Subgroups — C₂²×C₃×C₄×C₉×C₂₉ #72190874

Order 12528.

540 conjugacy classes of subgroups.

The subgroup lattice has 108 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₉×C₂₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×C₉×C₂₉	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃²×C₄×C₂₉	C₃	1	1	yes	maximal, characteristic
4	C₂²×C₄×C₉×C₂₉	C₃	1	3	yes	maximal, direct product factor
5	C₂²×C₃×C₉×C₂₉	C₄	1	4	yes	direct product factor
6	C₂²×C₃×C₉×C₂₉	C₂²	1	3	yes	—
7	C₃×C₄×C₉×C₂₉	C₂²	1	4	yes	direct product factor
8	C₂³×C₃²×C₂₉	C₂×C₃	1	1	yes	characteristic
9	C₂³×C₉×C₂₉	C₂×C₃	1	3	yes	—
10	C₂×C₃²×C₄×C₂₉	C₂×C₃	1	6	yes	—
11	C₂×C₄×C₉×C₂₉	C₂×C₃	1	18	yes	direct product factor
12	C₂×C₃×C₉×C₂₉	C₂×C₄	1	6	yes	direct product factor
13	C₂×C₃×C₉×C₂₉	C₂³	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₂₉	C₉	1	3	yes	direct product factor
15	C₂²×C₃×C₄×C₂₉	C₃²	1	1	yes	characteristic
16	C₂²×C₃²×C₂₉	C₃×C₄	1	4	yes	—
17	C₂²×C₃²×C₂₉	C₂²×C₃	1	3	yes	—
18	C₃²×C₄×C₂₉	C₂²×C₃	1	4	yes	—
19	C₂²×C₉×C₂₉	C₃×C₄	1	12	yes	direct product factor
20	C₂²×C₉×C₂₉	C₂²×C₃	1	9	yes	—
21	C₄×C₉×C₂₉	C₂²×C₃	1	12	yes	direct product factor
22	C₃×C₉×C₂₉	C₂²×C₄	1	1	yes	characteristic, direct product factor, Hall
23	C₂³×C₃×C₂₉	C₂×C₉	1	3	yes	—
24	C₂³×C₃×C₂₉	C₂×C₃²	1	1	yes	characteristic
25	C₂×C₃×C₄×C₂₉	C₂×C₉	1	18	yes	direct product factor
26	C₂×C₃×C₄×C₂₉	C₂×C₃²	1	6	yes	—
27	C₂×C₃²×C₂₉	C₂×C₃×C₄	1	6	yes	—
28	C₂×C₃²×C₂₉	C₂³×C₃	1	1	yes	characteristic
29	C₂×C₉×C₂₉	C₂×C₃×C₄	1	18	yes	direct product factor
30	C₂×C₉×C₂₉	C₂³×C₃	1	3	yes	—
31	C₂²×C₄×C₂₉	C₃×C₉	1	1	yes	characteristic, direct product factor, Hall
32	C₂²×C₃×C₄×C₉	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Hall
33	C₂²×C₃×C₂₉	C₄×C₉	1	12	yes	direct product factor
34	C₂²×C₃×C₂₉	C₂²×C₉	1	9	yes	—
35	C₂²×C₃×C₂₉	C₃²×C₄	1	4	yes	—
36	C₂²×C₃×C₂₉	C₂²×C₃²	1	3	yes	—
37	C₃×C₄×C₂₉	C₂²×C₉	1	12	yes	direct product factor
38	C₃×C₄×C₂₉	C₂²×C₃²	1	4	yes	—
39	C₃²×C₂₉	C₂²×C₃×C₄	1	1	yes	characteristic
40	C₉×C₂₉	C₂²×C₃×C₄	1	3	yes	direct product factor
41	C₂³×C₂₉	C₂×C₃×C₉	1	1	yes	characteristic
42	C₂×C₄×C₂₉	C₂×C₃×C₉	1	6	yes	direct product factor
43	C₂³×C₃×C₉	C₂×C₂₉	1	1	yes	characteristic
44	C₂×C₃×C₄×C₉	C₂×C₂₉	1	6	yes	direct product factor
45	C₂×C₃×C₂₉	C₂×C₄×C₉	1	18	yes	direct product factor
46	C₂×C₃×C₂₉	C₂³×C₉	1	3	yes	—
47	C₂×C₃×C₂₉	C₂×C₃²×C₄	1	6	yes	—
48	C₂×C₃×C₂₉	C₂³×C₃²	1	1	yes	characteristic
