Group #22622840 — subgroups

Subgroups — #22622840

Order 9344.

340 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 125 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	Q₈×Dic₁₄₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#17989298	C₂	1	1	yes	maximal, characteristic
3	#17988995	C₂	1	2	yes	maximal, complemented
4	#17988990	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988892	C₂	1	2	yes	maximal, complemented
6	#17988878	C₂	1	1	yes	maximal, characteristic
7	#17988866	C₂	1	2	yes	maximal
8	#17988714	C₂	1	1	yes	maximal, characteristic, complemented
9	#17988690	C₂	1	2	yes	maximal
10	C₄×#11924015	C₂	1	1	yes	maximal, characteristic
11	#477×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11924015	C₂²	1	1	yes	characteristic
13	#14556591	—	2	1	no	—
14	#14556591	C₂²	1	1	yes	characteristic
15	#14556590	—	2	2	no	—
16	#14556590	C₂²	1	2	yes	—
17	#14556582	C₂²	1	2	yes	—
18	#14556571	—	2	1	no	—
19	#14556567	—	2	2	no	—
20	Q₈×Dic₇₃	C₂²	1	2	yes	—
21	C₄×Dic₁₄₆	—	2	1	no	—
22	C₄×Dic₁₄₆	C₂²	1	1	yes	characteristic
23	C₄×D₂₉₂	C₂²	1	1	yes	characteristic
24	#14556553	C₂²	1	2	yes	—
25	C₄⋊C₄×D₇₃	—	2	1	no	—
26	#14556524	C₂²	1	2	yes	—
27	#14556524	C₂²	1	2	yes	—
28	#14556523	C₂²	1	2	yes	—
29	#14556522	C₂²	1	1	yes	characteristic
30	#14556520	C₂²	1	2	yes	—
31	#14556515	C₂²	1	1	yes	characteristic
32	#14556513	C₂²	1	2	yes	—
33	#14556511	C₂²	1	1	yes	characteristic
34	#14556501	C₂²	1	2	yes	—
35	C₄×#10596831	C₂²	1	1	yes	characteristic
36	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
37	C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
38	#107×C₇₃	C₂²	1	2	yes	—
39	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
40	#103×C₇₃	C₂²	1	2	yes	—
41	C₂×Dic₁₄₆	—	2	1	no	—
42	C₂×Dic₁₄₆	—	2	2	no	—
43	C₂×Dic₁₄₆	D₄	1	1	yes	characteristic
44	C₂×D₂₉₂	C₂³	1	1	yes	characteristic
45	C₂×C₄×D₇₃	—	2	1	no	—
46	C₂×C₄×D₇₃	—	2	2	no	—
47	#11924015	Q₈	1	4	yes	complemented
48	C₂×#10596831	C₂³	1	1	yes	characteristic
49	C₂×#10596831	C₂³	1	1	yes	characteristic
50	C₄⋊Dic₇₃	—	2	1	no	—
51	C₄⋊Dic₇₃	—	2	2	no	—
52	C₄⋊Dic₇₃	—	2	2	no	—
53	C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
54	C₄⋊Dic₇₃	C₂³	1	2	yes	—
55	C₄⋊Dic₇₃	C₂³	1	1	yes	characteristic
56	#11924007	—	2	2	no	—
57	#11924007	—	2	2	no	—
58	#11924007	D₄	1	2	yes	—
59	#11924005	—	2	1	no	—
