Group #32991056 — subgroups

Subgroups — #32991056

Order 18688.

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

normal subgroups only

This group has 494 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	#22610738	C₂	1	2	yes	maximal
2	#22610088	C₂	1	1	yes	maximal, characteristic, complemented
3	#22610058	C₂	1	2	yes	maximal, complemented
4	#22607758	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#1957	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556523	—	2	2	no	—
7	C₂×#14556512	—	2	4	no	—
8	C₂×C₄×#10596831	C₂²	1	2	yes	—
9	#17988457	C₂²	1	2	yes	—
10	#17988349	—	2	4	no	—
11	C₂×#127×C₇₃	C₂²	1	1	yes	characteristic
12	C₂×#102×C₇₃	—	2	2	no	—
13	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
14	#384×C₇₃	C₂²	1	1	yes	characteristic
15	C₂²×#10596831	D₄	1	4	yes	—
16	C₂×C₄⋊Dic₇₃	—	2	2	no	—
17	C₂×C₄⋊Dic₇₃	—	2	2	no	—
18	C₂×C₄⋊Dic₇₃	—	2	2	no	—
19	#14556523	—	4	4	no	—
20	#14556512	—	4	8	no	—
21	C₄×#10596831	—	2	2	no	—
22	C₄×#10596831	—	2	2	no	—
23	C₂²×D₄×C₇₃	—	2	1	no	—
24	C₂²×D₄×C₇₃	—	2	2	no	—
25	C₂²×C₈×C₇₃	D₄	1	2	yes	—
26	#127×C₇₃	—	2	2	no	—
27	#127×C₇₃	—	2	1	no	—
28	#127×C₇₃	Q₈	1	2	yes	—
29	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
30	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
31	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
32	#102×C₇₃	—	4	4	no	—
33	C₄×C₈×C₇₃	—	2	2	no	—
34	C₂²×Dic₇₃	—	4	2	no	—
35	C₂×#10596831	—	2	4	no	—
36	C₂×#10596831	—	2	4	no	—
37	C₂×#10596831	—	2	4	no	—
38	C₄⋊Dic₇₃	—	4	4	no	—
39	C₄⋊Dic₇₃	—	4	4	no	—
40	C₄⋊Dic₇₃	—	4	4	no	—
41	C₂⁴×C₇₃	—	4	1	no	—
42	C₂×D₄×C₇₃	—	2	4	no	—
43	C₂×D₄×C₇₃	—	2	2	no	—
44	C₂×D₄×C₇₃	—	4	1	no	—
45	C₂×D₄×C₇₃	—	4	2	no	—
46	C₂×D₄×C₇₃	—	4	2	no	—
47	C₂×D₄×C₇₃	—	4	4	no	—
48	C₂×D₄×C₇₃	—	4	2	no	—
49	C₂×D₄×C₇₃	—	4	4	no	—
50	C₂²×C₄×C₇₃	—	4	1	no	—
51	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₇₃	—	2	2	no	—
54	C₂×C₈×C₇₃	—	2	2	no	—
55	C₂×C₈×C₇₃	SD₁₆	1	4	yes	—
56	C₄⋊C₄×C₇₃	—	4	2	no	—
57	C₄⋊C₄×C₇₃	—	4	4	no	—
58	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₇₃	#41	1	2	yes	—
60	C₄²×C₇₃	#41	1	1	yes	characteristic
61	C₂³×C₇₃	—	4	2	no	—
62	C₂³×C₇₃	—	8	1	no	—
63	C₂³×C₇₃	—	8	2	no	—
64	C₂³×C₇₃	—	8	2	no	—
65	C₂³×C₇₃	—	8	1	no	—
66	C₂³×C₇₃	#127	1	1	yes	characteristic
67	D₄×C₇₃	—	4	4	no	—
68	D₄×C₇₃	—	4	8	no	—
69	D₄×C₇₃	—	8	2	no	—
70	D₄×C₇₃	—	8	4	no	—
71	C₂×C₄×C₇₃	—	8	1	no	—
72	C₂×C₄×C₇₃	—	8	2	no	—
73	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
74	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₇₃	#123	1	1	yes	characteristic
76	C₂×C₄×C₇₃	#123	1	2	yes	—
77	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
78	C₂×C₄×C₇₃	#135	1	4	yes	—
79	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	4	yes	—
80	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
81	C₂×Dic₇₃	—	8	2	no	—
82	C₂×Dic₇₃	—	8	2	no	—
83	C₂×Dic₇₃	—	8	2	no	—
84	C₈×C₇₃	—	2	4	no	—
85	#10596831	—	4	4	no	—
86	C₂²×C₇₃	—	8	4	no	—
87	C₂²×C₇₃	—	8	2	no	—
88	C₂²×C₇₃	—	16	1	no	—
89	C₂²×C₇₃	—	16	1	no	—
90	C₂²×C₇₃	—	16	2	no	—
91	C₂²×C₇₃	#399	1	1	yes	characteristic
92	C₂²×C₇₃	#492	1	1	yes	characteristic
93	C₂²×C₇₃	#495	1	2	yes	—
94	C₂²×C₇₃	#496	1	2	yes	—
95	C₂²×C₇₃	#496	1	1	yes	characteristic
96	C₄×C₇₃	—	16	1	no	—
97	C₄×C₇₃	#476	1	2	yes	—
98	C₄×C₇₃	#478	1	4	yes	—
99	C₄×C₇₃	#481	1	2	yes	—
