Group #34005537 — subgroups

Subgroups — #34005537

Order 18688.

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

normal subgroups only

This group has 410 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	#22626824	C₂	1	1	yes	maximal, characteristic
2	#22626100	C₂	1	1	yes	maximal, characteristic
3	#22625482	C₂	1	2	yes	maximal
4	#22624764	C₂	1	2	yes	maximal
5	#101×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	Q₈×#10596833	—	2	2	no	—
7	Q₈×#10596833	—	2	2	no	—
8	#17989603	—	2	2	no	—
9	#17989603	C₂²	1	2	yes	—
10	C₂×C₄⋊#10596833	C₂²	1	1	yes	characteristic
11	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
12	#17989453	—	2	2	no	—
13	#17989450	—	2	2	no	—
14	#17989448	—	2	2	no	—
15	C₂×Q₈×Dic₇₃	C₂²	1	1	yes	characteristic
16	C₂×#14556510	C₄	1	1	yes	characteristic
17	M₁₆×Dic₇₃	C₂²	1	2	yes	—
18	#17988355	C₄	1	2	yes	—
19	C₂×#101×C₇₃	C₄	1	1	yes	characteristic
20	C₂²×#10596833	—	2	1	no	—
21	C₂²×#10596833	—	2	1	no	—
22	C₄⋊#10596833	—	2	2	no	—
23	C₄⋊#10596833	—	2	2	no	—
24	C₄⋊#10596833	—	2	2	no	—
25	C₄⋊#10596833	D₄	1	2	yes	—
26	C₄⋊#10596833	D₄	1	2	yes	—
27	C₂²⋊#10596833	—	2	2	no	—
28	C₄×#10596833	—	2	4	no	—
29	C₄×#10596833	—	2	2	no	—
30	C₄×#10596833	—	2	2	no	—
31	C₂×#11924012	C₂×C₄	1	2	yes	—
32	Q₈×Dic₇₃	—	2	2	no	—
33	Q₈×Dic₇₃	—	2	2	no	—
34	C₂×C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₇₃	Q₈	1	2	yes	—
36	C₂×C₄×Dic₇₃	D₄	1	2	yes	—
37	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
38	#14556510	—	2	2	no	—
39	#14556501	—	2	2	no	—
40	C₈×Dic₇₃	—	2	2	no	—
41	C₂²×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
43	#101×C₇₃	C₈	1	4	yes	complemented
44	C₂×#10596833	—	2	2	no	—
45	C₂×#10596833	—	2	2	no	—
46	C₂×#10596833	—	2	2	no	—
47	C₂×#10596833	—	4	1	no	—
48	C₂×#10596833	—	4	1	no	—
49	C₂×#10596833	—	4	2	no	—
50	C₂×#10596833	—	4	1	no	—
51	C₂²×Dic₇₃	C₄⋊C₄	1	2	yes	—
52	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
54	#11924012	—	2	2	no	—
55	#11924012	—	2	2	no	—
56	C₂×#10596831	—	2	2	no	—
57	C₄⋊Dic₇₃	—	2	2	no	—
58	C₄⋊Dic₇₃	—	2	4	no	—
59	C₄×Dic₇₃	—	2	2	no	—
60	C₄×Dic₇₃	—	2	2	no	—
61	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
62	C₄×Dic₇₃	#41	1	2	yes	—
63	C₂×Q₈×C₇₃	—	2	2	no	—
64	C₂×Q₈×C₇₃	—	2	2	no	—
65	C₂×Q₈×C₇₃	C₂×C₈	1	2	yes	—