60	#11924005	—	2	2	no	—
61	C₄×Dic₇₃	—	2	2	no	—
62	C₄×Dic₇₃	C₂³	1	2	yes	—
63	C₂×Q₈×C₇₃	—	2	1	no	—
64	C₂×Q₈×C₇₃	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₇₃	C₂³	1	2	yes	—
66	C₄⋊C₄×C₇₃	—	2	1	no	—
67	C₄⋊C₄×C₇₃	—	2	1	no	—
68	C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
69	C₄⋊C₄×C₇₃	C₂³	1	1	yes	characteristic
70	C₄²×C₇₃	—	2	1	no	—
71	C₄²×C₇₃	C₂³	1	1	yes	characteristic
72	C₂²×D₇₃	—	2	1	no	—
73	Q₈×C₇₃	—	2	1	no	—
74	Q₈×C₇₃	—	8	1	no	—
75	Q₈×C₇₃	C₂×Q₈	1	2	yes	—
76	C₂×C₄×C₇₃	—	2	1	no	—
77	C₂×C₄×C₇₃	—	2	1	no	—
78	C₂×C₄×C₇₃	—	2	2	no	—
79	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
82	C₂×Dic₇₃	—	2	2	no	—
83	C₂×Dic₇₃	—	2	1	no	—
84	C₂×Dic₇₃	C₂×D₄	1	2	yes	—
85	C₂×Dic₇₃	C₂×D₄	1	2	yes	—
86	D₂₉₂	—	2	1	no	—
87	D₂₉₂	C₂×Q₈	1	2	yes	—
88	C₄×D₇₃	—	4	2	no	—
89	C₄×D₇₃	—	8	2	no	—
90	Dic₁₄₆	—	2	2	no	—
91	Dic₁₄₆	—	8	2	no	—
92	Dic₁₄₆	—	8	1	no	—
93	C₈×C₇₃	—	2	2	no	—
94	#10596831	—	2	1	no	—
95	#10596831	C₂×Q₈	1	2	yes	—
96	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
97	C₂×D₇₃	—	2	2	no	—
98	C₂×D₇₃	—	4	1	no	—
99	C₂×D₇₃	—	4	1	no	—
100	C₄×C₇₃	—	2	2	no	—
101	C₄×C₇₃	—	2	1	no	—
102	C₄×C₇₃	—	4	1	no	—
103	C₄×C₇₃	—	4	2	no	—
104	C₄×C₇₃	#137	1	2	yes	—
105	C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
106	C₄×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
107	Dic₇₃	—	2	2	no	—
108	Dic₇₃	—	2	2	no	—
109	Dic₇₃	—	4	1	no	—
110	Dic₇₃	—	4	2	no	—
111	C₂×C₇₃	D₄×Q₈	1	1	yes	characteristic
112	C₂×C₇₃	C₂×#137	1	1	yes	characteristic
113	C₂×C₇₃	#576	1	1	yes	characteristic
114	D₇₃	—	4	2	no	—
115	#3385	—	73	1	no	maximal, Sylow
116	C₇₃	#3385	1	1	yes	characteristic, complemented, Sylow
117	Q₈²	—	73	1	no	—
118	#554	—	73	1	no	—
119	#501	—	73	2	no	—
120	#500	—	73	1	no	—
121	#479	—	73	2	no	—
122	#477	D₇₃	1	1	yes	characteristic, complemented
123	#476	—	73	1	no	—
124	#474	—	73	2	no	—
125	#446	—	73	1	no	—
126	#440	—	73	2	no	—
127	C₄×SD₁₆	—	73	1	no	—
128	C₂×SD₁₆	—	73	1	no	—
129	C₄⋊Q₈	—	73	2	no	—
130	C₄⋊Q₈	—	73	1	no	—
131	C₄⋊Q₈	—	146	1	no	—
132	C₄⋊Q₈	—	146	2	no	—
133	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
134	#125	—	73	2	no	—
135	C₂²⋊Q₈	—	146	1	no	—
136	C₂²⋊Q₈	—	146	2	no	—
137	C₄×Q₈	—	73	1	no	—
138	C₄×Q₈	—	73	2	no	—
139	C₄×Q₈	—	146	1	no	—
140	C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
141	C₄×D₄	—	73	2	no	—
142	C₄×D₄	—	73	1	no	—
