Group #22610200 — subgroups

Subgroups — #22610200

Order 9344.

298 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 101 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×#14556556	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556522	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556516	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556514	C₂	1	1	yes	maximal, characteristic
5	#17988486	C₂	1	1	yes	maximal, characteristic
6	#17988341	C₂	1	1	yes	maximal, characteristic
7	#340×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₄₆	C₂²	1	1	yes	characteristic
9	C₂²×#10596831	C₂²	1	1	yes	characteristic
10	#14556556	—	2	2	no	—
11	#14556556	—	2	2	no	—
12	C₂×C₄⋊Dic₇₃	—	2	1	no	—
13	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
14	C₂×#11924007	—	2	1	no	—
15	C₂×C₄×Dic₇₃	—	2	1	no	—
16	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
17	#14556522	—	2	2	no	—
18	#14556516	—	2	2	no	—
19	#14556514	C₄	1	4	yes	complemented
20	#14556498	—	2	1	no	—
21	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
24	C₂×Dic₁₄₆	—	2	1	no	—
25	C₂×Dic₁₄₆	—	2	1	no	—
26	C₂×Dic₁₄₆	—	2	1	no	—
27	C₂×Dic₁₄₆	C₂×C₄	1	2	yes	—
28	C₂×Dic₁₄₆	D₄	1	2	yes	—
29	C₂×Dic₁₄₆	Q₈	1	2	yes	—
30	C₂²×Dic₇₃	—	2	1	no	—
31	C₂²×Dic₇₃	—	2	1	no	—
32	C₂²×Dic₇₃	D₄	1	1	yes	characteristic
33	C₂²×Dic₇₃	D₄	1	1	yes	characteristic
34	C₂×#10596831	—	2	1	no	—
35	C₂×#10596831	—	2	1	no	—
36	C₂×#10596831	C₂×C₄	1	2	yes	—
37	C₄⋊Dic₇₃	—	2	1	no	—
38	C₄⋊Dic₇₃	—	2	1	no	—
39	C₄⋊Dic₇₃	—	4	2	no	—
40	#11924007	—	2	4	no	—
41	C₄×Dic₇₃	—	2	4	no	—
42	C₄×Dic₇₃	—	2	1	no	—
43	C₄×Dic₇₃	—	2	1	no	—
44	C₂²×C₄×C₇₃	—	2	1	no	—
45	C₂²×C₄×C₇₃	—	2	1	no	—
46	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₇₃	—	2	1	no	—
48	C₂×C₈×C₇₃	—	2	1	no	—
49	C₂×C₈×C₇₃	—	2	1	no	—
50	C₄⋊C₄×C₇₃	—	2	1	no	—
51	C₄⋊C₄×C₇₃	—	2	1	no	—
52	C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
53	C₄⋊C₄×C₇₃	D₄	1	2	yes	—
54	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₇₃	—	2	2	no	—
56	C₂×C₄×C₇₃	—	2	2	no	—
57	C₂×C₄×C₇₃	—	4	1	no	—
58	C₂×C₄×C₇₃	—	4	1	no	—
59	C₂×C₄×C₇₃	—	4	1	no	—
60	C₂×C₄×C₇₃	—	4	1	no	—
61	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
66	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
