Group #32985717 — subgroups

Subgroups — #32985717

Order 18688.

526 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 199 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×#17988985	C₂	1	1	yes	maximal, characteristic
2	#22610891	C₂	1	2	yes	maximal
3	#22610732	C₂	1	2	yes	maximal
4	C₂×C₈×#10596831	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1844	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556591	C₂²	1	2	yes	—
7	#17988985	—	2	2	no	—
8	#17988985	C₄	1	4	yes	complemented
9	C₂×#14556523	—	2	2	no	—
10	C₂×#14556514	—	2	4	no	—
11	C₂×C₄×#10596831	C₂²	1	2	yes	—
12	#17988349	—	2	4	no	—
13	C₈×#10596831	—	2	2	no	—
14	C₂×#106×C₇₃	—	2	2	no	—
15	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
16	#384×C₇₃	C₂²	1	1	yes	characteristic
17	#384×C₇₃	C₂²	1	1	yes	characteristic
18	C₂²×Dic₁₄₆	—	2	2	no	—
19	C₂²×#10596831	D₄	1	4	yes	—
20	#14556591	—	2	2	no	—
21	#14556591	—	2	2	no	—
22	#14556591	—	2	2	no	—
23	#14556591	C₂×C₄	1	4	yes	—
24	C₂×C₄⋊Dic₇₃	—	2	2	no	—
25	C₂×C₄⋊Dic₇₃	—	2	2	no	—
26	#14556523	—	2	8	no	—
27	#14556514	—	4	8	no	—
28	C₄×#10596831	—	2	2	no	—
29	C₄×#10596831	—	2	2	no	—
30	C₂²×C₈×C₇₃	D₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
32	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
33	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
35	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
36	#106×C₇₃	—	4	4	no	—
37	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₇₃	D₄	1	2	yes	—
39	C₂×Dic₁₄₆	—	2	4	no	—
40	C₂×Dic₁₄₆	—	4	4	no	—
41	C₂×Dic₁₄₆	—	4	4	no	—
42	C₂×Dic₁₄₆	—	4	2	no	—
43	C₂²×Dic₇₃	—	4	2	no	—
44	C₂×#10596831	—	2	4	no	—
45	C₂×#10596831	—	2	4	no	—
46	C₂×#10596831	SD₁₆	1	8	yes	—
47	C₄⋊Dic₇₃	—	2	4	no	—
48	C₄⋊Dic₇₃	—	2	4	no	—
49	C₄⋊Dic₇₃	—	4	2	no	—
50	C₄⋊Dic₇₃	—	4	2	no	—
51	C₂²×C₄×C₇₃	—	4	1	no	—
52	C₂²×C₄×C₇₃	—	4	1	no	—
53	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₇₃	C₂²⋊C₄	1	4	yes	—
56	C₂×C₈×C₇₃	SD₁₆	1	4	yes	—
57	C₂×C₈×C₇₃	Q₁₆	1	4	yes	—
58	C₄⋊C₄×C₇₃	—	4	2	no	—
59	C₄⋊C₄×C₇₃	—	4	2	no	—
60	C₄⋊C₄×C₇₃	—	4	4	no	—
61	C₄⋊C₄×C₇₃	—	4	4	no	—
62	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₇₃	#41	1	2	yes	—
65	C₂³×C₇₃	#127	1	1	yes	characteristic
66	C₂×C₄×C₇₃	—	8	2	no	—
67	C₂×C₄×C₇₃	—	8	1	no	—
68	C₂×C₄×C₇₃	—	8	1	no	—
69	C₂×C₄×C₇₃	—	8	2	no	—
70	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
72	C₂×C₄×C₇₃	#121	1	2	yes	—
73	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
74	C₂×C₄×C₇₃	C₂×SD₁₆	1	4	yes	—
75	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
76	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
77	C₂×C₄×C₇₃	#135	1	4	yes	—
78	C₂×Dic₇₃	—	4	4	no	—
79	C₂×Dic₇₃	—	8	2	no	—
80	C₂×Dic₇₃	—	8	2	no	—
81	Dic₁₄₆	—	4	8	no	—
82	Dic₁₄₆	—	8	4	no	—
83	C₈×C₇₃	#103	1	8	yes	complemented
84	#10596831	—	2	8	no	—
85	C₂²×C₇₃	#390	1	1	yes	characteristic
86	C₂²×C₇₃	#492	1	2	yes	—
87	C₂²×C₇₃	#492	1	1	yes	characteristic
88	C₂²×C₇₃	#494	1	1	yes	characteristic
89	C₂²×C₇₃	#495	1	2	yes	—
90	C₄×C₇₃	—	16	1	no	—
91	C₄×C₇₃	—	16	1	no	—
92	C₄×C₇₃	C₂×#103	1	2	yes	—
93	C₄×C₇₃	C₄×SD₁₆	1	4	yes	—
94	C₄×C₇₃	#465	1	4	yes	—
95	C₄×C₇₃	#487	1	2	yes	—
96	Dic₇₃	—	8	4	no	—
97	#9819	—	73	1	no	maximal, Sylow
98	C₂×C₇₃	#1705	1	1	yes	characteristic