66	C₂×Q₈×C₇₃	M₁₆	1	2	yes	—
67	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
69	M₁₆×C₇₃	—	2	2	no	—
70	M₁₆×C₇₃	C₂×C₈	1	4	yes	—
71	C₂×C₈×C₇₃	—	2	2	no	—
72	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	Q₈×C₇₃	—	2	4	no	—
74	Q₈×C₇₃	—	2	4	no	—
75	C₂×C₄×C₇₃	—	2	2	no	—
76	C₂×C₄×C₇₃	—	2	2	no	—
77	C₂×C₄×C₇₃	C₄⋊C₈	1	4	yes	—
78	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₇₃	#131	1	2	yes	—
82	C₂×Dic₇₃	—	2	1	no	—
83	C₂×Dic₇₃	—	2	1	no	—
84	C₂×Dic₇₃	—	2	2	no	—
85	C₂×Dic₇₃	—	4	1	no	—
86	C₂×Dic₇₃	—	4	2	no	—
87	C₂×Dic₇₃	#120	1	1	yes	characteristic
88	C₂×Dic₇₃	C₂²⋊Q₈	1	2	yes	—
89	C₂×Dic₇₃	#123	1	1	yes	characteristic
90	#10596833	—	4	2	no	—
91	#10596833	—	4	2	no	—
92	#10596833	—	4	2	no	—
93	#10596833	—	8	1	no	—
94	C₈×C₇₃	—	2	4	no	—
95	#10596831	—	4	2	no	—
96	C₂²×C₇₃	—	2	1	no	—
97	C₂²×C₇₃	—	2	1	no	—
98	C₂²×C₇₃	#385	1	1	yes	characteristic
99	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
100	C₂²×C₇₃	#424	1	1	yes	characteristic
101	C₄×C₇₃	—	2	4	no	—
102	C₄×C₇₃	C₈×D₄	1	2	yes	—
103	C₄×C₇₃	#435	1	2	yes	—
104	Dic₇₃	—	2	1	no	—
105	Dic₇₃	—	2	2	no	—
106	Dic₇₃	—	4	2	no	—
107	Dic₇₃	#454	1	1	yes	characteristic
108	Dic₇₃	#456	1	1	yes	characteristic
109	#8836	—	73	1	no	maximal, Sylow
110	C₂×C₇₃	—	2	1	no	—
111	C₂×C₇₃	—	2	1	no	—
112	C₂×C₇₃	#1850	1	1	yes	characteristic
113	C₂×C₇₃	#1857	1	1	yes	characteristic
114	C₂×C₇₃	#1858	1	1	yes	characteristic
115	#2982	—	73	1	no	—
116	#1984	—	73	1	no	—
117	C₄×#101	—	73	1	no	—
118	#1565	—	73	2	no	—
119	#1278	—	73	2	no	—
120	C₇₃	#8836	1	1	yes	characteristic, complemented, Sylow
121	C₂×C₄×Q₈	—	73	1	no	—
122	C₈×Q₈	—	146	2	no	—
123	C₈×Q₈	—	146	2	no	—
124	#431	—	73	2	no	—
125	#431	—	146	2	no	—
126	C₂×C₄⋊C₈	—	73	1	no	—
127	C₂×#101	—	73	1	no	—
128	C₂×#101	#64408	1	1	yes	characteristic
129	C₄×M₁₆	—	73	2	no	—
130	C₂×C₄×C₈	—	73	1	no	—
131	#343	—	73	2	no	—
132	#336	—	146	2	no	—
133	#335	—	146	2	no	—
134	#334	—	146	2	no	—
135	C₂²×Q₈	C₂×#64408	1	1	yes	characteristic
136	C₂×M₁₆	—	73	2	no	—
137	C₂×M₁₆	C₂×#64408	1	2	yes	—
138	C₂²×C₈	—	146	1	no	—
139	C₂²×C₈	—	146	1	no	—
140	C₄×Q₈	—	146	2	no	—
141	C₄×Q₈	—	146	2	no	—
142	C₂×C₄⋊C₄	—	73	2	no	—
143	C₂×C₄⋊C₄	—	73	1	no	—
144	C₂×C₄²	—	73	1	no	—
145	C₂×C₄²	—	73	2	no	—
146	C₄⋊C₈	—	73	2	no	—
