Group #22625786 — subgroups

Subgroups — #22625786

Order 9344.

400 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#10596833	C₂	1	2	yes	maximal, complemented
2	C₂×#14556648	C₂	1	2	yes	maximal
3	#17989453	C₂	1	2	yes	maximal, complemented
4	C₂×C₄²×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10596833	C₂²	1	4	yes	—
6	C₄⋊#10596833	—	2	4	no	—
7	#14556648	—	2	4	no	—
8	C₂²×C₄×D₇₃	C₄	1	1	yes	characteristic
9	C₂²×C₄×D₇₃	C₄	1	1	yes	characteristic
10	C₂²×C₄×D₇₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₇₃	C₄	1	1	yes	characteristic
12	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
14	C₄²×D₇₃	—	2	4	no	—
15	C₂×C₄²×C₇₃	C₄	1	1	yes	characteristic
16	C₂×#10596833	—	2	4	no	—
17	C₂×#10596833	—	2	4	no	—
18	C₂×#10596833	—	2	4	no	—
19	C₂³×D₇₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×Dic₇₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×Dic₇₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×Dic₇₃	C₂³	1	1	yes	characteristic
23	C₂×C₄×D₇₃	—	2	2	no	—
24	C₂×C₄×D₇₃	—	2	2	no	—
25	C₂×C₄×D₇₃	—	2	2	no	—
26	C₂×C₄×D₇₃	—	2	2	no	—
27	C₂×C₄×D₇₃	—	2	2	no	—
28	C₂×C₄×D₇₃	—	2	2	no	—
29	C₂×C₄×D₇₃	—	2	2	no	—
30	C₂×C₄×D₇₃	C₈	1	4	yes	complemented
31	C₂×C₄×D₇₃	D₄	1	2	yes	—
32	C₂×C₄×D₇₃	Q₈	1	2	yes	—
33	C₄×Dic₇₃	—	2	2	no	—
34	C₄×Dic₇₃	—	2	2	no	—
35	C₄×Dic₇₃	D₄	1	4	yes	—
36	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₇₃	—	2	2	no	—
40	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×D₇₃	—	2	1	no	—
42	C₂²×D₇₃	—	2	1	no	—
43	C₂²×D₇₃	—	2	1	no	—
44	C₂²×D₇₃	—	2	1	no	—
45	C₂²×D₇₃	C₄⋊C₄	1	2	yes	—
46	C₂²×D₇₃	C₂×C₈	1	2	yes	—
47	C₂²×D₇₃	M₁₆	1	2	yes	—
48	C₂×C₄×C₇₃	—	2	1	no	—
49	C₂×C₄×C₇₃	—	2	1	no	—
50	C₂×C₄×C₇₃	—	2	1	no	—
51	C₂×C₄×C₇₃	—	2	1	no	—
52	C₂×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
53	C₂×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
54	C₂×C₄×C₇₃	C₂×C₈	1	2	yes	—
55	C₂×C₄×C₇₃	M₁₆	1	2	yes	—
56	C₂×C₄×C₇₃	M₁₆	1	2	yes	—
57	C₂×Dic₇₃	—	2	1	no	—
58	C₂×Dic₇₃	—	2	1	no	—
59	C₂×Dic₇₃	—	2	1	no	—
60	C₂×Dic₇₃	—	2	1	no	—
61	C₂×Dic₇₃	C₂²⋊C₄	1	2	yes	—
62	C₂×Dic₇₃	C₂×C₈	1	2	yes	—
63	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
64	C₂×Dic₇₃	C₂×D₄	1	2	yes	—
65	C₂×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
66	C₂×Dic₇₃	#41	1	2	yes	—
67	C₄×D₇₃	—	2	8	no	—
68	C₄×D₇₃	—	2	8	no	—
69	C₄×D₇₃	—	2	4	no	—
70	C₄×D₇₃	—	2	4	no	—
71	#10596833	—	4	4	no	—
72	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₇₃	#117	1	1	yes	characteristic
75	C₂²×C₇₃	C₂²×C₈	1	1	yes	characteristic
76	C₂²×C₇₃	C₂×M₁₆	1	1	yes	characteristic
77	C₂²×C₇₃	C₂×M₁₆	1	1	yes	characteristic
78	C₂²×C₇₃	C₂×M₁₆	1	1	yes	characteristic
79	C₂×D₇₃	—	2	2	no	—
80	C₂×D₇₃	—	2	2	no	—
81	C₂×D₇₃	—	2	2	no	—
82	C₂×D₇₃	—	2	2	no	—
83	C₂×D₇₃	—	2	2	no	—
84	C₂×D₇₃	—	2	2	no	—
85	C₂×D₇₃	C₄⋊C₈	1	4	yes	complemented
86	C₄×C₇₃	—	2	2	no	—
87	C₄×C₇₃	—	2	2	no	—
88	C₄×C₇₃	C₂²⋊C₈	1	4	yes	complemented
89	Dic₇₃	—	2	2	no	—
90	Dic₇₃	—	2	2	no	—
91	Dic₇₃	#121	1	2	yes	—
92	Dic₇₃	C₂²⋊Q₈	1	2	yes	—
93	C₂×C₇₃	#380	1	1	yes	characteristic
94	C₂×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
