Group #34014233 — subgroups

Subgroups — #34014233

Order 18688.

298 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 100 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×#17989520	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17989518	C₂	1	1	yes	maximal, characteristic, complemented
3	#22625045	C₂	1	2	yes	maximal, complemented
4	C₂²⋊#14556641	C₂	1	2	yes	maximal, complemented
5	C₂×#17989005	C₂	1	1	yes	maximal, characteristic
6	#17989520	—	2	2	no	—
7	#17989518	—	2	2	no	—
8	C₂×#14556641	C₂²	1	2	yes	—
9	C₂×#14556641	C₂²	1	2	yes	—
10	C₂×#14556597	C₄	1	1	yes	characteristic
11	#17989005	—	2	2	no	—
12	#17989005	—	2	2	no	—
13	C₂×#14556530	C₄	1	1	yes	characteristic
14	C₂×#14556530	C₄	1	1	yes	characteristic
15	C₂²×#11924003	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
18	C₂×M₃₂×C₇₃	C₄	1	1	yes	characteristic
19	#14556641	—	2	2	no	—
20	#14556641	—	2	2	no	—
21	C₂×#11924028	C₈	1	1	yes	characteristic
22	#14556597	—	2	2	no	—
23	#14556597	—	2	2	no	—
24	C₂×#11924012	C₈	1	1	yes	characteristic
25	C₂×#11924011	C₈	1	1	yes	characteristic
26	C₂×#11924011	C₈	1	1	yes	characteristic
27	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
29	#14556530	—	2	2	no	—
30	#14556530	—	2	2	no	—
31	C₂×#11924003	—	2	1	no	—
32	C₂×#11924003	—	2	1	no	—
33	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
34	C₂×#11924003	C₂³	1	1	yes	characteristic
35	C₁₆×D₇₃	—	2	1	no	—
36	C₁₆×D₇₃	—	2	1	no	—
37	C₁₆×D₇₃	D₄	1	2	yes	—
38	C₁₆×D₇₃	D₄	1	2	yes	—
39	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
40	M₃₂×C₇₃	—	2	2	no	—
41	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
43	#11924028	—	2	2	no	—
44	#11924028	—	2	2	no	—
45	C₂×Dic₁₄₆	C₁₆	1	1	yes	characteristic
46	C₂×D₂₉₂	C₁₆	1	1	yes	characteristic
47	C₂×C₂²⋊D₇₃	C₁₆	1	1	yes	characteristic
48	C₂×C₂²⋊D₇₃	C₁₆	1	1	yes	characteristic
49	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
50	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
51	#11924012	—	2	2	no	—
52	#11924011	—	2	2	no	—
53	#11924011	—	2	2	no	—
54	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
55	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
56	C₈×D₇₃	—	2	1	no	—
57	C₈×D₇₃	—	2	1	no	—
58	C₈×D₇₃	C₂²⋊C₄	1	2	yes	—
59	C₈×D₇₃	M₁₆	1	2	yes	—
60	#11924003	—	2	1	no	—
61	#11924003	C₂×D₄	1	1	yes	characteristic
62	#11924003	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
64	C₂×C₈×C₇₃	—	2	1	no	—
65	C₂×C₈×C₇₃	—	2	1	no	—
66	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
67	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
68	C₁₆×C₇₃	—	2	1	no	—
69	C₁₆×C₇₃	C₂²⋊C₄	1	2	yes	—
70	C₂³×C₇₃	C₂×C₁₆	1	1	yes	characteristic
71	C₂²×D₇₃	C₂×C₁₆	1	1	yes	characteristic
72	C₂²×D₇₃	C₂×C₁₆	1	1	yes	characteristic
73	C₂×C₄×C₇₃	—	2	1	no	—
74	C₂×C₄×C₇₃	—	2	1	no	—
75	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
77	C₂²⋊D₇₃	—	2	2	no	—
78	C₂²⋊D₇₃	—	2	2	no	—
79	C₂×Dic₇₃	C₂×C₁₆	1	1	yes	characteristic
80	C₂×Dic₇₃	C₂×C₁₆	1	1	yes	characteristic
81	D₂₉₂	—	2	2	no	—
82	C₄×D₇₃	—	2	1	no	—
83	C₄×D₇₃	—	2	1	no	—
84	C₄×D₇₃	C₂²⋊C₈	1	2	yes	—
85	C₄×D₇₃	M₃₂	1	2	yes	—
86	Dic₁₄₆	—	2	2	no	—
87	C₈×C₇₃	—	2	1	no	—
88	C₈×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
90	#10596831	—	2	1	no	—
91	#10596831	C₂²⋊C₈	1	2	yes	—
92	C₂²×C₇₃	—	2	1	no	—
93	C₂²×C₇₃	—	2	2	no	—
94	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
95	C₂×D₇₃	—	2	2	no	—
96	C₂×D₇₃	—	2	1	no	—