147	C₄⋊C₈	—	73	2	no	—
148	C₄⋊C₈	—	146	2	no	—
149	C₄⋊C₈	—	146	2	no	—
150	C₄⋊C₈	—	146	2	no	—
151	#101	—	146	2	no	—
152	#101	#10596833	1	4	yes	complemented
153	C₂²⋊C₈	—	146	2	no	—
154	#97	—	146	2	no	—
155	C₄×C₈	—	146	2	no	—
156	C₄×C₈	—	146	2	no	—
157	C₄×C₈	—	146	2	no	—
158	C₄×C₈	—	146	4	no	—
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	—	2	2	no	—
161	C₂×Q₈	C₂×#10596833	1	2	yes	—
162	C₂×Q₈	#11924038	1	2	yes	—
163	C₂²×C₄	—	73	1	no	—
164	C₂²×C₄	—	73	2	no	—
165	C₂²×C₄	—	73	1	no	—
166	C₂²×C₄	C₄⋊#64408	1	2	yes	—
167	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
168	M₁₆	—	2	2	no	—
169	M₁₆	—	146	2	no	—
170	M₁₆	—	146	2	no	—
171	M₁₆	C₂×#10596833	1	4	yes	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	292	1	no	—
174	C₂×C₈	—	292	1	no	—
175	C₂×C₈	—	292	1	no	—
176	C₂×C₈	—	292	2	no	—
177	C₂×C₈	—	146	2	no	—
178	C₂×C₈	—	146	2	no	—
179	C₂×C₈	—	146	2	no	—
180	C₂×C₈	—	146	2	no	—
181	C₄⋊C₄	—	146	4	no	—
182	C₄⋊C₄	—	146	2	no	—
183	C₄²	—	73	2	no	—
184	C₄²	—	73	2	no	—
185	C₄²	—	146	2	no	—
186	C₄²	—	146	2	no	—
187	C₂³	C₂×C₄⋊#64408	1	1	yes	characteristic
188	Q₈	—	2	4	no	—
189	Q₈	—	2	4	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	292	2	no	—
193	C₂×C₄	—	292	1	no	—
194	C₂×C₄	—	73	1	no	—
195	C₂×C₄	—	73	1	no	—
196	C₂×C₄	—	73	2	no	—
197	C₂×C₄	—	146	2	no	—
198	C₂×C₄	—	146	1	no	—
199	C₂×C₄	—	146	1	no	—
200	C₂×C₄	C₄⋊#10596833	1	4	yes	—
201	C₂×C₄	D₄×#64408	1	2	yes	—
202	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
203	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
204	C₂×C₄	#14556678	1	2	yes	—
205	C₈	—	2	4	no	—
206	C₈	—	292	2	no	—
207	C₈	—	292	2	no	—
208	C₈	—	292	2	no	—
209	C₈	—	292	2	no	—
210	C₈	—	584	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	#17989534	1	1	yes	characteristic
214	C₂²	C₂×C₄⋊#10596833	1	1	yes	characteristic
215	C₂²	#17989593	1	1	yes	characteristic
216	C₄	—	2	4	no	—
217	C₄	—	292	2	no	—
218	C₄	—	73	1	no	—
219	C₄	—	73	1	no	—
220	C₄	—	146	1	no	—
221	C₄	—	146	2	no	—
222	C₄	D₄×#10596833	1	2	yes	—
223	C₄	#17989616	1	2	yes	—
224	C₂	—	2	1	no	—
225	C₂	—	2	1	no	—
226	C₂	#22625874	1	1	yes	characteristic
227	C₂	#22625887	1	1	yes	characteristic
228	C₂	#22625889	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626824

