Group #32973138 — subgroups

Subgroups — #32973138

Order 18688.

320 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 99 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	#22609969	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988352	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11924012	—	2	1	no	—
5	C₂×#14556527	C₄	1	2	yes	—
6	#17988551	—	2	1	no	—
7	#17988352	—	2	1	no	—
8	#17988352	—	2	1	no	—
9	#17988352	C₄	1	2	yes	—
10	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₇₃	—	2	1	no	—
12	C₂²⋊C₁₆×C₇₃	—	2	1	no	—
13	C₂²⋊C₁₆×C₇₃	C₄	1	2	yes	—
14	C₂×#11924012	—	2	4	no	—
15	C₂×#11924012	—	2	2	no	—
16	C₂×#11924012	—	2	2	no	—
17	C₂×#11924012	—	4	2	no	—
18	C₂×#11924012	—	4	1	no	—
19	C₂²×#10596831	—	4	1	no	—
20	#14556527	—	4	2	no	—
21	C₂²⋊#10596831	—	4	1	no	—
22	C₂²⋊#10596831	—	4	1	no	—
23	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₇₃	—	4	2	no	—
25	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₃	Q₈	1	1	yes	characteristic
31	C₂×C₁₆×C₇₃	—	2	2	no	—
32	C₂×C₁₆×C₇₃	—	4	2	no	—
33	#11924012	—	2	4	no	—
34	#11924012	—	4	4	no	—
35	#11924012	—	4	2	no	—
36	#11924012	—	4	2	no	—
37	#11924012	—	4	2	no	—
38	C₂×#10596831	—	4	4	no	—
39	C₂×#10596831	—	4	2	no	—
40	C₂×#10596831	—	4	2	no	—
41	C₂⁴×C₇₃	Q₁₆	1	1	yes	characteristic
42	C₂²×C₄×C₇₃	—	2	2	no	—
43	C₂²×C₄×C₇₃	—	2	2	no	—
44	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₇₃	—	2	2	no	—
51	C₂×C₈×C₇₃	—	2	2	no	—
52	C₂×C₈×C₇₃	—	4	1	no	—
53	C₂×C₈×C₇₃	—	4	1	no	—
54	C₂×C₈×C₇₃	—	4	1	no	—
55	C₂×C₈×C₇₃	—	4	1	no	—
56	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
57	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
60	C₁₆×C₇₃	—	4	2	no	—
61	C₂³×C₇₃	—	2	2	no	—
62	C₂³×C₇₃	—	2	2	no	—
63	C₂³×C₇₃	—	4	1	no	—
64	C₂³×C₇₃	#103	1	1	yes	characteristic
65	C₂³×C₇₃	#103	1	1	yes	characteristic
66	C₂³×C₇₃	#107	1	1	yes	characteristic
67	C₂×C₄×C₇₃	—	2	2	no	—
68	C₂×C₄×C₇₃	—	2	2	no	—
69	C₂×C₄×C₇₃	—	2	2	no	—
70	C₂×C₄×C₇₃	—	2	1	no	—
71	C₂×C₄×C₇₃	—	2	1	no	—
72	C₂×C₄×C₇₃	—	2	1	no	—
73	C₂×C₄×C₇₃	—	2	1	no	—
74	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
75	C₂×C₄×C₇₃	#99	1	2	yes	—
76	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
77	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
78	C₂×C₄×C₇₃	#104	1	2	yes	—
79	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
80	C₈×C₇₃	—	2	1	no	—
81	C₈×C₇₃	—	2	1	no	—
82	C₈×C₇₃	—	2	1	no	—
83	C₈×C₇₃	—	2	1	no	—
84	#10596831	—	4	4	no	—
85	#10596831	—	8	2	no	—
86	C₂²×C₇₃	—	2	1	no	—
87	C₂²×C₇₃	—	2	1	no	—
88	C₂²×C₇₃	—	2	1	no	—
89	C₂²×C₇₃	—	2	1	no	—
90	C₂²×C₇₃	—	2	1	no	—
91	C₂²×C₇₃	—	2	1	no	—
92	C₂²×C₇₃	—	2	1	no	—
93	C₂²×C₇₃	—	4	4	no	—
94	C₂²×C₇₃	#328	1	2	yes	—
95	C₂²×C₇₃	#340	1	1	yes	characteristic
96	C₂²×C₇₃	#368	1	2	yes	—
97	C₄×C₇₃	—	2	1	no	—
98	C₄×C₇₃	—	2	1	no	—
99	C₄×C₇₃	#327	1	2	yes	—
100	C₄×C₇₃	#339	1	1	yes	characteristic
101	C₄×C₇₃	#342	1	1	yes	characteristic
102	#7261	—	73	1	no	maximal, Sylow
103	C₂×C₇₃	—	2	2	no	—
104	C₂×C₇₃	—	2	2	no	—
105	C₂×C₇₃	—	4	1	no	—
106	C₂×C₇₃	#1294	1	1	yes	characteristic
107	C₂×C₇₃	#1381	1	1	yes	characteristic
108	C₂×C₇₃	#1388	1	1	yes	characteristic
