Group #33241408 — subgroups

Subgroups — #33241408

Order 18688.

826 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 219 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×#17989019	C₂	1	4	yes	maximal, complemented
2	#22620359	C₂	1	2	yes	maximal, complemented
3	#22612202	C₂	1	4	yes	maximal, complemented
4	#22612197	C₂	1	4	yes	maximal, complemented
5	C₂×#346×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14556613	—	2	2	no	—
7	C₂×#14556613	C₂²	1	4	yes	—
8	#17989137	—	2	2	no	—
9	#17989019	—	2	16	no	—
10	C₂×Dic₅₈₄	C₂²	1	4	yes	—
11	C₂×#14556536	C₂²	1	8	yes	—
12	C₂×D₁₁₆₈	C₂²	1	4	yes	—
13	#17988971	—	2	2	no	—
14	#17988971	C₂²	1	4	yes	—
15	#17988967	C₂²	1	2	yes	—
16	#17988964	C₂²	1	2	yes	—
17	#17988960	—	2	2	no	—
18	C₂²×C₁₆×C₇₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
20	#346×C₇₃	C₂²	1	4	yes	—
21	C₂×#11924028	—	2	4	no	—
22	#14556613	—	2	16	no	—
23	#14556613	—	4	8	no	—
24	C₂×Dic₂₉₂	—	2	2	no	—
25	C₂×Dic₂₉₂	C₂³	1	4	yes	—
26	C₂×#11924017	—	2	4	no	—
27	C₂×#11924017	—	2	4	no	—
28	C₂×D₅₈₄	—	2	2	no	—
29	C₂×D₅₈₄	C₂³	1	4	yes	—
30	#14556591	—	2	2	no	—
31	#14556588	—	2	2	no	—
32	#14556579	—	2	4	no	—
33	#14556565	—	4	4	no	—
34	C₄×D₂₉₂	—	4	4	no	—
35	Dic₅₈₄	—	2	8	no	—
36	#14556536	—	2	16	no	—
37	D₁₁₆₈	—	2	8	no	—
38	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
41	C₂×C₁₆×C₇₃	D₄	1	8	yes	complemented
42	C₂×C₁₆×C₇₃	C₂³	1	4	yes	—
43	C₄×C₈×C₇₃	D₄	1	2	yes	—
44	C₄×C₈×C₇₃	C₂³	1	2	yes	—
45	#11924028	—	4	8	no	—
46	#11924028	—	4	8	no	—
47	C₂×Dic₁₄₆	—	2	4	no	—
48	C₂×D₂₉₂	—	2	4	no	—
49	C₂×D₂₉₂	—	4	4	no	—
50	C₂×C₂²⋊D₇₃	—	4	4	no	—
51	C₂×C₄×D₇₃	—	4	4	no	—
52	Dic₂₉₂	—	2	8	no	—
53	Dic₂₉₂	—	4	4	no	—
54	#11924017	—	4	8	no	—
55	#11924017	—	4	8	no	—
56	D₅₈₄	—	2	8	no	—
57	D₅₈₄	—	4	4	no	—
58	C₄⋊Dic₇₃	—	4	4	no	—
59	#11924005	—	4	8	no	—
60	C₂²×C₄×C₇₃	D₈	1	2	yes	—
61	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₇₃	D₈	1	4	yes	—
63	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₇₃	C₂×D₄	1	4	yes	—
65	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
67	C₄²×C₇₃	D₈	1	2	yes	—
68	C₄²×C₇₃	C₂×D₄	1	2	yes	—
69	C₁₆×C₇₃	C₂×D₄	1	8	yes	—
70	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
71	C₂²×D₇₃	—	4	4	no	—
72	C₂×C₄×C₇₃	—	4	2	no	—
73	C₂×C₄×C₇₃	#127	1	2	yes	—
74	C₂×C₄×C₇₃	C₂×D₈	1	4	yes	—
75	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
76	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
78	C₂²⋊D₇₃	—	8	8	no	—
79	C₂×Dic₇₃	—	4	4	no	—
80	D₂₉₂	—	4	4	no	—
81	D₂₉₂	—	4	4	no	—
82	D₂₉₂	—	8	8	no	—
83	C₄×D₇₃	—	8	8	no	—
84	Dic₁₄₆	—	4	4	no	—
85	Dic₁₄₆	—	4	4	no	—
86	C₈×C₇₃	#127	1	2	yes	—
87	C₈×C₇₃	C₂×D₈	1	4	yes	—
88	C₈×C₇₃	C₂²×D₄	1	2	yes	—
89	C₂²×C₇₃	—	2	2	no	—
90	C₂²×C₇₃	#493	1	2	yes	—
91	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
92	C₂×D₇₃	—	8	4	no	—
93	C₂×D₇₃	—	8	8	no	—
94	C₄×C₇₃	—	2	4	no	—
95	C₄×C₇₃	#493	1	2	yes	—
96	C₄×C₇₃	C₂×#127	1	1	yes	characteristic
97	C₄×C₇₃	C₂²×D₈	1	1	yes	characteristic
98	Dic₇₃	—	8	4	no	—
99	#33716	—	73	1	no	maximal, Sylow
100	C₂×C₇₃	—	2	2	no	—
101	C₂×C₇₃	C₂×#493	1	1	yes	characteristic
