Group #34007516 — subgroups

Subgroups — #34007516

Order 18688.

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

normal subgroups only

This group has 306 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	#22625482	C₂	1	2	yes	maximal
2	C₂×#17989448	C₂	1	1	yes	maximal, characteristic
3	#22625134	C₂	1	1	yes	maximal, characteristic
4	#22624777	C₂	1	2	yes	maximal
5	#108×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	#17989603	C₂²	1	1	yes	characteristic
7	#17989603	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#10596833	C₂²	1	2	yes	—
9	#17989450	—	2	2	no	—
10	#17989448	—	2	2	no	—
11	#17989448	—	2	1	no	—
12	#17989448	—	2	1	no	—
13	#17989428	—	2	1	no	—
14	C₈×#10596833	—	2	1	no	—
15	C₂×#14556527	C₄	1	1	yes	characteristic
16	M₁₆×Dic₇₃	C₂²	1	2	yes	—
17	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
18	#17988351	C₄	1	2	yes	—
19	C₂×#108×C₇₃	C₄	1	1	yes	characteristic
20	C₂²×#10596833	—	2	2	no	—
21	C₄⋊#10596833	—	2	2	no	—
22	C₄⋊#10596833	—	2	2	no	—
23	C₄⋊#10596833	—	2	1	no	—
24	C₄⋊#10596833	D₄	1	2	yes	—
25	C₂²⋊#10596833	—	2	1	no	—
26	C₂²⋊#10596833	Q₈	1	2	yes	—
27	C₄×#10596833	—	2	1	no	—
28	C₄×#10596833	—	2	1	no	—
29	C₂×#11924012	C₂×C₄	1	2	yes	—
30	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
32	#14556527	—	2	2	no	—
33	#14556501	—	2	2	no	—
34	C₈×Dic₇₃	—	2	2	no	—
35	C₈×Dic₇₃	D₄	1	1	yes	characteristic
36	C₈×Dic₇₃	D₄	1	1	yes	characteristic
37	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
38	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
39	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
41	#108×C₇₃	C₈	1	4	yes	complemented
42	C₂×#10596833	—	2	2	no	—
43	C₂×#10596833	—	2	2	no	—
44	C₂×#10596833	—	4	2	no	—
45	C₂×#10596833	—	8	2	no	—
46	C₂²×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
47	C₂²×Dic₇₃	#41	1	1	yes	characteristic
48	#11924012	—	2	2	no	—
49	#11924012	—	2	2	no	—
50	C₂×#10596831	—	2	1	no	—
51	C₂×#10596831	—	2	1	no	—
52	C₂×#10596831	—	2	2	no	—
53	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
54	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
55	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
56	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
57	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
58	C₄×Dic₇₃	#41	1	1	yes	characteristic
59	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
60	M₁₆×C₇₃	—	2	2	no	—
61	M₁₆×C₇₃	C₂×C₈	1	4	yes	—
62	C₂×C₈×C₇₃	—	2	2	no	—
63	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
66	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
67	C₂³×C₇₃	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
72	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
73	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
74	C₂×Dic₇₃	—	2	2	no	—
75	C₂×Dic₇₃	—	4	1	no	—
76	C₂×Dic₇₃	#121	1	1	yes	characteristic
77	C₂×Dic₇₃	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂×Dic₇₃	#125	1	1	yes	characteristic
79	C₂×Dic₇₃	C₄⋊Q₈	1	1	yes	characteristic
80	#10596833	—	4	2	no	—
81	#10596833	—	4	2	no	—
82	#10596833	—	8	2	no	—
83	C₈×C₇₃	—	2	4	no	—
84	C₈×C₇₃	C₄⋊C₈	1	2	yes	—
85	C₈×C₇₃	C₄⋊C₈	1	2	yes	—
86	#10596831	—	2	1	no	—
87	#10596831	—	2	1	no	—
88	#10596831	—	4	2	no	—
89	C₂²×C₇₃	—	2	1	no	—
90	C₂²×C₇₃	—	2	1	no	—
91	C₂²×C₇₃	#389	1	1	yes	characteristic
92	C₂²×C₇₃	C₈×Q₈	1	1	yes	characteristic
93	C₂²×C₇₃	#446	1	1	yes	characteristic
94	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
95	C₄×C₇₃	#424	1	1	yes	characteristic
