Group #33241297 — subgroups

Subgroups — C₂×#22612173 #33241297

Order 18688.

686 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 203 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×#17988989	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17988964	C₂	1	1	yes	maximal, characteristic
3	#22612173	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17988389	C₂	1	4	yes	maximal, complemented
5	C₂×#346×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₅₈₄	—	2	1	no	—
7	C₂²×D₅₈₄	C₂²	1	2	yes	—
8	C₂×#14556588	—	2	1	no	—
9	C₂×#14556583	—	2	1	no	—
10	#17988989	—	2	2	no	—
11	#17988989	C₂²	1	4	yes	complemented
12	#17988964	—	2	2	no	—
13	#17988964	C₂²	1	4	yes	—
14	C₂×#14556525	C₂²	1	2	yes	—
15	C₂×#14556523	—	2	1	no	—
16	#17988389	C₂²	1	16	yes	complemented
17	C₂²×C₁₆×C₇₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
19	#346×C₇₃	C₂²	1	4	yes	complemented
20	C₂²×D₂₉₂	—	2	2	no	—
21	C₂²×D₂₉₂	—	4	2	no	—
22	C₂×D₅₈₄	—	2	2	no	—
23	C₂×D₅₈₄	—	2	4	no	—
24	C₂×D₅₈₄	—	2	8	no	—
25	C₂×D₅₈₄	—	4	4	no	—
26	C₂×D₅₈₄	C₂³	1	4	yes	—
27	#14556588	—	2	4	no	—
28	#14556588	—	4	2	no	—
29	#14556583	—	2	4	no	—
30	#14556583	—	4	2	no	—
31	C₂×C₄⋊Dic₇₃	—	2	2	no	—
32	C₂×#11924007	—	4	2	no	—
33	#14556525	—	2	2	no	—
34	#14556525	C₂³	1	4	yes	—
35	#14556523	—	2	4	no	—
36	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
39	C₂×C₁₆×C₇₃	—	2	4	no	—
40	C₂×C₁₆×C₇₃	C₂³	1	4	yes	—
41	C₄×C₈×C₇₃	D₄	1	2	yes	—
42	C₄×C₈×C₇₃	C₂³	1	2	yes	—
43	C₂³×D₇₃	—	4	2	no	—
44	C₂×D₂₉₂	—	2	4	no	—
45	C₂×D₂₉₂	—	4	4	no	—
46	C₂×D₂₉₂	—	4	4	no	—
47	C₂×D₂₉₂	—	4	8	no	—
48	C₂×D₂₉₂	—	4	2	no	—
49	C₂×D₂₉₂	—	8	8	no	—
50	C₂²×Dic₇₃	—	4	2	no	—
51	D₅₈₄	—	2	8	no	—
52	D₅₈₄	—	4	4	no	—
53	D₅₈₄	—	4	8	no	—
54	C₄⋊Dic₇₃	—	2	4	no	—
55	C₄⋊Dic₇₃	—	4	2	no	—
56	#11924007	—	4	8	no	—
57	C₂²×C₄×C₇₃	D₈	1	2	yes	—
58	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₇₃	SD₁₆	1	4	yes	—
60	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₇₃	#41	1	4	yes	—
63	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
64	C₄²×C₇₃	D₈	1	2	yes	—
65	C₄²×C₇₃	C₂×D₄	1	2	yes	—
66	C₁₆×C₇₃	—	2	4	no	—
67	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
68	C₂²×D₇₃	—	4	4	no	—
69	C₂²×D₇₃	—	8	4	no	—
70	C₂²×D₇₃	—	8	8	no	—
71	C₂×C₄×C₇₃	—	2	4	no	—
72	C₂×C₄×C₇₃	#124	1	2	yes	—
73	C₂×C₄×C₇₃	C₂×D₈	1	4	yes	—
74	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
76	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
77	C₂×Dic₇₃	—	4	4	no	—
78	C₂×Dic₇₃	—	8	4	no	—
79	D₂₉₂	—	4	4	no	—
80	D₂₉₂	—	4	4	no	—
81	D₂₉₂	—	8	16	no	—
82	D₂₉₂	—	8	4	no	—
83	C₈×C₇₃	#124	1	2	yes	—
84	C₈×C₇₃	C₂×SD₁₆	1	4	yes	—
85	C₈×C₇₃	C₂×#41	1	2	yes	—
86	C₂²×C₇₃	#486	1	2	yes	—
87	C₂²×C₇₃	#509	1	4	yes	—
88	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
89	C₂×D₇₃	—	8	4	no	—
90	C₂×D₇₃	—	8	8	no	—
91	C₂×D₇₃	—	16	8	no	—
92	C₄×C₇₃	—	2	4	no	—
93	C₄×C₇₃	#486	1	2	yes	—
94	C₄×C₇₃	C₂×#124	1	1	yes	characteristic
95	C₄×C₇₃	C₂²×SD₁₆	1	1	yes	characteristic
96	Dic₇₃	—	8	4	no	—
97	C₂×#2244	—	73	1	no	maximal, Sylow
98	C₂×C₇₃	#2244	1	4	yes	complemented
99	C₂×C₇₃	C₂×#486	1	1	yes	characteristic