97	C₂×D₇₃	—	2	2	no	—
98	C₂×D₇₃	C₂²⋊C₁₆	1	2	yes	—
99	C₄×C₇₃	—	2	1	no	—
100	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
101	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
102	Dic₇₃	—	2	1	no	—
103	Dic₇₃	C₂²⋊C₁₆	1	2	yes	—
104	#13554	—	73	1	no	maximal, Sylow
105	C₂×C₇₃	—	4	1	no	—
106	C₂×C₇₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
107	C₂×C₇₃	#2258	1	2	yes	—
108	D₇₃	—	2	2	no	—
109	D₇₃	—	4	1	no	—
110	C₂×#504	—	73	1	no	—
111	C₂×M₆₄	—	73	1	no	—
112	C₂²×C₃₂	—	73	1	no	—
113	C₂²⋊C₃₂	—	73	2	no	—
114	C₂²⋊C₃₂	—	73	2	no	—
115	C₇₃	#13554	1	1	yes	characteristic, complemented, Sylow
116	C₂×#131	—	73	1	no	—
117	#504	—	146	2	no	—
118	#504	—	146	2	no	—
119	C₂×M₃₂	—	73	1	no	—
120	C₂×M₃₂	—	73	1	no	—
121	C₂×M₃₂	#64408	1	1	yes	characteristic
122	C₂²×C₁₆	—	73	1	no	—
123	C₂²×C₁₆	—	73	1	no	—
124	C₂²×C₁₆	—	73	1	no	—
125	M₆₄	—	146	2	no	—
126	C₂×C₃₂	—	73	2	no	—
127	C₂×C₃₂	—	73	2	no	—
128	C₂×C₃₂	—	146	2	no	—
129	C₂×#41	—	73	1	no	—
130	#131	—	146	2	no	—
131	#131	—	146	2	no	—
132	C₂×M₁₆	—	73	1	no	—
133	C₂×M₁₆	—	73	1	no	—
134	C₂×M₁₆	—	73	1	no	—
135	C₂²×C₈	—	73	1	no	—
136	C₂²×C₈	—	73	1	no	—
137	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
138	M₃₂	—	2	2	no	—
139	M₃₂	—	146	2	no	—
140	M₃₂	—	146	2	no	—
141	C₂×C₁₆	—	73	1	no	—
142	C₂×C₁₆	—	73	2	no	—
143	C₂×C₁₆	—	73	2	no	—
144	C₂×C₁₆	—	73	1	no	—
145	C₂×C₁₆	—	146	1	no	—
146	C₂×C₁₆	—	146	1	no	—
147	C₂×C₁₆	—	146	1	no	—
148	C₂×C₁₆	—	146	1	no	—
149	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
150	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
151	C₃₂	—	146	2	no	—
152	C₃₂	—	146	2	no	—
153	#41	—	146	2	no	—
154	#41	—	146	2	no	—
155	C₂×Q₈	—	73	1	no	—
156	C₂×D₄	—	73	1	no	—
157	C₂×D₄	—	73	1	no	—
158	C₂×D₄	—	73	1	no	—
159	C₂²×C₄	—	73	1	no	—
160	C₂²×C₄	—	73	1	no	—
161	C₂²×C₄	#11924036	1	1	yes	characteristic
162	M₁₆	—	146	2	no	—
163	M₁₆	—	146	2	no	—
164	M₁₆	—	146	2	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	73	1	no	—
168	C₂×C₈	—	73	2	no	—
169	C₂×C₈	—	73	2	no	—
170	C₂×C₈	—	73	1	no	—
171	C₂×C₈	—	146	1	no	—
172	C₂×C₈	—	146	1	no	—
173	C₂×C₈	#11924036	1	1	yes	characteristic
174	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
175	C₁₆	—	2	1	no	—
176	C₁₆	—	73	1	no	—
177	C₁₆	—	73	1	no	—
178	C₁₆	—	146	1	no	—
179	C₁₆	C₂²⋊#64408	1	2	yes	—
180	C₂³	—	73	1	no	—
181	C₂³	—	73	1	no	—
182	C₂³	#14556664	1	1	yes	characteristic
183	Q₈	—	146	2	no	—
184	D₄	—	146	2	no	—
185	D₄	—	146	2	no	—
186	D₄	—	146	2	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	73	1	no	—
190	C₂×C₄	—	73	2	no	—
191	C₂×C₄	—	73	2	no	—
192	C₂×C₄	—	73	1	no	—
193	C₂×C₄	—	146	1	no	—
194	C₂×C₄	—	146	1	no	—
195	C₂×C₄	#14556664	1	1	yes	characteristic
196	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
197	C₈	—	2	1	no	—
198	C₈	—	73	2	no	—
199	C₈	—	146	1	no	—
200	C₈	C₂×C₂²⋊#64408	1	1	yes	characteristic
201	C₈	D₇₃⋊M₁₆	1	1	yes	characteristic
202	C₂²	—	2	2	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	73	2	no	—
205	C₂²	—	146	2	no	—
206	C₂²	—	146	1	no	—
207	C₂²	—	146	2	no	—
208	C₂²	C₂×#14556664	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	73	2	no	—
211	C₄	—	146	1	no	—
212	C₄	#17989555	1	1	yes	characteristic
213	C₄	#17989643	1	1	yes	characteristic
214	C₂	—	4	1	no	—
215	C₂	—	292	1	no	—
216	C₂	—	146	2	no	—
217	C₂	#22626156	1	1	yes	characteristic
218	C₂	#22626355	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17989520