96	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
97	C₄×C₇₃	#436	1	1	yes	characteristic
98	Dic₇₃	—	2	1	no	—
99	Dic₇₃	—	2	1	no	—
100	Dic₇₃	—	2	1	no	—
101	Dic₇₃	#471	1	2	yes	—
102	#9607	—	73	1	no	maximal, Sylow
103	C₂×C₇₃	—	2	1	no	—
104	C₂×C₇₃	—	2	1	no	—
105	C₂×C₇₃	#1939	1	1	yes	characteristic
106	C₂×C₇₃	#1951	1	1	yes	characteristic
107	C₂×C₇₃	#1953	1	1	yes	characteristic
108	C₄×#108	—	73	1	no	—
109	#1565	—	73	2	no	—
110	C₂×#334	—	73	1	no	—
111	#1450	—	73	1	no	—
112	#1283	—	73	2	no	—
113	C₇₃	#9607	1	1	yes	characteristic, complemented, Sylow
114	#431	—	73	1	no	—
115	#431	—	73	1	no	—
116	C₂×#108	—	73	1	no	—
117	C₂×#108	#64408	1	1	yes	characteristic
118	C₂×C₄⋊C₈	—	73	2	no	—
119	C₄×M₁₆	—	73	2	no	—
120	C₂×C₄×C₈	—	73	1	no	—
121	#341	—	73	2	no	—
122	#335	—	146	2	no	—
123	#334	—	146	1	no	—
124	#334	—	146	1	no	—
125	#334	—	146	2	no	—
126	#322	—	146	1	no	—
127	C₈²	—	146	1	no	—
128	C₂×M₁₆	—	73	2	no	—
129	C₂×M₁₆	C₂×#64408	1	2	yes	—
130	C₂²×C₈	—	73	1	no	—
131	C₂²×C₈	—	146	2	no	—
132	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
133	C₂×C₄²	—	73	1	no	—
134	#108	—	146	2	no	—
135	#108	#10596833	1	4	yes	complemented
136	C₄⋊C₈	—	73	2	no	—
137	C₄⋊C₈	—	146	2	no	—
138	C₄⋊C₈	—	146	2	no	—
139	C₄⋊C₈	—	146	1	no	—
140	C₂²⋊C₈	—	73	2	no	—
141	C₂²⋊C₈	—	146	1	no	—
142	#97	—	146	2	no	—
143	C₄×C₈	—	73	1	no	—
144	C₄×C₈	—	73	1	no	—
145	C₄×C₈	—	73	1	no	—
146	C₄×C₈	—	73	1	no	—
147	C₄×C₈	—	146	2	no	—
148	C₄×C₈	—	146	1	no	—
149	C₄×C₈	—	146	1	no	—
150	C₂²×C₄	—	73	1	no	—
151	C₂²×C₄	—	73	1	no	—
152	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
153	M₁₆	—	2	2	no	—
154	M₁₆	—	146	2	no	—
155	M₁₆	—	146	2	no	—
156	M₁₆	C₂×#10596833	1	4	yes	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	292	2	no	—
159	C₂×C₈	—	584	2	no	—
160	C₂×C₈	—	73	1	no	—
161	C₂×C₈	—	73	1	no	—
162	C₂×C₈	—	146	2	no	—
163	C₂×C₈	—	146	2	no	—
164	C₂×C₈	—	146	2	no	—
165	C₂×C₈	—	146	1	no	—
166	C₂×C₈	—	146	1	no	—
167	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
168	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
169	C₂×C₈	C₂×#10596833	1	2	yes	—
170	C₂×C₈	#11924038	1	2	yes	—
171	C₄²	—	73	1	no	—
172	C₄²	—	73	1	no	—
173	C₄²	—	73	1	no	—
174	C₄²	—	73	1	no	—
175	C₂³	Q₈×#64408	1	1	yes	characteristic
176	C₂×C₄	—	292	1	no	—
177	C₂×C₄	—	73	1	no	—
178	C₂×C₄	—	73	1	no	—
179	C₂×C₄	—	73	1	no	—
180	C₂×C₄	—	73	1	no	—
181	C₂×C₄	—	146	2	no	—
182	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
183	C₂×C₄	D₄×#64408	1	1	yes	characteristic
184	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
185	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
186	C₂×C₄	#14556678	1	1	yes	characteristic
187	C₂×C₄	#14556679	1	1	yes	characteristic
188	C₈	—	2	4	no	—
189	C₈	—	292	2	no	—
190	C₈	—	292	2	no	—
191	C₈	—	292	2	no	—
192	C₈	—	584	2	no	—
193	C₈	—	146	1	no	—
194	C₈	—	146	1	no	—
195	C₈	C₄⋊#10596833	1	2	yes	—
196	C₈	C₄⋊#10596833	1	2	yes	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	#17989538	1	1	yes	characteristic
200	C₂²	Q₈×#10596833	1	1	yes	characteristic
201	C₂²	#17989637	1	1	yes	characteristic
202	C₄	—	73	2	no	—
203	C₄	—	146	1	no	—
204	C₄	—	146	1	no	—
205	C₄	—	146	1	no	—
206	C₄	C₂×C₄⋊#10596833	1	1	yes	characteristic
207	C₄	#17989594	1	1	yes	characteristic
208	C₄	D₄×#10596833	1	1	yes	characteristic
209	C₄	#17989621	1	1	yes	characteristic
210	C₂	—	2	1	no	—
211	C₂	—	2	1	no	—
212	C₂	#22626028	1	1	yes	characteristic
213	C₂	#22626048	1	1	yes	characteristic
214	C₂	#22626050	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625482