100	C₂×C₇₃	C₂×#509	1	2	yes	—
101	D₇₃	—	16	4	no	—
102	C₂×#499	—	73	1	no	—
103	C₂×#493	—	73	1	no	—
104	#2244	—	73	8	no	—
105	C₂×#357	—	73	4	no	—
106	C₂×#346	D₇₃	1	1	yes	characteristic, complemented
107	C₇₃	C₂×#2244	1	1	yes	characteristic, complemented, Sylow
108	C₂²×D₈	—	73	2	no	—
109	C₂²×D₈	—	146	1	no	—
110	C₂×#127	—	146	1	no	—
111	C₂×#125	—	146	1	no	—
112	C₂²×C₁₆	C₂×D₇₃	1	2	yes	—
113	#499	—	73	4	no	—
114	#499	—	146	2	no	—
115	#493	—	73	4	no	—
116	#493	—	146	2	no	—
117	C₂×#107	—	73	2	no	—
118	C₂×#106	—	146	1	no	—
119	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
120	#357	—	73	16	no	—
121	#346	C₂×D₇₃	1	4	yes	complemented
122	C₂²×D₄	—	292	2	no	—
123	C₂²×D₄	—	146	2	no	—
124	C₂×D₈	—	292	4	no	—
125	C₂×D₈	—	73	4	no	—
126	C₂×D₈	—	146	2	no	—
127	C₂×D₈	—	146	4	no	—
128	C₂×D₈	—	146	8	no	—
129	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
131	#127	—	292	2	no	—
132	#127	—	146	4	no	—
133	#125	—	292	2	no	—
134	#125	—	146	4	no	—
135	C₂×C₄⋊C₄	—	292	2	no	—
136	C₂×C₄⋊C₄	—	146	2	no	—
137	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
138	C₂×C₁₆	—	2	4	no	—
139	C₂×C₁₆	C₂²×D₇₃	1	4	yes	—
140	#107	—	73	4	no	—
141	#107	—	146	2	no	—
142	#106	—	146	4	no	—
143	C₄×C₈	D₂₉₂	1	2	yes	—
144	C₄×C₈	C₂²×D₇₃	1	2	yes	—
145	C₂⁴	—	292	2	no	—
146	C₂×D₄	—	292	4	no	—
147	C₂×D₄	—	292	2	no	—
148	C₂×D₄	—	292	8	no	—
149	C₂×D₄	—	292	4	no	—
150	C₂×D₄	—	584	8	no	—
151	C₂×D₄	—	146	4	no	—
152	C₂²×C₄	—	292	2	no	—
153	C₂²×C₄	D₅₈₄	1	2	yes	—
154	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
155	D₈	—	292	4	no	—
156	D₈	—	292	8	no	—
157	D₈	—	146	8	no	—
158	C₂×C₈	#11924017	1	4	yes	—
159	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
160	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
161	C₂×C₈	#11924028	1	4	yes	—
162	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
163	C₄⋊C₄	—	292	8	no	—
164	C₄⋊C₄	—	292	2	no	—
165	C₄⋊C₄	—	146	4	no	—
166	C₄²	D₅₈₄	1	2	yes	—
167	C₄²	C₂×D₂₉₂	1	2	yes	—
168	C₁₆	—	2	4	no	—
169	C₂³	—	292	4	no	—
170	C₂³	—	584	4	no	—
171	C₂³	—	584	8	no	—
172	C₂³	C₂×D₅₈₄	1	1	yes	characteristic
173	D₄	—	292	4	no	—
174	D₄	—	292	4	no	—
175	D₄	—	584	16	no	—
176	D₄	—	584	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	292	4	no	—
179	C₂×C₄	—	584	4	no	—
180	C₂×C₄	#14556579	1	2	yes	—
181	C₂×C₄	C₂×D₅₈₄	1	4	yes	—
182	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
183	C₂×C₄	C₂×#11924017	1	2	yes	—
184	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
185	C₈	#14556579	1	2	yes	—
186	C₈	C₂×#11924017	1	4	yes	—
187	C₈	C₂×#11924028	1	2	yes	—
188	C₂²	—	584	4	no	—
189	C₂²	—	584	8	no	—
190	C₂²	—	1168	8	no	—
191	C₂²	#17988934	1	2	yes	—
192	C₂²	#17989025	1	4	yes	—
193	C₂²	C₂²×D₅₈₄	1	1	yes	characteristic
194	C₄	—	2	4	no	—
195	C₄	—	584	4	no	—
196	C₄	#17988934	1	2	yes	—
197	C₄	C₂×#14556579	1	1	yes	characteristic
198	C₄	C₂²×#11924017	1	1	yes	characteristic
199	C₂	—	1168	4	no	—
200	C₂	#22612173	1	4	yes	direct product factor
201	C₂	C₂×#17988934	1	1	yes	characteristic
202	C₂	C₂×#17989025	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988989