C₂

yes

maximal, characteristic, complemented

C₂×#17989518

C₂

yes

maximal, characteristic, complemented

#22625045

C₂

yes

maximal, complemented

C₂²⋊#14556641

C₂

yes

maximal, complemented

C₂×#17989005

C₂

yes

maximal, characteristic

#17989520

—

#17989518

—

C₂×#14556641

C₂²

yes

—

C₂×#14556641

C₂²

yes

—

C₂×#14556597

C₄

yes

—

—

C₄

yes

characteristic

C₂×#14556530

C₄

yes

characteristic

C₂²×#11924003

C₂²

yes

characteristic

C₂×C₁₆×D₇₃

C₂²

yes

characteristic

C₂×C₁₆×D₇₃

C₂²

yes

characteristic

C₂×M₃₂×C₇₃

C₄

yes

—

—

C₈

yes

—

—

C₈

yes

characteristic

C₂×#11924011

C₈

yes

characteristic

C₂×#11924011

C₈

yes

characteristic

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₂×C₈×D₇₃

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#11924003

C₂³

yes

characteristic

C₁₆×D₇₃

—

C₁₆×D₇₃

—

C₁₆×D₇₃

D₄

yes

—

C₁₆×D₇₃

D₄

yes

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

#11924028

—

#11924028

—

C₂×Dic₁₄₆

C₁₆

yes

characteristic

C₂×D₂₉₂

C₁₆

yes

characteristic

C₂×C₂²⋊D₇₃

C₁₆

yes

characteristic

C₂×C₂²⋊D₇₃

C₁₆

yes

characteristic

C₂×C₄×D₇₃

C₂×C₈

yes

characteristic

C₂×C₄×D₇₃

C₂×C₈

yes

—

—

—

C₂×C₈

yes

characteristic

C₂×#10596831

C₂×C₈

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

C₂²⋊C₄

yes

—

C₈×D₇₃

M₁₆

yes

—

#11924003

—

#11924003

C₂×D₄

yes

characteristic

#11924003

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

C₂²⋊C₄

yes

—

C₂³×C₇₃

C₂×C₁₆

yes

characteristic

C₂²×D₇₃

C₂×C₁₆

yes

characteristic

C₂²×D₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂²⋊D₇₃

—

C₂²⋊D₇₃

—

C₂×Dic₇₃

C₂×C₁₆

yes

characteristic

C₂×Dic₇₃

C₂×C₁₆

yes

characteristic

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂²⋊C₈

yes

—

C₄×D₇₃

M₃₂

yes

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₇₃

C₂×M₁₆

yes

characteristic

#10596831

—

#10596831

C₂²⋊C₈

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂²×C₁₆

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

C₂²⋊C₁₆

yes

—

C₄×C₇₃

—

100

C₄×C₇₃

C₂×C₂²⋊C₈

yes

characteristic

101

C₄×C₇₃

C₂×M₃₂

yes

characteristic

102

Dic₇₃

—

103

Dic₇₃

C₂²⋊C₁₆

yes

—

104

#13554

—

maximal, Sylow

105

C₂×C₇₃

—

106

C₂×C₇₃

C₂×C₂²⋊C₁₆

yes

characteristic

107

C₂×C₇₃

#2258

yes

—

108

D₇₃

—

109

D₇₃

—

110

C₂×#504

—

111

C₂×M₆₄

—

112

C₂²×C₃₂

—

113

C₂²⋊C₃₂

—

114

C₂²⋊C₃₂

—

115

C₇₃

#13554

yes

characteristic, complemented, Sylow

116

C₂×#131

—

117

#504

—

146

—

118

#504

—

146

—

119

C₂×M₃₂

—

120

C₂×M₃₂

—

121

C₂×M₃₂

#64408

yes

characteristic

122

C₂²×C₁₆

—

123

C₂²×C₁₆

—

124

C₂²×C₁₆

—

125

M₆₄

—

146

—

126

C₂×C₃₂

—

127

C₂×C₃₂

—

128

C₂×C₃₂

—

146

—

129

C₂×#41

—

130

#131

—

146

—

131

#131

—

146

