Group #32978907 — subgroups

Subgroups — #32978907

Order 18688.

398 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 121 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×#17988711	C₂	1	1	yes	maximal, characteristic
2	#22609924	C₂	1	2	yes	maximal
3	#22608686	C₂	1	2	yes	maximal
4	#22607806	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1292	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₄₆	C₂²	1	1	yes	characteristic
7	#17988711	—	2	2	no	—
8	#17988711	—	2	2	no	—
9	#17988611	C₂²	1	2	yes	—
10	C₂×#14556520	C₄	1	2	yes	—
11	C₂×#14556509	—	2	2	no	—
12	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₇₃	C₄	1	2	yes	—
14	#17988355	—	2	2	no	—
15	#17988337	—	2	2	no	—
16	#17988337	—	2	2	no	—
17	#423×C₇₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
19	C₂²×Dic₁₄₆	D₄	1	1	yes	characteristic
20	C₂×#11924012	—	2	2	no	—
21	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
22	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
23	C₄×Dic₁₄₆	—	2	2	no	—
24	C₄×Dic₁₄₆	—	2	2	no	—
25	C₂×C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
26	C₂×#11924007	Q₈	1	2	yes	—
27	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
28	C₄⋊#10596831	—	2	2	no	—
29	C₄⋊#10596831	—	2	2	no	—
30	#14556520	—	2	4	no	—
31	#14556509	—	4	4	no	—
32	C₄×#10596831	D₄	1	2	yes	—
33	C₄×#10596831	D₄	1	2	yes	—
34	C₈×Dic₇₃	—	2	2	no	—
35	C₈×Dic₇₃	—	2	2	no	—
36	C₂×M₁₆×C₇₃	—	2	2	no	—
37	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₇₃	—	2	2	no	—
40	C₄⋊C₈×C₇₃	—	2	2	no	—
41	C₄⋊C₈×C₇₃	—	2	2	no	—
42	C₂×Dic₁₄₆	—	2	2	no	—
43	C₂×Dic₁₄₆	—	2	1	no	—
44	C₂×Dic₁₄₆	—	2	1	no	—
45	C₂×Dic₁₄₆	SD₁₆	1	4	yes	—
46	C₂²×Dic₇₃	C₄⋊C₄	1	2	yes	—
47	C₂²×Dic₇₃	C₄⋊C₄	1	2	yes	—
48	#11924012	—	4	2	no	—
49	#11924012	—	4	2	no	—
50	C₂×#10596831	—	2	2	no	—
51	C₂×#10596831	—	4	1	no	—
52	C₂×#10596831	—	4	2	no	—
53	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
55	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
56	C₄⋊Dic₇₃	—	2	2	no	—
57	#11924007	—	2	4	no	—
58	C₄×Dic₇₃	—	2	2	no	—
59	C₄×Dic₇₃	—	2	2	no	—
60	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
63	M₁₆×C₇₃	—	4	2	no	—
64	M₁₆×C₇₃	—	4	2	no	—
65	C₂×C₈×C₇₃	—	2	2	no	—
66	C₂×C₈×C₇₃	—	4	2	no	—
67	C₂×C₈×C₇₃	—	4	2	no	—
68	C₄²×C₇₃	C₂×D₄	1	2	yes	—
69	C₄²×C₇₃	#41	1	2	yes	—
70	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₇₃	—	2	1	no	—
72	C₂×C₄×C₇₃	—	2	1	no	—
73	C₂×C₄×C₇₃	—	4	1	no	—
74	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
76	C₂×C₄×C₇₃	#121	1	2	yes	—
77	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
78	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
79	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
80	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
81	C₂×Dic₇₃	—	2	2	no	—
82	C₂×Dic₇₃	—	2	2	no	—
83	C₂×Dic₇₃	—	2	2	no	—
84	C₂×Dic₇₃	—	4	2	no	—
85	C₂×Dic₇₃	#106	1	4	yes	—
86	Dic₁₄₆	—	2	4	no	—
87	Dic₁₄₆	—	2	4	no	—
88	C₈×C₇₃	—	4	2	no	—
89	C₈×C₇₃	—	8	2	no	—
90	#10596831	—	2	1	no	—
91	#10596831	—	2	2	no	—
92	#10596831	—	8	2	no	—
93	#10596831	#101	1	2	yes	complemented
94	C₂²×C₇₃	—	2	1	no	—
95	C₂²×C₇₃	—	2	1	no	—
96	C₂²×C₇₃	#380	1	1	yes	characteristic
97	C₂²×C₇₃	C₂×#106	1	1	yes	characteristic
98	C₂²×C₇₃	#427	1	1	yes	characteristic
99	C₄×C₇₃	—	2	1	no	—
100	C₄×C₇₃	—	2	2	no	—
101	C₄×C₇₃	C₂×#101	1	1	yes	characteristic
102	C₄×C₇₃	#413	1	1	yes	characteristic
103	C₄×C₇₃	#465	1	2	yes	—
104	C₄×C₇₃	#465	1	2	yes	—