95	C₂×C₇₃	#407	1	1	yes	characteristic
96	C₂×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
97	C₂×C₇₃	#423	1	1	yes	characteristic
98	C₂×C₇₃	#431	1	1	yes	characteristic
99	C₂×C₇₃	#432	1	1	yes	characteristic
100	D₇₃	—	2	4	no	—
101	#1797	—	73	1	no	maximal, Sylow
102	C₇₃	#1797	1	1	yes	characteristic, complemented, Sylow
103	C₂²×C₄²	—	73	1	no	—
104	C₂×C₄⋊C₈	—	73	2	no	—
105	C₂×C₂²⋊C₈	—	73	2	no	—
106	#336	—	73	2	no	—
107	C₂³×C₄	—	73	1	no	—
108	C₂³×C₄	—	73	1	no	—
109	C₂³×C₄	—	73	1	no	—
110	C₂²×C₈	—	73	4	no	—
111	C₂×C₄²	—	73	1	no	—
112	C₂×C₄²	—	73	1	no	—
113	C₂×C₄²	—	73	1	no	—
114	C₂×C₄²	—	146	4	no	—
115	C₂×C₄²	#64408	1	1	yes	characteristic
116	C₄⋊C₈	—	146	4	no	—
117	C₂²⋊C₈	—	146	4	no	—
118	C₂⁴	—	73	1	no	—
119	C₂²×C₄	—	73	1	no	—
120	C₂²×C₄	—	73	1	no	—
121	C₂²×C₄	—	73	1	no	—
122	C₂²×C₄	—	73	2	no	—
123	C₂²×C₄	—	73	2	no	—
124	C₂²×C₄	—	73	4	no	—
125	C₂²×C₄	—	146	2	no	—
126	C₂²×C₄	—	146	2	no	—
127	C₂²×C₄	—	146	2	no	—
128	C₂²×C₄	—	146	2	no	—
129	C₂²×C₄	—	146	2	no	—
130	C₂²×C₄	—	146	2	no	—
131	C₂²×C₄	—	146	2	no	—
132	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
133	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
134	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
135	C₂×C₈	—	146	4	no	—
136	C₂×C₈	—	146	4	no	—
137	C₂×C₈	—	146	4	no	—
138	C₄²	—	2	2	no	—
139	C₄²	—	73	4	no	—
140	C₄²	—	146	2	no	—
141	C₄²	—	146	2	no	—
142	C₂³	—	73	2	no	—
143	C₂³	—	73	2	no	—
144	C₂³	—	73	2	no	—
145	C₂³	—	146	1	no	—
146	C₂³	—	146	1	no	—
147	C₂³	—	146	1	no	—
148	C₂³	—	146	1	no	—
149	C₂³	C₂²×#64408	1	1	yes	characteristic
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	73	2	no	—
155	C₂×C₄	—	73	1	no	—
156	C₂×C₄	—	73	1	no	—
157	C₂×C₄	—	73	2	no	—
158	C₂×C₄	—	73	2	no	—
159	C₂×C₄	—	73	2	no	—
160	C₂×C₄	—	146	4	no	—
161	C₂×C₄	—	146	4	no	—
162	C₂×C₄	—	146	8	no	—
163	C₂×C₄	—	146	1	no	—
164	C₂×C₄	—	146	1	no	—
165	C₂×C₄	—	146	1	no	—
166	C₂×C₄	—	146	1	no	—
167	C₂×C₄	—	146	8	no	—
168	C₂×C₄	C₂²⋊#64408	1	2	yes	—
169	C₂×C₄	C₄⋊#64408	1	2	yes	—
170	C₂×C₄	#11924036	1	2	yes	—
171	C₂×C₄	#11924038	1	2	yes	—
172	C₂×C₄	#11924038	1	2	yes	—
173	C₈	—	292	4	no	—
174	C₂²	—	73	4	no	—
175	C₂²	—	146	2	no	—
176	C₂²	—	146	2	no	—
177	C₂²	—	146	2	no	—
178	C₂²	—	146	2	no	—
179	C₂²	—	146	2	no	—
180	C₂²	—	146	2	no	—
181	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
182	C₂²	C₂×C₄⋊#64408	1	1	yes	characteristic
183	C₂²	#14556670	1	1	yes	characteristic
184	C₂²	C₂×#11924036	1	1	yes	characteristic
185	C₂²	D₇₃⋊M₁₆	1	1	yes	characteristic
186	C₂²	C₂×#11924038	1	1	yes	characteristic
187	C₂²	C₂×#11924038	1	1	yes	characteristic
188	C₄	—	2	2	no	—
189	C₄	—	2	2	no	—
190	C₄	—	73	2	no	—
191	C₄	—	73	2	no	—
192	C₄	—	146	2	no	—
193	C₄	—	146	2	no	—
194	C₄	#14556648	1	4	yes	complemented
195	C₂	—	146	4	no	—
196	C₂	#17989529	1	1	yes	characteristic
197	C₂	C₂×#14556648	1	1	yes	characteristic
198	C₂	#17989558	1	1	yes	characteristic
199	C₂	#17989588	1	1	yes	characteristic
200	C₂	#17989591	1	1	yes	characteristic
201	C₂	#17989606	1	1	yes	characteristic
202	C₂	#17989609	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#10596833

