	C₂²×C₇	1	2	yes	—
34	C₅×Dic₃	C₂²×C₇	1	4	yes	—
35	C₇×Q₈	—	3	4	no	—
36	C₂×C₄×C₇	—	3	2	no	—
37	C₂×C₄×C₇	C₅×D₃	1	1	yes	characteristic
38	C₂×Dic₆	C₅×C₇	1	1	yes	characteristic, direct product factor, Hall
39	C₂×C₃×C₇	C₅×Q₈	1	2	yes	complemented
40	C₂×C₃×C₇	C₂³×C₅	1	1	yes	characteristic
41	C₅×Q₈	—	3	4	no	—
42	C₂×C₄×C₅	—	3	2	no	—
43	C₂×C₄×C₅	D₃×C₇	1	1	yes	characteristic
44	C₅×C₇	C₂×Dic₆	1	1	yes	characteristic, direct product factor, Hall
45	C₂×C₃×C₅	C₇×Q₈	1	2	yes	complemented
46	C₂×C₃×C₅	C₂³×C₇	1	1	yes	characteristic
47	C₂²×C₇	C₂×C₅×D₃	1	1	yes	characteristic
48	C₄×C₇	—	3	4	no	—
49	C₄×C₇	C₂×C₅×D₃	1	2	yes	—
50	C₂×C₃×C₄	C₂×C₅×C₇	1	1	yes	characteristic
51	C₂×Dic₃	C₂×C₅×C₇	1	2	yes	—
52	Dic₆	C₂×C₅×C₇	1	4	yes	direct product factor
53	C₃×C₇	C₂×C₅×Q₈	1	1	yes	characteristic, complemented, Hall
54	C₂²×C₅	C₂×D₃×C₇	1	1	yes	characteristic
55	C₄×C₅	—	3	4	no	—
56	C₄×C₅	C₂×D₃×C₇	1	2	yes	—
57	C₂×Q₈	—	3	1	no	Sylow
58	C₃×C₅	C₂×C₇×Q₈	1	1	yes	characteristic, complemented, Hall
59	C₂×C₇	C₅×Dic₆	1	2	yes	direct product factor
60	C₂×C₇	C₂²×C₅×D₃	1	1	yes	characteristic
61	C₂²×C₃	C₂²×C₅×C₇	1	1	yes	characteristic
62	C₃×C₄	C₂²×C₅×C₇	1	2	yes	—
63	Dic₃	C₂²×C₅×C₇	1	4	yes	—
64	C₂×C₅	C₇×Dic₆	1	2	yes	direct product factor
65	C₂×C₅	C₂²×D₃×C₇	1	1	yes	characteristic
66	Q₈	—	3	4	no	—
67	C₂×C₄	—	3	2	no	—
68	C₂×C₄	C₅×D₃×C₇	1	1	yes	characteristic
69	C₇	C₂×C₅×Dic₆	1	1	yes	characteristic, direct product factor, Sylow
70	C₂×C₃	C₅×C₇×Q₈	1	2	yes	complemented
71	C₂×C₃	C₂³×C₅×C₇	1	1	yes	characteristic
72	C₅	C₂×C₇×Dic₆	1	1	yes	characteristic, direct product factor, Sylow
73	C₂²	C₂×C₅×D₃×C₇	1	1	yes	characteristic
74	C₄	—	3	4	no	—
75	C₄	C₂×C₅×D₃×C₇	1	2	yes	—
76	C₃	C₂×C₅×C₇×Q₈	1	1	yes	characteristic, complemented, Sylow
77	C₂	C₅×C₇×Dic₆	1	2	yes	direct product factor
78	C₂	C₂²×C₅×D₃×C₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅×C₇	C₂×D₃
commutator, cosocle	C₂×C₃	C₂³×C₅×C₇
Frattini	C₂	C₂²×C₅×D₃×C₇
Fitting	C₂×C₃×C₄×C₅×C₇	C₂
socle, prime radical	C₂²×C₃×C₅×C₇	C₂²
hypercenter	C₂×C₄×C₅×C₇	D₃
hypocenter	C₃	C₂×C₅×C₇×Q₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅×C₇ ◁ C₂×C₄×C₅×C₇

Complements

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

N	H ≅ G/N	type
3. C₅×C₇×Dic₆	C₂	direct (N × H)
8. C₂×C₇×Dic₆	C₅	direct (N × H)
12. C₂×C₅×Dic₆	C₇	direct (N × H)
13. C₂×C₃×C₅×C₇	Q₈	semidirect
17. C₇×Dic₆	C₂×C₅	direct (N × H)
23. C₅×Dic₆	C₂×C₇	direct (N × H)
25. C₃×C₅×C₇	C₂×Q₈	semidirect
30. C₂×C₅×C₇	Dic₆	direct (N × H)
38. C₂×Dic₆	C₅×C₇	direct (N × H)
39. C₂×C₃×C₇	C₅×Q₈	semidirect
44. C₅×C₇	C₂×Dic₆	direct (N × H)
45. C₂×C₃×C₅	C₇×Q₈	semidirect
52. Dic₆	C₂×C₅×C₇	direct (N × H)
53. C₃×C₇	C₂×C₅×Q₈	semidirect
58. C₃×C₅	C₂×C₇×Q₈	semidirect
59. C₂×C₇	C₅×Dic₆	direct (N × H)
64. C₂×C₅	C₇×Dic₆	direct (N × H)
69. C₇	C₂×C₅×Dic₆	direct (N × H)
70. C₂×C₃	C₅×C₇×Q₈	semidirect
72. C₅	C₂×C₇×Dic₆	direct (N × H)
76. C₃	C₂×C₅×C₇×Q₈	semidirect
77. C₂	C₅×C₇×Dic₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₅×C₇	2
C₂×C₅×C₇×Dic₃	2
C₅×C₇×Dic₆	2
C₂×C₅×C₇×Q₈	3
C₂×C₇×Dic₆	5
C₂×C₅×Dic₆	7

Subgroups — C2×C5×C7×Dic6 #13132860

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×C₇×Dic₆ #13132860