143	C₂×C₄⋊C₄	—	146	1	no	—
144	#107	—	73	2	no	—
145	#107	—	73	2	no	—
146	#107	C₂×D₇₃	1	2	yes	—
147	#106	—	73	2	no	—
148	#106	—	73	1	no	—
149	C₄⋊C₈	—	73	2	no	—
150	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
151	#103	—	73	1	no	—
152	#103	C₂×D₇₃	1	2	yes	—
153	#102	—	73	2	no	—
154	#102	—	73	1	no	—
155	#97	—	73	2	no	—
156	C₄×C₈	—	73	1	no	—
157	C₂×Q₈	—	2	1	no	—
158	C₂×Q₈	—	73	1	no	—
159	C₂×Q₈	—	146	2	no	—
160	C₂×Q₈	—	146	1	no	—
161	C₂×Q₈	C₂²×D₇₃	1	1	yes	characteristic
162	C₂×D₄	—	73	1	no	—
163	C₂²×C₄	—	146	2	no	—
164	C₂²×C₄	—	146	1	no	—
165	SD₁₆	—	73	4	no	—
166	C₂×C₈	—	73	1	no	—
167	C₂×C₈	—	73	1	no	—
168	C₂×C₈	C₂²×D₇₃	1	2	yes	—
169	C₄⋊C₄	—	2	1	no	—
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	73	2	no	—
172	C₄⋊C₄	—	73	1	no	—
173	C₄⋊C₄	—	73	2	no	—
174	C₄⋊C₄	—	73	1	no	—
175	C₄⋊C₄	—	146	2	no	—
176	C₄⋊C₄	—	146	2	no	—
177	C₄⋊C₄	—	146	2	no	—
178	C₄⋊C₄	—	146	2	no	—
179	C₄⋊C₄	—	146	1	no	—
180	C₄⋊C₄	C₂²×D₇₃	1	1	yes	characteristic
181	C₄⋊C₄	C₂²×D₇₃	1	2	yes	—
182	C₂²⋊C₄	—	146	2	no	—
183	C₂²⋊C₄	—	146	1	no	—
184	C₄²	—	2	1	no	—
185	C₄²	—	73	2	no	—
186	C₄²	—	146	2	no	—
187	C₄²	C₂²×D₇₃	1	1	yes	characteristic
188	C₂³	—	146	1	no	—
189	Q₈	—	2	1	no	—
190	Q₈	—	8	1	no	—
191	Q₈	—	584	1	no	—
192	Q₈	—	584	2	no	—
193	Q₈	—	146	2	no	—
194	Q₈	Q₈×D₇₃	1	2	yes	—
195	D₄	—	73	2	no	—
196	D₄	—	146	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	292	2	no	—
201	C₂×C₄	—	584	2	no	—
202	C₂×C₄	—	73	2	no	—
203	C₂×C₄	—	73	2	no	—
204	C₂×C₄	—	146	2	no	—
205	C₂×C₄	—	146	1	no	—
206	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
207	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
208	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
209	C₈	—	2	2	no	—
210	C₈	—	73	2	no	—
211	C₈	—	146	1	no	—
212	C₂²	—	292	1	no	—
213	C₂²	—	292	1	no	—
214	C₂²	—	146	2	no	—
215	C₂²	C₂×D₄×D₇₃	1	1	yes	characteristic
216	C₄	—	2	1	no	—
217	C₄	—	2	2	no	—
218	C₄	—	4	2	no	—
219	C₄	—	4	1	no	—
220	C₄	—	292	1	no	—
221	C₄	—	292	2	no	—
222	C₄	—	146	2	no	—
223	C₄	—	146	2	no	—
224	C₄	#14556623	1	2	yes	—
225	C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
226	C₄	#14556638	1	1	yes	characteristic
227	C₂	—	292	2	no	—
228	C₂	#17989250	1	1	yes	characteristic
229	C₂	C₂×#14556623	1	1	yes	characteristic
230	C₂	#17989391	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₄₆