C₂

yes

maximal, complemented

C₂×#14556648

C₂

yes

maximal

#17989453

C₂

yes

maximal, complemented

C₂×C₄²×D₇₃

C₂

yes

maximal, characteristic

C₂²×#10596833

C₂²

yes

—

C₄⋊#10596833

—

#14556648

—

C₂²×C₄×D₇₃

C₄

yes

characteristic

C₂²×C₄×D₇₃

C₄

yes

characteristic

C₂²×C₄×D₇₃

C₂²

yes

characteristic

C₂×C₄×Dic₇₃

C₄

yes

characteristic

C₂×C₄×Dic₇₃

C₂²

yes

characteristic

C₂×C₄×Dic₇₃

C₂²

yes

characteristic

C₄²×D₇₃

—

C₂×C₄²×C₇₃

C₄

yes

—

—

—

C₂³×D₇₃

C₂×C₄

yes

characteristic

C₂²×Dic₇₃

C₂×C₄

yes

characteristic

C₂²×Dic₇₃

C₂×C₄

yes

characteristic

C₂²×Dic₇₃

C₂³

yes

characteristic

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

C₈

yes

complemented

C₂×C₄×D₇₃

D₄

yes

—

C₂×C₄×D₇₃

Q₈

yes

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

D₄

yes

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₄²×C₇₃

—

C₂³×C₇₃

C₂²×C₄

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

C₄⋊C₄

yes

—

C₂²×D₇₃

C₂×C₈

yes

—

C₂²×D₇₃

M₁₆

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₇₃

C₄⋊C₄

yes

—

C₂×C₄×C₇₃

C₂×C₈

yes

—

C₂×C₄×C₇₃

M₁₆

yes

—

C₂×C₄×C₇₃

M₁₆

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂²⋊C₄

yes

—

C₂×Dic₇₃

C₂×C₈

yes

—

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

—

C₂×Dic₇₃

C₂×Q₈

yes

characteristic

C₂×Dic₇₃

#41

yes

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

#10596833

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₃

#117

yes

characteristic

C₂²×C₇₃

C₂²×C₈

yes

characteristic

C₂²×C₇₃

C₂×M₁₆

yes

characteristic

C₂²×C₇₃

C₂×M₁₆

yes

characteristic

C₂²×C₇₃

C₂×M₁₆

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

C₄⋊C₈

yes

complemented

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂²⋊C₈

yes

complemented

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#121

yes

—

Dic₇₃

C₂²⋊Q₈

yes

—

C₂×C₇₃

#380

yes

characteristic

C₂×C₇₃

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₇₃

#407

yes

characteristic

C₂×C₇₃

C₂×C₄⋊C₈

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

100

D₇₃

—

101

#1797

—

maximal, Sylow

102

C₇₃

#1797

yes

characteristic, complemented, Sylow

103

C₂²×C₄²

—

104

C₂×C₄⋊C₈

—

105

C₂×C₂²⋊C₈

—

106

#336

—

107

C₂³×C₄

—

108

C₂³×C₄

—

109

C₂³×C₄

—

110

C₂²×C₈

—

111

C₂×C₄²

—

112

C₂×C₄²

—

113

C₂×C₄²

—

114

C₂×C₄²

—

146

—

115

C₂×C₄²

#64408

yes

characteristic

116

C₄⋊C₈

—

146

—

117

C₂²⋊C₈

—

146

—

118

C₂⁴

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

146

—

126

C₂²×C₄

—

146

—

127

C₂²×C₄

—

146

—

128

C₂²×C₄

—

146

—

129

C₂²×C₄

—

146

—

130

C₂²×C₄

—

146

—