C₂

yes

maximal, characteristic, complemented

C₂×#17988964

C₂

yes

maximal, characteristic

#22612173

C₂

yes

maximal, direct product factor

C₂×#17988389

C₂

yes

maximal, complemented

C₂×#346×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₅₈₄

—

C₂²×D₅₈₄

C₂²

yes

—

—

—

—

C₂²

yes

complemented

#17988964

—

#17988964

C₂²

yes

—

C₂×#14556525

C₂²

yes

—

C₂×#14556523

—

#17988389

C₂²

yes

complemented

C₂²×C₁₆×C₇₃

C₂²

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#346×C₇₃

C₂²

yes

complemented

C₂²×D₂₉₂

—

C₂²×D₂₉₂

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

C₂×D₅₈₄

C₂³

yes

—

—

—

—

—

C₂×C₄⋊Dic₇₃

—

C₂×#11924007

—

#14556525

—

#14556525

C₂³

yes

—

#14556523

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

C₂³

yes

—

C₄×C₈×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

C₂³

yes

—

C₂³×D₇₃

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂²×Dic₇₃

—

D₅₈₄

—

D₅₈₄

—

D₅₈₄

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

C₂²×C₄×C₇₃

D₈

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

SD₁₆

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

#41

yes

—

C₂×C₈×C₇₃

C₂⁴

yes

characteristic

C₄²×C₇₃

D₈

yes

—

C₄²×C₇₃

C₂×D₄

yes

—

C₁₆×C₇₃

—

C₂³×C₇₃

C₂×D₈

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#124

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×SD₁₆

yes

—

C₂×C₄×C₇₃

C₂²×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

#124

yes

—

C₈×C₇₃

C₂×SD₁₆

yes

—

C₈×C₇₃

C₂×#41

yes

—

C₂²×C₇₃

#486

yes

—

C₂²×C₇₃

#509

yes

—

C₂²×C₇₃

C₂²×D₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#486

yes

—

C₄×C₇₃

C₂×#124

yes

characteristic

C₄×C₇₃

C₂²×SD₁₆

yes

characteristic

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

100

C₂×C₇₃

C₂×#509

yes

—

101

D₇₃

—

102

C₂×#499

—

103

C₂×#493

—

104

#2244

—

105

C₂×#357

—

106

C₂×#346

D₇₃

yes

characteristic, complemented

107

C₇₃

C₂×#2244

yes

characteristic, complemented, Sylow

108

C₂²×D₈

—

109

C₂²×D₈

—

146

—

110

C₂×#127

—

146

—

111

C₂×#125

—

146

—

112

C₂²×C₁₆

C₂×D₇₃

yes

—

113

#499

—

114

#499

—

146

—

115

#493

—

116

#493

—

146

—

117

C₂×#107

—

118

C₂×#106

—

146

—

119

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

120

#357

—

121

#346

C₂×D₇₃

yes

complemented

122

C₂²×D₄

—

292

—

123

C₂²×D₄

—

146

—

124

C₂×D₈

—

292

—

125

C₂×D₈

—

126

C₂×D₈

—

146

—

127

C₂×D₈

—

146

—

128

C₂×D₈

—

146

—

129

C₂²×C₈

D₂₉₂

yes

characteristic

130

C₂²×C₈

C₂²×D₇₃

yes

characteristic

131

#127

—

292

—

132

#127

—

146

—

133

#125

—

292

—

134

#125

—

146

—

135

C₂×C₄⋊C₄

—

292

—

136

C₂×C₄⋊C₄

—

146

—

137

C₂×C₄²

D₂₉₂

yes

characteristic

138

C₂×C₁₆

—

139

C₂×C₁₆

C₂²×D₇₃

yes

—

140

#107

—

141

#107

—

146

—

142

#106

—

146

—

143

C₄×C₈

D₂₉₂

yes

—

144

C₄×C₈

C₂²×D₇₃

yes

—

145

C₂⁴

—

292

—

146

C₂×D₄

—

292

—

147

C₂×D₄

—

292

—

148

C₂×D₄

—

292

—

149

C₂×D₄

—

292

—

150