67	C₂×Dic₇₃	—	2	2	no	—
68	C₂×Dic₇₃	—	2	1	no	—
69	C₂×Dic₇₃	—	2	2	no	—
70	C₂×Dic₇₃	—	2	2	no	—
71	C₂×Dic₇₃	—	2	1	no	—
72	C₂×Dic₇₃	—	2	1	no	—
73	C₂×Dic₇₃	—	2	1	no	—
74	C₂×Dic₇₃	—	2	1	no	—
75	C₂×Dic₇₃	—	4	1	no	—
76	C₂×Dic₇₃	—	4	1	no	—
77	C₂×Dic₇₃	—	4	1	no	—
78	C₂×Dic₇₃	SD₁₆	1	2	yes	—
79	Dic₁₄₆	—	2	2	no	—
80	Dic₁₄₆	—	2	2	no	—
81	Dic₁₄₆	—	2	2	no	—
82	Dic₁₄₆	C₄⋊C₄	1	4	yes	complemented
83	C₈×C₇₃	—	4	1	no	—
84	#10596831	—	2	2	no	—
85	C₂²×C₇₃	C₄×D₄	1	1	yes	characteristic
86	C₂²×C₇₃	#120	1	1	yes	characteristic
87	C₂²×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
88	C₂²×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
89	C₂²×C₇₃	#135	1	1	yes	characteristic
90	C₂²×C₇₃	#137	1	1	yes	characteristic
91	C₂²×C₇₃	#137	1	1	yes	characteristic
92	C₄×C₇₃	—	4	2	no	—
93	C₄×C₇₃	—	4	2	no	—
94	C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
96	C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
97	C₄×C₇₃	#123	1	1	yes	characteristic
98	Dic₇₃	—	2	4	no	—
99	Dic₇₃	—	2	2	no	—
100	Dic₇₃	—	4	1	no	—
101	Dic₇₃	—	8	1	no	—
102	C₂×C₇₃	#385	1	1	yes	characteristic
103	C₂×C₇₃	C₄×SD₁₆	1	1	yes	characteristic
104	C₂×C₇₃	#441	1	1	yes	characteristic
105	C₂×C₇₃	#448	1	1	yes	characteristic
106	C₂×C₇₃	#452	1	1	yes	characteristic
107	C₂×C₇₃	#475	1	1	yes	characteristic
108	C₂×C₇₃	#479	1	1	yes	characteristic
109	#1865	—	73	1	no	maximal, Sylow
110	C₇₃	#1865	1	1	yes	characteristic, complemented, Sylow
111	C₂×C₄×Q₈	—	73	1	no	—
112	C₂×#106	—	73	1	no	—
113	C₂×#103	—	73	1	no	—
114	C₂×#103	—	73	1	no	—
115	#389	—	73	1	no	—
116	#340	D₇₃	1	1	yes	characteristic
117	#336	—	73	1	no	—
118	C₂²×Q₈	—	73	1	no	—
119	C₂²×C₈	—	73	1	no	—
120	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
121	C₄×Q₈	—	146	2	no	—
122	C₄×Q₈	—	146	2	no	—
123	C₂×C₄⋊C₄	—	73	1	no	—
124	C₂×C₄⋊C₄	—	146	1	no	—
125	C₂×C₄⋊C₄	—	146	1	no	—
126	C₂×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
127	C₂×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
128	C₂×C₄²	—	73	1	no	—
129	C₂×C₄²	—	146	1	no	—
130	#106	—	146	2	no	—
131	#103	—	73	4	no	—
132	#103	—	146	2	no	—
133	#95	—	146	1	no	—
134	C₂×Q₈	—	73	2	no	—
135	C₂×Q₈	—	73	2	no	—
136	C₂×Q₈	—	73	2	no	—
137	C₂×Q₈	—	146	1	no	—
138	C₂×Q₈	—	146	1	no	—
139	C₂×Q₈	—	146	1	no	—
