Subgroups — C₂×C₃³×C₄²×C₇ #72185232

Order 6048.

3024 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³×C₄×C₇	C₂	1	3	yes	maximal
2	C₃³×C₄²×C₇	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₃²×C₄²×C₇	C₃	1	13	yes	maximal, direct product factor
4	C₂³×C₃³×C₇	C₂²	1	1	yes	characteristic
5	C₂×C₃³×C₄×C₇	C₄	1	12	yes	direct product factor
6	C₂×C₃³×C₄×C₇	C₂²	1	6	yes	—
7	C₂²×C₃²×C₄×C₇	C₂×C₃	1	39	yes	—
8	C₃²×C₄²×C₇	C₂×C₃	1	52	yes	direct product factor
9	C₂×C₃³×C₄²	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂²×C₃³×C₇	C₂×C₄	1	6	yes	—
11	C₂²×C₃³×C₇	C₂³	1	1	yes	characteristic
12	C₃³×C₄×C₇	C₂×C₄	1	12	yes	direct product factor
13	C₂×C₃×C₄²×C₇	C₃²	1	13	yes	direct product factor
14	C₂³×C₃²×C₇	C₂²×C₃	1	13	yes	—
15	C₂×C₃²×C₄×C₇	C₃×C₄	1	156	yes	direct product factor
16	C₂×C₃²×C₄×C₇	C₂²×C₃	1	78	yes	—
17	C₂²×C₃³×C₄	C₂×C₇	1	3	yes	—
18	C₃³×C₄²	C₂×C₇	1	4	yes	direct product factor
19	C₂×C₃³×C₇	C₄²	1	4	yes	direct product factor
20	C₂×C₃³×C₇	C₂²×C₄	1	3	yes	—
21	C₂²×C₃×C₄×C₇	C₂×C₃²	1	39	yes	—
22	C₃×C₄²×C₇	C₂×C₃²	1	52	yes	direct product factor
23	C₂×C₃²×C₄²	C₃×C₇	1	13	yes	direct product factor
24	C₂²×C₃²×C₇	C₂×C₃×C₄	1	78	yes	—
25	C₂²×C₃²×C₇	C₂³×C₃	1	13	yes	—
26	C₃²×C₄×C₇	C₂×C₃×C₄	1	156	yes	direct product factor
27	C₂×C₄²×C₇	C₃³	1	1	yes	characteristic, direct product factor, Hall
28	C₂³×C₃³	C₂²×C₇	1	1	yes	characteristic
29	C₂×C₃³×C₄	C₄×C₇	1	12	yes	direct product factor
30	C₂×C₃³×C₄	C₂²×C₇	1	6	yes	—
31	C₃³×C₇	C₂×C₄²	1	1	yes	characteristic, direct product factor, Hall
32	C₂³×C₃×C₇	C₂²×C₃²	1	13	yes	—
33	C₂×C₃×C₄×C₇	C₃²×C₄	1	156	yes	direct product factor
34	C₂×C₃×C₄×C₇	C₂²×C₃²	1	78	yes	—
35	C₂²×C₃²×C₄	C₂×C₃×C₇	1	39	yes	—
36	C₃²×C₄²	C₂×C₃×C₇	1	52	yes	direct product factor
37	C₂×C₃²×C₇	C₃×C₄²	1	52	yes	direct product factor
38	C₂×C₃²×C₇	C₂²×C₃×C₄	1	39	yes	—
39	C₂²×C₄×C₇	C₂×C₃³	1	3	yes	—
40	C₄²×C₇	C₂×C₃³	1	4	yes	direct product factor
41	C₂²×C₃³	C₂×C₄×C₇	1	6	yes	—
42	C₂²×C₃³	C₂³×C₇	1	1	yes	characteristic
43	C₃³×C₄	C₂×C₄×C₇	1	12	yes	direct product factor
44	C₂×C₃×C₄²	C₃²×C₇	1	13	yes	direct product factor
45	C₂²×C₃×C₇	C₂×C₃²×C₄	1	78	yes	—