99	C₂×C₇₃	#1707	1	2	yes	—
100	C₂×C₇₃	#1708	1	1	yes	characteristic
101	C₂×C₇₃	#1956	1	1	yes	characteristic
102	C₂×C₇₃	#1968	1	2	yes	—
103	C₂×#498	—	73	1	no	—
104	#1990	—	73	2	no	—
105	#1956	—	73	2	no	—
106	#1844	D₇₃	1	1	yes	characteristic
107	C₂×C₈²	—	73	1	no	—
108	C₇₃	#9819	1	1	yes	characteristic, complemented, Sylow
109	C₂×C₄⋊Q₈	—	73	2	no	—
110	#498	—	73	4	no	—
111	#498	—	146	2	no	—
112	C₂×#106	—	2	2	no	—
113	C₂×#106	—	146	2	no	—
114	C₂×#103	—	146	4	no	—
115	C₂×C₄×C₈	—	73	2	no	—
116	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
117	#384	C₂×D₇₃	1	1	yes	characteristic
118	#384	C₂×D₇₃	1	1	yes	characteristic
119	#340	—	146	4	no	—
120	C₈²	—	146	2	no	—
121	C₂²×Q₈	—	146	2	no	—
122	C₂²×C₈	—	73	4	no	—
123	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
124	C₄⋊Q₈	—	73	4	no	—
125	C₄⋊Q₈	—	146	2	no	—
126	C₄⋊Q₈	—	146	2	no	—
127	C₄⋊Q₈	—	146	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	2	1	no	—
131	C₂×C₄⋊C₄	—	2	1	no	—
132	C₂×C₄⋊C₄	—	146	2	no	—
133	C₂×C₄⋊C₄	—	146	2	no	—
134	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
135	#106	—	4	4	no	—
136	#106	—	146	8	no	—
137	#103	—	292	8	no	—
138	C₄×C₈	—	146	2	no	—
139	C₄×C₈	—	146	2	no	—
140	C₄×C₈	C₄×D₇₃	1	2	yes	—
141	C₄×C₈	D₂₉₂	1	2	yes	—
142	C₂×Q₈	—	292	2	no	—
143	C₂×Q₈	—	292	4	no	—
144	C₂×Q₈	—	292	4	no	—
145	C₂×Q₈	—	146	4	no	—
146	C₂²×C₄	—	4	1	no	—
147	C₂²×C₄	—	4	1	no	—
148	C₂²×C₄	—	292	2	no	—
149	C₂²×C₄	D₄×D₇₃	1	2	yes	—
150	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
151	C₂×C₈	—	73	8	no	—
152	C₂×C₈	—	146	4	no	—
153	C₂×C₈	—	146	4	no	—
154	C₂×C₈	#11924005	1	4	yes	—
155	C₂×C₈	#11924016	1	4	yes	—
156	C₂×C₈	#11924018	1	4	yes	—
157	C₄⋊C₄	—	4	4	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	4	no	—
161	C₄⋊C₄	—	292	2	no	—
162	C₄⋊C₄	—	292	2	no	—
163	C₄⋊C₄	—	146	4	no	—
164	C₄⋊C₄	—	146	4	no	—
165	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
166	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
167	C₄²	#11924029	1	2	yes	—
168	C₂³	#14556587	1	1	yes	characteristic
169	Q₈	—	292	8	no	—
170	Q₈	—	584	4	no	—
171	C₂×C₄	—	8	2	no	—
172	C₂×C₄	—	8	1	no	—
173	C₂×C₄	—	8	1	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	—	292	4	no	—
176	C₂×C₄	—	584	2	no	—
177	C₂×C₄	—	584	2	no	—
178	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
179	C₂×C₄	#14556553	1	2	yes	—
180	C₂×C₄	#14556566	1	2	yes	—
181	C₂×C₄	#14556580	1	1	yes	characteristic
182	C₂×C₄	SD₁₆×D₇₃	1	4	yes	—
183	C₂×C₄	C₂×#11924016	1	2	yes	—
184	C₂×C₄	C₂×#11924018	1	2	yes	—
185	C₂×C₄	#14556612	1	4	yes	—
186	C₈	—	146	8	no	—
187	C₈	#14556514	1	8	yes	complemented
188	C₂²	#17988488	1	1	yes	characteristic
189	C₂²	#17988958	1	2	yes	—
190	C₂²	#17988959	1	1	yes	characteristic
191	C₂²	#17988966	1	1	yes	characteristic
192	C₂²	#17988968	1	2	yes	—
193	C₄	—	16	1	no	—
194	C₄	—	16	1	no	—
195	C₄	—	584	4	no	—
196	C₄	C₂×#14556514	1	2	yes	—
197	C₄	#17988674	1	4	yes	—
198	C₄	#17988817	1	4	yes	—
199	C₄	#17988939	1	2	yes	—
200	C₂	#22609517	1	1	yes	characteristic
201	C₂	#22609526	1	2	yes	—
202	C₂	#22609530	1	1	yes	characteristic
203	C₂	#22610732	1	1	yes	characteristic
204	C₂	#22610783	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988985

