Group #34010630 — subgroups

Subgroups — #34010630

Order 18688.

354 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 84 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	#22626293	C₂	1	1	yes	maximal, characteristic
2	#22625491	C₂	1	1	yes	maximal, characteristic
3	#22624892	C₂	1	4	yes	maximal, complemented
4	C₂×#17988681	C₂	1	1	yes	maximal, characteristic
5	#17989608	—	2	1	no	—
6	#17989508	—	2	1	no	—
7	C₂²⋊#11924031	—	2	1	no	—
8	#17989476	C₂²	1	2	yes	—
9	#17989451	—	2	1	no	—
10	#17989448	C₂²	1	2	yes	—
11	C₂²×Dic₂₉₂	C₄	1	1	yes	characteristic
12	C₂×#14556556	—	2	1	no	—
13	#17988681	—	2	1	no	—
14	#17988681	—	2	1	no	—
15	#17988681	C₄	1	2	yes	—
16	#17988681	C₂²	1	2	yes	—
17	C₂×#14556514	—	2	1	no	—
18	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
19	C₂×#107×C₇₃	C₄	1	1	yes	characteristic
20	C₂×#11924031	—	2	2	no	—
21	C₄⋊#10596833	—	2	2	no	—
22	C₂²⋊#10596833	—	4	1	no	—
23	#14556646	—	4	1	no	—
24	C₂²×Dic₁₄₆	—	2	1	no	—
25	C₂×Dic₂₉₂	—	2	4	no	—
26	C₂×Dic₂₉₂	—	2	1	no	—
27	C₂×Dic₂₉₂	C₂×C₄	1	2	yes	—
28	C₂²×#10596831	D₄	1	1	yes	characteristic
29	#14556556	—	2	2	no	—
30	#14556556	—	2	2	no	—
31	#14556556	—	4	1	no	—
32	#14556556	—	4	1	no	—
33	C₂×#11924007	—	4	1	no	—
34	C₂×C₄×Dic₇₃	—	4	1	no	—
35	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
36	#14556514	—	2	2	no	—
37	#14556514	—	2	2	no	—
38	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₇₃	D₄	1	1	yes	characteristic
40	C₈×Dic₇₃	D₄	1	1	yes	characteristic
41	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
44	#107×C₇₃	—	2	1	no	—
45	#107×C₇₃	C₂×C₄	1	2	yes	—
46	C₂×#10596833	—	4	2	no	—
47	#11924031	—	4	2	no	—
48	C₂×Dic₁₄₆	—	2	4	no	—
49	C₂×Dic₁₄₆	—	2	2	no	—
50	C₂×Dic₁₄₆	—	4	2	no	—
51	C₂×Dic₁₄₆	—	4	1	no	—
52	C₂²×Dic₇₃	—	4	1	no	—
53	C₂²×Dic₇₃	D₈	1	1	yes	characteristic
54	C₂²×Dic₇₃	SD₁₆	1	1	yes	characteristic
55	Dic₂₉₂	—	2	4	no	—
56	Dic₂₉₂	—	2	2	no	—
57	Dic₂₉₂	M₁₆	1	4	yes	—
58	C₂×#10596831	—	4	1	no	—
59	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
61	#11924007	—	4	2	no	—
62	#11924007	—	4	2	no	—
63	C₄×Dic₇₃	—	4	2	no	—
64	C₄×Dic₇₃	—	4	2	no	—
65	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₄×Dic₇₃	D₈	1	1	yes	characteristic
67	C₄×Dic₇₃	SD₁₆	1	1	yes	characteristic
68	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₃	—	4	1	no	—
70	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₇₃	—	2	2	no	—
72	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
74	C₄⋊C₄×C₇₃	—	2	2	no	—
75	C₄⋊C₄×C₇₃	—	4	1	no	—
76	C₂³×C₇₃	#102	1	1	yes	characteristic
77	C₂×C₄×C₇₃	—	2	1	no	—
78	C₂×C₄×C₇₃	—	2	1	no	—
79	C₂×C₄×C₇₃	—	4	2	no	—
80	C₂×C₄×C₇₃	—	8	2	no	—
81	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
82	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×Dic₇₃	—	2	2	no	—
84	C₂×Dic₇₃	—	4	1	no	—
85	C₂×Dic₇₃	—	4	4	no	—
86	C₂×Dic₇₃	—	4	2	no	—
87	C₂×Dic₇₃	#102	1	1	yes	characteristic
88	C₂×Dic₇₃	#102	1	1	yes	characteristic
89	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
90	C₂×Dic₇₃	C₂×SD₁₆	1	1	yes	characteristic
91	Dic₁₄₆	—	2	4	no	—
92	Dic₁₄₆	—	4	4	no	—
93	Dic₁₄₆	—	4	2	no	—
94	#10596833	—	8	2	no	—
95	C₈×C₇₃	—	2	1	no	—
96	C₈×C₇₃	C₂×M₁₆	1	2	yes	—
97	#10596831	—	2	1	no	—
98	#10596831	—	2	1	no	—
99	C₂²×C₇₃	—	2	1	no	—
100	C₂²×C₇₃	—	2	1	no	—
101	C₂²×C₇₃	#362	1	2	yes	—
102	C₂²×C₇₃	C₂×#102	1	1	yes	characteristic
103	C₄×C₇₃	—	2	1	no	—
104	C₄×C₇₃	—	8	2	no	—
105	C₄×C₇₃	#407	1	1	yes	characteristic