100	C₄×C₇₃	#482	1	4	yes	—
101	Dic₇₃	—	16	2	no	—
102	#10677	—	73	1	no	maximal, Sylow
103	C₂×C₇₃	—	16	2	no	—
104	C₂×C₇₃	#1708	1	1	yes	characteristic
105	C₂×C₇₃	#1715	1	1	yes	characteristic
106	C₂×C₇₃	#1717	1	2	yes	—
107	C₂×C₇₃	#2092	1	2	yes	—
108	C₂×C₇₃	#2094	1	1	yes	characteristic
109	#1957	—	73	2	no	—
110	#1957	D₇₃	1	1	yes	characteristic
111	#1844	—	73	1	no	—
112	#1834	—	73	2	no	—
113	#1273	—	73	1	no	—
114	C₇₃	#10677	1	1	yes	characteristic, complemented, Sylow
115	C₂×#127	C₂×D₇₃	1	1	yes	characteristic
116	C₂×#106	—	146	2	no	—
117	C₂×#102	—	2	2	no	—
118	C₂×#102	—	146	4	no	—
119	C₂×C₄×C₈	—	73	2	no	—
120	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
121	#384	—	73	2	no	—
122	#384	C₂×D₇₃	1	1	yes	characteristic
123	#340	—	146	4	no	—
124	C₂²×D₄	—	2	2	no	—
125	C₂²×D₄	—	2	1	no	—
126	C₂²×C₈	—	73	4	no	—
127	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
128	#127	—	2	2	no	—
129	#127	—	2	1	no	—
130	#127	Dic₁₄₆	1	2	yes	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	—	2	1	no	—
133	C₂×C₄⋊C₄	—	146	2	no	—
134	C₂×C₄⋊C₄	—	146	2	no	—
135	C₂×C₄⋊C₄	—	146	2	no	—
136	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
137	#106	—	292	4	no	—
138	#102	—	4	4	no	—
139	#102	—	292	8	no	—
140	C₄×C₈	—	2	2	no	—
141	C₄×C₈	—	146	2	no	—
142	C₄×C₈	—	146	2	no	—
143	C₂⁴	—	4	1	no	—
144	C₂×D₄	—	2	4	no	—
145	C₂×D₄	—	2	2	no	—
146	C₂×D₄	—	4	1	no	—
147	C₂×D₄	—	4	2	no	—
148	C₂×D₄	—	4	2	no	—
149	C₂×D₄	—	4	4	no	—
150	C₂×D₄	—	4	4	no	—
151	C₂×D₄	—	4	2	no	—
152	C₂²×C₄	—	4	1	no	—
153	C₂²×C₄	—	292	2	no	—
154	C₂²×C₄	D₄×D₇₃	1	2	yes	—
155	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	146	4	no	—
159	C₂×C₈	—	146	4	no	—
160	C₂×C₈	—	146	4	no	—
161	C₂×C₈	#11924016	1	4	yes	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	4	no	—
164	C₄⋊C₄	—	292	4	no	—
165	C₄⋊C₄	—	292	4	no	—
166	C₄⋊C₄	—	292	4	no	—
167	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
168	C₄²	#11924027	1	2	yes	—
169	C₄²	#11924028	1	1	yes	characteristic
170	C₂³	—	4	2	no	—
171	C₂³	—	8	1	no	—
172	C₂³	—	8	2	no	—
173	C₂³	—	8	2	no	—
174	C₂³	—	8	1	no	—
175	C₂³	#14556587	1	1	yes	characteristic
176	D₄	—	4	4	no	—
177	D₄	—	4	8	no	—
178	D₄	—	8	2	no	—
179	D₄	—	8	4	no	—
180	C₂×C₄	—	8	1	no	—
181	C₂×C₄	—	8	2	no	—
182	C₂×C₄	—	584	2	no	—
183	C₂×C₄	—	584	2	no	—
184	C₂×C₄	—	584	2	no	—
185	C₂×C₄	#14556568	1	2	yes	—
186	C₂×C₄	#14556570	1	1	yes	characteristic
187	C₂×C₄	#14556572	1	1	yes	characteristic
188	C₂×C₄	#14556573	1	2	yes	—
189	C₂×C₄	C₂×#11924016	1	2	yes	—
190	C₂×C₄	#14556612	1	4	yes	—
191	C₂×C₄	#14556615	1	4	yes	—
192	C₂×C₄	#14556616	1	2	yes	—
193	C₈	—	2	4	no	—
194	C₈	—	292	4	no	—
195	C₂²	—	8	2	no	—
196	C₂²	—	8	4	no	—
197	C₂²	—	16	1	no	—
198	C₂²	—	16	1	no	—
199	C₂²	—	16	2	no	—
200	C₂²	#17988523	1	1	yes	characteristic
201	C₂²	#17988959	1	1	yes	characteristic
202	C₂²	#17988968	1	2	yes	—
203	C₂²	#17988975	1	2	yes	—
204	C₂²	#17988977	1	1	yes	characteristic
205	C₄	—	16	1	no	—
206	C₄	—	1168	2	no	—
207	C₄	#17988877	1	2	yes	—
208	C₄	#17988889	1	4	yes	—
209	C₄	#17988903	1	2	yes	—
210	C₄	#17988910	1	4	yes	—
211	C₂	—	16	2	no	—
212	C₂	#22609530	1	1	yes	characteristic
213	C₂	#22609557	1	1	yes	characteristic
214	C₂	#22609572	1	2	yes	—
215	C₂	#22611454	1	2	yes	—
216	C₂	#22611464	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22610738

