Subgroups — C₂²×C₄²×C₇²×C₉ #72205239

Order 28224.

7470 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×C₇²×C₉	C₂	1	3	yes	maximal
2	C₂×C₄²×C₇²×C₉	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₃×C₄²×C₇²	C₃	1	1	yes	maximal, characteristic
4	C₂⁴×C₇²×C₉	C₂²	1	1	yes	characteristic
5	C₂²×C₄×C₇²×C₉	C₄	1	24	yes	direct product factor
6	C₂²×C₄×C₇²×C₉	C₂²	1	18	yes	—
7	C₄²×C₇²×C₉	C₂²	1	16	yes	direct product factor
8	C₂³×C₃×C₄×C₇²	C₂×C₃	1	3	yes	—
9	C₂×C₃×C₄²×C₇²	C₂×C₃	1	12	yes	—
10	C₂²×C₄²×C₇×C₉	C₇	1	8	yes	maximal, direct product factor
11	C₂³×C₇²×C₉	C₂×C₄	1	12	yes	—
12	C₂³×C₇²×C₉	C₂³	1	3	yes	—
13	C₂×C₄×C₇²×C₉	C₂×C₄	1	72	yes	direct product factor
14	C₂×C₄×C₇²×C₉	C₂³	1	12	yes	—
15	C₂²×C₄²×C₇²	C₉	1	1	yes	characteristic, direct product factor, Hall
16	C₂⁴×C₃×C₇²	C₂²×C₃	1	1	yes	characteristic
17	C₂²×C₃×C₄×C₇²	C₃×C₄	1	24	yes	—
18	C₂²×C₃×C₄×C₇²	C₂²×C₃	1	18	yes	—
19	C₃×C₄²×C₇²	C₂²×C₃	1	16	yes	—
20	C₂³×C₄×C₇×C₉	C₂×C₇	1	24	yes	—
21	C₂×C₄²×C₇×C₉	C₂×C₇	1	96	yes	direct product factor
22	C₂²×C₇²×C₉	C₄²	1	16	yes	direct product factor
23	C₂²×C₇²×C₉	C₂²×C₄	1	18	yes	—
24	C₂²×C₇²×C₉	C₂⁴	1	1	yes	characteristic
25	C₄×C₇²×C₉	C₂²×C₄	1	24	yes	direct product factor
26	C₂³×C₄×C₇²	C₂×C₉	1	3	yes	—
27	C₂×C₄²×C₇²	C₂×C₉	1	12	yes	direct product factor
28	C₂²×C₃×C₄²×C₇	C₃×C₇	1	8	yes	—
29	C₂³×C₃×C₇²	C₂×C₃×C₄	1	12	yes	—
30	C₂³×C₃×C₇²	C₂³×C₃	1	3	yes	—
31	C₂×C₃×C₄×C₇²	C₂×C₃×C₄	1	72	yes	—
32	C₂×C₃×C₄×C₇²	C₂³×C₃	1	12	yes	—
33	C₂⁴×C₇×C₉	C₂²×C₇	1	8	yes	—
34	C₂²×C₄×C₇×C₉	C₄×C₇	1	192	yes	direct product factor
35	C₂²×C₄×C₇×C₉	C₂²×C₇	1	144	yes	—
36	C₄²×C₇×C₉	C₂²×C₇	1	128	yes	direct product factor
37	C₂×C₇²×C₉	C₂×C₄²	1	12	yes	direct product factor
38	C₂×C₇²×C₉	C₂³×C₄	1	3	yes	—
39	C₂⁴×C₇²	C₂²×C₉	1	1	yes	characteristic
40	C₂²×C₄×C₇²	C₄×C₉	1	24	yes	direct product factor
41	C₂²×C₄×C₇²	C₂²×C₉	1	18	yes	—
