Subgroups — C₃×C₅×Q₈×D₇ #13132836

Order 1680.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₄×C₅×D₇	C₂	1	3	yes	maximal
3	C₃×C₅×Dic₁₄	C₂	1	3	yes	maximal, complemented
4	C₅×Q₈×D₇	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₃×C₅×D₇	C₂²	1	1	yes	characteristic
6	C₃×C₄×C₅×C₇	C₂²	1	3	yes	—
7	C₃×C₅×Dic₇	C₂²	1	3	yes	—
8	C₃×Q₈×D₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₅×C₇×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
10	C₄×C₅×D₇	C₂×C₃	1	3	yes	—
11	C₅×Dic₁₄	C₂×C₃	1	3	yes	complemented
12	C₂×C₃×C₅×Q₈	—	7	1	no	maximal, Hall
13	C₂×C₃×C₅×C₇	C₂³	1	1	yes	characteristic
14	C₃×C₅×D₇	Q₈	1	2	yes	direct product factor
15	C₃×C₇×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
16	C₃×C₄×D₇	C₂×C₅	1	3	yes	—
17	C₃×Dic₁₄	C₂×C₅	1	3	yes	complemented
18	C₂×C₅×D₇	C₂²×C₃	1	1	yes	characteristic
19	C₄×C₅×C₇	C₂²×C₃	1	3	yes	—
20	C₅×Dic₇	C₂²×C₃	1	3	yes	—
21	C₃×C₅×Q₈	—	7	3	no	—
22	C₃×C₅×Q₈	D₇	1	1	yes	characteristic, direct product factor
23	C₂×C₃×C₄×C₅	—	7	3	no	—
24	Q₈×D₇	C₃×C₅	1	1	yes	characteristic, direct product factor, Hall
25	C₃×C₅×C₇	C₂×Q₈	1	1	yes	characteristic, complemented, Hall
26	C₂×C₃×D₇	C₂²×C₅	1	1	yes	characteristic
27	C₃×C₄×C₇	C₂²×C₅	1	3	yes	—
28	C₃×Dic₇	C₂²×C₅	1	3	yes	—
29	C₂×C₅×Q₈	—	7	1	no	Hall
30	C₂×C₅×C₇	C₂³×C₃	1	1	yes	characteristic
31	C₅×D₇	C₃×Q₈	1	2	yes	direct product factor
32	C₂²×C₃×C₅	—	7	1	no	—
33	C₃×C₄×C₅	—	7	3	no	—
34	C₃×C₄×C₅	C₂×D₇	1	3	yes	—
35	C₇×Q₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
36	C₄×D₇	C₂×C₃×C₅	1	3	yes	—
37	Dic₁₄	C₂×C₃×C₅	1	3	yes	complemented
38	C₂×C₃×Q₈	—	7	1	no	Hall
39	C₂×C₃×C₇	C₂³×C₅	1	1	yes	characteristic
40	C₃×D₇	C₅×Q₈	1	2	yes	direct product factor
41	C₅×Q₈	—	7	3	no	—
42	C₅×Q₈	C₃×D₇	1	1	yes	characteristic, direct product factor
43	C₂×C₄×C₅	—	7	3	no	—
44	C₅×C₇	C₂×C₃×Q₈	1	1	yes	characteristic, complemented, Hall
45	C₂×C₃×C₅	—	7	2	no	—
46	C₂×C₃×C₅	C₂²×D₇	1	1	yes	characteristic
47	C₂×D₇	C₂²×C₃×C₅	1	1	yes	characteristic
48	C₄×C₇	C₂²×C₃×C₅	1	3	yes	—
49	Dic₇	C₂²×C₃×C₅	1	3	yes	—
50	C₃×Q₈	—	7	3	no	—
51	C₃×Q₈	C₅×D₇	1	1	yes	characteristic, direct product factor
52	C₂×C₃×C₄	—	7	3	no	—
53	C₃×C₇	C₂×C₅×Q₈	1	1	yes	characteristic, complemented, Hall
54	C₂²×C₅	—	7	1	no	—
55	C₄×C₅	—	7	3	no	—