46	C₂²×C₃×C₇	C₂³×C₃²	1	13	yes	—
47	C₃×C₄×C₇	C₂×C₃²×C₄	1	156	yes	direct product factor
48	C₂³×C₃²	C₂²×C₃×C₇	1	13	yes	—
49	C₂×C₃²×C₄	C₃×C₄×C₇	1	156	yes	direct product factor
50	C₂×C₃²×C₄	C₂²×C₃×C₇	1	78	yes	—
51	C₃²×C₇	C₂×C₃×C₄²	1	13	yes	direct product factor
52	C₂³×C₇	C₂²×C₃³	1	1	yes	characteristic
53	C₂×C₄×C₇	C₃³×C₄	1	12	yes	direct product factor
54	C₂×C₄×C₇	C₂²×C₃³	1	6	yes	—
55	C₂×C₃³	C₄²×C₇	1	4	yes	direct product factor
56	C₂×C₃³	C₂²×C₄×C₇	1	3	yes	—
57	C₂²×C₃×C₄	C₂×C₃²×C₇	1	39	yes	—
58	C₃×C₄²	C₂×C₃²×C₇	1	52	yes	direct product factor
59	C₂×C₃×C₇	C₃²×C₄²	1	52	yes	direct product factor
60	C₂×C₃×C₇	C₂²×C₃²×C₄	1	39	yes	—
61	C₂²×C₃²	C₂×C₃×C₄×C₇	1	78	yes	—
62	C₂²×C₃²	C₂³×C₃×C₇	1	13	yes	—
63	C₃²×C₄	C₂×C₃×C₄×C₇	1	156	yes	direct product factor
64	C₂×C₄²	C₃³×C₇	1	1	yes	characteristic, direct product factor, Sylow
65	C₂²×C₇	C₂×C₃³×C₄	1	6	yes	—
66	C₂²×C₇	C₂³×C₃³	1	1	yes	characteristic
67	C₄×C₇	C₂×C₃³×C₄	1	12	yes	direct product factor
68	C₃³	C₂×C₄²×C₇	1	1	yes	characteristic, direct product factor, Sylow
69	C₂³×C₃	C₂²×C₃²×C₇	1	13	yes	—
70	C₂×C₃×C₄	C₃²×C₄×C₇	1	156	yes	direct product factor
71	C₂×C₃×C₄	C₂²×C₃²×C₇	1	78	yes	—
72	C₃×C₇	C₂×C₃²×C₄²	1	13	yes	direct product factor
73	C₂×C₃²	C₃×C₄²×C₇	1	52	yes	direct product factor
74	C₂×C₃²	C₂²×C₃×C₄×C₇	1	39	yes	—
75	C₂²×C₄	C₂×C₃³×C₇	1	3	yes	—
76	C₄²	C₂×C₃³×C₇	1	4	yes	direct product factor
77	C₂×C₇	C₃³×C₄²	1	4	yes	direct product factor
78	C₂×C₇	C₂²×C₃³×C₄	1	3	yes	—
79	C₂²×C₃	C₂×C₃²×C₄×C₇	1	78	yes	—
80	C₂²×C₃	C₂³×C₃²×C₇	1	13	yes	—
81	C₃×C₄	C₂×C₃²×C₄×C₇	1	156	yes	direct product factor
82	C₃²	C₂×C₃×C₄²×C₇	1	13	yes	direct product factor
83	C₂³	C₂²×C₃³×C₇	1	1	yes	characteristic
84	C₂×C₄	C₃³×C₄×C₇	1	12	yes	direct product factor
85	C₂×C₄	C₂²×C₃³×C₇	1	6	yes	—
86	C₇	C₂×C₃³×C₄²	1	1	yes	characteristic, direct product factor, Sylow
87	C₂×C₃	C₃²×C₄²×C₇	1	52	yes	direct product factor
88	C₂×C₃	C₂²×C₃²×C₄×C₇	1	39	yes	—
89	C₂²	C₂×C₃³×C₄×C₇	1	6	yes	—
90	C₂²	C₂³×C₃³×C₇	1	1	yes	characteristic
91	C₄	C₂×C₃³×C₄×C₇	1	12	yes	direct product factor
92	C₃	C₂×C₃²×C₄²×C₇	1	13	yes	direct product factor
