Group #22610693 — subgroups

Subgroups — #22610693

Order 9344.

256 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 107 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×#14556523	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556522	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556520	C₂	1	1	yes	maximal, characteristic
4	#17988456	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₄×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	#17988349	C₂	1	1	yes	maximal, characteristic
7	#340×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10596831	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
11	C₂×#11924007	—	2	1	no	—
12	C₂×#11924007	—	2	1	no	—
13	C₂×C₄×Dic₇₃	—	2	1	no	—
14	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
15	#14556523	—	2	2	no	—
16	#14556522	—	2	2	no	—
17	#14556520	C₄	1	4	yes	complemented
18	#14556498	—	2	1	no	—
19	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
22	C₂²×Dic₇₃	—	2	1	no	—
23	C₂²×Dic₇₃	—	2	1	no	—
24	C₂²×Dic₇₃	D₄	1	1	yes	characteristic
25	C₂²×Dic₇₃	D₄	1	1	yes	characteristic
26	C₂×#10596831	—	2	1	no	—
27	C₂×#10596831	—	2	1	no	—
28	C₂×#10596831	C₂×C₄	1	2	yes	—
29	C₄⋊Dic₇₃	—	2	1	no	—
30	C₄⋊Dic₇₃	—	2	1	no	—
31	C₄⋊Dic₇₃	—	2	1	no	—
32	C₄⋊Dic₇₃	D₄	1	2	yes	—
33	#11924007	—	4	2	no	—
34	#11924007	—	4	2	no	—
35	C₄×Dic₇₃	—	4	2	no	—
36	C₄×Dic₇₃	C₂×C₄	1	2	yes	—
37	C₄×Dic₇₃	Q₈	1	2	yes	—
38	C₂²×C₄×C₇₃	—	2	1	no	—
39	C₂²×C₄×C₇₃	—	2	1	no	—
40	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₇₃	—	2	1	no	—
42	C₂×C₈×C₇₃	—	2	1	no	—
43	C₂×C₈×C₇₃	C₂×C₄	1	2	yes	—
44	C₄⋊C₄×C₇₃	—	2	1	no	—
45	C₄⋊C₄×C₇₃	—	2	1	no	—
46	C₄⋊C₄×C₇₃	—	2	1	no	—
47	C₄⋊C₄×C₇₃	Q₈	1	2	yes	—
48	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇₃	—	2	2	no	—
50	C₂×C₄×C₇₃	—	4	1	no	—
51	C₂×C₄×C₇₃	—	4	1	no	—
52	C₂×C₄×C₇₃	—	4	1	no	—
53	C₂×C₄×C₇₃	—	4	1	no	—
54	C₂×C₄×C₇₃	—	4	1	no	—
55	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
60	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
61	C₂×Dic₇₃	—	2	1	no	—
62	C₂×Dic₇₃	—	2	1	no	—
63	C₂×Dic₇₃	—	2	2	no	—
64	C₂×Dic₇₃	—	4	1	no	—
65	C₂×Dic₇₃	—	4	1	no	—
66	C₂×Dic₇₃	—	4	1	no	—
67	C₂×Dic₇₃	—	4	1	no	—
68	C₂×Dic₇₃	—	4	1	no	—
69	C₂×Dic₇₃	C₄⋊C₄	1	2	yes	—
70	C₂×Dic₇₃	C₄⋊C₄	1	2	yes	—
71	C₂×Dic₇₃	SD₁₆	1	2	yes	—
72	C₂×Dic₇₃	SD₁₆	1	2	yes	—
73	C₈×C₇₃	—	2	2	no	—
74	#10596831	—	2	2	no	—
75	C₂²×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₇₃	#121	1	1	yes	characteristic
77	C₂²×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂²×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
79	C₂²×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
80	C₂²×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂²×C₇₃	#137	1	1	yes	characteristic
82	C₄×C₇₃	—	4	2	no	—
83	C₄×C₇₃	—	8	1	no	—
84	C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
85	C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
86	C₄×C₇₃	#125	1	1	yes	characteristic
87	C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
88	Dic₇₃	—	2	2	no	—
89	Dic₇₃	—	4	2	no	—
90	Dic₇₃	—	8	1	no	—
91	Dic₇₃	#106	1	4	yes	complemented
92	C₂×C₇₃	#389	1	1	yes	characteristic
93	C₂×C₇₃	C₂×#106	1	1	yes	characteristic
