Group #33004147 — subgroups

Subgroups — #33004147

Order 18688.

450 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 147 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	#22618644	C₂	1	1	yes	maximal, characteristic
2	#22611687	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#17988370	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14556597	C₂²	1	2	yes	—
6	#17989137	C₄	1	1	yes	characteristic
7	C₂×#14556531	C₂²	1	2	yes	—
8	#17988666	—	2	2	no	—
9	C₈×D₂₉₂	—	2	2	no	—
10	#17988611	C₄	1	1	yes	characteristic
11	#17988553	C₄	1	2	yes	—
12	C₄⋊#11924003	—	2	2	no	—
13	#17988370	—	2	8	no	—
14	C₂²×C₁₆×C₇₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
16	C₄×C₁₆×C₇₃	—	2	2	no	—
17	C₂×#11924028	C₂×C₄	1	2	yes	—
18	#14556597	—	2	4	no	—
19	#14556597	—	2	4	no	—
20	C₂×#11924012	C₂×C₄	1	2	yes	—
21	C₂×#11924011	—	2	2	no	—
22	C₂×C₈×D₇₃	—	2	2	no	—
23	#14556591	C₈	1	1	yes	characteristic
24	#14556588	C₈	1	1	yes	characteristic
25	#14556579	C₈	1	2	yes	—
26	#14556565	—	2	2	no	—
27	C₄×D₂₉₂	—	2	2	no	—
28	#14556531	—	2	4	no	—
29	C₂×#11924003	—	2	2	no	—
30	C₄⋊#10596831	—	2	2	no	—
31	#14556503	—	2	4	no	—
32	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₇₃	—	2	2	no	—
36	C₂×C₁₆×C₇₃	—	2	2	no	—
37	C₂×C₁₆×C₇₃	D₄	1	4	yes	complemented
38	C₄×C₈×C₇₃	D₄	1	2	yes	—
39	C₄×C₈×C₇₃	D₄	1	2	yes	—
40	#11924028	—	2	4	no	—
41	#11924028	—	2	4	no	—
42	C₂×Dic₁₄₆	C₂×C₈	1	2	yes	—
43	C₂×D₂₉₂	—	2	2	no	—
44	C₂×D₂₉₂	C₂×C₈	1	2	yes	—
45	C₂×C₂²⋊D₇₃	—	2	2	no	—
46	C₂×C₄×D₇₃	—	2	2	no	—
47	#11924012	—	2	4	no	—
48	#11924011	—	4	4	no	—
49	C₂×#10596831	—	2	2	no	—
50	C₈×D₇₃	—	4	2	no	—
51	C₈×D₇₃	—	4	2	no	—
52	C₄⋊Dic₇₃	—	2	2	no	—
53	#11924005	—	2	4	no	—
54	#11924003	—	4	2	no	—
55	C₂²×C₄×C₇₃	M₁₆	1	2	yes	—
56	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₇₃	—	2	2	no	—
58	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
59	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₇₃	#41	1	2	yes	—
63	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
65	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
66	C₁₆×C₇₃	—	2	4	no	—
67	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
68	C₂²×D₇₃	—	2	2	no	—
69	C₂×C₄×C₇₃	—	2	4	no	—
70	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
71	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
72	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₇₃	#131	1	2	yes	—
76	C₂²⋊D₇₃	—	4	4	no	—
77	C₂×Dic₇₃	—	2	2	no	—
78	D₂₉₂	—	2	4	no	—
79	D₂₉₂	—	4	4	no	—
80	C₄×D₇₃	—	4	2	no	—
81	C₄×D₇₃	—	4	2	no	—
82	Dic₁₄₆	—	2	4	no	—
83	C₈×C₇₃	—	2	2	no	—
84	C₈×C₇₃	#121	1	2	yes	—
85	C₈×C₇₃	#124	1	1	yes	characteristic
86	C₈×C₇₃	#127	1	1	yes	characteristic
87	#10596831	—	4	2	no	—
88	C₂²×C₇₃	—	2	2	no	—
89	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
90	C₂²×C₇₃	#436	1	2	yes	—
91	C₂×D₇₃	—	4	4	no	—
92	C₂×D₇₃	—	4	2	no	—
93	C₄×C₇₃	—	2	2	no	—
94	C₄×C₇₃	#350	1	4	yes	complemented
95	C₄×C₇₃	#390	1	1	yes	characteristic
96	C₄×C₇₃	#408	1	1	yes	characteristic
97	C₄×C₇₃	C₈×D₄	1	2	yes	—
98	Dic₇₃	—	4	2	no	—
99	#13191	—	73	1	no	maximal, Sylow
100	C₂×C₇₃	—	2	2	no	—
101	C₂×C₇₃	#1954	1	1	yes	characteristic
102	C₂×C₇₃	C₂×#350	1	2	yes	—
103	D₇₃	—	8	2	no	—
104	#2931	—	73	1	no	—