140	C₂²×C₄	—	2	1	no	—
141	C₂²×C₄	—	2	1	no	—
142	C₂²×C₄	—	73	1	no	—
143	C₂²×C₄	—	73	1	no	—
144	C₂²×C₄	—	146	1	no	—
145	C₂²×C₄	—	146	1	no	—
146	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	73	2	no	—
151	C₂×C₈	—	146	1	no	—
152	C₂×C₈	—	146	1	no	—
153	C₄⋊C₄	—	2	1	no	—
154	C₄⋊C₄	—	2	1	no	—
155	C₄⋊C₄	—	292	2	no	—
156	C₄⋊C₄	—	146	4	no	—
157	C₄⋊C₄	—	146	1	no	—
158	C₄⋊C₄	—	146	1	no	—
159	C₄⋊C₄	C₄×D₇₃	1	2	yes	—
160	C₄⋊C₄	D₂₉₂	1	2	yes	—
161	C₄²	—	146	4	no	—
162	C₄²	—	146	1	no	—
163	C₄²	—	146	1	no	—
164	C₂³	D₄×D₇₃	1	1	yes	characteristic
165	Q₈	—	73	4	no	—
166	Q₈	—	146	2	no	—
167	Q₈	—	146	2	no	—
168	Q₈	—	146	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	—	292	1	no	—
176	C₂×C₄	—	292	1	no	—
177	C₂×C₄	—	292	1	no	—
178	C₂×C₄	—	73	2	no	—
179	C₂×C₄	—	146	1	no	—
180	C₂×C₄	—	146	1	no	—
181	C₂×C₄	—	146	1	no	—
182	C₂×C₄	—	146	2	no	—
183	C₂×C₄	—	146	1	no	—
184	C₂×C₄	—	146	2	no	—
185	C₂×C₄	—	146	2	no	—
186	C₂×C₄	—	146	1	no	—
187	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
188	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
189	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
190	C₂×C₄	Q₈×D₇₃	1	1	yes	characteristic
191	C₂×C₄	#11924027	1	1	yes	characteristic
192	C₂×C₄	#11924028	1	1	yes	characteristic
193	C₈	—	4	1	no	—
194	C₈	—	146	2	no	—
195	C₂²	#14556551	1	1	yes	characteristic
196	C₂²	#14556561	1	1	yes	characteristic
197	C₂²	#14556569	1	1	yes	characteristic
198	C₂²	SD₁₆×D₇₃	1	1	yes	characteristic
199	C₂²	#14556610	1	1	yes	characteristic
200	C₂²	#14556620	1	1	yes	characteristic
201	C₂²	#14556623	1	1	yes	characteristic
202	C₄	—	4	2	no	—
203	C₄	—	4	2	no	—
204	C₄	—	292	1	no	—
205	C₄	—	584	1	no	—
206	C₄	—	146	4	no	—
207	C₄	—	146	2	no	—
208	C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
209	C₄	C₄×D₂₉₂	1	1	yes	characteristic
210	C₄	#14556571	1	1	yes	characteristic
211	C₄	#14556573	1	1	yes	characteristic
212	C₂	#17988461	1	1	yes	characteristic
213	C₂	#17988678	1	1	yes	characteristic
214	C₂	#17988691	1	1	yes	characteristic
215	C₂	#17988727	1	1	yes	characteristic
216	C₂	#17988750	1	1	yes	characteristic
217	C₂	#17988871	1	1	yes	characteristic
218	C₂	#17988893	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556556