102	C₂×C₇₃	#3414	1	2	yes	—
103	D₇₃	—	16	4	no	—
104	C₂×#508	—	73	4	no	—
105	#3147	—	73	2	no	—
106	#2251	—	73	4	no	—
107	#2250	—	73	4	no	—
108	C₂×#346	D₇₃	1	1	yes	characteristic, complemented
109	C₇₃	#33716	1	1	yes	characteristic, complemented, Sylow
110	C₂×#135	—	73	4	no	—
111	C₂×#135	—	146	2	no	—
112	#532	—	146	2	no	—
113	#508	—	146	16	no	—
114	C₂×Q₃₂	—	73	4	no	—
115	C₂×SD₃₂	—	73	8	no	—
116	C₂×D₁₆	—	73	4	no	—
117	C₂²×C₁₆	C₂×D₇₃	1	2	yes	—
118	#495	—	73	4	no	—
119	#495	—	146	2	no	—
120	#494	—	73	2	no	—
121	#493	—	73	2	no	—
122	#492	—	146	2	no	—
123	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
124	#346	C₂×D₇₃	1	4	yes	—
125	C₂×#41	—	146	4	no	—
126	#135	—	292	8	no	—
127	#135	—	146	16	no	—
128	C₂×Q₁₆	—	73	4	no	—
129	C₂×Q₁₆	—	146	2	no	—
130	C₂×SD₁₆	—	146	4	no	—
131	C₂×SD₁₆	—	146	4	no	—
132	C₂×D₈	—	73	4	no	—
133	C₂×D₈	—	146	2	no	—
134	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
136	C₄⋊Q₈	—	146	2	no	—
137	#127	—	146	2	no	—
138	#124	—	146	4	no	—
139	#121	—	292	4	no	—
140	C₄×D₄	—	292	4	no	—
141	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
142	Q₃₂	—	146	8	no	—
143	SD₃₂	—	146	16	no	—
144	D₁₆	—	146	8	no	—
145	C₂×C₁₆	D₂₉₂	1	8	yes	complemented
146	C₂×C₁₆	C₂²×D₇₃	1	4	yes	—
147	C₄×C₈	D₂₉₂	1	2	yes	—
148	C₄×C₈	C₂²×D₇₃	1	2	yes	—
149	#41	—	292	8	no	—
150	#41	—	292	8	no	—
151	C₂×Q₈	—	146	4	no	—
152	C₂×D₄	—	292	4	no	—
153	C₂×D₄	—	292	4	no	—
154	C₂×D₄	—	146	4	no	—
155	C₂²×C₄	—	292	4	no	—
156	C₂²×C₄	D₅₈₄	1	2	yes	—
157	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
158	Q₁₆	—	292	4	no	—
159	Q₁₆	—	146	8	no	—
160	SD₁₆	—	292	8	no	—
161	SD₁₆	—	292	8	no	—
162	D₈	—	292	4	no	—
163	D₈	—	146	8	no	—
164	C₂×C₈	D₅₈₄	1	4	yes	—
165	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
166	C₂×C₈	C₂×D₂₉₂	1	4	yes	—
167	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
168	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
169	C₄⋊C₄	—	292	4	no	—
170	C₂²⋊C₄	—	292	8	no	—
171	C₄²	D₅₈₄	1	2	yes	—
172	C₄²	C₂×D₂₉₂	1	2	yes	—
173	C₁₆	C₂×D₂₉₂	1	8	yes	—
174	C₂³	—	292	4	no	—
175	C₂³	C₂×D₅₈₄	1	1	yes	characteristic
176	Q₈	—	292	4	no	—
177	Q₈	—	292	4	no	—
178	D₄	—	292	4	no	—
179	D₄	—	292	4	no	—
180	D₄	—	584	8	no	—
181	D₄	—	584	8	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	292	4	no	—
184	C₂×C₄	—	584	8	no	—
185	C₂×C₄	#14556588	1	2	yes	—
186	C₂×C₄	C₂×D₅₈₄	1	4	yes	—
187	C₂×C₄	C₂×D₅₈₄	1	2	yes	—
188	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
189	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
190	C₈	#14556588	1	2	yes	—
191	C₈	C₂×D₅₈₄	1	4	yes	—
192	C₈	C₂²×D₂₉₂	1	2	yes	—
193	C₂²	—	2	2	no	—
194	C₂²	—	584	8	no	—
195	C₂²	—	584	4	no	—
196	C₂²	#17988964	1	2	yes	—
197	C₂²	C₂²×D₅₈₄	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	—	584	4	no	—
200	C₄	#17988964	1	2	yes	—
201	C₄	C₂×#14556588	1	1	yes	characteristic
202	C₄	C₂²×D₅₈₄	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	—	1168	4	no	—
205	C₂	C₂×#17988964	1	1	yes	characteristic
206	C₂	#22623067	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17989019

