Group #32985613 — subgroups

Subgroups — #32985613

Order 18688.

410 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 139 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×#17988992	C₂	1	1	yes	maximal, characteristic
2	#22610920	C₂	1	2	yes	maximal
3	#22608690	C₂	1	1	yes	maximal, characteristic
4	#22608615	C₂	1	2	yes	maximal
5	C₈×M₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556591	C₂²	1	2	yes	—
7	#17988992	—	2	2	no	—
8	#17988992	—	2	1	no	—
9	#17988992	—	2	1	no	—
10	C₂×#14556525	C₄	1	4	yes	—
11	#17988611	C₂²	1	2	yes	—
12	#17988355	—	2	4	no	—
13	#17988340	—	2	2	no	—
14	#17988334	—	2	4	no	—
15	#431×C₇₃	C₂²	1	1	yes	characteristic
16	C₄×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
18	#322×C₇₃	—	2	1	no	—
19	C₈²×C₇₃	—	2	1	no	—
20	C₂²×Dic₁₄₆	—	2	2	no	—
21	C₂×#11924012	—	2	2	no	—
22	#14556591	—	2	2	no	—
23	#14556591	—	2	2	no	—
24	#14556591	—	2	2	no	—
25	#14556591	—	2	2	no	—
26	C₂×C₄⋊Dic₇₃	—	2	2	no	—
27	C₂×C₄⋊Dic₇₃	C₂×C₄	1	4	yes	—
28	#14556525	—	2	4	no	—
29	#14556525	—	2	4	no	—
30	C₄⋊#10596831	—	2	2	no	—
31	C₄⋊#10596831	—	2	2	no	—
32	C₂×M₁₆×C₇₃	D₄	1	2	yes	—
33	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₇₃	D₄	1	2	yes	—
36	C₂²⋊C₈×C₇₃	—	2	1	no	—
37	#97×C₇₃	—	2	1	no	—
38	C₄×C₈×C₇₃	—	2	1	no	—
39	C₄×C₈×C₇₃	—	2	1	no	—
40	C₄×C₈×C₇₃	D₄	1	2	yes	—
41	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
42	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
43	C₂×Dic₁₄₆	—	4	2	no	—
44	C₂×Dic₁₄₆	—	4	2	no	—
45	C₂×Dic₁₄₆	—	8	4	no	—
46	C₂²×Dic₇₃	—	2	4	no	—
47	#11924012	—	4	4	no	—
48	C₂×#10596831	—	4	2	no	—
49	C₄⋊Dic₇₃	—	2	4	no	—
50	C₄⋊Dic₇₃	—	2	4	no	—
51	C₄⋊Dic₇₃	—	4	2	no	—
52	C₄⋊Dic₇₃	—	4	2	no	—
53	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
56	M₁₆×C₇₃	—	2	2	no	—
57	M₁₆×C₇₃	Q₁₆	1	4	yes	—
58	C₂×C₈×C₇₃	—	2	2	no	—
59	C₂×C₈×C₇₃	—	2	4	no	—
60	C₂×C₈×C₇₃	—	2	2	no	—
61	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₇₃	#41	1	1	yes	characteristic
67	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₇₃	—	2	2	no	—
69	C₂×C₄×C₇₃	—	2	2	no	—
70	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
73	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
74	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
75	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
77	C₂×C₄×C₇₃	#135	1	2	yes	—
78	C₂×Dic₇₃	—	4	4	no	—
79	C₂×Dic₇₃	—	8	4	no	—
80	Dic₁₄₆	—	8	4	no	—
81	Dic₁₄₆	—	8	4	no	—
82	C₈×C₇₃	—	2	2	no	—
83	C₈×C₇₃	—	2	4	no	—
84	C₈×C₇₃	—	4	2	no	—
85	C₈×C₇₃	#101	1	4	yes	—
86	#10596831	—	8	2	no	—
87	C₂²×C₇₃	—	2	1	no	—
88	C₂²×C₇₃	—	2	1	no	—
89	C₂²×C₇₃	#390	1	1	yes	characteristic
90	C₂²×C₇₃	C₄×Q₁₆	1	2	yes	—
91	C₄×C₇₃	—	2	1	no	—
92	C₄×C₇₃	—	2	1	no	—
93	C₄×C₇₃	C₂×#101	1	2	yes	—
94	C₄×C₇₃	#467	1	2	yes	—
95	C₄×C₇₃	#472	1	2	yes	—
96	C₄×C₇₃	#494	1	1	yes	characteristic
97	C₄×C₇₃	#495	1	1	yes	characteristic
98	Dic₇₃	—	8	4	no	—
99	#9801	—	73	1	no	maximal, Sylow
100	C₂×C₇₃	—	2	2	no	—
101	C₂×C₇₃	#1966	1	1	yes	characteristic
102	C₂×C₇₃	#1969	1	1	yes	characteristic
103	C₂×C₇₃	#1978	1	1	yes	characteristic
104	C₂×#500	—	73	1	no	—
105	#1997	—	73	2	no	—
106	#1567	—	73	1	no	—