C₂

yes

maximal, characteristic

C₂×#14556522

C₂

yes

maximal, characteristic

C₂×#14556516

C₂

yes

maximal, characteristic

C₂×#14556514

C₂

yes

maximal, characteristic

#17988486

C₂

yes

maximal, characteristic

#17988341

C₂

yes

maximal, characteristic

#340×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₆

C₂²

yes

characteristic

C₂²×#10596831

C₂²

yes

characteristic

#14556556

—

#14556556

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

C₂×#11924007

—

C₂×C₄×Dic₇₃

—

C₂×C₄×Dic₇₃

C₂²

yes

—

—

C₄

yes

complemented

#14556498

—

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

C₂×C₄

yes

—

C₂×Dic₁₄₆

D₄

yes

—

C₂×Dic₁₄₆

Q₈

yes

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

D₄

yes

characteristic

C₂²×Dic₇₃

D₄

yes

—

—

C₂×C₄

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₄⋊C₄×C₇₃

D₄

yes

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₈

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

SD₁₆

yes

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

C₄⋊C₄

yes

complemented

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

C₄×D₄

yes

characteristic

C₂²×C₇₃

#120

yes

characteristic

C₂²×C₇₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₃

C₂×SD₁₆

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₃

C₄×D₄

yes

characteristic

C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₄×C₇₃

#123

yes

characteristic

Dic₇₃

—

Dic₇₃

—

100

Dic₇₃

—

101

Dic₇₃

—

102

C₂×C₇₃

#385

yes

characteristic

103

C₂×C₇₃

C₄×SD₁₆

yes

characteristic

104

C₂×C₇₃

#441

yes

characteristic

105

C₂×C₇₃

#448

yes

characteristic

106

C₂×C₇₃

#452

yes

characteristic

107

C₂×C₇₃

#475

yes

characteristic

108

C₂×C₇₃

yes

109

—

110

C₇₃

#1865

yes

characteristic, complemented, Sylow

111

C₂×C₄×Q₈

—

112

C₂×#106

—

113

C₂×#103

—

114

C₂×#103

—

115

#389

—

116

#340

D₇₃

yes

characteristic

117

#336

—

118

C₂²×Q₈

—

119

C₂²×C₈

—

120

C₂²×C₈

C₂×D₇₃

yes

characteristic

121

C₄×Q₈

—

146

—

122

C₄×Q₈

—

146

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

146

—

125

C₂×C₄⋊C₄

—

146

—

126

C₂×C₄⋊C₄

C₂×D₇₃

yes

characteristic

127

C₂×C₄⋊C₄

C₂×D₇₃

yes

characteristic

128

C₂×C₄²

—

129

C₂×C₄²

—

146

—

130

#106

—

146

—

131

#103

—

132

#103

—

146

—

133

#95

—

146

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂×Q₈

—

137

C₂×Q₈

—

146

—

138

C₂×Q₈

—

146

—

139

C₂×Q₈

—

146

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

146

—

145

C₂²×C₄

—

146

—

146

C₂²×C₄

C₂²×D₇₃

yes

characteristic

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

146

—

152

C₂×C₈

—

146

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

292

—

156

C₄⋊C₄

—

146

—

157

C₄⋊C₄

—

146

—

158

C₄⋊C₄

—

146

—

159

C₄⋊C₄

C₄×D₇₃

yes

—

160

C₄⋊C₄

D₂₉₂

yes

—

161

C₄²

—

146

—

162

C₄²

—

146

—

163

C₄²

—

146

—

164

C₂³

D₄×D₇₃

yes

characteristic

165

Q₈

—

166

Q₈

—

146

—

167

Q₈

—

146

—

168

Q₈

—

146

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

292

—

176

C₂×C₄

—

292

—

177

C₂×C₄

—

292

—

178

C₂×C₄

—

179

C₂×C₄

—

146

—

180

C₂×C₄

—

146

—

181

C₂×C₄

—

146

—

182

C₂×C₄

—

146

—

183

C₂×C₄

—

146

—

184

C₂×C₄

—

146

—

185

C₂×C₄

—

146

—

186

C₂×C₄

—

146

—

187

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

188

C₂×C₄

D₄×D₇₃

yes

characteristic

189

C₂×C₄

C₂×D₂₉₂

yes

characteristic

190

C₂×C₄

Q₈×D₇₃

yes

characteristic

191

C₂×C₄

#11924027

yes

characteristic

192

C₂×C₄

#11924028

yes

characteristic

193

C₈

—

194

C₈

—

146

—

195

C₂²

#14556551

yes

characteristic

196

C₂²

#14556561

yes

characteristic

197

C₂²

#14556569

yes

characteristic

198

C₂²

SD₁₆×D₇₃

yes

characteristic

199

C₂²

#14556610

yes

characteristic

200

C₂²

#14556620

yes

characteristic

201

C₂²

#14556623

yes

characteristic

202

C₄

—

203

C₄

—

204

C₄

—

292

—

205

C₄

—

584

—

206

C₄

—

146

—

207

C₄

—

146

—

208

C₄

C₄⋊C₄×D₇₃

yes

characteristic

209

C₄

C₄×D₂₉₂

yes

characteristic

210

C₄

#14556571

yes

characteristic

211

C₄

#14556573

yes

characteristic

212

C₂

#17988461

yes

characteristic

213

C₂

#17988678

yes

characteristic

214

C₂

#17988691

yes

characteristic

215

C₂

#17988727

yes

characteristic

216

C₂

#17988750

yes

characteristic

217

C₂

#17988871

yes

characteristic

218

C₂

#17988893

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	#340×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₄
hypercenter	#340	D₇₃
hypocenter	C₇₃	#1865
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₇₃

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

#340×C₇₃

C₂

socle

C₂³×C₇₃

C₂×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
19. #14556514	C₄	semidirect
82. Dic₁₄₆	C₄⋊C₄	semidirect
110. C₇₃	#1865	semidirect

H ≅ G/N

type

19. #14556514

C₄

semidirect

82. Dic₁₄₆

C₄⋊C₄

semidirect

110. C₇₃

#1865

semidirect

Subgroups — #22610200

Conjugacy classes of subgroups

Complements