94	C₂×C₇₃	#428	1	1	yes	characteristic
95	C₂×C₇₃	#460	1	1	yes	characteristic
96	C₂×C₇₃	#461	1	1	yes	characteristic
97	C₂×C₇₃	#475	1	1	yes	characteristic
98	C₂×C₇₃	#476	1	1	yes	characteristic
99	#1948	—	73	1	no	maximal, Sylow
100	C₇₃	#1948	1	1	yes	characteristic, complemented, Sylow
101	C₂×#106	—	73	1	no	—
102	C₂×#106	—	73	1	no	—
103	C₂×C₄⋊C₈	—	73	1	no	—
104	#384	—	73	1	no	—
105	C₄×C₄⋊C₄	—	73	1	no	—
106	#340	—	73	1	no	—
107	#340	D₇₃	1	1	yes	characteristic
108	C₂²×C₈	—	73	1	no	—
109	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
110	C₂×C₄⋊C₄	—	73	1	no	—
111	C₂×C₄⋊C₄	—	73	1	no	—
112	C₂×C₄⋊C₄	—	146	1	no	—
113	C₂×C₄⋊C₄	—	146	1	no	—
114	C₂×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
115	C₂×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
116	C₂×C₄²	—	73	1	no	—
117	C₂×C₄²	—	146	1	no	—
118	#106	—	146	2	no	—
119	#106	—	146	2	no	—
120	C₄⋊C₈	—	73	4	no	—
121	#95	—	146	1	no	—
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	—	73	1	no	—
125	C₂²×C₄	—	73	1	no	—
126	C₂²×C₄	—	146	1	no	—
127	C₂²×C₄	—	146	1	no	—
128	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	73	2	no	—
132	C₂×C₈	—	146	1	no	—
133	C₂×C₈	—	146	1	no	—
134	C₂×C₈	C₄×D₇₃	1	2	yes	—
135	C₄⋊C₄	—	2	1	no	—
136	C₄⋊C₄	—	2	1	no	—
137	C₄⋊C₄	—	2	1	no	—
138	C₄⋊C₄	—	292	2	no	—
139	C₄⋊C₄	—	292	2	no	—
140	C₄⋊C₄	—	73	2	no	—
141	C₄⋊C₄	—	146	1	no	—
142	C₄⋊C₄	—	146	1	no	—
143	C₄⋊C₄	—	146	1	no	—
144	C₄⋊C₄	Dic₁₄₆	1	2	yes	—
145	C₄²	—	292	2	no	—
146	C₄²	—	73	2	no	—
147	C₄²	—	73	2	no	—
148	C₂³	D₄×D₇₃	1	1	yes	characteristic
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	292	1	no	—
156	C₂×C₄	—	292	1	no	—
157	C₂×C₄	—	292	1	no	—
158	C₂×C₄	—	292	1	no	—
159	C₂×C₄	—	292	1	no	—
160	C₂×C₄	—	73	2	no	—
161	C₂×C₄	—	73	2	no	—
162	C₂×C₄	—	73	2	no	—
163	C₂×C₄	—	73	2	no	—
164	C₂×C₄	—	146	2	no	—
165	C₂×C₄	—	146	1	no	—
166	C₂×C₄	—	146	1	no	—
167	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
168	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
169	C₂×C₄	Q₈×D₇₃	1	1	yes	characteristic
170	C₂×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
171	C₂×C₄	#11924027	1	1	yes	characteristic
172	C₂×C₄	#11924028	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	146	2	no	—
175	C₂²	C₄⋊C₄×D₇₃	1	1	yes	characteristic
176	C₂²	#14556564	1	1	yes	characteristic
177	C₂²	#14556568	1	1	yes	characteristic
178	C₂²	SD₁₆×D₇₃	1	1	yes	characteristic
179	C₂²	SD₁₆×D₇₃	1	1	yes	characteristic
180	C₂²	#14556615	1	1	yes	characteristic
181	C₂²	#14556623	1	1	yes	characteristic
182	C₄	—	4	2	no	—
183	C₄	—	8	1	no	—
184	C₄	—	292	2	no	—
185	C₄	—	584	1	no	—
186	C₄	—	73	4	no	—
187	C₄	—	146	2	no	—
188	C₄	#14556551	1	1	yes	characteristic
189	C₄	C₄×Dic₁₄₆	1	1	yes	characteristic
190	C₄	#14556581	1	1	yes	characteristic
191	C₄	#14556590	1	1	yes	characteristic
192	C₂	#17988484	1	1	yes	characteristic
193	C₂	#106×D₇₃	1	1	yes	characteristic
194	C₂	#17988628	1	1	yes	characteristic
195	C₂	#17988786	1	1	yes	characteristic
196	C₂	#17988791	1	1	yes	characteristic
197	C₂	#17988870	1	1	yes	characteristic
198	C₂	#17988875	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556523