105	Dic₇₃	—	2	4	no	—
106	Dic₇₃	—	4	2	no	—
107	#8615	—	73	1	no	maximal, Sylow
108	C₂×C₇₃	—	2	1	no	—
109	C₂×C₇₃	—	2	1	no	—
110	C₂×C₇₃	#1801	1	1	yes	characteristic
111	C₂×C₇₃	#1807	1	1	yes	characteristic
112	C₂×C₇₃	#1813	1	1	yes	characteristic
113	C₂×C₈×Q₈	—	73	1	no	—
114	#1801	—	73	2	no	—
115	#1566	—	73	2	no	—
116	#1292	—	73	1	no	—
117	#1292	D₇₃	1	1	yes	characteristic
118	C₇₃	#8615	1	1	yes	characteristic, complemented, Sylow
119	C₂×C₄×Q₈	—	73	1	no	—
120	C₈×Q₈	—	146	2	no	—
121	C₈×Q₈	—	146	2	no	—
122	#423	—	73	2	no	—
123	#423	C₂×D₇₃	1	2	yes	—
124	C₂×C₄⋊C₈	—	73	2	no	—
125	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
126	C₂×#101	—	146	2	no	—
127	C₂×C₄×C₈	—	73	1	no	—
128	C₂×C₄×C₈	—	73	2	no	—
129	#343	—	146	2	no	—
130	#334	—	146	2	no	—
131	#334	—	146	2	no	—
132	C₂²×Q₈	—	73	1	no	—
133	C₂×M₁₆	—	2	2	no	—
134	C₂×M₁₆	—	146	2	no	—
135	C₂²×C₈	—	73	1	no	—
136	C₂²×C₈	—	73	1	no	—
137	C₂²×C₈	C₄×D₇₃	1	2	yes	—
138	C₄×Q₈	—	146	2	no	—
139	C₄×Q₈	—	146	2	no	—
140	C₂×C₄⋊C₄	—	73	1	no	—
141	C₂×C₄⋊C₄	—	73	2	no	—
142	C₂×C₄²	—	73	2	no	—
143	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
144	C₄⋊C₈	—	2	2	no	—
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	146	4	no	—
148	C₄⋊C₈	—	146	2	no	—
149	C₄⋊C₈	—	146	2	no	—
150	#101	—	292	4	no	—
151	C₄×C₈	—	73	2	no	—
152	C₄×C₈	—	73	2	no	—
153	C₄×C₈	—	146	2	no	—
154	C₄×C₈	—	146	2	no	—
155	C₂×Q₈	—	73	4	no	—
156	C₂×Q₈	—	146	2	no	—
157	C₂×Q₈	—	146	1	no	—
158	C₂×Q₈	—	146	1	no	—
159	C₂²×C₄	—	73	2	no	—
160	C₂²×C₄	—	73	2	no	—
161	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
162	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
163	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
164	M₁₆	—	4	2	no	—
165	M₁₆	—	4	2	no	—
166	M₁₆	—	292	2	no	—
167	M₁₆	—	292	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	—	4	2	no	—
171	C₂×C₈	—	292	2	no	—
172	C₂×C₈	—	292	1	no	—
173	C₂×C₈	—	73	1	no	—
174	C₂×C₈	—	73	1	no	—
175	C₂×C₈	—	73	2	no	—
176	C₂×C₈	—	146	2	no	—
177	C₄⋊C₄	—	146	2	no	—
178	C₄⋊C₄	—	146	4	no	—
179	C₄²	—	146	2	no	—
180	C₄²	—	146	2	no	—
181	C₄²	D₄×D₇₃	1	2	yes	—
182	C₄²	#11924028	1	2	yes	—
183	C₂³	C₄⋊C₄×D₇₃	1	1	yes	characteristic
184	Q₈	—	146	4	no	—
185	Q₈	—	146	4	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	4	1	no	—
189	C₂×C₄	—	292	2	no	—
190	C₂×C₄	—	73	4	no	—
191	C₂×C₄	—	146	2	no	—
192	C₂×C₄	—	146	2	no	—
193	C₂×C₄	—	146	2	no	—
194	C₂×C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
195	C₂×C₄	#14556548	1	1	yes	characteristic
196	C₂×C₄	#14556564	1	2	yes	—
197	C₂×C₄	#14556569	1	2	yes	—
198	C₂×C₄	SD₁₆×D₇₃	1	2	yes	—
199	C₂×C₄	#14556610	1	1	yes	characteristic
200	C₂×C₄	#14556614	1	1	yes	characteristic
201	C₈	—	4	2	no	—
202	C₈	—	8	2	no	—
203	C₈	—	584	2	no	—
204	C₈	—	73	2	no	—
205	C₈	—	146	1	no	—
206	C₈	—	146	2	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	#17988446	1	1	yes	characteristic
210	C₂²	#106×D₇₃	1	1	yes	characteristic
211	C₂²	#17988624	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	2	no	—
214	C₄	—	292	2	no	—
215	C₄	—	146	4	no	—
216	C₄	#101×D₇₃	1	1	yes	characteristic
217	C₄	#17988568	1	1	yes	characteristic
218	C₄	#17988815	1	2	yes	—
219	C₄	#17988819	1	2	yes	—
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	#22609926	1	1	yes	characteristic
223	C₂	#22609955	1	1	yes	characteristic
224	C₂	#22609980	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988711