C₂

yes

maximal, characteristic

#22610891

C₂

yes

maximal

#22610732

C₂

yes

maximal

C₂×C₈×#10596831

C₂

yes

maximal, characteristic

C₇₃×#1844

C₂

yes

maximal, characteristic

C₂×#14556591

C₂²

yes

—

#17988985

—

#17988985

C₄

yes

—

—

C₂²

yes

—

#17988349

—

C₈×#10596831

—

C₂×#106×C₇₃

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#384×C₇₃

C₂²

yes

characteristic

#384×C₇₃

C₂²

yes

characteristic

C₂²×Dic₁₄₆

—

C₂²×#10596831

D₄

yes

—

—

—

—

C₂×C₄

yes

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

—

—

—

—

C₂²×C₈×C₇₃

D₄

yes

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

#106×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

D₄

yes

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

SD₁₆

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂²⋊C₄

yes

—

C₂×C₈×C₇₃

SD₁₆

yes

—

C₂×C₈×C₇₃

Q₁₆

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

#41

yes

—

C₂³×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄×D₄

yes

—

C₂×C₄×C₇₃

#121

yes

—

C₂×C₄×C₇₃

#124

yes

characteristic

C₂×C₄×C₇₃

C₂×SD₁₆

yes

—

C₂×C₄×C₇₃

C₂×SD₁₆

yes

—

C₂×C₄×C₇₃

C₂×Q₁₆

yes

—

C₂×C₄×C₇₃

#135

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

#103

yes

complemented

#10596831

—

C₂²×C₇₃

#390

yes

characteristic

C₂²×C₇₃

#492

yes

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×#103

yes

—

C₄×C₇₃

C₄×SD₁₆

yes

—

C₄×C₇₃

#465

yes

—

C₄×C₇₃

#487

yes

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

100

C₂×C₇₃

#1708

yes

characteristic

101

C₂×C₇₃

#1956

yes

characteristic

102

C₂×C₇₃

#1968

yes

—

103

C₂×#498

—

104

#1990

—

105

#1956

—

106

#1844

D₇₃

yes

characteristic

107

C₂×C₈²

—

108

C₇₃

#9819

yes

characteristic, complemented, Sylow

109

C₂×C₄⋊Q₈

—

110

#498

—

111

#498

—

146

—

112

C₂×#106

—

113

C₂×#106

—

146

—

114

C₂×#103

—

146

—

115

C₂×C₄×C₈

—

116

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

117

#384

C₂×D₇₃

yes

characteristic

118

#384

C₂×D₇₃

yes

characteristic

119

#340

—

146

—

120

C₈²

—

146

—

121

C₂²×Q₈

—

146

—

122

C₂²×C₈

—

123

C₂²×C₈

C₂²⋊D₇₃

yes

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

146

—

126

C₄⋊Q₈

—

146

—

127

C₄⋊Q₈

—

146

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

—

146

—

133

C₂×C₄⋊C₄

—

146

—

134

C₂×C₄²

C₂²×D₇₃

yes

characteristic

135

#106

—

136

#106

—

146

—

137

#103

—

292

—

138

C₄×C₈

—

146

—

139

C₄×C₈

—

146

—

140

C₄×C₈

C₄×D₇₃

yes

—

141

C₄×C₈

D₂₉₂

yes

—

142

C₂×Q₈

—

292

—

143

C₂×Q₈

—

292

—

144

C₂×Q₈

—