106	C₄×C₇₃	#421	1	1	yes	characteristic
107	Dic₇₃	—	2	1	no	—
108	Dic₇₃	—	2	1	no	—
109	Dic₇₃	—	2	1	no	—
110	Dic₇₃	—	2	1	no	—
111	Dic₇₃	—	4	4	no	—
112	#11668	—	73	1	no	maximal, Sylow
113	C₂×C₇₃	—	2	2	no	—
114	C₂×C₇₃	#1486	1	1	yes	characteristic
115	C₂×C₇₃	#2145	1	1	yes	characteristic
116	C₂×C₇₃	C₂×#362	1	1	yes	characteristic
117	C₂×C₄×Q₁₆	—	73	1	no	—
118	#2163	—	73	1	no	—
119	#1568	—	73	1	no	—
120	#1334	—	73	4	no	—
121	C₇₃	#11668	1	1	yes	characteristic, complemented, Sylow
122	C₂²×Q₁₆	—	73	1	no	—
123	C₂×C₄×Q₈	—	146	1	no	—
124	C₄×Q₁₆	—	73	2	no	—
125	C₄×Q₁₆	—	73	2	no	—
126	C₄×Q₁₆	—	146	1	no	—
127	C₄×Q₁₆	—	146	1	no	—
128	#432	—	146	1	no	—
129	C₂×#107	#64408	1	1	yes	characteristic
130	C₂×#103	—	146	1	no	—
131	C₂×C₄×C₈	—	73	1	no	—
132	C₄⋊C₁₆	—	146	1	no	—
133	C₂²⋊C₁₆	—	146	1	no	—
134	#346	—	73	2	no	—
135	#335	—	146	1	no	—
136	#334	—	73	2	no	—
137	C₂²×Q₈	—	146	1	no	—
138	C₂×Q₁₆	—	73	2	no	—
139	C₂×Q₁₆	—	146	4	no	—
140	C₂×Q₁₆	—	146	1	no	—
141	C₂²×C₈	—	73	1	no	—
142	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
143	C₄×Q₈	—	292	1	no	—
144	C₄×Q₈	—	292	1	no	—
145	C₄×Q₈	—	146	2	no	—
146	C₄×Q₈	—	146	2	no	—
147	C₂×C₄⋊C₄	—	2	1	no	—
148	C₂×C₄⋊C₄	—	292	1	no	—
149	C₂×C₄²	—	292	1	no	—
150	C₂×C₄²	—	73	1	no	—
151	C₂×C₁₆	—	146	2	no	—
152	#107	—	2	1	no	—
153	#107	C₂×#64408	1	2	yes	—
154	C₄⋊C₈	—	146	2	no	—
155	#103	—	146	2	no	—
156	#103	—	146	2	no	—
157	C₂²⋊C₈	—	292	1	no	—
158	#97	—	292	1	no	—
159	C₄×C₈	—	73	1	no	—
160	C₄×C₈	—	73	1	no	—
161	C₄×C₈	—	73	1	no	—
162	C₄×C₈	—	73	1	no	—
163	C₂×Q₈	—	292	2	no	—
164	C₂×Q₈	—	292	1	no	—
165	C₂×Q₈	—	146	4	no	—
166	C₂×Q₈	—	146	2	no	—
167	C₂²×C₄	—	4	1	no	—
168	C₂²×C₄	—	292	1	no	—
169	C₂²×C₄	—	73	1	no	—
170	C₂²×C₄	—	73	1	no	—
171	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
172	Q₁₆	—	73	4	no	—
173	Q₁₆	—	146	4	no	—
174	Q₁₆	—	146	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	292	2	no	—
177	C₂×C₈	—	292	1	no	—
178	C₂×C₈	—	73	1	no	—
179	C₂×C₈	—	73	1	no	—
180	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
181	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	4	1	no	—
184	C₄⋊C₄	—	292	2	no	—
185	C₄⋊C₄	—	292	2	no	—
186	C₄²	—	292	2	no	—
187	C₄²	—	292	2	no	—
188	C₄²	—	73	1	no	—
189	C₄²	—	73	1	no	—
190	C₄²	—	73	1	no	—
191	C₄²	—	73	1	no	—
192	C₁₆	—	292	2	no	—
193	C₂³	#14556653	1	1	yes	characteristic
194	Q₈	—	292	4	no	—
195	Q₈	—	292	2	no	—
196	Q₈	—	146	4	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	8	2	no	—
201	C₂×C₄	—	292	4	no	—
202	C₂×C₄	—	292	2	no	—
203	C₂×C₄	—	292	1	no	—
204	C₂×C₄	—	73	1	no	—
205	C₂×C₄	—	73	1	no	—
206	C₂×C₄	—	73	1	no	—
207	C₂×C₄	—	73	1	no	—
208	C₂×C₄	—	146	2	no	—
209	C₂×C₄	#14556653	1	1	yes	characteristic
210	C₂×C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
211	C₈	—	2	1	no	—
212	C₈	—	584	2	no	—
213	C₈	—	146	1	no	—
214	C₈	—	146	1	no	—
215	C₈	D₇₃⋊M₁₆	1	2	yes	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	#17989506	1	2	yes	—
219	C₂²	C₂×#14556653	1	1	yes	characteristic
220	C₄	—	2	1	no	—
221	C₄	—	8	2	no	—
222	C₄	—	292	4	no	—
223	C₄	—	146	1	no	—
224	C₄	—	146	1	no	—
225	C₄	—	146	1	no	—
226	C₄	—	146	1	no	—
227	C₄	#17989559	1	1	yes	characteristic
228	C₄	#17989586	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	#22625258	1	1	yes	characteristic
231	C₂	#22626252	1	1	yes	characteristic
232	C₂	C₂×#17989506	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626293

