Subgroups — C₂²×C₃²×C₇×C₈ #72182003

Order 2016.

456 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₄×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×C₇×C₈	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃×C₇×C₈	C₃	1	4	yes	maximal, direct product factor
4	C₂³×C₃²×C₇	C₄	1	1	yes	characteristic
5	C₂×C₃²×C₄×C₇	C₄	1	3	yes	—
6	C₂×C₃²×C₄×C₇	C₂²	1	3	yes	—
7	C₃²×C₇×C₈	C₂²	1	4	yes	direct product factor
8	C₂²×C₃×C₄×C₇	C₂×C₃	1	4	yes	—
9	C₂×C₃×C₇×C₈	C₂×C₃	1	24	yes	direct product factor
10	C₂²×C₃²×C₈	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂²×C₃²×C₇	C₈	1	4	yes	direct product factor
12	C₂²×C₃²×C₇	C₂×C₄	1	3	yes	—
13	C₃²×C₄×C₇	C₂×C₄	1	3	yes	—
14	C₃²×C₄×C₇	C₂³	1	1	yes	characteristic
15	C₂²×C₇×C₈	C₃²	1	1	yes	characteristic, direct product factor, Hall
16	C₂³×C₃×C₇	C₃×C₄	1	4	yes	—
17	C₂×C₃×C₄×C₇	C₃×C₄	1	12	yes	—
18	C₂×C₃×C₄×C₇	C₂²×C₃	1	12	yes	—
19	C₃×C₇×C₈	C₂²×C₃	1	16	yes	direct product factor
20	C₂²×C₃²×C₄	C₂×C₇	1	1	yes	characteristic
21	C₂×C₃²×C₈	C₂×C₇	1	6	yes	direct product factor
22	C₂×C₃²×C₇	C₂×C₈	1	6	yes	direct product factor
23	C₂×C₃²×C₇	C₂²×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₇	C₂×C₃²	1	1	yes	characteristic
25	C₂×C₇×C₈	C₂×C₃²	1	6	yes	direct product factor
26	C₂²×C₃×C₈	C₃×C₇	1	4	yes	direct product factor
27	C₂²×C₃×C₇	C₃×C₈	1	16	yes	direct product factor
28	C₂²×C₃×C₇	C₂×C₃×C₄	1	12	yes	—
29	C₃×C₄×C₇	C₂×C₃×C₄	1	12	yes	—
30	C₃×C₄×C₇	C₂³×C₃	1	4	yes	—
31	C₂³×C₃²	C₄×C₇	1	1	yes	characteristic
32	C₂×C₃²×C₄	C₄×C₇	1	3	yes	—
33	C₂×C₃²×C₄	C₂²×C₇	1	3	yes	—
34	C₃²×C₈	C₂²×C₇	1	4	yes	direct product factor
35	C₃²×C₇	C₂²×C₈	1	1	yes	characteristic, direct product factor, Hall
36	C₂³×C₇	C₃²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₇	C₃²×C₄	1	3	yes	—
38	C₂×C₄×C₇	C₂²×C₃²	1	3	yes	—
39	C₇×C₈	C₂²×C₃²	1	4	yes	direct product factor
40	C₂²×C₃×C₄	C₂×C₃×C₇	1	4	yes	—
41	C₂×C₃×C₈	C₂×C₃×C₇	1	24	yes	direct product factor
42	C₂×C₃×C₇	C₂×C₃×C₈	1	24	yes	direct product factor
43	C₂×C₃×C₇	C₂²×C₃×C₄	1	4	yes	—
44	C₂²×C₃²	C₇×C₈	1	4	yes	direct product factor
45	C₂²×C₃²	C₂×C₄×C₇	1	3	yes	—
46	C₃²×C₄	C₂×C₄×C₇	1	3	yes	—
47	C₃²×C₄	C₂³×C₇	1	1	yes	characteristic
48	C₂²×C₈	C₃²×C₇	1	1	yes	characteristic, direct product factor, Sylow
49	C₂²×C₇	C₃²×C₈	1	4	yes	direct product factor
50	C₂²×C₇	C₂×C₃²×C₄	1	3	yes	—
51	C₄×C₇	C₂×C₃²×C₄	1	3	yes	—
52	C₄×C₇	C₂³×C₃²	1	1	yes	characteristic
53	C₂³×C₃	C₃×C₄×C₇	1	4	yes	—
54	C₂×C₃×C₄	C₃×C₄×C₇	1	12	yes	—
55	C₂×C₃×C₄	C₂²×C₃×C₇	1	12	yes	—
56	C₃×C₈	C₂²×C₃×C₇	1	16	yes	direct product factor
57	C₃×C₇	C₂²×C₃×C₈	1	4	yes	direct product factor
58	C₂×C₃²	C₂×C₇×C₈	1	6	yes	direct product factor