105	#2150	—	73	1	no	—
106	C₂×#350	—	73	4	no	—
107	C₂×C₄×C₁₆	D₇₃	1	1	yes	characteristic, complemented
108	C₇₃	#13191	1	1	yes	characteristic, complemented, Sylow
109	C₂×#131	—	73	2	no	—
110	#532	—	73	1	no	—
111	C₂×M₃₂	—	73	2	no	—
112	C₂²×C₁₆	C₂×D₇₃	1	2	yes	—
113	#436	—	146	2	no	—
114	C₈×D₄	—	146	2	no	—
115	#423	—	73	1	no	—
116	#408	—	73	2	no	—
117	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
118	C₄⋊C₁₆	—	146	2	no	—
119	#350	—	146	8	no	—
120	C₄×C₁₆	—	2	2	no	—
121	C₂×#41	—	73	2	no	—
122	#131	—	146	4	no	—
123	#131	—	146	4	no	—
124	C₂×M₁₆	—	73	2	no	—
125	C₂×M₁₆	—	146	2	no	—
126	C₂²×C₈	—	146	2	no	—
127	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
128	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
129	C₄⋊Q₈	—	73	1	no	—
130	#127	—	73	1	no	—
131	#124	—	73	2	no	—
132	#121	—	146	2	no	—
133	C₄×D₄	—	146	2	no	—
134	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
135	M₃₂	—	146	4	no	—
136	C₂×C₁₆	—	2	2	no	—
137	C₂×C₁₆	—	2	2	no	—
138	C₂×C₁₆	—	146	2	no	—
139	C₂×C₁₆	D₂₉₂	1	4	yes	complemented
140	C₄⋊C₈	—	146	2	no	—
141	C₂²⋊C₈	—	146	4	no	—
142	C₄×C₈	D₂₉₂	1	2	yes	—
143	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
144	#41	—	146	4	no	—
145	#41	—	146	4	no	—
146	C₂×Q₈	—	73	2	no	—
147	C₂×D₄	—	73	2	no	—
148	C₂×D₄	—	146	2	no	—
149	C₂×D₄	—	146	2	no	—
150	C₂²×C₄	—	146	2	no	—
151	C₂²×C₄	#11924011	1	2	yes	—
152	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
153	M₁₆	—	292	4	no	—
154	M₁₆	—	146	4	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	292	2	no	—
157	C₂×C₈	—	292	2	no	—
158	C₂×C₈	—	146	2	no	—
159	C₂×C₈	#11924005	1	2	yes	—
160	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
161	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
162	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
163	C₂×C₈	#11924028	1	2	yes	—
164	C₄⋊C₄	—	146	2	no	—
165	C₂²⋊C₄	—	146	4	no	—
166	C₄²	#11924005	1	2	yes	—
167	C₄²	C₈×D₇₃	1	1	yes	characteristic
168	C₄²	C₈×D₇₃	1	1	yes	characteristic
169	C₁₆	—	2	4	no	—
170	C₁₆	—	292	2	no	—
171	C₂³	—	146	2	no	—
172	C₂³	C₂×#11924011	1	1	yes	characteristic
173	Q₈	—	146	4	no	—
174	D₄	—	292	4	no	—
175	D₄	—	292	4	no	—
176	D₄	—	146	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	292	2	no	—
179	C₂×C₄	—	292	2	no	—
180	C₂×C₄	—	146	2	no	—
181	C₂×C₄	#14556503	1	2	yes	—
182	C₂×C₄	#14556503	1	2	yes	—
183	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
184	C₂×C₄	C₄×D₂₉₂	1	2	yes	—
185	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
186	C₂×C₄	#14556597	1	2	yes	—
187	C₈	—	2	2	no	—
188	C₈	—	292	2	no	—
189	C₈	#14556565	1	2	yes	—
190	C₈	#14556579	1	1	yes	characteristic
191	C₈	#14556588	1	1	yes	characteristic
192	C₂²	—	2	2	no	—
193	C₂²	—	292	2	no	—
194	C₂²	—	292	4	no	—
195	C₂²	C₂×#14556503	1	1	yes	characteristic
196	C₂²	#17988666	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	292	2	no	—
199	C₄	#17988370	1	4	yes	complemented
200	C₄	#17988489	1	1	yes	characteristic
201	C₄	#17988553	1	1	yes	characteristic
202	C₄	C₈×D₂₉₂	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	—	584	2	no	—
205	C₂	#22610725	1	1	yes	characteristic
206	C₂	C₂×#17988370	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22618644