107	#1549	—	73	2	no	—
108	C₈×M₁₆	D₇₃	1	1	yes	characteristic
109	C₇₃	#9801	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₄⋊Q₈	—	73	2	no	—
111	#500	—	146	2	no	—
112	#500	—	146	1	no	—
113	#500	—	146	1	no	—
114	#431	C₂×D₇₃	1	1	yes	characteristic
115	C₂×#107	—	73	4	no	—
116	#423	—	73	2	no	—
117	C₄×M₁₆	C₂×D₇₃	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
119	#343	—	146	4	no	—
120	#335	—	146	2	no	—
121	#332	—	146	4	no	—
122	#322	—	2	1	no	—
123	C₈²	—	2	1	no	—
124	C₂²×Q₈	—	146	2	no	—
125	C₂×M₁₆	—	146	2	no	—
126	C₂×M₁₆	D₂₉₂	1	2	yes	—
127	C₂²×C₈	C₄×D₇₃	1	2	yes	—
128	C₄⋊Q₈	—	146	2	no	—
129	C₄⋊Q₈	—	146	2	no	—
130	C₄⋊Q₈	—	146	2	no	—
131	C₄⋊Q₈	—	146	2	no	—
132	C₂×C₄⋊C₄	—	73	4	no	—
133	C₂×C₄⋊C₄	—	146	2	no	—
134	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
135	#107	—	146	4	no	—
136	#107	—	146	4	no	—
137	C₄⋊C₈	—	146	2	no	—
138	C₄⋊C₈	—	146	2	no	—
139	C₄⋊C₈	D₂₉₂	1	2	yes	—
140	C₂²⋊C₈	—	2	1	no	—
141	#97	—	2	1	no	—
142	C₄×C₈	—	2	1	no	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	D₂₉₂	1	2	yes	—
145	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
146	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
147	C₂×Q₈	—	292	2	no	—
148	C₂×Q₈	—	292	2	no	—
149	C₂×Q₈	—	584	4	no	—
150	C₂²×C₄	—	146	4	no	—
151	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
152	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
153	C₂²×C₄	#11924028	1	1	yes	characteristic
154	M₁₆	—	2	2	no	—
155	M₁₆	—	292	4	no	—
156	M₁₆	Dic₂₉₂	1	4	yes	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	292	2	no	—
161	C₂×C₈	#11924005	1	2	yes	—
162	C₂×C₈	#11924005	1	2	yes	—
163	C₄⋊C₄	—	292	2	no	—
164	C₄⋊C₄	—	292	2	no	—
165	C₄⋊C₄	—	146	4	no	—
166	C₄⋊C₄	—	146	4	no	—
167	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
168	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
169	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
170	C₄²	#11924028	1	1	yes	characteristic
171	C₂³	C₄×D₂₉₂	1	1	yes	characteristic
172	Q₈	—	584	4	no	—
173	Q₈	—	584	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	292	4	no	—
177	C₂×C₄	—	584	4	no	—
178	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
179	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
180	C₂×C₄	#14556565	1	1	yes	characteristic
181	C₂×C₄	#14556565	1	1	yes	characteristic
182	C₂×C₄	#14556579	1	1	yes	characteristic
183	C₂×C₄	#14556588	1	1	yes	characteristic
184	C₂×C₄	C₂×Dic₂₉₂	1	2	yes	—
185	C₂×C₄	#14556613	1	2	yes	—
186	C₈	—	2	2	no	—
187	C₈	—	2	4	no	—
188	C₈	—	4	2	no	—
189	C₈	—	584	2	no	—
190	C₈	#14556509	1	4	yes	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	#17988489	1	1	yes	characteristic
194	C₂²	C₄×Dic₂₉₂	1	2	yes	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	584	4	no	—
198	C₄	C₂×#14556509	1	2	yes	—
199	C₄	#17988828	1	2	yes	—
200	C₄	#17988857	1	2	yes	—
201	C₄	#17988967	1	1	yes	characteristic
202	C₄	#17988971	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	#22610775	1	1	yes	characteristic
205	C₂	#22610792	1	1	yes	characteristic
206	C₂	#22610841	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988992