C₂

yes

maximal, characteristic

#22609924

C₂

yes

maximal

#22608686

C₂

yes

maximal

#22607806

C₂

yes

maximal, characteristic

C₇₃×#1292

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₄₆

C₂²

yes

—

—

C₂²

yes

—

C₂×#14556520

C₄

yes

—

C₂×#14556509

—

C₂×C₄×#10596831

C₂²

yes

characteristic

C₂×C₈×Dic₇₃

C₄

yes

—

#17988355

—

#17988337

—

#17988337

—

#423×C₇₃

C₂²

yes

—

C₂×C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₂²×Dic₁₄₆

D₄

yes

characteristic

C₂×#11924012

—

C₂²×#10596831

C₂×C₄

yes

characteristic

C₂²×#10596831

C₂×C₄

yes

characteristic

C₄×Dic₁₄₆

—

C₄×Dic₁₄₆

—

C₂×C₄⋊Dic₇₃

D₄

yes

characteristic

C₂×#11924007

Q₈

yes

—

C₂×C₄×Dic₇₃

C₂×C₄

yes

—

—

—

—

—

D₄

yes

—

C₄×#10596831

D₄

yes

—

C₈×Dic₇₃

—

C₈×Dic₇₃

—

C₂×M₁₆×C₇₃

—

C₂²×C₈×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

SD₁₆

yes

—

C₂²×Dic₇₃

C₄⋊C₄

yes

—

C₂²×Dic₇₃

C₄⋊C₄

yes

—

—

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#10596831

C₂²⋊C₄

yes

—

C₂×#10596831

C₂²⋊C₄

yes

characteristic

C₄⋊Dic₇₃

—

#11924007

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×Q₈

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

—

C₄²×C₇₃

#41

yes

—

C₂³×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₂²⋊C₄

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

—

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₃

C₂×SD₁₆

yes

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

#106

yes

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

—

—

—

yes

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

100

C₄×C₇₃

—

101

C₄×C₇₃

C₂×#101

yes

characteristic

102

C₄×C₇₃

#413

yes

characteristic

103

C₄×C₇₃

#465

yes

—

104

C₄×C₇₃

#465

yes

—

105

Dic₇₃

—

106

Dic₇₃

—

107

#8615

—

maximal, Sylow

108

C₂×C₇₃

—

109

C₂×C₇₃

—

110

C₂×C₇₃

#1801

yes

characteristic

111

C₂×C₇₃

#1807

yes

characteristic

112

C₂×C₇₃

#1813

yes

characteristic

113

C₂×C₈×Q₈

—

114

#1801

—

115

#1566

—

116

#1292

—

117

#1292

D₇₃

yes

characteristic

118

C₇₃

#8615

yes

characteristic, complemented, Sylow

119

C₂×C₄×Q₈

—

120

C₈×Q₈

—

146

—

121

C₈×Q₈

—

146

—

122

#423

—

123

#423

C₂×D₇₃

yes

—

124

C₂×C₄⋊C₈

—

125

C₂×C₄⋊C₈

C₂×D₇₃

yes

characteristic

126

C₂×#101

—

146

—

127

C₂×C₄×C₈

—

128

C₂×C₄×C₈

—

129

#343

—

146

—

130

#334

—

146

—

131

#334

—

146

—

132

C₂²×Q₈

—

133

C₂×M₁₆

—

134

C₂×M₁₆

—

146

—

135

C₂²×C₈

—

136

C₂²×C₈

—

137

C₂²×C₈

C₄×D₇₃

yes

—

138

C₄×Q₈

—

146

—

139

C₄×Q₈

—

146

—

140

C₂×C₄⋊C₄

—

141

C₂×C₄⋊C₄

—

142

C₂×C₄²

—

143

C₂×C₄²

C₂²×D₇₃

yes

characteristic

144

C₄⋊C₈

—

145

C₄⋊C₈

—

146

C₄⋊C₈

—

147

C₄⋊C₈

—

146