C₂

yes

maximal

#22610088

C₂

yes

maximal, characteristic, complemented

#22610058

C₂

yes

maximal, complemented

#22607758

C₂

yes

maximal, characteristic, complemented

C₇₃×#1957

C₂

yes

maximal, characteristic

C₂×#14556523

—

C₂×#14556512

—

C₂×C₄×#10596831

C₂²

yes

—

#17988457

C₂²

yes

—

#17988349

—

C₂×#127×C₇₃

C₂²

yes

characteristic

C₂×#102×C₇₃

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#384×C₇₃

C₂²

yes

characteristic

C₂²×#10596831

D₄

yes

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

—

—

—

—

C₂²×D₄×C₇₃

—

C₂²×D₄×C₇₃

—

C₂²×C₈×C₇₃

D₄

yes

—

#127×C₇₃

—

#127×C₇₃

—

#127×C₇₃

Q₈

yes

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

#102×C₇₃

—

C₄×C₈×C₇₃

—

C₂²×Dic₇₃

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂⁴×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

SD₁₆

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×Q₈

yes

characteristic

C₄²×C₇₃

#41

yes

—

C₄²×C₇₃

#41

yes

characteristic

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

#127

yes

characteristic

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

—

C₂×C₄×C₇₃

C₂×SD₁₆

yes

—

C₂×C₄×C₇₃

#135

yes

—

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₂²×C₇₃

#496

yes

—

C₂²×C₇₃

#496

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

#476

yes

—

C₄×C₇₃

#478

yes

—

C₄×C₇₃

#481

yes

—

100

C₄×C₇₃

#482

yes

—

101

Dic₇₃

—

102

#10677

—

maximal, Sylow

103

C₂×C₇₃

—

104

C₂×C₇₃

#1708

yes

characteristic

105

C₂×C₇₃

#1715

yes

characteristic

106

C₂×C₇₃

#1717

yes

—

107

C₂×C₇₃

#2092

yes

—

108

C₂×C₇₃

#2094

yes

characteristic

109

#1957

—

110

#1957

D₇₃

yes

characteristic

111

#1844

—

112

#1834

—

113

#1273

—

114

C₇₃

#10677

yes

characteristic, complemented, Sylow

115

C₂×#127

C₂×D₇₃

yes

characteristic

116

C₂×#106

—

146

—

117

C₂×#102

—

118

C₂×#102

—

146

—

119

C₂×C₄×C₈

—

120

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

121

#384

—

122

#384

C₂×D₇₃

yes

characteristic

123

#340

—

146

—

124

C₂²×D₄

—

125

C₂²×D₄

—

126

C₂²×C₈

—

127

C₂²×C₈

C₂²⋊D₇₃

yes

—

128

#127

—

129

#127

—

130

#127

Dic₁₄₆

yes

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

146

—

134

C₂×C₄⋊C₄

—

146

—

135

C₂×C₄⋊C₄

—

146

—

136

C₂×C₄²

C₂²×D₇₃

yes

characteristic

137

#106

—

292

—

138

#102

—

139

#102

—

292

—

140

C₄×C₈

—

141

C₄×C₈

—

146

—

142

C₄×C₈

—

146

—

143

C₂⁴

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂²×C₄

—

153

C₂²×C₄

—

292

—

154