42	C₄²×C₇²	C₂²×C₉	1	16	yes	direct product factor
43	C₂³×C₃×C₄×C₇	C₂×C₃×C₇	1	24	yes	—
44	C₂×C₃×C₄²×C₇	C₂×C₃×C₇	1	96	yes	—
45	C₂²×C₃×C₇²	C₃×C₄²	1	16	yes	—
46	C₂²×C₃×C₇²	C₂²×C₃×C₄	1	18	yes	—
47	C₂²×C₃×C₇²	C₂⁴×C₃	1	1	yes	characteristic
48	C₃×C₄×C₇²	C₂²×C₃×C₄	1	24	yes	—
49	C₂²×C₄²×C₉	C₇²	1	1	yes	characteristic, direct product factor, Hall
50	C₂³×C₇×C₉	C₂×C₄×C₇	1	96	yes	—
51	C₂³×C₇×C₉	C₂³×C₇	1	24	yes	—
52	C₂×C₄×C₇×C₉	C₂×C₄×C₇	1	576	yes	direct product factor
53	C₂×C₄×C₇×C₉	C₂³×C₇	1	96	yes	—
54	C₂²×C₄²×C₇	C₇×C₉	1	8	yes	direct product factor
55	C₇²×C₉	C₂²×C₄²	1	1	yes	characteristic, direct product factor, Hall
56	C₂³×C₇²	C₂×C₄×C₉	1	12	yes	—
57	C₂³×C₇²	C₂³×C₉	1	3	yes	—
58	C₂×C₄×C₇²	C₂×C₄×C₉	1	72	yes	direct product factor
59	C₂×C₄×C₇²	C₂³×C₉	1	12	yes	—
60	C₂⁴×C₃×C₇	C₂²×C₃×C₇	1	8	yes	—
61	C₂²×C₃×C₄×C₇	C₃×C₄×C₇	1	192	yes	—
62	C₂²×C₃×C₄×C₇	C₂²×C₃×C₇	1	144	yes	—
63	C₃×C₄²×C₇	C₂²×C₃×C₇	1	128	yes	—
64	C₂×C₃×C₇²	C₂×C₃×C₄²	1	12	yes	—
65	C₂×C₃×C₇²	C₂³×C₃×C₄	1	3	yes	—
66	C₂³×C₄×C₉	C₂×C₇²	1	3	yes	—
67	C₂×C₄²×C₉	C₂×C₇²	1	12	yes	direct product factor
68	C₂²×C₇×C₉	C₄²×C₇	1	128	yes	direct product factor
69	C₂²×C₇×C₉	C₂²×C₄×C₇	1	144	yes	—
70	C₂²×C₇×C₉	C₂⁴×C₇	1	8	yes	—
71	C₄×C₇×C₉	C₂²×C₄×C₇	1	192	yes	direct product factor
72	C₂³×C₄×C₇	C₂×C₇×C₉	1	24	yes	—
73	C₂×C₄²×C₇	C₂×C₇×C₉	1	96	yes	direct product factor
74	C₂²×C₇²	C₄²×C₉	1	16	yes	direct product factor
75	C₂²×C₇²	C₂²×C₄×C₉	1	18	yes	—
76	C₂²×C₇²	C₂⁴×C₉	1	1	yes	characteristic
77	C₄×C₇²	C₂²×C₄×C₉	1	24	yes	direct product factor
78	C₂²×C₃×C₄²	C₃×C₇²	1	1	yes	characteristic
79	C₂³×C₃×C₇	C₂×C₃×C₄×C₇	1	96	yes	—
80	C₂³×C₃×C₇	C₂³×C₃×C₇	1	24	yes	—
81	C₂×C₃×C₄×C₇	C₂×C₃×C₄×C₇	1	576	yes	—
82	C₂×C₃×C₄×C₇	C₂³×C₃×C₇	1	96	yes	—
83	C₃×C₇²	C₂²×C₃×C₄²	1	1	yes	characteristic
84	C₂⁴×C₉	C₂²×C₇²	1	1	yes	characteristic
85	C₂²×C₄×C₉	C₄×C₇²	1	24	yes	direct product factor
