Subgroups — C₃×C₇×Q₈×D₅ #13132846

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×C₇×D₅	C₂	1	3	yes	maximal
3	C₃×C₇×Dic₁₀	C₂	1	3	yes	maximal, complemented
4	C₇×Q₈×D₅	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₃×C₇×D₅	C₂²	1	1	yes	characteristic
6	C₃×C₄×C₅×C₇	C₂²	1	3	yes	—
7	C₃×C₇×Dic₅	C₂²	1	3	yes	—
8	C₂×C₃×C₇×Q₈	—	5	1	no	maximal, Hall
9	C₅×C₇×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
10	C₄×C₇×D₅	C₂×C₃	1	3	yes	—
11	C₇×Dic₁₀	C₂×C₃	1	3	yes	complemented
12	C₃×Q₈×D₅	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₅×C₇	C₂³	1	1	yes	characteristic
14	C₃×C₇×D₅	Q₈	1	2	yes	direct product factor
15	C₃×C₇×Q₈	—	5	3	no	—
16	C₃×C₇×Q₈	D₅	1	1	yes	characteristic, direct product factor
17	C₂×C₃×C₄×C₇	—	5	3	no	—
18	C₂×C₇×D₅	C₂²×C₃	1	1	yes	characteristic
19	C₄×C₅×C₇	C₂²×C₃	1	3	yes	—
20	C₇×Dic₅	C₂²×C₃	1	3	yes	—
21	C₃×C₅×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
22	C₃×C₄×D₅	C₂×C₇	1	3	yes	—
23	C₃×Dic₁₀	C₂×C₇	1	3	yes	complemented
24	C₂×C₇×Q₈	—	5	1	no	Hall
25	C₃×C₅×C₇	C₂×Q₈	1	1	yes	characteristic, complemented, Hall
26	C₂²×C₃×C₇	—	5	1	no	—
27	C₃×C₄×C₇	—	5	3	no	—
28	C₃×C₄×C₇	C₂×D₅	1	3	yes	—
29	Q₈×D₅	C₃×C₇	1	1	yes	characteristic, direct product factor, Hall
30	C₂×C₅×C₇	C₂³×C₃	1	1	yes	characteristic
31	C₇×D₅	C₃×Q₈	1	2	yes	direct product factor
32	C₂×C₃×D₅	C₂²×C₇	1	1	yes	characteristic
33	C₃×C₄×C₅	C₂²×C₇	1	3	yes	—
34	C₃×Dic₅	C₂²×C₇	1	3	yes	—
35	C₇×Q₈	—	5	3	no	—
36	C₇×Q₈	C₃×D₅	1	1	yes	characteristic, direct product factor
37	C₂×C₄×C₇	—	5	3	no	—
38	C₂×C₃×Q₈	—	5	1	no	Hall
39	C₂×C₃×C₇	—	5	2	no	—
40	C₂×C₃×C₇	C₂²×D₅	1	1	yes	characteristic
41	C₅×Q₈	C₂×C₃×C₇	1	1	yes	characteristic, complemented
42	C₄×D₅	C₂×C₃×C₇	1	3	yes	—
43	Dic₁₀	C₂×C₃×C₇	1	3	yes	complemented
44	C₅×C₇	C₂×C₃×Q₈	1	1	yes	characteristic, complemented, Hall
45	C₂×C₃×C₅	C₂³×C₇	1	1	yes	characteristic
46	C₃×D₅	C₇×Q₈	1	2	yes	direct product factor
47	C₂²×C₇	—	5	1	no	—
48	C₄×C₇	—	5	3	no	—
49	C₄×C₇	C₂×C₃×D₅	1	3	yes	—
50	C₃×Q₈	—	5	3	no	—
51	C₃×Q₈	C₇×D₅	1	1	yes	characteristic, direct product factor
52	C₂×C₃×C₄	—	5	3	no	—
53	C₃×C₇	Q₈×D₅	1	1	yes	characteristic, direct product factor, Hall
54	C₂×D₅	C₂²×C₃×C₇	1	1	yes	characteristic