C₂²×C₄

D₄×D₇₃

yes

—

155

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

146

—

159

C₂×C₈

—

146

—

160

C₂×C₈

—

146

—

161

C₂×C₈

#11924016

yes

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

292

—

165

C₄⋊C₄

—

292

—

166

C₄⋊C₄

—

292

—

167

C₄²

C₂×Dic₁₄₆

yes

characteristic

168

C₄²

#11924027

yes

—

169

C₄²

#11924028

yes

characteristic

170

C₂³

—

171

C₂³

—

172

C₂³

—

173

C₂³

—

174

C₂³

—

175

C₂³

#14556587

yes

characteristic

176

D₄

—

177

D₄

—

178

D₄

—

179

D₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

584

—

183

C₂×C₄

—

584

—

184

C₂×C₄

—

584

—

185

C₂×C₄

#14556568

yes

—

186

C₂×C₄

#14556570

yes

characteristic

187

C₂×C₄

#14556572

yes

characteristic

188

C₂×C₄

#14556573

yes

—

189

C₂×C₄

C₂×#11924016

yes

—

190

C₂×C₄

#14556612

yes

—

191

C₂×C₄

#14556615

yes

—

192

C₂×C₄

#14556616

yes

—

193

C₈

—

194

C₈

—

292

—

195

C₂²

—

196

C₂²

—

197

C₂²

—

198

C₂²

—

199

C₂²

—

200

C₂²

#17988523

yes

characteristic

201

C₂²

#17988959

yes

characteristic

202

C₂²

#17988968

yes

—

203

C₂²

#17988975

yes

—

204

C₂²

#17988977

yes

characteristic

205

C₄

—

206

C₄

—

1168

—

207

C₄

#17988877

yes

—

208

C₄

#17988889

yes

—

209

C₄

#17988903

yes

—

210

C₄

#17988910

yes

—

211

C₂

—

212

C₂

#22609530

yes

characteristic

213

C₂

#22609557

yes

characteristic

214

C₂

#22609572

yes

—

215

C₂

#22611454

yes

—

216

C₂

#22611464

yes

characteristic

	subgroup	G/H
center	C₂³	#14556587
commutator, cosocle	C₂×C₄²×C₇₃	C₂³
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1957	C₂
socle	C₂³×C₇₃	#127
prime radical	C₂×#127×C₇₃	C₂²
hypercenter	#1957	D₇₃
hypocenter	C₇₃	#10677
LCS	G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1957

subgroup

G/H

center

C₂³

#14556587

commutator, cosocle

C₂×C₄²×C₇₃

C₂³

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1957

C₂

socle

C₂³×C₇₃

#127

prime radical

C₂×#127×C₇₃

C₂²

D₇₃

C₇₃

G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1957

N	H ≅ G/N	type
2. #22610088	C₂	semidirect
3. #22610058	C₂	semidirect
4. #22607758	C₂	semidirect
114. C₇₃	#10677	semidirect

H ≅ G/N

type

2. #22610088

C₂

semidirect

3. #22610058

C₂

semidirect

4. #22607758

C₂

semidirect

114. C₇₃

#10677

semidirect

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

subgroup	index
#22610738	2
#22610088	2
#22610058	2
#22607758	2
C₇₃×#1957	2
#10677	73

subgroup

index

C₇₃×#1957

Subgroups — #32991056

Conjugacy classes of subgroups

Complements

Maximal subgroups