93	C₂	C₃³×C₄²×C₇	1	4	yes	direct product factor
94	C₂	C₂²×C₃³×C₄×C₇	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²	C₂³×C₃³×C₇
socle, prime radical	C₂³×C₃³×C₇	C₂²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃³×C₄²×C₇	C₂	direct (N × H)
3. C₂×C₃²×C₄²×C₇	C₃	direct (N × H)
5. C₂×C₃³×C₄×C₇	C₄	direct (N × H)
8. C₃²×C₄²×C₇	C₂×C₃	direct (N × H)
9. C₂×C₃³×C₄²	C₇	direct (N × H)
12. C₃³×C₄×C₇	C₂×C₄	direct (N × H)
13. C₂×C₃×C₄²×C₇	C₃²	direct (N × H)
15. C₂×C₃²×C₄×C₇	C₃×C₄	direct (N × H)
18. C₃³×C₄²	C₂×C₇	direct (N × H)
19. C₂×C₃³×C₇	C₄²	direct (N × H)
22. C₃×C₄²×C₇	C₂×C₃²	direct (N × H)
23. C₂×C₃²×C₄²	C₃×C₇	direct (N × H)
26. C₃²×C₄×C₇	C₂×C₃×C₄	direct (N × H)
27. C₂×C₄²×C₇	C₃³	direct (N × H)
29. C₂×C₃³×C₄	C₄×C₇	direct (N × H)
31. C₃³×C₇	C₂×C₄²	direct (N × H)
33. C₂×C₃×C₄×C₇	C₃²×C₄	direct (N × H)
36. C₃²×C₄²	C₂×C₃×C₇	direct (N × H)
37. C₂×C₃²×C₇	C₃×C₄²	direct (N × H)
40. C₄²×C₇	C₂×C₃³	direct (N × H)
43. C₃³×C₄	C₂×C₄×C₇	direct (N × H)
44. C₂×C₃×C₄²	C₃²×C₇	direct (N × H)
47. C₃×C₄×C₇	C₂×C₃²×C₄	direct (N × H)
49. C₂×C₃²×C₄	C₃×C₄×C₇	direct (N × H)
51. C₃²×C₇	C₂×C₃×C₄²	direct (N × H)
53. C₂×C₄×C₇	C₃³×C₄	direct (N × H)
55. C₂×C₃³	C₄²×C₇	direct (N × H)
58. C₃×C₄²	C₂×C₃²×C₇	direct (N × H)
59. C₂×C₃×C₇	C₃²×C₄²	direct (N × H)
63. C₃²×C₄	C₂×C₃×C₄×C₇	direct (N × H)
64. C₂×C₄²	C₃³×C₇	direct (N × H)
67. C₄×C₇	C₂×C₃³×C₄	direct (N × H)
68. C₃³	C₂×C₄²×C₇	direct (N × H)
70. C₂×C₃×C₄	C₃²×C₄×C₇	direct (N × H)
72. C₃×C₇	C₂×C₃²×C₄²	direct (N × H)
73. C₂×C₃²	C₃×C₄²×C₇	direct (N × H)
76. C₄²	C₂×C₃³×C₇	direct (N × H)
77. C₂×C₇	C₃³×C₄²	direct (N × H)
81. C₃×C₄	C₂×C₃²×C₄×C₇	direct (N × H)
82. C₃²	C₂×C₃×C₄²×C₇	direct (N × H)
84. C₂×C₄	C₃³×C₄×C₇	direct (N × H)
86. C₇	C₂×C₃³×C₄²	direct (N × H)
87. C₂×C₃	C₃²×C₄²×C₇	direct (N × H)
91. C₄	C₂×C₃³×C₄×C₇	direct (N × H)
92. C₃	C₂×C₃²×C₄²×C₇	direct (N × H)
93. C₂	C₃³×C₄²×C₇	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×C₃³×C₄×C₇	2
C₃³×C₄²×C₇	2
C₂×C₃²×C₄²×C₇	3
C₂×C₃³×C₄²	7

Subgroups — C2×C33×C42×C7 #72185232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃³×C₄²×C₇ #72185232