Group #32991009 — subgroups

Subgroups — #32991009

Order 18688.

296 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 91 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	#22610913	C₂	1	1	yes	maximal, characteristic
2	#22610050	C₂	1	1	yes	maximal, characteristic
3	#22608705	C₂	1	1	yes	maximal, characteristic
4	#22608702	C₂	1	1	yes	maximal, characteristic
5	#22608334	C₂	1	1	yes	maximal, characteristic
6	#22607759	C₂	1	1	yes	maximal, characteristic
7	C₇₃×#1481	C₂	1	1	yes	maximal, characteristic
8	#17988640	C₂²	1	1	yes	characteristic
9	#17988611	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
11	#17988433	C₂²	1	1	yes	characteristic
12	#17988346	—	2	2	no	—
13	#17988341	—	2	2	no	—
14	#17988339	—	2	2	no	—
15	#17988338	—	2	2	no	—
16	Q₈⋊#10596831	—	2	2	no	—
17	#533×C₇₃	C₂²	1	1	yes	characteristic
18	#431×C₇₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
20	Q₈⋊C₈×C₇₃	—	2	2	no	—
21	C₂×#11924012	—	2	1	no	—
22	C₂²×#10596831	—	2	1	no	—
23	C₂×C₄×Dic₇₃	D₄	1	2	yes	—
24	C₄⋊#10596831	—	2	1	no	—
25	C₄⋊#10596831	—	2	1	no	—
26	C₄⋊#10596831	—	2	1	no	—
27	C₄⋊#10596831	—	2	1	no	—
28	C₄⋊#10596831	—	2	1	no	—
29	C₂²⋊#10596831	—	2	1	no	—
30	C₄×#10596831	D₄	1	2	yes	—
31	#14556498	—	2	1	no	—
32	#14556498	—	2	1	no	—
33	C₂²×C₈×C₇₃	—	2	1	no	—
34	C₄⋊Q₈×C₇₃	Q₈	1	1	yes	characteristic
35	C₄⋊Q₈×C₇₃	Q₈	1	1	yes	characteristic
36	#125×C₇₃	—	2	1	no	—
37	C₂²⋊Q₈×C₇₃	—	2	1	no	—
38	C₂²⋊Q₈×C₇₃	—	2	1	no	—
39	C₄×Q₈×C₇₃	—	2	1	no	—
40	C₄×Q₈×C₇₃	—	2	1	no	—
41	#117×C₇₃	—	2	1	no	—
42	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₇₃	—	2	1	no	—
44	C₄⋊C₈×C₇₃	—	2	1	no	—
45	C₄⋊C₈×C₇₃	—	2	1	no	—
46	C₂²⋊C₈×C₇₃	—	2	1	no	—
47	C₄×C₈×C₇₃	D₄	1	2	yes	—
48	C₂²×Dic₇₃	—	2	2	no	—
49	#11924012	—	4	2	no	—
50	C₂×#10596831	—	2	2	no	—
51	C₂×#10596831	—	4	1	no	—
52	C₂×#10596831	—	4	1	no	—
53	C₂×#10596831	—	4	2	no	—
54	C₄×Dic₇₃	—	2	2	no	—
55	C₄×Dic₇₃	—	2	2	no	—
56	C₂×Q₈×C₇₃	—	2	1	no	—
57	C₂×Q₈×C₇₃	—	2	1	no	—
58	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
60	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
61	C₂×C₈×C₇₃	—	2	2	no	—
62	C₂×C₈×C₇₃	—	4	1	no	—
63	C₂×C₈×C₇₃	—	4	2	no	—
64	C₄⋊C₄×C₇₃	—	2	1	no	—
65	C₄⋊C₄×C₇₃	—	2	1	no	—
66	C₄⋊C₄×C₇₃	—	2	1	no	—
67	C₄⋊C₄×C₇₃	—	2	1	no	—
68	C₄⋊C₄×C₇₃	—	4	1	no	—
69	C₄⋊C₄×C₇₃	—	4	1	no	—
70	C₂²⋊C₄×C₇₃	—	4	1	no	—
71	C₄²×C₇₃	—	4	1	no	—
72	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
75	C₄²×C₇₃	#41	1	1	yes	characteristic
76	C₂³×C₇₃	#123	1	1	yes	characteristic
77	Q₈×C₇₃	—	4	2	no	—
78	Q₈×C₇₃	—	4	2	no	—
79	C₂×C₄×C₇₃	—	4	1	no	—
80	C₂×C₄×C₇₃	—	4	1	no	—
81	C₂×C₄×C₇₃	—	4	1	no	—
82	C₂×C₄×C₇₃	—	4	1	no	—
83	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
84	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₇₃	#123	1	1	yes	characteristic
87	C₂×C₄×C₇₃	#123	1	1	yes	characteristic
88	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
89	C₂×C₄×C₇₃	#135	1	2	yes	—
90	C₂×C₄×C₇₃	#137	1	2	yes	—
91	C₂×Dic₇₃	—	4	2	no	—
92	C₂×Dic₇₃	—	4	2	no	—
93	C₂×Dic₇₃	—	4	2	no	—
94	C₈×C₇₃	—	4	2	no	—
95	C₈×C₇₃	—	8	1	no	—
96	#10596831	—	4	2	no	—
97	#10596831	—	8	1	no	—
98	#10596831	—	8	1	no	—
99	C₂²×C₇₃	—	2	1	no	—
100	C₂²×C₇₃	—	2	1	no	—
101	C₂²×C₇₃	#399	1	1	yes	characteristic
102	C₂²×C₇₃	#485	1	2	yes	—
103	C₄×C₇₃	—	2	1	no	—
104	C₄×C₇₃	—	2	1	no	—
105	C₄×C₇₃	—	2	1	no	—
106	C₄×C₇₃	—	2	1	no	—
107	C₄×C₇₃	—	8	1	no	—
108	C₄×C₇₃	—	8	1	no	—
109	C₄×C₇₃	#479	1	2	yes	—
110	C₄×C₇₃	#495	1	1	yes	characteristic
111	C₄×C₇₃	#497	1	1	yes	characteristic
112	Dic₇₃	—	8	2	no	—