56	C₄×C₅	C₂×C₃×D₇	1	3	yes	—
57	C₂×Q₈	—	7	1	no	Sylow
58	C₃×C₅	Q₈×D₇	1	1	yes	characteristic, direct product factor, Hall
59	C₂×C₇	C₂³×C₃×C₅	1	1	yes	characteristic
60	D₇	C₃×C₅×Q₈	1	2	yes	direct product factor
61	C₂²×C₃	—	7	1	no	—
62	C₃×C₄	—	7	3	no	—
63	C₃×C₄	C₂×C₅×D₇	1	3	yes	—
64	C₂×C₅	—	7	2	no	—
65	C₂×C₅	C₂²×C₃×D₇	1	1	yes	characteristic
66	Q₈	—	7	3	no	—
67	Q₈	C₃×C₅×D₇	1	1	yes	characteristic, direct product factor
68	C₂×C₄	—	7	3	no	—
69	C₇	C₂×C₃×C₅×Q₈	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₃	—	7	2	no	—
71	C₂×C₃	C₂²×C₅×D₇	1	1	yes	characteristic
72	C₅	C₃×Q₈×D₇	1	1	yes	characteristic, direct product factor, Sylow
73	C₂²	—	7	1	no	—
74	C₄	—	7	3	no	—
75	C₄	C₂×C₃×C₅×D₇	1	3	yes	—
76	C₃	C₅×Q₈×D₇	1	1	yes	characteristic, direct product factor, Sylow
77	C₂	—	7	2	no	—
78	C₂	C₂²×C₃×C₅×D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₅	C₂²×D₇
commutator, cosocle	C₂×C₇	C₂³×C₃×C₅
Frattini	C₂	C₂²×C₃×C₅×D₇
Fitting	C₃×C₅×C₇×Q₈	C₂
socle	C₂×C₃×C₅×C₇	C₂³
prime radical	C₂×C₃×C₅×D₇	C₂²
hypercenter	C₃×C₅×Q₈	D₇
hypocenter	C₇	C₂×C₃×C₅×Q₈
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃×C₅ ◁ C₃×C₅×Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×C₇×Q₈	C₂	semidirect
3. C₃×C₅×Dic₁₄	C₂	semidirect
4. C₅×Q₈×D₇	C₃	direct (N × H)
8. C₃×Q₈×D₇	C₅	direct (N × H)
9. C₅×C₇×Q₈	C₂×C₃	semidirect
11. C₅×Dic₁₄	C₂×C₃	semidirect
14. C₃×C₅×D₇	Q₈	direct (N × H)
15. C₃×C₇×Q₈	C₂×C₅	semidirect
17. C₃×Dic₁₄	C₂×C₅	semidirect
22. C₃×C₅×Q₈	D₇	direct (N × H)
24. Q₈×D₇	C₃×C₅	direct (N × H)
25. C₃×C₅×C₇	C₂×Q₈	semidirect
31. C₅×D₇	C₃×Q₈	direct (N × H)
35. C₇×Q₈	C₂×C₃×C₅	semidirect
37. Dic₁₄	C₂×C₃×C₅	semidirect
40. C₃×D₇	C₅×Q₈	direct (N × H)
42. C₅×Q₈	C₃×D₇	direct (N × H)
44. C₅×C₇	C₂×C₃×Q₈	semidirect
51. C₃×Q₈	C₅×D₇	direct (N × H)
53. C₃×C₇	C₂×C₅×Q₈	semidirect
58. C₃×C₅	Q₈×D₇	direct (N × H)
60. D₇	C₃×C₅×Q₈	direct (N × H)
67. Q₈	C₃×C₅×D₇	direct (N × H)
69. C₇	C₂×C₃×C₅×Q₈	semidirect
72. C₅	C₃×Q₈×D₇	direct (N × H)
76. C₃	C₅×Q₈×D₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×C₇×Q₈	2
C₃×C₄×C₅×D₇	2
C₃×C₅×Dic₁₄	2
C₅×Q₈×D₇	3
C₃×Q₈×D₇	5
C₂×C₃×C₅×Q₈	7

Subgroups — C3×C5×Q8×D7 #13132836

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×Q₈×D₇ #13132836