C₂

yes

maximal, characteristic

#22610920

C₂

yes

maximal

#22608690

C₂

yes

maximal, characteristic

#22608615

C₂

yes

maximal

C₈×M₁₆×C₇₃

C₂

yes

maximal, characteristic

C₂×#14556591

C₂²

yes

—

—

—

—

C₄

yes

—

#17988611

C₂²

yes

—

#17988355

—

#17988340

—

#17988334

—

#431×C₇₃

C₂²

yes

characteristic

C₄×M₁₆×C₇₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#322×C₇₃

—

C₈²×C₇₃

—

C₂²×Dic₁₄₆

—

—

—

—

—

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

C₂×C₄

yes

—

—

—

—

—

C₂×M₁₆×C₇₃

D₄

yes

—

C₂²×C₈×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

D₄

yes

—

C₂²⋊C₈×C₇₃

—

#97×C₇₃

—

C₄×C₈×C₇₃

—

C₄×C₈×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

#11924012

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

#41

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

Q₁₆

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

#41

yes

characteristic

C₂³×C₇₃

C₄×D₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄×D₄

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

C₂×Q₁₆

yes

—

C₂×C₄×C₇₃

#135

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

#101

yes

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#390

yes

characteristic

C₂²×C₇₃

C₄×Q₁₆

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×#101

yes

—

C₄×C₇₃

#467

yes

—

C₄×C₇₃

#472

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

#9801

—

maximal, Sylow

100

C₂×C₇₃

—

101

C₂×C₇₃

#1966

yes

characteristic

102

C₂×C₇₃

#1969

yes

characteristic

103

C₂×C₇₃

#1978

yes

characteristic

104

C₂×#500

—

105

#1997

—

106

#1567

—

107

#1549

—

108

C₈×M₁₆

D₇₃

yes

characteristic

109

C₇₃

#9801

yes

characteristic, complemented, Sylow

110

C₂×C₄⋊Q₈

—

111

#500

—

146

—

112

#500

—

146

—

113

#500

—

146

—

114

#431

C₂×D₇₃

yes

characteristic

115

C₂×#107

—

116

#423

—

117

C₄×M₁₆

C₂×D₇₃

yes

characteristic

118

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

119

#343

—

146

—

120

#335

—

146

—

121

#332

—

146

—

122

#322

—

123

C₈²

—

124

C₂²×Q₈

—

146

—

125

C₂×M₁₆

—

146

—

126

C₂×M₁₆

D₂₉₂

yes

—

127

C₂²×C₈

C₄×D₇₃

yes

—

128

C₄⋊Q₈

—

146

—

129

C₄⋊Q₈

—

146

—

130

C₄⋊Q₈

—

146

—

131

C₄⋊Q₈

—

146

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

146

—

134

C₂×C₄²

C₂²×D₇₃

yes

characteristic

135

#107

—

146

—

136

#107

—

146

—

137

C₄⋊C₈

—

146

—

138

C₄⋊C₈

—

146

—

139

C₄⋊C₈

D₂₉₂

yes

—

140

C₂²⋊C₈

—

141

#97

—

142

C₄×C₈

—

143

C₄×C₈

—

144

C₄×C₈

D₂₉₂

yes

—

145

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

146