C₂

yes

maximal, characteristic

#22626100

C₂

yes

maximal, characteristic

#22625482

C₂

yes

maximal

#22624764

C₂

yes

maximal

#101×Dic₇₃

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

C₂×C₄⋊#10596833

C₂²

yes

characteristic

C₂×C₄×#10596833

C₂²

yes

—

—

—

C₂×Q₈×Dic₇₃

C₂²

yes

characteristic

C₂×#14556510

C₄

yes

characteristic

M₁₆×Dic₇₃

C₂²

yes

—

#17988355

C₄

yes

—

C₂×#101×C₇₃

C₄

yes

—

—

—

—

—

D₄

yes

—

C₄⋊#10596833

D₄

yes

—

—

—

—

—

C₂×C₄

yes

—

Q₈×Dic₇₃

—

Q₈×Dic₇₃

—

C₂×C₄⋊Dic₇₃

C₂×C₄

yes

characteristic

C₂×C₄⋊Dic₇₃

Q₈

yes

—

C₂×C₄×Dic₇₃

D₄

yes

—

C₂×C₄×Dic₇₃

C₂³

yes

characteristic

#14556510

—

#14556501

—

C₈×Dic₇₃

—

C₂²×Q₈×C₇₃

C₂×C₄

yes

characteristic

C₂×M₁₆×C₇₃

C₂×C₄

yes

—

#101×C₇₃

C₈

yes

—

—

—

—

—

—

—

C₂²×Dic₇₃

C₄⋊C₄

yes

—

C₂²×Dic₇₃

C₂×D₄

yes

characteristic

C₂²×Dic₇₃

C₂×Q₈

yes

—

—

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂×D₄

yes

—

C₄×Dic₇₃

#41

yes

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

C₂×C₈

yes

—

C₂×Q₈×C₇₃

M₁₆

yes

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

C₂×C₈

yes

—

C₂×C₈×C₇₃

—

C₂³×C₇₃

C₂×C₄⋊C₄

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄⋊C₈

yes

—

C₂×C₄×C₇₃

C₄×D₄

yes

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₃

#131

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

#120

yes

characteristic

C₂×Dic₇₃

C₂²⋊Q₈

yes

—

C₂×Dic₇₃

yes

—

—

—

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#385

yes

characteristic

C₂²×C₇₃

C₂×C₄⋊C₈

yes

characteristic

100

C₂²×C₇₃

#424

yes

characteristic

101

C₄×C₇₃

—

102

C₄×C₇₃

C₈×D₄

yes

—

103

C₄×C₇₃

#435

yes

—

104

Dic₇₃

—

105

Dic₇₃

—

106

Dic₇₃

—

107

Dic₇₃

#454

yes

characteristic

108

Dic₇₃

yes

109

—

110

C₂×C₇₃

—

111

C₂×C₇₃

—

112

C₂×C₇₃

#1850

yes

characteristic

113

C₂×C₇₃

#1857

yes

characteristic

114

C₂×C₇₃

#1858

yes

characteristic

115

#2982

—

116

#1984

—

117

C₄×#101

—

118

#1565

—

119

#1278

—

120

C₇₃

#8836

yes

characteristic, complemented, Sylow

121

C₂×C₄×Q₈

—

122

C₈×Q₈

—

146

—

123

C₈×Q₈

—

146

—

124

#431

—

125

#431

—

146

—

126

C₂×C₄⋊C₈

—

127

C₂×#101

—

128

C₂×#101

#64408

yes

characteristic

129

C₄×M₁₆

—

130

C₂×C₄×C₈

—

131

#343

—

132

#336

—

146

—

133

#335

—

146

—

134

#334

—

146

—

135

C₂²×Q₈

C₂×#64408

yes

characteristic

136

C₂×M₁₆

—

137

C₂×M₁₆

C₂×#64408

yes

—

138

C₂²×C₈

—

146

—

139

C₂²×C₈

—

146

—

140

C₄×Q₈

—

146

—

141

C₄×Q₈

—

146

—

142

C₂×C₄⋊C₄

—

143

C₂×C₄⋊C₄

—

144

C₂×C₄²

—

145

C₂×C₄²

—

146

C₄⋊C₈

—

147

C₄⋊C₈

—

148

C₄⋊C₈

—

146

—

149

C₄⋊C₈

—

146

—

150

C₄⋊C₈

—

146

—

151

#101

—

146

—

152

#101

#10596833