C₂

yes

maximal, characteristic

#22625491

C₂

yes

maximal, characteristic

#22624892

C₂

yes

maximal, complemented

C₂×#17988681

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

#17989451

—

#17989448

C₂²

yes

—

C₂²×Dic₂₉₂

C₄

yes

—

—

—

C₄

yes

—

#17988681

C₂²

yes

—

C₂×#14556514

—

C₂×C₈×Dic₇₃

C₂²

yes

characteristic

C₂×#107×C₇₃

C₄

yes

—

—

—

—

C₂²×Dic₁₄₆

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

C₂×C₄

yes

—

C₂²×#10596831

D₄

yes

—

—

—

—

—

C₂×C₄×Dic₇₃

—

C₂×C₄×Dic₇₃

D₄

yes

characteristic

#14556514

—

#14556514

—

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

D₄

yes

characteristic

C₈×Dic₇₃

D₄

yes

characteristic

C₈×Dic₇₃

C₂³

yes

characteristic

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₄⋊C₄×C₇₃

—

#107×C₇₃

—

#107×C₇₃

C₂×C₄

yes

—

C₂×#10596833

—

#11924031

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

D₈

yes

characteristic

C₂²×Dic₇₃

SD₁₆

yes

characteristic

Dic₂₉₂

—

Dic₂₉₂

—

Dic₂₉₂

M₁₆

yes

—

C₂×#10596831

—

C₂×#10596831

C₂²⋊C₄

yes

characteristic

C₂×#10596831

C₂×D₄

yes

characteristic

#11924007

—

#11924007

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₇₃

D₈

yes

characteristic

C₄×Dic₇₃

SD₁₆

yes

characteristic

C₄×Dic₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂³×C₇₃

#102

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#102

yes

characteristic

C₂×C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

C₂×D₈

yes

characteristic

C₂×Dic₇₃

C₂×SD₁₆

yes

characteristic

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

C₂×M₁₆

yes

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

100

C₂²×C₇₃

—

101

C₂²×C₇₃

#362

yes

—

102

C₂²×C₇₃

C₂×#102

yes

characteristic

103

C₄×C₇₃

—

104

C₄×C₇₃

—

105

C₄×C₇₃

#407

yes

characteristic

106

C₄×C₇₃

#421

yes

characteristic

107

Dic₇₃

—

108

Dic₇₃

—

109

Dic₇₃

—

110

Dic₇₃

—

111

Dic₇₃

—

112

#11668

—

maximal, Sylow

113

C₂×C₇₃

—

114

C₂×C₇₃

#1486

yes

characteristic

115

C₂×C₇₃

#2145

yes

characteristic

116

C₂×C₇₃

C₂×#362

yes

characteristic

117

C₂×C₄×Q₁₆

—

118

#2163

—

119

#1568

—

120

#1334

—

121

C₇₃

#11668

yes

characteristic, complemented, Sylow

122

C₂²×Q₁₆

—

123

C₂×C₄×Q₈

—

146

—

124

C₄×Q₁₆

—

125

C₄×Q₁₆

—

126

C₄×Q₁₆

—

146

—

127

C₄×Q₁₆

—

146

—

128

#432

—

146

—

129

yes

130

—

146

—

131

C₂×C₄×C₈

—

132

C₄⋊C₁₆

—

146

—

133

C₂²⋊C₁₆

—

146

—

134

#346

—

135

#335

—

146

—

136

#334

—

137

C₂²×Q₈

—

146

—

138

C₂×Q₁₆

—

139

C₂×Q₁₆

—

146

—

140

C₂×Q₁₆

—

146

—

141

C₂²×C₈

—

142

C₂²×C₈

C₂×#64408

yes

characteristic

143

C₄×Q₈

—

292

—

144

C₄×Q₈

—

292

—

145

C₄×Q₈

—

146

—

146

C₄×Q₈

—

146

—

147