109	C₂×C₂²⋊C₁₆	D₇₃	1	1	yes	characteristic
110	#1810	—	73	1	no	—
111	C₂×#341	—	73	1	no	—
112	C₇₃	#7261	1	1	yes	characteristic, complemented, Sylow
113	C₂²×M₁₆	—	146	1	no	—
114	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
115	C₂²×C₁₆	—	2	1	no	—
116	C₂×#108	—	73	2	no	—
117	#407	—	146	1	no	—
118	C₂²⋊C₁₆	—	2	1	no	—
119	C₂²⋊C₁₆	Dic₇₃	1	2	yes	—
120	#341	—	73	2	no	—
121	#341	—	146	1	no	—
122	#341	—	146	1	no	—
123	C₂³×C₄	D₂₉₂	1	1	yes	characteristic
124	C₂×M₁₆	—	292	2	no	—
125	C₂×M₁₆	—	292	1	no	—
126	C₂×M₁₆	—	146	2	no	—
127	C₂×M₁₆	—	146	4	no	—
128	C₂×M₁₆	—	146	2	no	—
129	C₂²×C₈	—	4	2	no	—
130	C₂²×C₈	—	292	1	no	—
131	C₂²×C₈	Dic₁₄₆	1	1	yes	characteristic
132	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
133	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
134	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
135	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
136	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
137	C₂×C₁₆	—	2	2	no	—
138	C₂×C₁₆	—	4	2	no	—
139	#108	—	292	2	no	—
140	C₂²⋊C₈	—	292	1	no	—
141	C₂²⋊C₈	—	292	1	no	—
142	C₂⁴	Dic₂₉₂	1	1	yes	characteristic
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	#11924005	1	1	yes	characteristic
146	C₂²×C₄	#11924005	1	1	yes	characteristic
147	C₂²×C₄	C₄⋊Dic₇₃	1	1	yes	characteristic
148	C₂²×C₄	D₅₈₄	1	1	yes	characteristic
149	C₂²×C₄	#11924017	1	1	yes	characteristic
150	C₂²×C₄	#11924017	1	1	yes	characteristic
151	M₁₆	—	292	4	no	—
152	M₁₆	—	292	2	no	—
153	M₁₆	—	292	2	no	—
154	M₁₆	—	292	2	no	—
155	M₁₆	—	146	4	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	4	1	no	—
159	C₂×C₈	—	4	1	no	—
160	C₂×C₈	—	4	1	no	—
161	C₂×C₈	—	4	1	no	—
162	C₂×C₈	—	292	4	no	—
163	C₂×C₈	—	292	2	no	—
164	C₂×C₈	—	292	2	no	—
165	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
166	C₂×C₈	C₂²⋊Dic₇₃	1	1	yes	characteristic
167	C₂×C₈	#11924007	1	1	yes	characteristic
168	C₂×C₈	#11924007	1	1	yes	characteristic
169	C₁₆	—	4	2	no	—
170	C₂³	—	2	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	—	4	1	no	—
173	C₂³	#14556516	1	1	yes	characteristic
174	C₂³	#14556516	1	1	yes	characteristic
175	C₂³	#14556525	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	#14556498	1	1	yes	characteristic
184	C₂×C₄	#14556505	1	2	yes	—
185	C₂×C₄	#14556513	1	1	yes	characteristic
186	C₂×C₄	#14556513	1	1	yes	characteristic
187	C₂×C₄	#14556517	1	2	yes	—
188	C₂×C₄	#14556523	1	1	yes	characteristic
189	C₈	—	2	1	no	—
190	C₈	—	2	1	no	—
191	C₈	—	2	1	no	—
192	C₈	—	2	1	no	—
193	C₈	—	292	4	no	—
194	C₈	—	584	2	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	—	2	1	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	4	4	no	—
203	C₂²	#17988325	1	2	yes	—
204	C₂²	#17988350	1	1	yes	characteristic
205	C₂²	#17988417	1	2	yes	—
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	#17988322	1	2	yes	—
209	C₄	#17988348	1	1	yes	characteristic
210	C₄	#17988353	1	1	yes	characteristic
211	C₂	—	2	2	no	—
212	C₂	—	2	2	no	—
213	C₂	—	4	1	no	—
214	C₂	#22607811	1	1	yes	characteristic
215	C₂	#22608025	1	1	yes	characteristic
216	C₂	#22608043	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22609969