—

148

C₄⋊C₈

—

146

—

149

C₄⋊C₈

—

146

—

150

#101

—

292

—

151

C₄×C₈

—

152

C₄×C₈

—

153

C₄×C₈

—

146

—

154

C₄×C₈

—

146

—

155

C₂×Q₈

—

156

C₂×Q₈

—

146

—

157

C₂×Q₈

—

146

—

158

C₂×Q₈

—

146

—

159

C₂²×C₄

—

160

C₂²×C₄

—

161

C₂²×C₄

C₂×C₄×D₇₃

yes

characteristic

162

C₂²×C₄

D₄×D₇₃

yes

characteristic

163

C₂²×C₄

Q₈×D₇₃

yes

characteristic

164

M₁₆

—

165

M₁₆

—

166

M₁₆

—

292

—

167

M₁₆

—

292

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

292

—

172

C₂×C₈

—

292

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

176

C₂×C₈

—

146

—

177

C₄⋊C₄

—

146

—

178

C₄⋊C₄

—

146

—

179

C₄²

—

146

—

180

C₄²

—

146

—

181

C₄²

D₄×D₇₃

yes

—

182

C₄²

#11924028

yes

—

183

C₂³

C₄⋊C₄×D₇₃

yes

characteristic

184

Q₈

—

146

—

185

Q₈

—

146

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

292

—

190

C₂×C₄

—

191

C₂×C₄

—

146

—

192

C₂×C₄

—

146

—

193

C₂×C₄

—

146

—

194

C₂×C₄

C₂²⋊C₄×D₇₃

yes

characteristic

195

C₂×C₄

#14556548

yes

characteristic

196

C₂×C₄

#14556564

yes

—

197

C₂×C₄

#14556569

yes

—

198

C₂×C₄

SD₁₆×D₇₃

yes

—

199

C₂×C₄

#14556610

yes

characteristic

200

C₂×C₄

#14556614

yes

characteristic

201

C₈

—

202

C₈

—

203

C₈

—

584

—

204

C₈

—

205

C₈

—

146

—

206

C₈

—

146

—

207

C₂²

—

208

C₂²

—

209

C₂²

#17988446

yes

characteristic

210

C₂²

#106×D₇₃

yes

characteristic

211

C₂²

#17988624

yes

characteristic

212

C₄

—

213

C₄

—

214

C₄

—

292

—

215

C₄

—

146

—

216

C₄

#101×D₇₃

yes

characteristic

217

C₄

#17988568

yes

characteristic

218

C₄

#17988815

yes

—

219

C₄

#17988819

yes

—

220

C₂

—

221

C₂

—

222

C₂

#22609926

yes

characteristic

223

C₂

#22609955

yes

characteristic

224

C₂

#22609980

yes

characteristic

	subgroup	G/H
center	C₂²	#17988446
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1292	C₂
socle	C₂²×C₇₃	#380
prime radical	C₂³×C₇₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1292	D₇₃
hypocenter	C₇₃	#8615
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1292

subgroup

G/H

center

C₂²

#17988446

commutator

C₂²×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1292

C₂

socle

C₂²×C₇₃

#380

prime radical

C₂³×C₇₃

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1292

N	H ≅ G/N	type
93. #10596831	#101	semidirect
118. C₇₃	#8615	semidirect

H ≅ G/N

type

93. #10596831

#101

semidirect

118. C₇₃

#8615

semidirect

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

subgroup	index
C₂×#17988711	2
#22609924	2
#22608686	2
#22607806	2
C₇₃×#1292	2
#8615	73

subgroup

index

C₇₃×#1292

Subgroups — #32978907

Conjugacy classes of subgroups

Complements

Maximal subgroups