—

132

C₂×M₁₆

—

133

C₂×M₁₆

—

134

C₂×M₁₆

—

135

C₂²×C₈

—

136

C₂²×C₈

—

137

C₂²×C₈

C₂×#64408

yes

characteristic

138

M₃₂

—

139

M₃₂

—

146

—

140

M₃₂

—

146

—

141

C₂×C₁₆

—

142

C₂×C₁₆

—

143

C₂×C₁₆

—

144

C₂×C₁₆

—

145

C₂×C₁₆

—

146

—

146

C₂×C₁₆

—

146

—

147

C₂×C₁₆

—

146

—

148

C₂×C₁₆

—

146

—

149

C₂×C₁₆

C₂×#64408

yes

characteristic

150

C₂×C₁₆

C₂×#64408

yes

characteristic

151

C₃₂

—

146

—

152

C₃₂

—

146

—

153

#41

—

146

—

154

#41

—

146

—

155

C₂×Q₈

—

156

C₂×D₄

—

157

C₂×D₄

—

158

C₂×D₄

—

159

C₂²×C₄

—

160

C₂²×C₄

—

161

C₂²×C₄

#11924036

yes

characteristic

162

M₁₆

—

146

—

163

M₁₆

—

146

—

164

M₁₆

—

146

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

146

—

172

C₂×C₈

—

146

—

173

C₂×C₈

#11924036

yes

characteristic

174

C₂×C₈

C₂²×#64408

yes

characteristic

175

C₁₆

—

176

C₁₆

—

177

C₁₆

—

178

C₁₆

—

146

—

179

C₁₆

C₂²⋊#64408

yes

—

180

C₂³

—

181

C₂³

—

182

C₂³

#14556664

yes

characteristic

183

Q₈

—

146

—

184

D₄

—

146

—

185

D₄

—

146

—

186

D₄

—

146

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

146

—

194

C₂×C₄

—

146

—

195

C₂×C₄

#14556664

yes

characteristic

196

C₂×C₄

C₂×#11924036

yes

characteristic

197

C₈

—

198

C₈

—

199

C₈

—

146

—

200

C₈

C₂×C₂²⋊#64408

yes

characteristic

201

C₈

D₇₃⋊M₁₆

yes

characteristic

202

C₂²

—

203

C₂²

—

204

C₂²

—

205

C₂²

—

146

—

206

C₂²

—

146

—

207

C₂²

—

146

—

208

C₂²

C₂×#14556664

yes

characteristic

209

C₄

—

210

C₄

—

211

C₄

—

146

—

212

C₄

#17989555

yes

characteristic

213

C₄

#17989643

yes

characteristic

214

C₂

—

215

C₂

—

292

—

216

C₂

—

146

—

217

C₂

#22626156

yes

characteristic

218

C₂

#22626355

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#64408
commutator, socle	C₂²×C₇₃	C₂²×C₁₆
Fitting	C₂×M₃₂×C₇₃	C₄
prime radical	C₂×#11924028	C₈
cosocle	C₂×#11924003	C₂³
hypercenter	C₂×M₃₂	#64408
hypocenter	C₇₃	#13554
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×#64408

commutator, socle

C₂²×C₇₃

C₂²×C₁₆

Fitting

C₂×M₃₂×C₇₃

C₄

C₈

C₂³

C₂×M₃₂

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
1. C₂×#17989520	C₂	semidirect
2. C₂×#17989518	C₂	semidirect
3. #22625045	C₂	semidirect
4. C₂²⋊#14556641	C₂	semidirect
115. C₇₃	#13554	semidirect

H ≅ G/N

type

1. C₂×#17989520

C₂

semidirect

2. C₂×#17989518

C₂

semidirect

3. #22625045

C₂

semidirect

4. C₂²⋊#14556641

C₂

semidirect

115. C₇₃

#13554

semidirect

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

subgroup	index
C₂×#17989520	2
C₂×#17989518	2
#22625045	2
C₂²⋊#14556641	2
C₂×#17989005	2
#13554	73

subgroup

index

Subgroups — #34014233

Conjugacy classes of subgroups

Complements

Maximal subgroups