C₂×C₄⋊C₄

—

148

C₂×C₄⋊C₄

—

292

—

149

C₂×C₄²

—

292

—

150

C₂×C₄²

—

151

C₂×C₁₆

—

146

—

152

#107

—

153

#107

C₂×#64408

yes

—

154

C₄⋊C₈

—

146

—

155

#103

—

146

—

156

#103

—

146

—

157

C₂²⋊C₈

—

292

—

158

#97

—

292

—

159

C₄×C₈

—

160

C₄×C₈

—

161

C₄×C₈

—

162

C₄×C₈

—

163

C₂×Q₈

—

292

—

164

C₂×Q₈

—

292

—

165

C₂×Q₈

—

146

—

166

C₂×Q₈

—

146

—

167

C₂²×C₄

—

168

C₂²×C₄

—

292

—

169

C₂²×C₄

—

170

C₂²×C₄

—

171

C₂²×C₄

C₂²⋊#64408

yes

characteristic

172

Q₁₆

—

173

Q₁₆

—

146

—

174

Q₁₆

—

146

—

175

C₂×C₈

—

176

C₂×C₈

—

292

—

177

C₂×C₈

—

292

—

178

C₂×C₈

—

179

C₂×C₈

—

180

C₂×C₈

C₂²⋊#64408

yes

characteristic

181

C₂×C₈

C₂²×#64408

yes

characteristic

182

C₄⋊C₄

—

183

C₄⋊C₄

—

184

C₄⋊C₄

—

292

—

185

C₄⋊C₄

—

292

—

186

C₄²

—

292

—

187

C₄²

—

292

—

188

C₄²

—

189

C₄²

—

190

C₄²

—

191

C₄²

—

192

C₁₆

—

292

—

193

C₂³

#14556653

yes

characteristic

194

Q₈

—

292

—

195

Q₈

—

292

—

196

Q₈

—

146

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

292

—

202

C₂×C₄

—

292

—

203

C₂×C₄

—

292

—

204

C₂×C₄

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

—

208

C₂×C₄

—

146

—

209

C₂×C₄

#14556653

yes

characteristic

210

C₂×C₄

C₂×C₂²⋊#64408

yes

characteristic

211

C₈

—

212

C₈

—

584

—

213

C₈

—

146

—

214

C₈

—

146

—

215

C₈

D₇₃⋊M₁₆

yes

—

216

C₂²

—

217

C₂²

—

218

C₂²

#17989506

yes

—

219

C₂²

C₂×#14556653

yes

characteristic

220

C₄

—

221

C₄

—

222

C₄

—

292

—

223

C₄

—

146

—

224

C₄

—

146

—

225

C₄

—

146

—

226

C₄

—

146

—

227

C₄

#17989559

yes

characteristic

228

C₄

#17989586

yes

characteristic

229

C₂

—

230

C₂

#22625258

yes

characteristic

231

C₂

#22626252

yes

characteristic

232

C₂

C₂×#17989506

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#14556653
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×#64408
Fitting	C₂×#107×C₇₃	C₄
socle	C₂²×C₇₃	C₂×#102
prime radical	C₂³×C₇₃	#102
cosocle	C₈×Dic₇₃	C₂³
hypercenter	C₂×#107	#64408
hypocenter	C₇₃	#11668
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#107

subgroup

G/H

center

C₂²

C₂×#14556653

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×#64408

Fitting

C₂×#107×C₇₃

C₄

C₂²×C₇₃

C₂³×C₇₃

C₈×Dic₇₃

C₂³

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#107

N	H ≅ G/N	type
3. #22624892	C₂	semidirect
121. C₇₃	#11668	semidirect

H ≅ G/N

type

3. #22624892

C₂

semidirect

121. C₇₃

#11668

semidirect

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

subgroup	index
#22626293	2
#22625491	2
#22624892	2
C₂×#17988681	2
#11668	73

subgroup

index

Subgroups — #34010630

Conjugacy classes of subgroups

Complements

Maximal subgroups