Subgroups — C₂²×C₃×C₅×Dic₇ #13132838

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

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×Dic₇	C₂	direct (N × H)
3. C₂²×C₅×Dic₇	C₃	direct (N × H)
4. C₂²×C₃×C₅×C₇	C₄	semidirect
6. C₃×C₅×Dic₇	C₂²	direct (N × H)
7. C₂²×C₃×Dic₇	C₅	direct (N × H)
9. C₂×C₅×Dic₇	C₂×C₃	direct (N × H)
11. C₂×C₃×C₅×C₇	C₂×C₄	semidirect
14. C₂×C₃×Dic₇	C₂×C₅	direct (N × H)
15. C₂²×C₅×C₇	C₃×C₄	semidirect
17. C₅×Dic₇	C₂²×C₃	direct (N × H)
20. C₂²×Dic₇	C₃×C₅	direct (N × H)
21. C₃×C₅×C₇	C₂²×C₄	semidirect
22. C₂²×C₃×C₇	C₄×C₅	semidirect
24. C₃×Dic₇	C₂²×C₅	direct (N × H)
26. C₂×C₅×C₇	C₂×C₃×C₄	semidirect
28. C₂²×C₃×C₅	Dic₇	direct (N × H)
32. C₂×Dic₇	C₂×C₃×C₅	direct (N × H)
34. C₂×C₃×C₇	C₂×C₄×C₅	semidirect
38. C₅×C₇	C₂²×C₃×C₄	semidirect
39. C₂×C₃×C₅	C₂×Dic₇	direct (N × H)
41. C₂²×C₇	C₃×C₄×C₅	semidirect
43. Dic₇	C₂²×C₃×C₅	direct (N × H)
46. C₃×C₇	C₂²×C₄×C₅	semidirect
47. C₂²×C₅	C₃×Dic₇	direct (N × H)
51. C₃×C₅	C₂²×Dic₇	direct (N × H)
52. C₂×C₇	C₂×C₃×C₄×C₅	semidirect
54. C₂²×C₃	C₅×Dic₇	direct (N × H)
57. C₂×C₅	C₂×C₃×Dic₇	direct (N × H)
61. C₇	C₂²×C₃×C₄×C₅	semidirect
62. C₂×C₃	C₂×C₅×Dic₇	direct (N × H)
64. C₅	C₂²×C₃×Dic₇	direct (N × H)
65. C₂²	C₃×C₅×Dic₇	direct (N × H)
68. C₃	C₂²×C₅×Dic₇	direct (N × H)
69. C₂	C₂×C₃×C₅×Dic₇	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×C₃×C₅×Dic₇	2
C₂²×C₅×Dic₇	3
C₂²×C₃×Dic₇	5
C₂²×C₃×C₄×C₅	7

Subgroups — C22×C3×C5×Dic7 #13132838

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₅×Dic₇ #13132838