C₂×D₄

—

584

—

151

C₂×D₄

—

146

—

152

C₂²×C₄

—

292

—

153

C₂²×C₄

D₅₈₄

yes

—

154

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

155

D₈

—

292

—

156

D₈

—

292

—

157

D₈

—

146

—

158

C₂×C₈

#11924017

yes

—

159

C₂×C₈

C₂×D₂₉₂

yes

characteristic

160

C₂×C₈

C₂×D₂₉₂

yes

—

161

C₂×C₈

#11924028

yes

—

162

C₂×C₈

C₂³×D₇₃

yes

characteristic

163

C₄⋊C₄

—

292

—

164

C₄⋊C₄

—

292

—

165

C₄⋊C₄

—

146

—

166

C₄²

D₅₈₄

yes

—

167

C₄²

C₂×D₂₉₂

yes

—

168

C₁₆

—

169

C₂³

—

292

—

170

C₂³

—

584

—

171

C₂³

—

584

—

172

C₂³

C₂×D₅₈₄

yes

characteristic

173

D₄

—

292

—

174

D₄

—

292

—

175

D₄

—

584

—

176

D₄

—

584

—

177

C₂×C₄

—

178

C₂×C₄

—

292

—

179

C₂×C₄

—

584

—

180

C₂×C₄

#14556579

yes

—

181

C₂×C₄

C₂×D₅₈₄

yes

—

182

C₂×C₄

C₂×D₅₈₄

yes

characteristic

183

C₂×C₄

C₂×#11924017

yes

—

184

C₂×C₄

C₂²×D₂₉₂

yes

characteristic

185

C₈

#14556579

yes

—

186

C₈

C₂×#11924017

yes

—

187

C₈

C₂×#11924028

yes

—

188

C₂²

—

584

—

189

C₂²

—

584

—

190

C₂²

—

1168

—

191

C₂²

#17988934

yes

—

192

C₂²

#17989025

yes

—

193

C₂²

C₂²×D₅₈₄

yes

characteristic

194

C₄

—

195

C₄

—

584

—

196

C₄

#17988934

yes

—

197

C₄

C₂×#14556579

yes

characteristic

198

C₄

C₂²×#11924017

yes

characteristic

199

C₂

—

1168

—

200

C₂

#22612173

yes

direct product factor

201

C₂

C₂×#17988934

yes

characteristic

202

C₂

C₂×#17989025

yes

—

	subgroup	G/H
center	C₂³	C₂×D₅₈₄
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₂×#346×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₈
prime radical	C₂×#17988964	C₂
hypercenter	C₂×#346	D₇₃
hypocenter	C₇₃	C₂×#2244
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂³

C₂×D₅₈₄

commutator, cosocle

C₂×C₈×C₇₃

C₂⁴

Frattini

C₂×C₈

C₂³×D₇₃

Fitting

C₂×#346×C₇₃

C₂

socle

C₂³×C₇₃

C₂×D₈

C₂

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
1. C₂×#17988989	C₂	semidirect
3. #22612173	C₂	direct (N × H)
4. C₂×#17988389	C₂	semidirect
5. C₂×#346×C₇₃	C₂	semidirect
11. #17988989	C₂²	semidirect
16. #17988389	C₂²	semidirect
19. #346×C₇₃	C₂²	semidirect
98. C₂×C₇₃	#2244	semidirect
106. C₂×#346	D₇₃	semidirect
107. C₇₃	C₂×#2244	semidirect
121. #346	C₂×D₇₃	semidirect
200. C₂	#22612173	direct (N × H)

H ≅ G/N

type

1. C₂×#17988989

C₂

semidirect

3. #22612173

C₂

direct (N × H)

4. C₂×#17988389

C₂

semidirect

5. C₂×#346×C₇₃

C₂

semidirect

11. #17988989

C₂²

semidirect

16. #17988389

C₂²

semidirect

19. #346×C₇₃

C₂²

semidirect

98. C₂×C₇₃

#2244

semidirect

106. C₂×#346

D₇₃

semidirect

107. C₇₃

C₂×#2244

semidirect

121. #346

C₂×D₇₃

semidirect

200. C₂

#22612173

direct (N × H)

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

subgroup	index
C₂×#17988989	2
C₂×#17988964	2
#22612173	2
C₂×#17988389	2
C₂×#346×C₇₃	2
C₂×#2244	73

subgroup

index

C₂×#346×C₇₃

Subgroups — C₂×#22612173 #33241297

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22612173 #33241297

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22612173 #33241297