Subgroups — C₅×A₄×Dic₇ #13132587

Order 1680.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₇×A₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×Dic₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅×C₇×A₄	C₄	1	1	yes	characteristic, complemented
4	C₃×C₅×Dic₇	—	4	1	no	maximal
5	A₄×Dic₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂³×C₅×C₇	C₂×C₃	1	1	yes	characteristic
7	C₂×C₅×Dic₇	—	3	1	no	—
8	C₂×C₅×Dic₇	—	3	1	no	—
9	C₄×C₅×A₄	—	7	1	no	maximal, Hall
10	C₂×C₃×C₅×C₇	—	4	1	no	—
11	C₂×C₇×A₄	C₂×C₅	1	1	yes	characteristic
12	C₂²×C₅×C₇	—	3	1	no	—
13	C₂²×C₅×C₇	—	3	1	no	—
14	C₂²×C₅×C₇	C₃×C₄	1	1	yes	characteristic, complemented
15	C₅×Dic₇	—	3	1	no	—
16	C₅×Dic₇	A₄	1	1	yes	characteristic, direct product factor
17	C₂×C₅×A₄	D₇	1	1	yes	characteristic
18	C₂²×Dic₇	C₃×C₅	1	1	yes	characteristic, complemented, Hall
19	C₃×C₅×C₇	—	4	1	no	Hall
20	C₇×A₄	C₄×C₅	1	1	yes	characteristic, complemented
21	C₃×Dic₇	—	4	1	no	—
22	C₂²×C₄×C₅	—	7	1	no	Hall
23	C₂×C₅×C₇	—	3	1	no	—
24	C₂×C₅×C₇	—	3	1	no	—
25	C₂×C₅×C₇	C₂×A₄	1	1	yes	characteristic
26	C₅×A₄	Dic₇	1	1	yes	characteristic, direct product factor
27	C₃×C₄×C₅	—	28	1	no	—
28	C₂³×C₇	C₂×C₃×C₅	1	1	yes	characteristic
29	C₂×Dic₇	—	3	1	no	—
30	C₂×Dic₇	—	3	1	no	—
31	C₄×A₄	—	7	1	no	Hall
32	C₂×C₃×C₇	—	4	1	no	—
33	C₂³×C₅	C₃×D₇	1	1	yes	characteristic
34	C₂×C₄×C₅	—	21	1	no	—
35	C₂×C₄×C₅	—	21	1	no	—
36	C₅×C₇	C₄×A₄	1	1	yes	characteristic, complemented, Hall
37	C₂×C₃×C₅	—	4	1	no	—
38	C₂²×C₇	—	3	1	no	—
39	C₂²×C₇	—	3	1	no	—
40	C₂²×C₇	C₃×C₄×C₅	1	1	yes	characteristic, complemented
41	Dic₇	—	3	1	no	—
42	Dic₇	C₅×A₄	1	1	yes	characteristic, direct product factor
43	C₂×A₄	C₅×D₇	1	1	yes	characteristic
44	C₃×C₇	—	4	1	no	Hall
45	C₂²×C₅	—	3	1	no	—
46	C₂²×C₅	—	3	1	no	—
47	C₂²×C₅	C₃×Dic₇	1	1	yes	characteristic, complemented
48	C₄×C₅	—	7	1	no	—
49	C₄×C₅	—	21	1	no	—
50	C₂²×C₄	—	7	1	no	Sylow
51	C₃×C₅	—	4	1	no	Hall
52	C₂×C₇	—	3	1	no	—
53	C₂×C₇	—	3	1	no	—
54	C₂×C₇	C₂×C₅×A₄	1	1	yes	characteristic
55	A₄	C₅×Dic₇	1	1	yes	characteristic, direct product factor
56	C₃×C₄	—	28	1	no	—
57	C₂×C₅	—	3	1	no	—
58	C₂×C₅	—	3	1	no	—
59	C₂×C₅	A₄×D₇	1	1	yes	characteristic
60	C₂³	C₃×C₅×D₇	1	1	yes	characteristic
61	C₂×C₄	—	21	1	no	—
62	C₂×C₄	—	21	1	no	—
63	C₇	C₄×C₅×A₄	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₃	—	4	1	no	—
65	C₅	A₄×Dic₇	1	1	yes	characteristic, direct product factor, Sylow
66	C₂²	—	3	1	no	—
67	C₂²	—	3	1	no	—
68	C₂²	C₃×C₅×Dic₇	1	1	yes	characteristic, complemented
69	C₄	—	7	1	no	—
70	C₄	—	21	1	no	—
71	C₃	—	4	1	no	Sylow
72	C₂	—	3	1	no	—
73	C₂	—	3	1	no	—
74	C₂	C₅×A₄×D₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₅	A₄×D₇
commutator, hypocenter	C₂²×C₇	C₃×C₄×C₅
Frattini	C₂	C₅×A₄×D₇
Fitting, socle	C₂³×C₅×C₇	C₂×C₃
prime radical	C₂×C₅×C₇×A₄	C₂
cosocle	C₂³×C₇	C₂×C₃×C₅
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₂×C₅

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×Dic₇	C₃	semidirect
3. C₅×C₇×A₄	C₄	semidirect
5. A₄×Dic₇	C₅	direct (N × H)
14. C₂²×C₅×C₇	C₃×C₄	semidirect
16. C₅×Dic₇	A₄	direct (N × H)
18. C₂²×Dic₇	C₃×C₅	semidirect
20. C₇×A₄	C₄×C₅	semidirect
26. C₅×A₄	Dic₇	direct (N × H)
36. C₅×C₇	C₄×A₄	semidirect
40. C₂²×C₇	C₃×C₄×C₅	semidirect
42. Dic₇	C₅×A₄	direct (N × H)
47. C₂²×C₅	C₃×Dic₇	semidirect
55. A₄	C₅×Dic₇	direct (N × H)
63. C₇	C₄×C₅×A₄	semidirect
65. C₅	A₄×Dic₇	direct (N × H)
68. C₂²	C₃×C₅×Dic₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×C₇×A₄	2
C₂²×C₅×Dic₇	3
C₃×C₅×Dic₇	4
A₄×Dic₇	5
C₄×C₅×A₄	7

Subgroups — C5×A4×Dic7 #13132587

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×A₄×Dic₇ #13132587