C₂

yes

maximal, characteristic

#22611687

C₂

yes

maximal, characteristic, complemented

C₂×#17988370

C₂

yes

maximal, complemented

C₂×C₄×C₁₆×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#14556597

C₂²

yes

—

#17989137

C₄

yes

characteristic

C₂×#14556531

C₂²

yes

—

#17988666

—

C₈×D₂₉₂

—

#17988611

C₄

yes

characteristic

#17988553

C₄

yes

—

C₄⋊#11924003

—

#17988370

—

C₂²×C₁₆×C₇₃

C₂²

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

C₄×C₁₆×C₇₃

—

C₂×#11924028

C₂×C₄

yes

—

#14556597

—

#14556597

—

C₂×#11924012

C₂×C₄

yes

—

C₂×#11924011

—

C₂×C₈×D₇₃

—

#14556591

C₈

yes

characteristic

#14556588

C₈

yes

characteristic

#14556579

C₈

yes

—

#14556565

—

C₄×D₂₉₂

—

—

—

—

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

complemented

C₄×C₈×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

D₄

yes

—

#11924028

—

#11924028

—

C₂×Dic₁₄₆

C₂×C₈

yes

—

C₂×D₂₉₂

—

C₂×D₂₉₂

C₂×C₈

yes

—

C₂×C₂²⋊D₇₃

—

C₂×C₄×D₇₃

—

#11924012

—

#11924011

—

C₂×#10596831

—

C₈×D₇₃

—

C₈×D₇₃

—

C₄⋊Dic₇₃

—

#11924005

—

#11924003

—

C₂²×C₄×C₇₃

M₁₆

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

#41

yes

—

C₄²×C₇₃

C₂²⋊C₄

yes

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

C₂×M₁₆

yes

characteristic

C₂²×D₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

—

C₂×C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄×D₄

yes

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

#131

yes

—

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

#121

yes

—

C₈×C₇₃

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₇₃

#436

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

C₈×D₄

yes

—

Dic₇₃

—

#13191

—

maximal, Sylow

100

C₂×C₇₃

—

101

C₂×C₇₃

#1954

yes

characteristic

102

C₂×C₇₃

C₂×#350

yes

—

103

D₇₃

—

104

#2931

—

105

#2150

—

106

C₂×#350

—

107

C₂×C₄×C₁₆

D₇₃

yes

characteristic, complemented

108

C₇₃

#13191

yes

characteristic, complemented, Sylow

109

C₂×#131

—

110

#532

—

111

C₂×M₃₂

—

112

C₂²×C₁₆

C₂×D₇₃

yes

—

113

#436

—

146

—

114

C₈×D₄

—

146

—

115

#423

—

116

#408

—

117

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

118

C₄⋊C₁₆

—

146

—

119

#350

—

146

—

120

C₄×C₁₆

—

121

C₂×#41

—

122

#131

—

146

—

123

#131

—

146

—

124

C₂×M₁₆

—

125

C₂×M₁₆

—

146

—

126

C₂²×C₈

—

146

—

127

C₂²×C₈

C₄×D₇₃

yes

characteristic

128

C₂²×C₈

C₂²×D₇₃

yes

characteristic

129

C₄⋊Q₈

—

130

#127

—

131

#124

—

132

#121

—

146

—

133

C₄×D₄

—

146

—

134

C₂×C₄²

C₄×D₇₃

yes

characteristic

135

M₃₂

—

146

—

136

C₂×C₁₆

—

137

C₂×C₁₆

—

138

C₂×C₁₆

—

146

—

139

C₂×C₁₆

D₂₉₂

yes

complemented

140

C₄⋊C₈

—

146

—

141

C₂²⋊C₈

—

146

—

142

C₄×C₈

D₂₉₂

yes

—

143

C₄×C₈

C₂²⋊D₇₃

yes

—

144

#41

—

146

—

145

#41

—

146

—

146

C₂×Q₈

—

147

C₂×D₄

—

148

C₂×D₄

—

146

—

149

C₂×D₄

—

146

—

150

C₂²×C₄

—

146

—