C₂

yes

maximal

C₂×#17989448

C₂

yes

maximal, characteristic

#22625134

C₂

yes

maximal, characteristic

#22624777

C₂

yes

maximal

#108×Dic₇₃

C₂

yes

maximal, characteristic

#17989603

C₂²

yes

characteristic

#17989603

C₂²

yes

characteristic

C₂×C₄⋊#10596833

C₂²

yes

—

—

—

—

—

—

—

C₄

yes

characteristic

M₁₆×Dic₇₃

C₂²

yes

—

C₂×C₈×Dic₇₃

C₂²

yes

characteristic

#17988351

C₄

yes

—

C₂×#108×C₇₃

C₄

yes

—

—

—

—

D₄

yes

—

C₂²⋊#10596833

—

C₂²⋊#10596833

Q₈

yes

—

C₄×#10596833

—

C₄×#10596833

—

C₂×#11924012

C₂×C₄

yes

—

C₂²×#10596831

C₂×C₄

yes

characteristic

C₂×C₄×Dic₇₃

C₂³

yes

characteristic

#14556527

—

#14556501

—

C₈×Dic₇₃

—

C₈×Dic₇₃

D₄

yes

characteristic

C₈×Dic₇₃

D₄

yes

characteristic

C₈×Dic₇₃

Q₈

yes

characteristic

C₈×Dic₇₃

Q₈

yes

characteristic

C₂×M₁₆×C₇₃

C₂×C₄

yes

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

#108×C₇₃

C₈

yes

—

—

—

—

C₂²×Dic₇₃

C₂×Q₈

yes

C₂²×Dic₇₃

yes

—

—

—

—

—

C₄⋊C₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₄×Dic₇₃

C₂×D₄

yes

characteristic

C₄×Dic₇₃

C₂×D₄

yes

characteristic

C₄×Dic₇₃

C₂×Q₈

yes

characteristic

C₄×Dic₇₃

#41

yes

characteristic

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

C₂×C₈

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

—

C₂×C₈×C₇₃

M₁₆

yes

—

C₂³×C₇₃

C₄×Q₈

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×M₁₆

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

#121

yes

characteristic

C₂×Dic₇₃

C₂²⋊Q₈

yes

characteristic

C₂×Dic₇₃

#125

yes

characteristic

C₂×Dic₇₃

C₄⋊Q₈

yes

—

—

—

C₈×C₇₃

—

C₈×C₇₃

C₄⋊C₈

yes

—

C₈×C₇₃

C₄⋊C₈

yes

—

#10596831

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#389

yes

characteristic

C₂²×C₇₃

C₈×Q₈

yes

characteristic

C₂²×C₇₃

#446

yes

characteristic

C₄×C₇₃

C₂×C₄⋊C₈

yes

characteristic

C₄×C₇₃

#424

yes

characteristic

C₄×C₇₃

C₈×D₄

yes

characteristic

C₄×C₇₃

#436

yes

characteristic

Dic₇₃

—

Dic₇₃

—

100

Dic₇₃

—

101

Dic₇₃

#471

yes

—

102

#9607

—

maximal, Sylow

103

C₂×C₇₃

—

104

C₂×C₇₃

—

105

C₂×C₇₃

#1939

yes

characteristic

106

C₂×C₇₃

#1951

yes

characteristic

107

C₂×C₇₃

#1953

yes

characteristic

108

C₄×#108

—

109

#1565

—

110

C₂×#334

—

111

#1450

—

112

#1283

—

113

C₇₃

#9607

yes

characteristic, complemented, Sylow

114

#431

—

115

#431

—

116

C₂×#108

—

117

C₂×#108

#64408

yes

characteristic

118

C₂×C₄⋊C₈

—

119

C₄×M₁₆

—

120

C₂×C₄×C₈

—

121

#341

—

122

#335

—

146

—

123

#334

—

146

—

124

#334

—

146

—

125

#334

—

146

—

126

#322

—

146

—

127

C₈²

—

146

—

128

C₂×M₁₆

—

129

C₂×M₁₆

C₂×#64408

yes

—

130

C₂²×C₈

—

131

C₂²×C₈

—

146

—

132

C₂²×C₈

C₂×#64408

yes

characteristic

133

C₂×C₄²

—

134

#108

—

146

—

135

#108

#10596833

yes

complemented

136

C₄⋊C₈

—

137

C₄⋊C₈

—

146

—

138

C₄⋊C₈

—

146

—

139

C₄⋊C₈