131

C₂²×C₄

—

146

—

132

C₂²×C₄

C₂×#64408

yes

characteristic

133

C₂²×C₄

C₂×#64408

yes

characteristic

134

C₂²×C₄

C₂×#64408

yes

characteristic

135

C₂×C₈

—

146

—

136

C₂×C₈

—

146

—

137

C₂×C₈

—

146

—

138

C₄²

—

139

C₄²

—

140

C₄²

—

146

—

141

C₄²

—

146

—

142

C₂³

—

143

C₂³

—

144

C₂³

—

145

C₂³

—

146

—

146

C₂³

—

146

—

147

C₂³

—

146

—

148

C₂³

—

146

—

149

C₂³

C₂²×#64408

yes

characteristic

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

146

—

161

C₂×C₄

—

146

—

162

C₂×C₄

—

146

—

163

C₂×C₄

—

146

—

164

C₂×C₄

—

146

—

165

C₂×C₄

—

146

—

166

C₂×C₄

—

146

—

167

C₂×C₄

—

146

—

168

C₂×C₄

C₂²⋊#64408

yes

—

169

C₂×C₄

C₄⋊#64408

yes

—

170

C₂×C₄

#11924036

yes

—

171

C₂×C₄

#11924038

yes

—

172

C₂×C₄

#11924038

yes

—

173

C₈

—

292

—

174

C₂²

—

175

C₂²

—

146

—

176

C₂²

—

146

—

177

C₂²

—

146

—

178

C₂²

—

146

—

179

C₂²

—

146

—

180

C₂²

—

146

—

181

C₂²

C₂×C₂²⋊#64408

yes

characteristic

182

C₂²

C₂×C₄⋊#64408

yes

characteristic

183

C₂²

#14556670

yes

characteristic

184

C₂²

C₂×#11924036

yes

characteristic

185

C₂²

D₇₃⋊M₁₆

yes

characteristic

186

C₂²

C₂×#11924038

yes

characteristic

187

C₂²

C₂×#11924038

yes

characteristic

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

146

—

193

C₄

—

146

—

194

C₄

#14556648

yes

complemented

195

C₂

—

146

—

196

C₂

#17989529

yes

characteristic

197

C₂

C₂×#14556648

yes

characteristic

198

C₂

#17989558

yes

characteristic

199

C₂

#17989588

yes

characteristic

200

C₂

#17989591

yes

characteristic

201

C₂

#17989606

yes

characteristic

202

C₂

#17989609

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×#64408
commutator	C₂²×C₇₃	C₂²×C₈
Fitting	C₂×C₄²×C₇₃	C₄
socle	C₂³×C₇₃	C₂²×C₄
hypercenter	C₂×C₄²	#64408
hypocenter	C₇₃	#1797
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

subgroup

G/H

center, Frattini

C₂³

C₂²×#64408

commutator

C₂²×C₇₃

C₂²×C₈

Fitting

C₂×C₄²×C₇₃

C₄

socle

C₂³×C₇₃

C₂²×C₄

C₂×C₄²

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂³ ◁ C₂×C₄²

N	H ≅ G/N	type
1. C₂×C₄⋊#10596833	C₂	semidirect
3. #17989453	C₂	semidirect
30. C₂×C₄×D₇₃	C₈	semidirect
85. C₂×D₇₃	C₄⋊C₈	semidirect
88. C₄×C₇₃	C₂²⋊C₈	semidirect
102. C₇₃	#1797	semidirect
194. C₄	#14556648	semidirect

H ≅ G/N

type

1. C₂×C₄⋊#10596833

C₂

semidirect

3. #17989453

C₂

semidirect

30. C₂×C₄×D₇₃

C₈

semidirect

85. C₂×D₇₃

C₄⋊C₈

semidirect

88. C₄×C₇₃

C₂²⋊C₈

semidirect

102. C₇₃

#1797

semidirect

194. C₄

#14556648

semidirect

Subgroups — #22625786

Conjugacy classes of subgroups

Complements