113	#10669	—	73	1	no	maximal, Sylow
114	C₂×C₇₃	—	2	2	no	—
115	C₂×C₇₃	#2097	1	1	yes	characteristic
116	C₂×C₇₃	#2098	1	1	yes	characteristic
117	C₂×C₇₃	#2098	1	1	yes	characteristic
118	#1995	—	73	1	no	—
119	#1832	—	73	1	no	—
120	#1571	—	73	1	no	—
121	#1570	—	73	1	no	—
122	#1481	—	73	1	no	—
123	#1481	D₇₃	1	1	yes	characteristic
124	#1274	—	73	1	no	—
125	C₇₃	#10669	1	1	yes	characteristic, complemented, Sylow
126	#533	C₂×D₇₃	1	1	yes	characteristic
127	#431	—	73	1	no	—
128	#431	C₂×D₇₃	1	1	yes	characteristic
129	#423	—	73	1	no	—
130	C₂×C₄⋊C₈	—	73	1	no	—
131	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
132	#376	—	73	1	no	—
133	#339	—	146	2	no	—
134	#336	—	146	2	no	—
135	#335	—	146	2	no	—
136	#334	—	146	2	no	—
137	Q₈⋊C₈	—	2	2	no	—
138	Q₈⋊C₈	—	146	2	no	—
139	C₂×M₁₆	—	146	1	no	—
140	C₂²×C₈	—	2	1	no	—
141	C₂²×C₈	—	146	1	no	—
142	C₄⋊Q₈	Dic₁₄₆	1	1	yes	characteristic
143	C₄⋊Q₈	Dic₁₄₆	1	1	yes	characteristic
144	#125	—	2	1	no	—
145	C₂²⋊Q₈	—	2	1	no	—
146	C₂²⋊Q₈	—	2	1	no	—
147	C₄×Q₈	—	2	1	no	—
148	C₄×Q₈	—	2	1	no	—
149	#117	—	2	1	no	—
150	C₂×C₄²	—	73	2	no	—
151	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
152	C₄⋊C₈	—	2	1	no	—
153	C₄⋊C₈	—	2	1	no	—
154	C₄⋊C₈	—	2	1	no	—
155	C₄⋊C₈	—	146	1	no	—
156	C₄⋊C₈	—	146	1	no	—
157	C₄⋊C₈	—	146	1	no	—
158	C₄⋊C₈	—	146	1	no	—
159	C₄⋊C₈	—	146	1	no	—
160	C₂²⋊C₈	—	2	1	no	—
161	C₂²⋊C₈	—	146	1	no	—
162	C₄×C₈	—	73	2	no	—
163	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
164	#95	—	146	1	no	—
165	#95	—	146	1	no	—
166	C₂×Q₈	—	2	1	no	—
167	C₂×Q₈	—	2	1	no	—
168	C₂²×C₄	—	146	2	no	—
169	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
170	C₂²×C₄	#11924027	1	1	yes	characteristic
171	C₂²×C₄	#11924028	1	1	yes	characteristic
172	M₁₆	—	292	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	—	4	2	no	—
176	C₂×C₈	—	292	1	no	—
177	C₂×C₈	—	292	1	no	—
178	C₂×C₈	—	292	2	no	—
179	C₂×C₈	—	146	2	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	1	no	—
183	C₄⋊C₄	—	2	1	no	—
184	C₄⋊C₄	—	4	1	no	—
185	C₄⋊C₄	—	4	1	no	—
186	C₂²⋊C₄	—	4	1	no	—
187	C₄²	—	4	1	no	—
188	C₄²	—	146	2	no	—
189	C₄²	—	146	2	no	—
190	C₄²	D₄×D₇₃	1	1	yes	characteristic
191	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
192	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
193	C₄²	#11924027	1	1	yes	characteristic
194	C₂³	#14556573	1	1	yes	characteristic
195	Q₈	—	4	2	no	—
196	Q₈	—	4	2	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	4	1	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	292	2	no	—
202	C₂×C₄	—	292	2	no	—
203	C₂×C₄	—	292	2	no	—
204	C₂×C₄	#14556568	1	1	yes	characteristic
205	C₂×C₄	#14556568	1	1	yes	characteristic
206	C₂×C₄	#14556570	1	1	yes	characteristic
207	C₂×C₄	#14556572	1	1	yes	characteristic
208	C₂×C₄	#14556573	1	1	yes	characteristic
209	C₂×C₄	#14556587	1	1	yes	characteristic
210	C₂×C₄	#14556612	1	2	yes	—
211	C₂×C₄	#14556623	1	2	yes	—
212	C₈	—	4	2	no	—
213	C₈	—	8	1	no	—
214	C₈	—	292	2	no	—
215	C₈	—	584	1	no	—
216	C₈	—	584	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	2	1	no	—
219	C₂²	#17988523	1	1	yes	characteristic
220	C₂²	#17988926	1	2	yes	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	8	1	no	—
226	C₄	—	8	1	no	—
227	C₄	—	584	2	no	—
228	C₄	#17988892	1	2	yes	—
229	C₄	#17988968	1	1	yes	characteristic
230	C₄	#17988980	1	1	yes	characteristic
231	C₂	—	2	2	no	—
232	C₂	#22611481	1	1	yes	characteristic
233	C₂	#22611484	1	1	yes	characteristic
234	C₂	#22611486	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22610913