55	C₄×C₅	C₂²×C₃×C₇	1	3	yes	—
56	Dic₅	C₂²×C₃×C₇	1	3	yes	—
57	C₂×Q₈	—	5	1	no	Sylow
58	C₃×C₅	C₂×C₇×Q₈	1	1	yes	characteristic, complemented, Hall
59	C₂×C₇	—	5	2	no	—
60	C₂×C₇	C₂²×C₃×D₅	1	1	yes	characteristic
61	C₂²×C₃	—	5	1	no	—
62	C₃×C₄	—	5	3	no	—
63	C₃×C₄	C₂×C₇×D₅	1	3	yes	—
64	C₂×C₅	C₂³×C₃×C₇	1	1	yes	characteristic
65	D₅	C₃×C₇×Q₈	1	2	yes	direct product factor
66	Q₈	—	5	3	no	—
67	Q₈	C₃×C₇×D₅	1	1	yes	characteristic, direct product factor
68	C₂×C₄	—	5	3	no	—
69	C₇	C₃×Q₈×D₅	1	1	yes	characteristic, direct product factor, Sylow
70	C₂×C₃	—	5	2	no	—
71	C₂×C₃	C₂²×C₇×D₅	1	1	yes	characteristic
72	C₅	C₂×C₃×C₇×Q₈	1	1	yes	characteristic, complemented, Sylow
73	C₂²	—	5	1	no	—
74	C₄	—	5	3	no	—
75	C₄	C₂×C₃×C₇×D₅	1	3	yes	—
76	C₃	C₇×Q₈×D₅	1	1	yes	characteristic, direct product factor, Sylow
77	C₂	—	5	2	no	—
78	C₂	C₂²×C₃×C₇×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₇	C₂²×D₅
commutator, cosocle	C₂×C₅	C₂³×C₃×C₇
Frattini	C₂	C₂²×C₃×C₇×D₅
Fitting	C₃×C₅×C₇×Q₈	C₂
socle	C₂×C₃×C₅×C₇	C₂³
prime radical	C₂×C₃×C₇×D₅	C₂²
hypercenter	C₃×C₇×Q₈	D₅
hypocenter	C₅	C₂×C₃×C₇×Q₈
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₇ ◁ C₃×C₇×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×C₇×Dic₁₀	C₂	semidirect
4. C₇×Q₈×D₅	C₃	direct (N × H)
9. C₅×C₇×Q₈	C₂×C₃	semidirect
11. C₇×Dic₁₀	C₂×C₃	semidirect
12. C₃×Q₈×D₅	C₇	direct (N × H)
14. C₃×C₇×D₅	Q₈	direct (N × H)
16. C₃×C₇×Q₈	D₅	direct (N × H)
21. C₃×C₅×Q₈	C₂×C₇	semidirect
23. C₃×Dic₁₀	C₂×C₇	semidirect
25. C₃×C₅×C₇	C₂×Q₈	semidirect
29. Q₈×D₅	C₃×C₇	direct (N × H)
31. C₇×D₅	C₃×Q₈	direct (N × H)
36. C₇×Q₈	C₃×D₅	direct (N × H)
41. C₅×Q₈	C₂×C₃×C₇	semidirect
43. Dic₁₀	C₂×C₃×C₇	semidirect
44. C₅×C₇	C₂×C₃×Q₈	semidirect
46. C₃×D₅	C₇×Q₈	direct (N × H)
51. C₃×Q₈	C₇×D₅	direct (N × H)
53. C₃×C₇	Q₈×D₅	direct (N × H)
58. C₃×C₅	C₂×C₇×Q₈	semidirect
65. D₅	C₃×C₇×Q₈	direct (N × H)
67. Q₈	C₃×C₇×D₅	direct (N × H)
69. C₇	C₃×Q₈×D₅	direct (N × H)
72. C₅	C₂×C₃×C₇×Q₈	semidirect
76. C₃	C₇×Q₈×D₅	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×C₇×D₅	2
C₃×C₇×Dic₁₀	2
C₇×Q₈×D₅	3
C₂×C₃×C₇×Q₈	5
C₃×Q₈×D₅	7

Subgroups — C3×C7×Q8×D5 #13132846

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×Q₈×D₅ #13132846