C₂

yes

maximal, characteristic

C₂×#17988352

C₂

yes

maximal, characteristic

C₂×C₂²⋊C₁₆×C₇₃

C₂

yes

maximal, characteristic

C₂²×#11924012

—

C₂×#14556527

C₄

yes

—

—

—

—

C₄

yes

—

C₂³×C₈×C₇₃

C₂²

yes

characteristic

C₂²×C₁₆×C₇₃

—

C₂²⋊C₁₆×C₇₃

—

C₂²⋊C₁₆×C₇₃

C₄

yes

—

—

—

—

—

—

—

—

—

—

C₂³×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

Q₈

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

—

—

—

—

—

—

—

—

C₂⁴×C₇₃

Q₁₆

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

D₈

yes

characteristic

C₂²×C₄×C₇₃

SD₁₆

yes

characteristic

C₂²×C₄×C₇₃

SD₁₆

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

yes

C₂³×C₇₃

yes

C₂³×C₇₃

yes

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#95

yes

characteristic

C₂×C₄×C₇₃

#99

yes

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

—

C₂×C₄×C₇₃

#106

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#328

yes

—

C₂²×C₇₃

#340

yes

characteristic

C₂²×C₇₃

#368

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#327

yes

—

100

C₄×C₇₃

#339

yes

characteristic

101

C₄×C₇₃

yes

102

—

103

C₂×C₇₃

—

104

C₂×C₇₃

—

105

C₂×C₇₃

—

106

C₂×C₇₃

#1294

yes

characteristic

107

C₂×C₇₃

#1381

yes

characteristic

108

C₂×C₇₃

#1388

yes

characteristic

109

C₂×C₂²⋊C₁₆

D₇₃

yes

characteristic

110

#1810

—

111

C₂×#341

—

112

C₇₃

#7261

yes

characteristic, complemented, Sylow

113

C₂²×M₁₆

—

146

—

114

C₂³×C₈

C₂×D₇₃

yes

characteristic

115

C₂²×C₁₆

—

116

C₂×#108

—

117

#407

—

146

—

118

C₂²⋊C₁₆

—

119

C₂²⋊C₁₆

Dic₇₃

yes

—

120

#341

—

121

#341

—

146

—

122

#341

—

146

—

123

C₂³×C₄

D₂₉₂

yes

characteristic

124

C₂×M₁₆

—

292

—

125

C₂×M₁₆

—

292

—

126

C₂×M₁₆

—

146

—

127

C₂×M₁₆

—

146

—

128

C₂×M₁₆

—

146

—

129

C₂²×C₈

—

130

C₂²×C₈

—

292

—

131

C₂²×C₈

Dic₁₄₆

yes

characteristic

132

C₂²×C₈

C₄×D₇₃

yes

characteristic

133

C₂²×C₈

C₄×D₇₃

yes

characteristic

134

C₂²×C₈

C₂×Dic₇₃

yes

characteristic

135

C₂²×C₈

C₂²⋊D₇₃

yes

characteristic

136

C₂²×C₈

C₂²⋊D₇₃

yes

characteristic

137

C₂×C₁₆

—

138

C₂×C₁₆

—

139

#108

—

292

—

140

C₂²⋊C₈

—

292

—

141

C₂²⋊C₈

—

292

—

142

C₂⁴

Dic₂₉₂

yes

characteristic

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

#11924005

yes

characteristic

146

C₂²×C₄

#11924005

yes

characteristic

147

C₂²×C₄

C₄⋊Dic₇₃

yes

characteristic

148

C₂²×C₄

D₅₈₄

yes

characteristic

149