49	C₂²×C₃²×C₄	C₃×C₂₉	1	1	yes	characteristic
50	C₂²×C₄×C₉	C₃×C₂₉	1	3	yes	direct product factor
51	C₂²×C₂₉	C₃×C₄×C₉	1	4	yes	direct product factor
52	C₂²×C₂₉	C₂²×C₃×C₉	1	3	yes	—
53	C₄×C₂₉	C₂²×C₃×C₉	1	4	yes	direct product factor
54	C₂²×C₃×C₉	C₄×C₂₉	1	4	yes	direct product factor
55	C₂²×C₃×C₉	C₂²×C₂₉	1	3	yes	—
56	C₃×C₄×C₉	C₂²×C₂₉	1	4	yes	direct product factor
57	C₃×C₂₉	C₂²×C₄×C₉	1	3	yes	direct product factor
58	C₃×C₂₉	C₂²×C₃²×C₄	1	1	yes	characteristic
59	C₂³×C₃²	C₂×C₃×C₂₉	1	1	yes	characteristic
60	C₂×C₃²×C₄	C₂×C₃×C₂₉	1	6	yes	—
61	C₂³×C₉	C₂×C₃×C₂₉	1	3	yes	—
62	C₂×C₄×C₉	C₂×C₃×C₂₉	1	18	yes	direct product factor
63	C₂×C₂₉	C₂×C₃×C₄×C₉	1	6	yes	direct product factor
64	C₂×C₂₉	C₂³×C₃×C₉	1	1	yes	characteristic
65	C₂×C₃×C₉	C₂×C₄×C₂₉	1	6	yes	direct product factor
66	C₂×C₃×C₉	C₂³×C₂₉	1	1	yes	characteristic
67	C₂²×C₃×C₄	C₉×C₂₉	1	3	yes	direct product factor
68	C₂²×C₃×C₄	C₃²×C₂₉	1	1	yes	characteristic
69	C₂²×C₃²	C₃×C₄×C₂₉	1	4	yes	—
70	C₂²×C₃²	C₂²×C₃×C₂₉	1	3	yes	—
71	C₃²×C₄	C₂²×C₃×C₂₉	1	4	yes	—
72	C₂²×C₉	C₃×C₄×C₂₉	1	12	yes	direct product factor
73	C₂²×C₉	C₂²×C₃×C₂₉	1	9	yes	—
74	C₄×C₉	C₂²×C₃×C₂₉	1	12	yes	direct product factor
75	C₂₉	C₂²×C₃×C₄×C₉	1	1	yes	characteristic, direct product factor, Sylow
76	C₃×C₉	C₂²×C₄×C₂₉	1	1	yes	characteristic, direct product factor, Sylow
77	C₂³×C₃	C₂×C₉×C₂₉	1	3	yes	—
78	C₂³×C₃	C₂×C₃²×C₂₉	1	1	yes	characteristic
79	C₂×C₃×C₄	C₂×C₉×C₂₉	1	18	yes	direct product factor
80	C₂×C₃×C₄	C₂×C₃²×C₂₉	1	6	yes	—
81	C₂×C₃²	C₂×C₃×C₄×C₂₉	1	6	yes	—
82	C₂×C₃²	C₂³×C₃×C₂₉	1	1	yes	characteristic
83	C₂×C₉	C₂×C₃×C₄×C₂₉	1	18	yes	direct product factor
84	C₂×C₉	C₂³×C₃×C₂₉	1	3	yes	—
85	C₂²×C₄	C₃×C₉×C₂₉	1	1	yes	characteristic, direct product factor, Sylow
86	C₂²×C₃	C₄×C₉×C₂₉	1	12	yes	direct product factor
87	C₂²×C₃	C₂²×C₉×C₂₉	1	9	yes	—
88	C₂²×C₃	C₃²×C₄×C₂₉	1	4	yes	—
89	C₂²×C₃	C₂²×C₃²×C₂₉	1	3	yes	—
90	C₃×C₄	C₂²×C₉×C₂₉	1	12	yes	direct product factor
91	C₃×C₄	C₂²×C₃²×C₂₉	1	4	yes	—
92	C₃²	C₂²×C₃×C₄×C₂₉	1	1	yes	characteristic
93	C₉	C₂²×C₃×C₄×C₂₉	1	3	yes	direct product factor
94	C₂³	C₂×C₃×C₉×C₂₉	1	1	yes	characteristic
95	C₂×C₄	C₂×C₃×C₉×C₂₉	1	6	yes	direct product factor
96	C₂×C₃	C₂×C₄×C₉×C₂₉	1	18	yes	direct product factor
97	C₂×C₃	C₂×C₃²×C₄×C₂₉	1	6	yes	—
98	C₂×C₃	C₂³×C₉×C₂₉	1	3	yes	—
99	C₂×C₃	C₂³×C₃²×C₂₉	1	1	yes	characteristic
100	C₂²	C₃×C₄×C₉×C₂₉	1	4	yes	direct product factor