C₂

yes

maximal, characteristic

#22610050

C₂

yes

maximal, characteristic

#22608705

C₂

yes

maximal, characteristic

#22608702

C₂

yes

maximal, characteristic

#22608334

C₂

yes

maximal, characteristic

#22607759

C₂

yes

maximal, characteristic

C₇₃×#1481

C₂

yes

maximal, characteristic

#17988640

C₂²

yes

characteristic

#17988611

C₂²

yes

characteristic

C₂×C₄⋊#10596831

C₂²

yes

characteristic

#17988433

C₂²

yes

—

—

—

—

—

#533×C₇₃

C₂²

yes

characteristic

#431×C₇₃

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₇₃

C₂²

yes

characteristic

Q₈⋊C₈×C₇₃

—

C₂×#11924012

—

C₂²×#10596831

—

C₂×C₄×Dic₇₃

D₄

yes

—

—

—

—

—

—

—

D₄

yes

—

#14556498

—

#14556498

—

C₂²×C₈×C₇₃

—

C₄⋊Q₈×C₇₃

Q₈

yes

characteristic

C₄⋊Q₈×C₇₃

Q₈

yes

characteristic

#125×C₇₃

—

C₂²⋊Q₈×C₇₃

—

C₂²⋊Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

#117×C₇₃

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

—

C₂²⋊C₈×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

—

C₂²×Dic₇₃

—

—

—

—

—

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

C₂²×C₄×C₇₃

yes

C₂²×C₄×C₇₃

yes

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

C₂×Q₈

yes

C₄²×C₇₃

yes

C₂³×C₇₃

yes

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

—

C₂×C₄×C₇₃

#137

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

100

C₂²×C₇₃

—

101

C₂²×C₇₃

#399

yes

characteristic

102

C₂²×C₇₃

#485

yes

—

103

C₄×C₇₃

—

104

C₄×C₇₃

—

105

C₄×C₇₃

—

106

C₄×C₇₃

—

107

C₄×C₇₃

—

108

C₄×C₇₃

—

109

C₄×C₇₃

#479

yes

—

110

C₄×C₇₃

#495

yes

characteristic

111

C₄×C₇₃

#497

yes

characteristic

112

Dic₇₃

—

113

#10669

—

maximal, Sylow

114

C₂×C₇₃

—

115

C₂×C₇₃

#2097

yes

characteristic

116

C₂×C₇₃

#2098

yes

characteristic

117

C₂×C₇₃

#2098

yes

characteristic

118

#1995

—

119

#1832

—

120

#1571

—

121

#1570

—

122

#1481

—

123

#1481

D₇₃

yes

characteristic

124

#1274

—

125

C₇₃

#10669

yes

characteristic, complemented, Sylow

126

#533

C₂×D₇₃

yes

characteristic

127

#431

—

128

#431

C₂×D₇₃

yes

characteristic

129

#423

—

130

C₂×C₄⋊C₈

—

131

C₂×C₄⋊C₈

C₂×D₇₃

yes

characteristic

132

#376

—

133

#339

—

146

—

134

#336

—

146

—

135

#335

—

146

—

136

#334

—

146

—

137

Q₈⋊C₈

—

138

Q₈⋊C₈

—

146

—

139

C₂×M₁₆

—

146

—

140

C₂²×C₈

—

141

C₂²×C₈

—

146

—

142

C₄⋊Q₈

Dic₁₄₆

yes

characteristic

143

C₄⋊Q₈

Dic₁₄₆

yes

characteristic

144

#125

—

145

C₂²⋊Q₈

—

146

C₂²⋊Q₈

—

147

C₄×Q₈

—

148

C₄×Q₈

—

149

#117

—

150

C₂×C₄²

—

151

C₂×C₄²

C₂²×D₇₃

yes

characteristic

152

C₄⋊C₈

—

153

C₄⋊C₈

—

154

C₄⋊C₈

—

155

C₄⋊C₈

—

146

—

156

C₄⋊C₈

—

146

—

157

C₄⋊C₈

—

146

—

158

C₄⋊C₈

—