C₂

yes

maximal, complemented

#22620359

C₂

yes

maximal, complemented

#22612202

C₂

yes

maximal, complemented

#22612197

C₂

yes

maximal, complemented

C₂×#346×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#14556613

—

C₂×#14556613

C₂²

yes

—

#17989137

—

#17989019

—

C₂×Dic₅₈₄

C₂²

yes

—

C₂×#14556536

C₂²

yes

—

C₂×D₁₁₆₈

C₂²

yes

—

#17988971

—

#17988971

C₂²

yes

—

#17988967

C₂²

yes

—

#17988964

C₂²

yes

—

#17988960

—

C₂²×C₁₆×C₇₃

C₂²

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#346×C₇₃

C₂²

yes

—

C₂×#11924028

—

#14556613

—

#14556613

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

C₂³

yes

—

C₂×#11924017

—

C₂×#11924017

—

C₂×D₅₈₄

—

C₂×D₅₈₄

C₂³

yes

—

—

—

—

—

C₄×D₂₉₂

—

Dic₅₈₄

—

#14556536

—

D₁₁₆₈

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

D₄

yes

complemented

C₂×C₁₆×C₇₃

C₂³

yes

—

C₄×C₈×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

C₂³

yes

—

#11924028

—

#11924028

—

C₂×Dic₁₄₆

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×C₂²⋊D₇₃

—

C₂×C₄×D₇₃

—

Dic₂₉₂

—

Dic₂₉₂

—

#11924017

—

#11924017

—

D₅₈₄

—

D₅₈₄

—

C₄⋊Dic₇₃

—

#11924005

—

C₂²×C₄×C₇₃

D₈

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

C₂⁴

yes

characteristic

C₄²×C₇₃

D₈

yes

—

C₄²×C₇₃

C₂×D₄

yes

—

C₁₆×C₇₃

C₂×D₄

yes

—

C₂³×C₇₃

C₂×D₈

yes

characteristic

C₂²×D₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#127

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

C₂²×D₄

yes

characteristic

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

#127

yes

—

C₈×C₇₃

C₂×D₈

yes

—

C₈×C₇₃

C₂²×D₄

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

#493

yes

—

C₂²×C₇₃

C₂²×D₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#493

yes

—

C₄×C₇₃

C₂×#127

yes

characteristic

C₄×C₇₃

C₂²×D₈

yes

characteristic

Dic₇₃

—

#33716

—

maximal, Sylow

100

C₂×C₇₃

—

101

C₂×C₇₃

C₂×#493

yes

characteristic

102

C₂×C₇₃

#3414

yes

—

103

D₇₃

—

104

C₂×#508

—

105

#3147

—

106

#2251

—

107

#2250

—

108

C₂×#346

D₇₃

yes

characteristic, complemented

109

C₇₃

#33716

yes

characteristic, complemented, Sylow

110

C₂×#135

—

111

C₂×#135

—

146

—

112

#532

—

146

—

113

#508

—

146

—

114

C₂×Q₃₂

—

115

C₂×SD₃₂

—

116

C₂×D₁₆

—

117

C₂²×C₁₆

C₂×D₇₃

yes

—

118

#495

—

119

#495

—

146

—

120

#494

—

121

#493

—

122

#492

—

146

—

123

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

124

#346

C₂×D₇₃

yes

—

125

C₂×#41

—

146

—

126

#135

—

292

—

127

#135

—

146

—

128

C₂×Q₁₆

—

129

C₂×Q₁₆

—

146

—

130

C₂×SD₁₆

—

146

—

131

C₂×SD₁₆

—

146

—

132

C₂×D₈

—

133

C₂×D₈

—

146

—

134

C₂²×C₈

D₂₉₂

yes

characteristic

135

C₂²×C₈

C₂²×D₇₃

yes

characteristic

136

C₄⋊Q₈

—

146

—

137

#127

—

146

—

138

#124

—

146

—

139

#121

—

292

—

140

C₄×D₄

—

292

—

141

C₂×C₄²

D₂₉₂

yes

characteristic

142

Q₃₂