86	C₂²×C₄×C₉	C₂²×C₇²	1	18	yes	—
87	C₄²×C₉	C₂²×C₇²	1	16	yes	direct product factor
88	C₂×C₇×C₉	C₂×C₄²×C₇	1	96	yes	direct product factor
89	C₂×C₇×C₉	C₂³×C₄×C₇	1	24	yes	—
90	C₂⁴×C₇	C₂²×C₇×C₉	1	8	yes	—
91	C₂²×C₄×C₇	C₄×C₇×C₉	1	192	yes	direct product factor
92	C₂²×C₄×C₇	C₂²×C₇×C₉	1	144	yes	—
93	C₄²×C₇	C₂²×C₇×C₉	1	128	yes	direct product factor
94	C₂×C₇²	C₂×C₄²×C₉	1	12	yes	direct product factor
95	C₂×C₇²	C₂³×C₄×C₉	1	3	yes	—
96	C₂³×C₃×C₄	C₂×C₃×C₇²	1	3	yes	—
97	C₂×C₃×C₄²	C₂×C₃×C₇²	1	12	yes	—
98	C₂²×C₃×C₇	C₃×C₄²×C₇	1	128	yes	—
99	C₂²×C₃×C₇	C₂²×C₃×C₄×C₇	1	144	yes	—
100	C₂²×C₃×C₇	C₂⁴×C₃×C₇	1	8	yes	—
101	C₃×C₄×C₇	C₂²×C₃×C₄×C₇	1	192	yes	—
102	C₂³×C₉	C₂×C₄×C₇²	1	12	yes	—
103	C₂³×C₉	C₂³×C₇²	1	3	yes	—
104	C₂×C₄×C₉	C₂×C₄×C₇²	1	72	yes	direct product factor
105	C₂×C₄×C₉	C₂³×C₇²	1	12	yes	—
106	C₂²×C₄²	C₇²×C₉	1	1	yes	characteristic, direct product factor, Sylow
107	C₇×C₉	C₂²×C₄²×C₇	1	8	yes	direct product factor
108	C₂³×C₇	C₂×C₄×C₇×C₉	1	96	yes	—
109	C₂³×C₇	C₂³×C₇×C₉	1	24	yes	—
110	C₂×C₄×C₇	C₂×C₄×C₇×C₉	1	576	yes	direct product factor
111	C₂×C₄×C₇	C₂³×C₇×C₉	1	96	yes	—
112	C₇²	C₂²×C₄²×C₉	1	1	yes	characteristic, direct product factor, Sylow
113	C₂⁴×C₃	C₂²×C₃×C₇²	1	1	yes	characteristic
114	C₂²×C₃×C₄	C₃×C₄×C₇²	1	24	yes	—
115	C₂²×C₃×C₄	C₂²×C₃×C₇²	1	18	yes	—
116	C₃×C₄²	C₂²×C₃×C₇²	1	16	yes	—
117	C₂×C₃×C₇	C₂×C₃×C₄²×C₇	1	96	yes	—
118	C₂×C₃×C₇	C₂³×C₃×C₄×C₇	1	24	yes	—
119	C₂²×C₉	C₄²×C₇²	1	16	yes	direct product factor
120	C₂²×C₉	C₂²×C₄×C₇²	1	18	yes	—
121	C₂²×C₉	C₂⁴×C₇²	1	1	yes	characteristic
122	C₄×C₉	C₂²×C₄×C₇²	1	24	yes	direct product factor
123	C₂³×C₄	C₂×C₇²×C₉	1	3	yes	—
124	C₂×C₄²	C₂×C₇²×C₉	1	12	yes	direct product factor
125	C₂²×C₇	C₄²×C₇×C₉	1	128	yes	direct product factor
126	C₂²×C₇	C₂²×C₄×C₇×C₉	1	144	yes	—
127	C₂²×C₇	C₂⁴×C₇×C₉	1	8	yes	—
128	C₄×C₇	C₂²×C₄×C₇×C₉	1	192	yes	direct product factor