101	C₂²	C₂²×C₃×C₉×C₂₉	1	3	yes	—
102	C₄	C₂²×C₃×C₉×C₂₉	1	4	yes	direct product factor
103	C₃	C₂²×C₄×C₉×C₂₉	1	3	yes	direct product factor
104	C₃	C₂²×C₃²×C₄×C₂₉	1	1	yes	characteristic
105	C₂	C₂×C₃×C₄×C₉×C₂₉	1	6	yes	direct product factor
106	C₂	C₂³×C₃×C₉×C₂₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₃	C₂³×C₃²×C₂₉
socle, prime radical	C₂³×C₃²×C₂₉	C₂×C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₄×C₉×C₂₉	C₂	direct (N × H)
4. C₂²×C₄×C₉×C₂₉	C₃	direct (N × H)
5. C₂²×C₃×C₉×C₂₉	C₄	direct (N × H)
7. C₃×C₄×C₉×C₂₉	C₂²	direct (N × H)
11. C₂×C₄×C₉×C₂₉	C₂×C₃	direct (N × H)
12. C₂×C₃×C₉×C₂₉	C₂×C₄	direct (N × H)
14. C₂²×C₃×C₄×C₂₉	C₉	direct (N × H)
19. C₂²×C₉×C₂₉	C₃×C₄	direct (N × H)
21. C₄×C₉×C₂₉	C₂²×C₃	direct (N × H)
22. C₃×C₉×C₂₉	C₂²×C₄	direct (N × H)
25. C₂×C₃×C₄×C₂₉	C₂×C₉	direct (N × H)
29. C₂×C₉×C₂₉	C₂×C₃×C₄	direct (N × H)
31. C₂²×C₄×C₂₉	C₃×C₉	direct (N × H)
32. C₂²×C₃×C₄×C₉	C₂₉	direct (N × H)
33. C₂²×C₃×C₂₉	C₄×C₉	direct (N × H)
37. C₃×C₄×C₂₉	C₂²×C₉	direct (N × H)
40. C₉×C₂₉	C₂²×C₃×C₄	direct (N × H)
42. C₂×C₄×C₂₉	C₂×C₃×C₉	direct (N × H)
44. C₂×C₃×C₄×C₉	C₂×C₂₉	direct (N × H)
45. C₂×C₃×C₂₉	C₂×C₄×C₉	direct (N × H)
50. C₂²×C₄×C₉	C₃×C₂₉	direct (N × H)
51. C₂²×C₂₉	C₃×C₄×C₉	direct (N × H)
53. C₄×C₂₉	C₂²×C₃×C₉	direct (N × H)
54. C₂²×C₃×C₉	C₄×C₂₉	direct (N × H)
56. C₃×C₄×C₉	C₂²×C₂₉	direct (N × H)
57. C₃×C₂₉	C₂²×C₄×C₉	direct (N × H)
62. C₂×C₄×C₉	C₂×C₃×C₂₉	direct (N × H)
63. C₂×C₂₉	C₂×C₃×C₄×C₉	direct (N × H)
65. C₂×C₃×C₉	C₂×C₄×C₂₉	direct (N × H)
67. C₂²×C₃×C₄	C₉×C₂₉	direct (N × H)
72. C₂²×C₉	C₃×C₄×C₂₉	direct (N × H)
74. C₄×C₉	C₂²×C₃×C₂₉	direct (N × H)
75. C₂₉	C₂²×C₃×C₄×C₉	direct (N × H)
76. C₃×C₉	C₂²×C₄×C₂₉	direct (N × H)
79. C₂×C₃×C₄	C₂×C₉×C₂₉	direct (N × H)
83. C₂×C₉	C₂×C₃×C₄×C₂₉	direct (N × H)
85. C₂²×C₄	C₃×C₉×C₂₉	direct (N × H)
86. C₂²×C₃	C₄×C₉×C₂₉	direct (N × H)
90. C₃×C₄	C₂²×C₉×C₂₉	direct (N × H)
93. C₉	C₂²×C₃×C₄×C₂₉	direct (N × H)
95. C₂×C₄	C₂×C₃×C₉×C₂₉	direct (N × H)
96. C₂×C₃	C₂×C₄×C₉×C₂₉	direct (N × H)
100. C₂²	C₃×C₄×C₉×C₂₉	direct (N × H)
102. C₄	C₂²×C₃×C₉×C₂₉	direct (N × H)
103. C₃	C₂²×C₄×C₉×C₂₉	direct (N × H)
105. C₂	C₂×C₃×C₄×C₉×C₂₉	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×C₉×C₂₉	2
C₂×C₃×C₄×C₉×C₂₉	2
C₂²×C₃²×C₄×C₂₉	3
C₂²×C₄×C₉×C₂₉	3
C₂²×C₃×C₄×C₉	29

Subgroups — C22×C3×C4×C9×C29 #72190874

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₄×C₉×C₂₉ #72190874