59	C₂×C₃²	C₂²×C₄×C₇	1	1	yes	characteristic
60	C₂²×C₄	C₂×C₃²×C₇	1	1	yes	characteristic
61	C₂×C₈	C₂×C₃²×C₇	1	6	yes	direct product factor
62	C₂×C₇	C₂×C₃²×C₈	1	6	yes	direct product factor
63	C₂×C₇	C₂²×C₃²×C₄	1	1	yes	characteristic
64	C₂²×C₃	C₃×C₇×C₈	1	16	yes	direct product factor
65	C₂²×C₃	C₂×C₃×C₄×C₇	1	12	yes	—
66	C₃×C₄	C₂×C₃×C₄×C₇	1	12	yes	—
67	C₃×C₄	C₂³×C₃×C₇	1	4	yes	—
68	C₃²	C₂²×C₇×C₈	1	1	yes	characteristic, direct product factor, Sylow
69	C₂³	C₃²×C₄×C₇	1	1	yes	characteristic
70	C₂×C₄	C₃²×C₄×C₇	1	3	yes	—
71	C₂×C₄	C₂²×C₃²×C₇	1	3	yes	—
72	C₈	C₂²×C₃²×C₇	1	4	yes	direct product factor
73	C₇	C₂²×C₃²×C₈	1	1	yes	characteristic, direct product factor, Sylow
74	C₂×C₃	C₂×C₃×C₇×C₈	1	24	yes	direct product factor
75	C₂×C₃	C₂²×C₃×C₄×C₇	1	4	yes	—
76	C₂²	C₃²×C₇×C₈	1	4	yes	direct product factor
77	C₂²	C₂×C₃²×C₄×C₇	1	3	yes	—
78	C₄	C₂×C₃²×C₄×C₇	1	3	yes	—
79	C₄	C₂³×C₃²×C₇	1	1	yes	characteristic
80	C₃	C₂²×C₃×C₇×C₈	1	4	yes	direct product factor
81	C₂	C₂×C₃²×C₇×C₈	1	6	yes	direct product factor
82	C₂	C₂²×C₃²×C₄×C₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂³×C₃²×C₇
socle, prime radical	C₂³×C₃²×C₇	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃²×C₇×C₈	C₂	direct (N × H)
3. C₂²×C₃×C₇×C₈	C₃	direct (N × H)
7. C₃²×C₇×C₈	C₂²	direct (N × H)
9. C₂×C₃×C₇×C₈	C₂×C₃	direct (N × H)
10. C₂²×C₃²×C₈	C₇	direct (N × H)
11. C₂²×C₃²×C₇	C₈	direct (N × H)
15. C₂²×C₇×C₈	C₃²	direct (N × H)
19. C₃×C₇×C₈	C₂²×C₃	direct (N × H)
21. C₂×C₃²×C₈	C₂×C₇	direct (N × H)
22. C₂×C₃²×C₇	C₂×C₈	direct (N × H)
25. C₂×C₇×C₈	C₂×C₃²	direct (N × H)
26. C₂²×C₃×C₈	C₃×C₇	direct (N × H)
27. C₂²×C₃×C₇	C₃×C₈	direct (N × H)
34. C₃²×C₈	C₂²×C₇	direct (N × H)
35. C₃²×C₇	C₂²×C₈	direct (N × H)
39. C₇×C₈	C₂²×C₃²	direct (N × H)
41. C₂×C₃×C₈	C₂×C₃×C₇	direct (N × H)
42. C₂×C₃×C₇	C₂×C₃×C₈	direct (N × H)
44. C₂²×C₃²	C₇×C₈	direct (N × H)
48. C₂²×C₈	C₃²×C₇	direct (N × H)
49. C₂²×C₇	C₃²×C₈	direct (N × H)
56. C₃×C₈	C₂²×C₃×C₇	direct (N × H)
57. C₃×C₇	C₂²×C₃×C₈	direct (N × H)
58. C₂×C₃²	C₂×C₇×C₈	direct (N × H)
61. C₂×C₈	C₂×C₃²×C₇	direct (N × H)
62. C₂×C₇	C₂×C₃²×C₈	direct (N × H)
64. C₂²×C₃	C₃×C₇×C₈	direct (N × H)
68. C₃²	C₂²×C₇×C₈	direct (N × H)
72. C₈	C₂²×C₃²×C₇	direct (N × H)
73. C₇	C₂²×C₃²×C₈	direct (N × H)
74. C₂×C₃	C₂×C₃×C₇×C₈	direct (N × H)
76. C₂²	C₃²×C₇×C₈	direct (N × H)
80. C₃	C₂²×C₃×C₇×C₈	direct (N × H)
81. C₂	C₂×C₃²×C₇×C₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃²×C₄×C₇	2
C₂×C₃²×C₇×C₈	2
C₂²×C₃×C₇×C₈	3
C₂²×C₃²×C₈	7

Subgroups — C22×C32×C7×C8 #72182003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃²×C₇×C₈ #72182003