129	C₂³×C₃	C₂×C₃×C₄×C₇²	1	12	yes	—
130	C₂³×C₃	C₂³×C₃×C₇²	1	3	yes	—
131	C₂×C₃×C₄	C₂×C₃×C₄×C₇²	1	72	yes	—
132	C₂×C₃×C₄	C₂³×C₃×C₇²	1	12	yes	—
133	C₃×C₇	C₂²×C₃×C₄²×C₇	1	8	yes	—
134	C₂×C₉	C₂×C₄²×C₇²	1	12	yes	direct product factor
135	C₂×C₉	C₂³×C₄×C₇²	1	3	yes	—
136	C₂⁴	C₂²×C₇²×C₉	1	1	yes	characteristic
137	C₂²×C₄	C₄×C₇²×C₉	1	24	yes	direct product factor
138	C₂²×C₄	C₂²×C₇²×C₉	1	18	yes	—
139	C₄²	C₂²×C₇²×C₉	1	16	yes	direct product factor
140	C₂×C₇	C₂×C₄²×C₇×C₉	1	96	yes	direct product factor
141	C₂×C₇	C₂³×C₄×C₇×C₉	1	24	yes	—
142	C₂²×C₃	C₃×C₄²×C₇²	1	16	yes	—
143	C₂²×C₃	C₂²×C₃×C₄×C₇²	1	18	yes	—
144	C₂²×C₃	C₂⁴×C₃×C₇²	1	1	yes	characteristic
145	C₃×C₄	C₂²×C₃×C₄×C₇²	1	24	yes	—
146	C₉	C₂²×C₄²×C₇²	1	1	yes	characteristic, direct product factor, Sylow
147	C₂³	C₂×C₄×C₇²×C₉	1	12	yes	—
148	C₂³	C₂³×C₇²×C₉	1	3	yes	—
149	C₂×C₄	C₂×C₄×C₇²×C₉	1	72	yes	direct product factor
150	C₂×C₄	C₂³×C₇²×C₉	1	12	yes	—
151	C₇	C₂²×C₄²×C₇×C₉	1	8	yes	direct product factor
152	C₂×C₃	C₂×C₃×C₄²×C₇²	1	12	yes	—
153	C₂×C₃	C₂³×C₃×C₄×C₇²	1	3	yes	—
154	C₂²	C₄²×C₇²×C₉	1	16	yes	direct product factor
155	C₂²	C₂²×C₄×C₇²×C₉	1	18	yes	—
156	C₂²	C₂⁴×C₇²×C₉	1	1	yes	characteristic
157	C₄	C₂²×C₄×C₇²×C₉	1	24	yes	direct product factor
158	C₃	C₂²×C₃×C₄²×C₇²	1	1	yes	characteristic
159	C₂	C₂×C₄²×C₇²×C₉	1	12	yes	direct product factor
160	C₂	C₂³×C₄×C₇²×C₉	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₃	C₂⁴×C₃×C₇²
socle, prime radical	C₂⁴×C₃×C₇²	C₂²×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×C₄²×C₇²×C₉	C₂	direct (N × H)
5. C₂²×C₄×C₇²×C₉	C₄	direct (N × H)
7. C₄²×C₇²×C₉	C₂²	direct (N × H)
10. C₂²×C₄²×C₇×C₉	C₇	direct (N × H)
13. C₂×C₄×C₇²×C₉	C₂×C₄	direct (N × H)
15. C₂²×C₄²×C₇²	C₉	direct (N × H)
21. C₂×C₄²×C₇×C₉	C₂×C₇	direct (N × H)
22. C₂²×C₇²×C₉	C₄²	direct (N × H)
25. C₄×C₇²×C₉	C₂²×C₄	direct (N × H)