292

—

145

C₂×Q₈

—

146

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

292

—

149

C₂²×C₄

D₄×D₇₃

yes

—

150

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

151

C₂×C₈

—

152

C₂×C₈

—

146

—

153

C₂×C₈

—

146

—

154

C₂×C₈

#11924005

yes

—

155

C₂×C₈

#11924016

yes

—

156

C₂×C₈

#11924018

yes

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

292

—

162

C₄⋊C₄

—

292

—

163

C₄⋊C₄

—

146

—

164

C₄⋊C₄

—

146

—

165

C₄²

C₂×C₄×D₇₃

yes

characteristic

166

C₄²

C₂×D₂₉₂

yes

characteristic

167

C₄²

#11924029

yes

—

168

C₂³

#14556587

yes

characteristic

169

Q₈

—

292

—

170

Q₈

—

584

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

292

—

176

C₂×C₄

—

584

—

177

C₂×C₄

—

584

—

178

C₂×C₄

C₂×#11924005

yes

characteristic

179

C₂×C₄

#14556553

yes

—

180

C₂×C₄

#14556566

yes

—

181

C₂×C₄

#14556580

yes

characteristic

182

C₂×C₄

SD₁₆×D₇₃

yes

—

183

C₂×C₄

C₂×#11924016

yes

—

184

C₂×C₄

C₂×#11924018

yes

—

185

C₂×C₄

#14556612

yes

—

186

C₈

—

146

—

187

C₈

#14556514

yes

complemented

188

C₂²

#17988488

yes

characteristic

189

C₂²

#17988958

yes

—

190

C₂²

#17988959

yes

characteristic

191

C₂²

#17988966

yes

characteristic

192

C₂²

#17988968

yes

—

193

C₄

—

194

C₄

—

195

C₄

—

584

—

196

C₄

C₂×#14556514

yes

—

197

C₄

#17988674

yes

—

198

C₄

#17988817

yes

—

199

C₄

#17988939

yes

—

200

C₂

#22609517

yes

characteristic

201

C₂

#22609526

yes

—

202

C₂

#22609530

yes

characteristic

203

C₂

#22610732

yes

characteristic

204

C₂

#22610783

yes

—

	subgroup	G/H
center	C₂³	#14556587
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1844	C₂
socle, prime radical	C₂³×C₇₃	#127
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1844	D₇₃
hypocenter	C₇₃	#9819
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

subgroup

G/H

center

C₂³

#14556587

commutator

C₄²×C₇₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1844

C₂

socle, prime radical

C₂³×C₇₃

#127

cosocle

C₂×C₄²×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

N	H ≅ G/N	type
8. #17988985	C₄	semidirect
83. C₈×C₇₃	#103	semidirect
108. C₇₃	#9819	semidirect
187. C₈	#14556514	semidirect

H ≅ G/N

type

8. #17988985

C₄

semidirect

83. C₈×C₇₃

#103

semidirect

108. C₇₃

#9819

semidirect

187. C₈

#14556514

semidirect

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

subgroup	index
C₂×#17988985	2
#22610891	2
#22610732	2
C₂×C₈×#10596831	2
C₇₃×#1844	2
#9819	73

subgroup

index

C₇₃×#1844

Subgroups — #32985717

Conjugacy classes of subgroups

Complements

Maximal subgroups