—

146

—

143

SD₃₂

—

146

—

144

D₁₆

—

146

—

145

C₂×C₁₆

D₂₉₂

yes

complemented

146

C₂×C₁₆

C₂²×D₇₃

yes

—

147

C₄×C₈

D₂₉₂

yes

—

148

C₄×C₈

C₂²×D₇₃

yes

—

149

#41

—

292

—

150

#41

—

292

—

151

C₂×Q₈

—

146

—

152

C₂×D₄

—

292

—

153

C₂×D₄

—

292

—

154

C₂×D₄

—

146

—

155

C₂²×C₄

—

292

—

156

C₂²×C₄

D₅₈₄

yes

—

157

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

158

Q₁₆

—

292

—

159

Q₁₆

—

146

—

160

SD₁₆

—

292

—

161

SD₁₆

—

292

—

162

D₈

—

292

—

163

D₈

—

146

—

164

C₂×C₈

D₅₈₄

yes

—

165

C₂×C₈

C₂×D₂₉₂

yes

characteristic

166

C₂×C₈

C₂×D₂₉₂

yes

—

167

C₂×C₈

C₂×D₂₉₂

yes

—

168

C₂×C₈

C₂³×D₇₃

yes

characteristic

169

C₄⋊C₄

—

292

—

170

C₂²⋊C₄

—

292

—

171

C₄²

D₅₈₄

yes

—

172

C₄²

C₂×D₂₉₂

yes

—

173

C₁₆

C₂×D₂₉₂

yes

—

174

C₂³

—

292

—

175

C₂³

C₂×D₅₈₄

yes

characteristic

176

Q₈

—

292

—

177

Q₈

—

292

—

178

D₄

—

292

—

179

D₄

—

292

—

180

D₄

—

584

—

181

D₄

—

584

—

182

C₂×C₄

—

183

C₂×C₄

—

292

—

184

C₂×C₄

—

584

—

185

C₂×C₄

#14556588

yes

—

186

C₂×C₄

C₂×D₅₈₄

yes

—

187

C₂×C₄

C₂×D₅₈₄

yes

—

188

C₂×C₄

C₂×D₅₈₄

yes

characteristic

189

C₂×C₄

C₂²×D₂₉₂

yes

characteristic

190

C₈

#14556588

yes

—

191

C₈

C₂×D₅₈₄

yes

—

192

C₈

C₂²×D₂₉₂

yes

—

193

C₂²

—

194

C₂²

—

584

—

195

C₂²

—

584

—

196

C₂²

#17988964

yes

—

197

C₂²

C₂²×D₅₈₄

yes

characteristic

198

C₄

—

199

C₄

—

584

—

200

C₄

#17988964

yes

—

201

C₄

C₂×#14556588

yes

characteristic

202

C₄

C₂²×D₅₈₄

yes

characteristic

203

C₂

—

204

C₂

—

1168

—

205

C₂

C₂×#17988964

yes

characteristic

206

C₂

#22623067

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₅₈₄
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₂×#346×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₈
hypercenter	C₂×#346	D₇₃
hypocenter	C₇₃	#33716
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

C₂×D₅₈₄

commutator, cosocle

C₂×C₈×C₇₃

C₂⁴

Frattini

C₂×C₈

C₂³×D₇₃

Fitting

C₂×#346×C₇₃

C₂

socle

C₂²×C₇₃

C₂²×D₈

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
1. C₂×#17989019	C₂	semidirect
2. #22620359	C₂	semidirect
3. #22612202	C₂	semidirect
4. #22612197	C₂	semidirect
5. C₂×#346×C₇₃	C₂	semidirect
41. C₂×C₁₆×C₇₃	D₄	semidirect
108. C₂×#346	D₇₃	semidirect
109. C₇₃	#33716	semidirect
145. C₂×C₁₆	D₂₉₂	semidirect

H ≅ G/N

type

1. C₂×#17989019

C₂

semidirect

2. #22620359

C₂

semidirect

3. #22612202

C₂

semidirect

4. #22612197

C₂

semidirect

5. C₂×#346×C₇₃

C₂

semidirect

41. C₂×C₁₆×C₇₃

D₄

semidirect

108. C₂×#346

D₇₃

semidirect

109. C₇₃

#33716

semidirect

145. C₂×C₁₆

D₂₉₂

semidirect

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

subgroup	index
C₂×#17989019	2
#22620359	2
#22612202	2
#22612197	2
C₂×#346×C₇₃	2
#33716	73

subgroup

index

C₂×#346×C₇₃

Subgroups — #33241408

Conjugacy classes of subgroups

Complements

Maximal subgroups