C₂²×C₄

#11924017

yes

characteristic

150

C₂²×C₄

#11924017

yes

characteristic

151

M₁₆

—

292

—

152

M₁₆

—

292

—

153

M₁₆

—

292

—

154

M₁₆

—

292

—

155

M₁₆

—

146

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

292

—

163

C₂×C₈

—

292

—

164

C₂×C₈

—

292

—

165

C₂×C₈

C₄×Dic₇₃

yes

characteristic

166

C₂×C₈

C₂²⋊Dic₇₃

yes

characteristic

167

C₂×C₈

#11924007

yes

characteristic

168

C₂×C₈

#11924007

yes

characteristic

169

C₁₆

—

170

C₂³

—

171

C₂³

—

172

C₂³

—

173

C₂³

#14556516

yes

characteristic

174

C₂³

#14556516

yes

characteristic

175

C₂³

#14556525

yes

characteristic

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

#14556498

yes

characteristic

184

C₂×C₄

#14556505

yes

—

185

C₂×C₄

#14556513

yes

characteristic

186

C₂×C₄

#14556513

yes

characteristic

187

C₂×C₄

#14556517

yes

—

188

C₂×C₄

#14556523

yes

characteristic

189

C₈

—

190

C₈

—

191

C₈

—

192

C₈

—

193

C₈

—

292

—

194

C₈

—

584

—

195

C₂²

—

196

C₂²

—

197

C₂²

—

198

C₂²

—

199

C₂²

—

200

C₂²

—

201

C₂²

—

202

C₂²

—

203

C₂²

#17988325

yes

—

204

C₂²

#17988350

yes

characteristic

205

C₂²

#17988417

yes

—

206

C₄

—

207

C₄

—

208

C₄

#17988322

yes

—

209

C₄

#17988348

yes

characteristic

210

C₄

#17988353

yes

characteristic

211

C₂

—

212

C₂

—

213

C₂

—

214

C₂

#22607811

yes

characteristic

215

C₂

#22608025

yes

characteristic

216

C₂

#22608043

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556513
commutator	C₂×C₈×C₇₃	C₄²
Frattini	C₂³×C₈	C₂×D₇₃
Fitting	C₂×C₂²⋊C₁₆×C₇₃	C₂
socle	C₂²×C₇₃	#340
prime radical	C₂⁴×C₇₃	Q₁₆
cosocle	C₂³×C₈×C₇₃	C₂²
hypercenter	C₂×C₂²⋊C₁₆	D₇₃
hypocenter	C₇₃	#7261
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆

subgroup

G/H

center

C₂×C₄

#14556513

commutator

C₂×C₈×C₇₃

C₄²

Frattini

C₂³×C₈

C₂×D₇₃

Fitting

C₂×C₂²⋊C₁₆×C₇₃

C₂

socle

C₂²×C₇₃

#340

prime radical

C₂⁴×C₇₃

Q₁₆

cosocle

C₂³×C₈×C₇₃

C₂²

hypercenter

C₂×C₂²⋊C₁₆

D₇₃

hypocenter

C₇₃

#7261

LCS

G ▷ C₂×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆

N	H ≅ G/N	type
112. C₇₃	#7261	semidirect

H ≅ G/N

type

112. C₇₃

#7261

semidirect

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

subgroup	index
#22609969	2
C₂×#17988352	2
C₂×C₂²⋊C₁₆×C₇₃	2
#7261	73

subgroup

index

#22609969

C₂×#17988352

C₂×C₂²⋊C₁₆×C₇₃

#7261

Subgroups — #32973138

Conjugacy classes of subgroups

Complements

Maximal subgroups