C₂

yes

maximal, characteristic, complemented

#17989298

C₂

yes

maximal, characteristic

#17988995

C₂

yes

maximal, complemented

#17988990

C₂

yes

maximal, characteristic, complemented

#17988892

C₂

yes

maximal, complemented

#17988878

C₂

yes

maximal, characteristic

#17988866

C₂

yes

maximal

#17988714

C₂

yes

maximal, characteristic, complemented

#17988690

C₂

yes

maximal

C₄×#11924015

C₂

yes

maximal, characteristic

#477×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924015

C₂²

yes

characteristic

#14556591

—

#14556591

C₂²

yes

characteristic

#14556590

—

#14556590

C₂²

yes

—

#14556582

C₂²

yes

—

#14556571

—

#14556567

—

Q₈×Dic₇₃

C₂²

yes

—

C₄×Dic₁₄₆

—

C₄×Dic₁₄₆

C₂²

yes

characteristic

C₄×D₂₉₂

C₂²

yes

characteristic

#14556553

C₂²

yes

—

C₄⋊C₄×D₇₃

—

#14556524

C₂²

yes

—

#14556524

C₂²

yes

—

#14556523

C₂²

yes

—

#14556522

C₂²

yes

characteristic

#14556520

C₂²

yes

—

#14556515

C₂²

yes

characteristic

#14556513

C₂²

yes

—

#14556511

C₂²

yes

characteristic

#14556501

C₂²

yes

—

C₄×#10596831

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

C₄×Q₈×C₇₃

C₂²

yes

characteristic

#107×C₇₃

C₂²

yes

—

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

#103×C₇₃

C₂²

yes

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

D₄

yes

characteristic

C₂×D₂₉₂

C₂³

yes

characteristic

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

#11924015

Q₈

yes

complemented

C₂×#10596831

C₂³

yes

characteristic

C₂×#10596831

C₂³

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

D₄

yes

characteristic

C₄⋊Dic₇₃

C₂³

yes

—

C₄⋊Dic₇₃

C₂³

yes

—

—

D₄

yes

—

#11924005

—

#11924005

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂³

yes

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

C₂³

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂³

yes

—

C₄⋊C₄×C₇₃

C₂³

yes

characteristic

C₄²×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂²×D₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