151

C₂²×C₄

#11924011

yes

—

152

C₂²×C₄

C₂×C₄×D₇₃

yes

characteristic

153

M₁₆

—

292

—

154

M₁₆

—

146

—

155

C₂×C₈

—

156

C₂×C₈

—

292

—

157

C₂×C₈

—

292

—

158

C₂×C₈

—

146

—

159

C₂×C₈

#11924005

yes

—

160

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

161

C₂×C₈

C₂×D₂₉₂

yes

—

162

C₂×C₈

C₂×D₂₉₂

yes

characteristic

163

C₂×C₈

#11924028

yes

—

164

C₄⋊C₄

—

146

—

165

C₂²⋊C₄

—

146

—

166

C₄²

#11924005

yes

—

167

C₄²

C₈×D₇₃

yes

characteristic

168

C₄²

C₈×D₇₃

yes

characteristic

169

C₁₆

—

170

C₁₆

—

292

—

171

C₂³

—

146

—

172

C₂³

C₂×#11924011

yes

characteristic

173

Q₈

—

146

—

174

D₄

—

292

—

175

D₄

—

292

—

176

D₄

—

146

—

177

C₂×C₄

—

178

C₂×C₄

—

292

—

179

C₂×C₄

—

292

—

180

C₂×C₄

—

146

—

181

C₂×C₄

#14556503

yes

—

182

C₂×C₄

#14556503

yes

—

183

C₂×C₄

C₂×#11924005

yes

characteristic

184

C₂×C₄

C₄×D₂₉₂

yes

—

185

C₂×C₄

C₂×C₈×D₇₃

yes

characteristic

186

C₂×C₄

#14556597

yes

—

187

C₈

—

188

C₈

—

292

—

189

C₈

#14556565

yes

—

190

C₈

#14556579

yes

characteristic

191

C₈

#14556588

yes

characteristic

192

C₂²

—

193

C₂²

—

292

—

194

C₂²

—

292

—

195

C₂²

C₂×#14556503

yes

characteristic

196

C₂²

#17988666

yes

—

197

C₄

—

198

C₄

—

292

—

199

C₄

#17988370

yes

complemented

200

C₄

#17988489

yes

characteristic

201

C₄

#17988553

yes

characteristic

202

C₄

C₈×D₂₉₂

yes

—

203

C₂

—

204

C₂

—

584

—

205

C₂

#22610725

yes

characteristic

206

C₂

C₂×#17988370

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₂×C₄×C₁₆×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₂²⋊C₈
prime radical	#17989137	C₄
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₃
hypocenter	C₇₃	#13191
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₂×C₈

C₂×D₂₉₂

commutator

C₂×C₄×C₇₃

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₂×C₄×C₁₆×C₇₃

C₂

socle

C₂²×C₇₃

C₂×C₂²⋊C₈

prime radical

#17989137

C₄

cosocle

C₂²×C₈×C₇₃

C₂³

hypercenter

C₂×C₄×C₁₆

D₇₃

hypocenter

C₇₃

#13191

LCS

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

N	H ≅ G/N	type
2. #22611687	C₂	semidirect
3. C₂×#17988370	C₂	semidirect
4. C₂×C₄×C₁₆×C₇₃	C₂	semidirect
37. C₂×C₁₆×C₇₃	D₄	semidirect
94. C₄×C₇₃	#350	semidirect
107. C₂×C₄×C₁₆	D₇₃	semidirect
108. C₇₃	#13191	semidirect
139. C₂×C₁₆	D₂₉₂	semidirect
199. C₄	#17988370	semidirect

H ≅ G/N

type

2. #22611687

C₂

semidirect

3. C₂×#17988370

C₂

semidirect

4. C₂×C₄×C₁₆×C₇₃

C₂

semidirect

37. C₂×C₁₆×C₇₃

D₄

semidirect

94. C₄×C₇₃

#350

semidirect

107. C₂×C₄×C₁₆

D₇₃

semidirect

108. C₇₃

#13191

semidirect

139. C₂×C₁₆

D₂₉₂

semidirect

199. C₄

#17988370

semidirect

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

subgroup	index
#22618644	2
#22611687	2
C₂×#17988370	2
C₂×C₄×C₁₆×C₇₃	2
#13191	73

subgroup

index

C₂×C₄×C₁₆×C₇₃

Subgroups — #33004147

Conjugacy classes of subgroups

Complements

Maximal subgroups