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

147

C₂×Q₈

—

292

—

148

C₂×Q₈

—

292

—

149

C₂×Q₈

—

584

—

150

C₂²×C₄

—

146

—

151

C₂²×C₄

C₂×C₄×D₇₃

yes

characteristic

152

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

153

C₂²×C₄

#11924028

yes

characteristic

154

M₁₆

—

155

M₁₆

—

292

—

156

M₁₆

Dic₂₉₂

yes

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

292

—

161

C₂×C₈

#11924005

yes

—

162

C₂×C₈

#11924005

yes

—

163

C₄⋊C₄

—

292

—

164

C₄⋊C₄

—

292

—

165

C₄⋊C₄

—

146

—

166

C₄⋊C₄

—

146

—

167

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

168

C₄²

C₂×D₂₉₂

yes

characteristic

169

C₄²

C₂×D₂₉₂

yes

characteristic

170

C₄²

#11924028

yes

characteristic

171

C₂³

C₄×D₂₉₂

yes

characteristic

172

Q₈

—

584

—

173

Q₈

—

584

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

292

—

177

C₂×C₄

—

584

—

178

C₂×C₄

C₂×#11924005

yes

characteristic

179

C₂×C₄

C₄×D₂₉₂

yes

characteristic

180

C₂×C₄

#14556565

yes

characteristic

181

C₂×C₄

#14556565

yes

characteristic

182

C₂×C₄

#14556579

yes

characteristic

183

C₂×C₄

#14556588

yes

characteristic

184

C₂×C₄

C₂×Dic₂₉₂

yes

—

185

C₂×C₄

#14556613

yes

—

186

C₈

—

187

C₈

—

188

C₈

—

189

C₈

—

584

—

190

C₈

#14556509

yes

—

191

C₂²

—

192

C₂²

—

193

C₂²

#17988489

yes

characteristic

194

C₂²

C₄×Dic₂₉₂

yes

—

195

C₄

—

196

C₄

—

197

C₄

—

584

—

198

C₄

C₂×#14556509

yes

—

199

C₄

#17988828

yes

—

200

C₄

#17988857

yes

—

201

C₄

#17988967

yes

characteristic

202

C₄

#17988971

yes

characteristic

203

C₂

—

204

C₂

#22610775

yes

characteristic

205

C₂

#22610792

yes

characteristic

206

C₂

#22610841

yes

characteristic

	subgroup	G/H
center	C₂²	#17988489
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₈×M₁₆×C₇₃	C₂
socle	C₂²×C₇₃	#390
prime radical	C₂³×C₇₃	C₄×D₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₈×M₁₆	D₇₃
hypocenter	C₇₃	#9801
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆

subgroup

G/H

center

C₂²

#17988489

commutator

C₂²×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₈×M₁₆×C₇₃

C₂

socle

C₂²×C₇₃

#390

prime radical

C₂³×C₇₃

C₄×D₄

cosocle

C₂×C₄²×C₇₃

C₂³

hypercenter

C₈×M₁₆

D₇₃

hypocenter

C₇₃

#9801

LCS

G ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆

N	H ≅ G/N	type
109. C₇₃	#9801	semidirect

H ≅ G/N

type

109. C₇₃

#9801

semidirect

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

subgroup	index
C₂×#17988992	2
#22610920	2
#22608690	2
#22608615	2
C₈×M₁₆×C₇₃	2
#9801	73

subgroup

index

C₈×M₁₆×C₇₃

Subgroups — #32985613

Conjugacy classes of subgroups

Complements

Maximal subgroups