—

146

—

140

C₂²⋊C₈

—

141

C₂²⋊C₈

—

146

—

142

#97

—

146

—

143

C₄×C₈

—

144

C₄×C₈

—

145

C₄×C₈

—

146

C₄×C₈

—

147

C₄×C₈

—

146

—

148

C₄×C₈

—

146

—

149

C₄×C₈

—

146

—

150

C₂²×C₄

—

151

C₂²×C₄

—

152

C₂²×C₄

C₂²×#64408

yes

characteristic

153

M₁₆

—

154

M₁₆

—

146

—

155

M₁₆

—

146

—

156

M₁₆

C₂×#10596833

yes

—

157

C₂×C₈

—

158

C₂×C₈

—

292

—

159

C₂×C₈

—

584

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

146

—

163

C₂×C₈

—

146

—

164

C₂×C₈

—

146

—

165

C₂×C₈

—

146

—

166

C₂×C₈

—

146

—

167

C₂×C₈

C₄⋊#64408

yes

characteristic

168

C₂×C₈

C₄⋊#64408

yes

characteristic

169

C₂×C₈

C₂×#10596833

yes

—

170

C₂×C₈

#11924038

yes

—

171

C₄²

—

172

C₄²

—

173

C₄²

—

174

C₄²

—

175

C₂³

Q₈×#64408

yes

characteristic

176

C₂×C₄

—

292

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

146

—

182

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

183

C₂×C₄

D₄×#64408

yes

characteristic

184

C₂×C₄

C₂²×#10596833

yes

characteristic

185

C₂×C₄

C₂×#11924038

yes

characteristic

186

C₂×C₄

#14556678

yes

characteristic

187

C₂×C₄

#14556679

yes

characteristic

188

C₈

—

189

C₈

—

292

—

190

C₈

—

292

—

191

C₈

—

292

—

192

C₈

—

584

—

193

C₈

—

146

—

194

C₈

—

146

—

195

C₈

C₄⋊#10596833

yes

—

196

C₈

C₄⋊#10596833

yes

—

197

C₂²

—

198

C₂²

—

199

C₂²

#17989538

yes

characteristic

200

C₂²

Q₈×#10596833

yes

characteristic

201

C₂²

#17989637

yes

characteristic

202

C₄

—

203

C₄

—

146

—

204

C₄

—

146

—

205

C₄

—

146

—

206

C₄

C₂×C₄⋊#10596833

yes

characteristic

207

C₄

#17989594

yes

characteristic

208

C₄

D₄×#10596833

yes

characteristic

209

C₄

#17989621

yes

characteristic

210

C₂

—

211

C₂

—

212

C₂

#22626028

yes

characteristic

213

C₂

#22626048

yes

characteristic

214

C₂

#22626050

yes

characteristic

	subgroup	G/H
center	C₂²	#17989538
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×#108×C₇₃	C₄
socle	C₂²×C₇₃	#389
prime radical	C₂³×C₇₃	C₄×Q₈
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	C₂×#108	#64408
hypocenter	C₇₃	#9607
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂²

#17989538

commutator

C₂×C₄×C₇₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#64408

Fitting

C₂×#108×C₇₃

C₄

socle

C₂²×C₇₃

#389

prime radical

C₂³×C₇₃

C₄×Q₈

cosocle

C₂×C₄×Dic₇₃

C₂³

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
41. #108×C₇₃	C₈	semidirect
113. C₇₃	#9607	semidirect
135. #108	#10596833	semidirect

H ≅ G/N

type

41. #108×C₇₃

C₈

semidirect

113. C₇₃

#9607

semidirect

135. #108

#10596833

semidirect

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

subgroup	index
#22625482	2
C₂×#17989448	2
#22625134	2
#22624777	2
#108×Dic₇₃	2
#9607	73

subgroup

index

#108×Dic₇₃

Subgroups — #34007516

Conjugacy classes of subgroups

Complements

Maximal subgroups