C₂×Q₈

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂⁴

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×D₄

yes

—

C₂×Dic₇₃

C₂×D₄

yes

—

D₂₉₂

—

D₂₉₂

C₂×Q₈

yes

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

#10596831

—

#10596831

C₂×Q₈

yes

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

100

C₄×C₇₃

—

101

C₄×C₇₃

—

102

C₄×C₇₃

—

103

C₄×C₇₃

—

104

C₄×C₇₃

#137

yes

—

105

C₄×C₇₃

C₂²×Q₈

yes

characteristic

106

C₄×C₇₃

2₋¹⁺⁴

yes

characteristic

107

Dic₇₃

—

108

Dic₇₃

—

109

Dic₇₃

—

110

Dic₇₃

—

111

C₂×C₇₃

D₄×Q₈

yes

characteristic

112

C₂×C₇₃

C₂×#137

yes

characteristic

113

C₂×C₇₃

#576

yes

characteristic

114

D₇₃

—

115

#3385

—

maximal, Sylow

116

C₇₃

#3385

yes

characteristic, complemented, Sylow

117

Q₈²

—

118

#554

—

119

#501

—

120

#500

—

121

#479

—

122

#477

D₇₃

yes

characteristic, complemented

123

#476

—

124

#474

—

125

#446

—

126

#440

—

127

C₄×SD₁₆

—

128

C₂×SD₁₆

—

129

C₄⋊Q₈

—

130

C₄⋊Q₈

—

131

C₄⋊Q₈

—

146

—

132

C₄⋊Q₈

—

146

—

133

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

134

#125

—

135

C₂²⋊Q₈

—

146

—

136

C₂²⋊Q₈

—

146

—

137

C₄×Q₈

—

138

C₄×Q₈

—

139

C₄×Q₈

—

146

—

140

C₄×Q₈

C₂×D₇₃

yes

characteristic

141

C₄×D₄

—

142

C₄×D₄

—

143

C₂×C₄⋊C₄

—

146

—

144

#107

—

145

#107

—

146

#107

C₂×D₇₃

yes

—

147

#106

—

148

#106

—

149

C₄⋊C₈

—

150

C₄⋊C₈

C₂×D₇₃

yes

characteristic

151

#103

—

152

#103

C₂×D₇₃

yes

—

153

#102

—

154

#102

—

155

#97

—

156

C₄×C₈

—

157

C₂×Q₈

—

158

C₂×Q₈

—

159

C₂×Q₈

—

146

—

160

C₂×Q₈

—

146

—

161

C₂×Q₈

C₂²×D₇₃

yes

characteristic

162

C₂×D₄

—

163

C₂²×C₄

—

146

—

164

C₂²×C₄

—

146

—

165

SD₁₆

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

C₂²×D₇₃

yes

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

146

—

176

C₄⋊C₄

—

146

—

177

C₄⋊C₄

—

146

—

178

C₄⋊C₄

—

146

—

179

C₄⋊C₄

—

146

—

180

C₄⋊C₄

C₂²×D₇₃

yes

characteristic

181

C₄⋊C₄

C₂²×D₇₃

yes

—

182

C₂²⋊C₄

—

146

—

183

C₂²⋊C₄

—

146

—

184

C₄²

—

185

C₄²

—

186

C₄²

—

146

—

187

C₄²

C₂²×D₇₃

yes

characteristic

188

C₂³

—

146

—

189

Q₈

—

190

Q₈

—

191

Q₈

—

584

—

192

Q₈

—

584

—

193

Q₈

—

146

—

194

Q₈

Q₈×D₇₃

yes

—

195

D₄

—

196

D₄

—

146

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

292

—

201

C₂×C₄

—

584

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

146

—

205

C₂×C₄

—

146

—

206

C₂×C₄

D₄×D₇₃

yes

characteristic

207

C₂×C₄

D₄×D₇₃

yes

characteristic

208

C₂×C₄

C₂³×D₇₃

yes

characteristic

209

C₈

—

210

C₈

—

211

C₈

—

146

—

212

C₂²

—

292

—

213

C₂²

—

292

—

214

C₂²

—

146

—

215

C₂²

C₂×D₄×D₇₃

yes

characteristic

216

C₄

—

217

C₄

—

218

C₄

—

219

C₄

—

220

C₄

—

292

—

221

C₄

—

292

—

222

C₄

—

146

—

223

C₄

—

146

—

224

C₄

#14556623

yes

—

225

C₄

C₂×Q₈×D₇₃

yes

characteristic

226

C₄

#14556638

yes

characteristic

227

C₂

—

292

—

228

C₂

#17989250

yes

characteristic

229

C₂

C₂×#14556623

yes

characteristic

230

C₂

#17989391

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₃
Fitting	#477×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₄
hypercenter	#477	D₇₃
hypocenter	C₇₃	#3385
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

subgroup

G/H

center

C₂²

C₂×D₄×D₇₃

commutator

C₂×C₄×C₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₃

Fitting

#477×C₇₃

C₂

socle

C₂²×C₇₃

C₂²×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #477

N	H ≅ G/N	type
1. Q₈×Dic₁₄₆	C₂	semidirect
3. #17988995	C₂	semidirect
4. #17988990	C₂	semidirect
5. #17988892	C₂	semidirect
8. #17988714	C₂	semidirect
11. #477×C₇₃	C₂	semidirect
47. #11924015	Q₈	semidirect
116. C₇₃	#3385	semidirect
122. #477	D₇₃	semidirect

H ≅ G/N

type

1. Q₈×Dic₁₄₆

C₂

semidirect

3. #17988995

C₂

semidirect

4. #17988990

C₂

semidirect

5. #17988892

C₂

semidirect

8. #17988714

C₂

semidirect

11. #477×C₇₃

C₂

semidirect

47. #11924015

Q₈

semidirect

116. C₇₃

#3385

semidirect

122. #477

D₇₃

semidirect

Subgroups — #22622840

Conjugacy classes of subgroups

Complements