C₂

yes

maximal, characteristic

C₂×#14556522

C₂

yes

maximal, characteristic

C₂×#14556520

C₂

yes

maximal, characteristic

#17988456

C₂

yes

maximal, characteristic

C₄⋊C₄×Dic₇₃

C₂

yes

maximal, characteristic

#17988349

C₂

yes

maximal, characteristic

#340×C₇₃

C₂

yes

maximal, characteristic

C₂²×#10596831

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

C₂×#11924007

—

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

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

D₄

yes

characteristic

C₂²×Dic₇₃

D₄

yes

—

—

C₂×C₄

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

D₄

yes

—

#11924007

—

#11924007

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂×C₄

yes

—

C₄×Dic₇₃

Q₈

yes

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

Q₈

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×Q₈

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

C₄⋊C₄

yes

—

C₂×Dic₇₃

C₄⋊C₄

yes

—

C₂×Dic₇₃

SD₁₆

yes

—

C₂×Dic₇₃

SD₁₆

yes

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₃

#121

yes

characteristic

C₂²×C₇₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₃

C₂×SD₁₆

yes

characteristic

C₂²×C₇₃

C₂×SD₁₆

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₄×D₄

yes

characteristic

C₄×C₇₃

C₄×Q₈

yes

characteristic

C₄×C₇₃

#125

yes

characteristic

C₄×C₇₃

C₄⋊Q₈

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

100

C₇₃

#1948

yes

characteristic, complemented, Sylow

101

C₂×#106

—

102

C₂×#106

—

103

C₂×C₄⋊C₈

—

104

#384

—

105

C₄×C₄⋊C₄

—

106

#340

—

107

#340

D₇₃

yes

characteristic

108

C₂²×C₈

—

109

C₂²×C₈

C₂×D₇₃

yes

characteristic

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

146

—

113

C₂×C₄⋊C₄

—

146

—

114

C₂×C₄⋊C₄

C₂×D₇₃

yes

characteristic

115

C₂×C₄⋊C₄

C₂×D₇₃

yes

characteristic

116

C₂×C₄²

—

117

C₂×C₄²

—

146

—

118

#106

—

146

—

119

#106

—

146

—

120

C₄⋊C₈

—

121

#95

—

146

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

146

—

127

C₂²×C₄

—

146

—

128

C₂²×C₄

C₂²×D₇₃

yes

characteristic

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

146

—

133

C₂×C₈

—

146

—

134

C₂×C₈

C₄×D₇₃

yes

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

292

—

139

C₄⋊C₄

—

292

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

146

—

142

C₄⋊C₄

—

146

—

143

C₄⋊C₄

—

146

—

144

C₄⋊C₄

Dic₁₄₆

yes

—

145

C₄²

—

292

—

146

C₄²

—

147

C₄²

—

148

C₂³

D₄×D₇₃

yes

characteristic

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

292

—

156

C₂×C₄

—

292

—

157

C₂×C₄

—

292

—

158

C₂×C₄

—

292

—

159

C₂×C₄

—

292

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

146

—

165

C₂×C₄

—

146

—

166

C₂×C₄

—

146

—

167

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

168

C₂×C₄

D₄×D₇₃

yes

characteristic

169

C₂×C₄

Q₈×D₇₃

yes

characteristic

170

C₂×C₄

C₂×Dic₁₄₆

yes

characteristic

171

C₂×C₄

#11924027

yes

characteristic

172

C₂×C₄

#11924028

yes

characteristic

173

C₈

—

174

C₈

—

146

—

175

C₂²

C₄⋊C₄×D₇₃

yes

characteristic

176

C₂²

#14556564

yes

characteristic

177

C₂²

#14556568

yes

characteristic

178

C₂²

SD₁₆×D₇₃

yes

characteristic

179

C₂²

SD₁₆×D₇₃

yes

characteristic

180

C₂²

#14556615

yes

characteristic

181

C₂²

#14556623

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

292

—

185

C₄

—

584

—

186

C₄

—

187

C₄

—

146

—

188

C₄

#14556551

yes

characteristic

189

C₄

C₄×Dic₁₄₆

yes

characteristic

190

C₄

#14556581

yes

characteristic

191

C₄

#14556590

yes

characteristic

192

C₂

#17988484

yes

characteristic

193

C₂

#106×D₇₃

yes

characteristic

194

C₂

#17988628

yes

characteristic

195

C₂

#17988786

yes

characteristic

196

C₂

#17988791

yes

characteristic

197

C₂

#17988870

yes

characteristic

198

C₂

#17988875

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	#1948
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
17. #14556520	C₄	semidirect
91. Dic₇₃	#106	semidirect
100. C₇₃	#1948	semidirect

H ≅ G/N

type

17. #14556520

C₄

semidirect

91. Dic₇₃

#106

semidirect

100. C₇₃

#1948

semidirect

Subgroups — #22610693

Conjugacy classes of subgroups

Complements