146

—

159

C₄⋊C₈

—

146

—

160

C₂²⋊C₈

—

161

C₂²⋊C₈

—

146

—

162

C₄×C₈

—

163

C₄×C₈

C₂²⋊D₇₃

yes

—

164

#95

—

146

—

165

#95

—

146

—

166

C₂×Q₈

—

167

C₂×Q₈

—

168

C₂²×C₄

—

146

—

169

C₂²×C₄

D₄×D₇₃

yes

characteristic

170

C₂²×C₄

#11924027

yes

characteristic

171

C₂²×C₄

#11924028

yes

characteristic

172

M₁₆

—

292

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

176

C₂×C₈

—

292

—

177

C₂×C₈

—

292

—

178

C₂×C₈

—

292

—

179

C₂×C₈

—

146

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₄⋊C₄

—

183

C₄⋊C₄

—

184

C₄⋊C₄

—

185

C₄⋊C₄

—

186

C₂²⋊C₄

—

187

C₄²

—

188

C₄²

—

146

—

189

C₄²

—

146

—

190

C₄²

D₄×D₇₃

yes

characteristic

191

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

192

C₄²

C₂×Dic₁₄₆

yes

characteristic

193

C₄²

#11924027

yes

characteristic

194

C₂³

#14556573

yes

characteristic

195

Q₈

—

196

Q₈

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

292

—

202

C₂×C₄

—

292

—

203

C₂×C₄

—

292

—

204

C₂×C₄

#14556568

yes

characteristic

205

C₂×C₄

#14556568

yes

characteristic

206

C₂×C₄

#14556570

yes

characteristic

207

C₂×C₄

#14556572

yes

characteristic

208

C₂×C₄

#14556573

yes

characteristic

209

C₂×C₄

#14556587

yes

characteristic

210

C₂×C₄

#14556612

yes

—

211

C₂×C₄

#14556623

yes

—

212

C₈

—

213

C₈

—

214

C₈

—

292

—

215

C₈

—

584

—

216

C₈

—

584

—

217

C₂²

—

218

C₂²

—

219

C₂²

#17988523

yes

characteristic

220

C₂²

#17988926

yes

—

221

C₄

—

222

C₄

—

223

C₄

—

224

C₄

—

225

C₄

—

226

C₄

—

227

C₄

—

584

—

228

C₄

#17988892

yes

—

229

C₄

#17988968

yes

characteristic

230

C₄

#17988980

yes

characteristic

231

C₂

—

232

C₂

#22611481

yes

characteristic

233

C₂

#22611484

yes

characteristic

234

C₂

#22611486

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556587
commutator, cosocle	C₂×C₄²×C₇₃	C₂³
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1481	C₂
socle	C₂²×C₇₃	#399
prime radical	C₂³×C₇₃	#123
hypercenter	#1481	D₇₃
hypocenter	C₇₃	#10669
LCS	G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1481

subgroup

G/H

center

C₂×C₄

#14556587

commutator, cosocle

C₂×C₄²×C₇₃

C₂³

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1481

C₂

C₂²×C₇₃

C₂³×C₇₃

D₇₃

C₇₃

G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1481

N	H ≅ G/N	type
125. C₇₃	#10669	semidirect

H ≅ G/N

type

125. C₇₃

#10669

semidirect

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

subgroup	index
#22610913	2
#22610050	2
#22608705	2
#22608702	2
#22608334	2
#22607759	2
C₇₃×#1481	2
#10669	73

subgroup

index

C₇₃×#1481

Subgroups — #32991009

Conjugacy classes of subgroups

Complements

Maximal subgroups