yes

complemented

153

C₂²⋊C₈

—

146

—

154

#97

—

146

—

155

C₄×C₈

—

146

—

156

C₄×C₈

—

146

—

157

C₄×C₈

—

146

—

158

C₄×C₈

—

146

—

159

C₂×Q₈

—

160

C₂×Q₈

—

161

C₂×Q₈

C₂×#10596833

yes

—

162

C₂×Q₈

#11924038

yes

—

163

C₂²×C₄

—

164

C₂²×C₄

—

165

C₂²×C₄

—

166

C₂²×C₄

C₄⋊#64408

yes

—

167

C₂²×C₄

C₂²×#64408

yes

characteristic

168

M₁₆

—

169

M₁₆

—

146

—

170

M₁₆

—

146

—

171

M₁₆

C₂×#10596833

yes

—

172

C₂×C₈

—

173

C₂×C₈

—

292

—

174

C₂×C₈

—

292

—

175

C₂×C₈

—

292

—

176

C₂×C₈

—

292

—

177

C₂×C₈

—

146

—

178

C₂×C₈

—

146

—

179

C₂×C₈

—

146

—

180

C₂×C₈

—

146

—

181

C₄⋊C₄

—

146

—

182

C₄⋊C₄

—

146

—

183

C₄²

—

184

C₄²

—

185

C₄²

—

146

—

186

C₄²

—

146

—

187

C₂³

C₂×C₄⋊#64408

yes

characteristic

188

Q₈

—

189

Q₈

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

292

—

193

C₂×C₄

—

292

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

146

—

198

C₂×C₄

—

146

—

199

C₂×C₄

—

146

—

200

C₂×C₄

C₄⋊#10596833

yes

—

201

C₂×C₄

D₄×#64408

yes

—

202

C₂×C₄

C₂²×#10596833

yes

characteristic

203

C₂×C₄

C₂×#11924038

yes

characteristic

204

C₂×C₄

#14556678

yes

—

205

C₈

—

206

C₈

—

292

—

207

C₈

—

292

—

208

C₈

—

292

—

209

C₈

—

292

—

210

C₈

—

584

—

211

C₂²

—

212

C₂²

—

213

C₂²

#17989534

yes

characteristic

214

C₂²

C₂×C₄⋊#10596833

yes

characteristic

215

C₂²

#17989593

yes

characteristic

216

C₄

—

217

C₄

—

292

—

218

C₄

—

219

C₄

—

220

C₄

—

146

—

221

C₄

—

146

—

222

C₄

D₄×#10596833

yes

—

223

C₄

#17989616

yes

—

224

C₂

—

225

C₂

—

226

C₂

#22625874

yes

characteristic

227

C₂

#22625887

yes

characteristic

228

C₂

#22625889

yes

characteristic

	subgroup	G/H
center	C₂²	#17989534
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×#101×C₇₃	C₄
socle	C₂²×C₇₃	#385
prime radical	C₂³×C₇₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	C₂×#101	#64408
hypocenter	C₇₃	#8836
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂²

#17989534

commutator

C₂×C₄×C₇₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#64408

Fitting

C₂×#101×C₇₃

C₄

socle

C₂²×C₇₃

#385

prime radical

C₂³×C₇₃

C₂×C₄⋊C₄

cosocle

C₂×C₄×Dic₇₃

C₂³

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
43. #101×C₇₃	C₈	semidirect
120. C₇₃	#8836	semidirect
152. #101	#10596833	semidirect

H ≅ G/N

type

43. #101×C₇₃

C₈

semidirect

120. C₇₃

#8836

semidirect

152. #101

#10596833

semidirect

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

subgroup	index
#22626824	2
#22626100	2
#22625482	2
#22624764	2
#101×Dic₇₃	2
#8836	73

subgroup

index

#101×Dic₇₃

Subgroups — #34005537

Conjugacy classes of subgroups

Complements

Maximal subgroups