27. C₂×C₄²×C₇²	C₂×C₉	direct (N × H)
34. C₂²×C₄×C₇×C₉	C₄×C₇	direct (N × H)
36. C₄²×C₇×C₉	C₂²×C₇	direct (N × H)
37. C₂×C₇²×C₉	C₂×C₄²	direct (N × H)
40. C₂²×C₄×C₇²	C₄×C₉	direct (N × H)
42. C₄²×C₇²	C₂²×C₉	direct (N × H)
49. C₂²×C₄²×C₉	C₇²	direct (N × H)
52. C₂×C₄×C₇×C₉	C₂×C₄×C₇	direct (N × H)
54. C₂²×C₄²×C₇	C₇×C₉	direct (N × H)
55. C₇²×C₉	C₂²×C₄²	direct (N × H)
58. C₂×C₄×C₇²	C₂×C₄×C₉	direct (N × H)
67. C₂×C₄²×C₉	C₂×C₇²	direct (N × H)
68. C₂²×C₇×C₉	C₄²×C₇	direct (N × H)
71. C₄×C₇×C₉	C₂²×C₄×C₇	direct (N × H)
73. C₂×C₄²×C₇	C₂×C₇×C₉	direct (N × H)
74. C₂²×C₇²	C₄²×C₉	direct (N × H)
77. C₄×C₇²	C₂²×C₄×C₉	direct (N × H)
85. C₂²×C₄×C₉	C₄×C₇²	direct (N × H)
87. C₄²×C₉	C₂²×C₇²	direct (N × H)
88. C₂×C₇×C₉	C₂×C₄²×C₇	direct (N × H)
91. C₂²×C₄×C₇	C₄×C₇×C₉	direct (N × H)
93. C₄²×C₇	C₂²×C₇×C₉	direct (N × H)
94. C₂×C₇²	C₂×C₄²×C₉	direct (N × H)
104. C₂×C₄×C₉	C₂×C₄×C₇²	direct (N × H)
106. C₂²×C₄²	C₇²×C₉	direct (N × H)
107. C₇×C₉	C₂²×C₄²×C₇	direct (N × H)
110. C₂×C₄×C₇	C₂×C₄×C₇×C₉	direct (N × H)
112. C₇²	C₂²×C₄²×C₉	direct (N × H)
119. C₂²×C₉	C₄²×C₇²	direct (N × H)
122. C₄×C₉	C₂²×C₄×C₇²	direct (N × H)
124. C₂×C₄²	C₂×C₇²×C₉	direct (N × H)
125. C₂²×C₇	C₄²×C₇×C₉	direct (N × H)
128. C₄×C₇	C₂²×C₄×C₇×C₉	direct (N × H)
134. C₂×C₉	C₂×C₄²×C₇²	direct (N × H)
137. C₂²×C₄	C₄×C₇²×C₉	direct (N × H)
139. C₄²	C₂²×C₇²×C₉	direct (N × H)
140. C₂×C₇	C₂×C₄²×C₇×C₉	direct (N × H)
146. C₉	C₂²×C₄²×C₇²	direct (N × H)
149. C₂×C₄	C₂×C₄×C₇²×C₉	direct (N × H)
151. C₇	C₂²×C₄²×C₇×C₉	direct (N × H)
154. C₂²	C₄²×C₇²×C₉	direct (N × H)
157. C₄	C₂²×C₄×C₇²×C₉	direct (N × H)
159. C₂	C₂×C₄²×C₇²×C₉	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×C₄×C₇²×C₉	2
C₂×C₄²×C₇²×C₉	2
C₂²×C₃×C₄²×C₇²	3
C₂²×C₄²×C₇×C₉	7

Subgroups — C22×C42×C72×C9 #72205239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄²×C₇²×C₉ #72205239