Group #22611190 — subgroups

Subgroups — #22611190

Order 9344.

276 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₂²×#11924015	C₂	1	1	yes	maximal, characteristic
2	#17988615	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988596	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988593	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556507	C₂	1	1	yes	maximal, characteristic
6	#17988351	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#101×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₂₉₂	C₂²	1	1	yes	characteristic
9	C₂×#11924015	—	2	2	no	—
10	C₂×#11924015	—	2	2	no	—
11	C₂×#11924015	—	2	2	no	—
12	C₂×#11924012	C₂²	1	1	yes	characteristic
13	C₂²×#10596831	C₂²	1	1	yes	characteristic
14	#14556550	C₂²	1	1	yes	characteristic
15	#14556520	—	2	2	no	—
16	#14556515	—	2	2	no	—
17	#14556513	—	2	2	no	—
18	#14556507	—	2	2	no	—
19	C₂²×Q₈×C₇₃	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
21	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
22	#101×C₇₃	—	2	2	no	—
23	C₂³×D₇₃	—	2	1	no	—
24	C₂×D₂₉₂	—	2	1	no	—
25	C₂×D₂₉₂	—	2	1	no	—
26	C₂×D₂₉₂	—	2	1	no	—
27	C₂×D₂₉₂	—	2	1	no	—
28	C₂×D₂₉₂	—	2	1	no	—
29	C₂×D₂₉₂	D₄	1	2	yes	—
30	#11924015	—	2	8	no	—
31	#11924012	—	2	1	no	—
32	#11924012	—	2	1	no	—
33	C₂×#10596831	—	2	1	no	—
34	C₂×#10596831	—	2	1	no	—
35	C₂×#10596831	—	2	1	no	—
36	C₂×#10596831	D₄	1	2	yes	—
37	C₄⋊Dic₇₃	D₄	1	2	yes	—
38	C₂²⋊Dic₇₃	—	2	1	no	—
39	C₄×Dic₇₃	—	2	1	no	—
40	C₂×Q₈×C₇₃	—	2	1	no	—
41	C₂×Q₈×C₇₃	—	2	1	no	—
42	C₂×Q₈×C₇₃	—	2	1	no	—
43	C₂×Q₈×C₇₃	—	2	1	no	—
44	C₂×Q₈×C₇₃	—	2	1	no	—
45	C₂×Q₈×C₇₃	D₄	1	2	yes	—
46	C₂²×C₄×C₇₃	—	2	1	no	—
47	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
48	M₁₆×C₇₃	—	2	1	no	—
49	M₁₆×C₇₃	—	2	1	no	—
50	M₁₆×C₇₃	—	2	1	no	—
51	M₁₆×C₇₃	—	2	1	no	—
52	C₂×C₈×C₇₃	—	2	1	no	—
53	C₂×C₈×C₇₃	—	2	1	no	—
54	C₂³×C₇₃	#41	1	1	yes	characteristic
55	C₂²×D₇₃	—	2	2	no	—
56	C₂²×D₇₃	—	4	1	no	—
57	C₂²×D₇₃	—	4	1	no	—
58	C₂²×D₇₃	—	8	2	no	—
59	Q₈×C₇₃	—	2	2	no	—
60	Q₈×C₇₃	—	2	2	no	—
61	Q₈×C₇₃	—	4	1	no	—
62	Q₈×C₇₃	—	4	1	no	—
63	C₂×C₄×C₇₃	—	2	2	no	—
64	C₂×C₄×C₇₃	—	4	1	no	—
65	C₂×C₄×C₇₃	—	4	1	no	—
66	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
71	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
72	C₂×Dic₇₃	—	2	2	no	—
73	D₂₉₂	—	2	2	no	—
74	D₂₉₂	—	2	2	no	—
75	D₂₉₂	—	4	1	no	—
76	D₂₉₂	—	4	1	no	—
77	C₈×C₇₃	—	4	1	no	—
78	C₈×C₇₃	—	4	1	no	—
79	#10596831	—	2	2	no	—
80	#10596831	—	4	1	no	—
81	C₂²×C₇₃	—	2	2	no	—
82	C₂²×C₇₃	#121	1	1	yes	characteristic
83	C₂²×C₇₃	#123	1	1	yes	characteristic
84	C₂²×C₇₃	#123	1	1	yes	characteristic
85	C₂×D₇₃	—	4	2	no	—
86	C₂×D₇₃	—	8	2	no	—
87	C₂×D₇₃	—	8	2	no	—
88	C₂×D₇₃	—	8	2	no	—
89	C₄×C₇₃	—	4	2	no	—
90	C₄×C₇₃	#120	1	1	yes	characteristic
91	C₄×C₇₃	#121	1	1	yes	characteristic
92	C₄×C₇₃	#121	1	1	yes	characteristic
93	C₄×C₇₃	#124	1	1	yes	characteristic
94	Dic₇₃	—	4	2	no	—
95	C₂×C₇₃	—	2	1	no	—
96	C₂×C₇₃	—	2	1	no	—
97	C₂×C₇₃	#394	1	1	yes	characteristic
98	C₂×C₇₃	#471	1	2	yes	—
99	D₇₃	—	8	2	no	—
100	#2046	—	73	1	no	maximal, Sylow
101	C₇₃	#2046	1	1	yes	characteristic, complemented, Sylow
102	C₂²×SD₁₆	—	73	1	no	—
103	#424	—	73	1	no	—
104	#419	—	73	1	no	—
105	#418	—	73	1	no	—
106	C₂×#101	D₇₃	1	1	yes	characteristic, complemented
107	C₂×#100	—	73	1	no	—
108	#341	—	73	1	no	—
109	C₂²×Q₈	C₂×D₇₃	1	1	yes	characteristic
110	C₂²×D₄	—	73	1	no	—
111	C₂×SD₁₆	—	146	2	no	—
112	C₂×SD₁₆	—	146	2	no	—
113	C₂×SD₁₆	—	146	2	no	—
114	C₂×M₁₆	—	73	1	no	—
115	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
116	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
117	C₂²×C₈	—	73	1	no	—
118	#117	—	73	1	no	—
119	C₄⋊C₈	—	146	2	no	—
120	#103	—	146	2	no	—
121	#102	—	146	2	no	—
122	#101	—	2	2	no	—
123	#100	—	146	2	no	—
124	C₂⁴	—	146	1	no	—
125	C₂×Q₈	—	2	1	no	—
126	C₂×Q₈	—	2	1	no	—
127	C₂×Q₈	—	2	1	no	—
128	C₂×Q₈	—	2	1	no	—
129	C₂×Q₈	—	2	1	no	—
130	C₂×Q₈	C₂²⋊D₇₃	1	2	yes	—
131	C₂×D₄	—	73	2	no	—
132	C₂×D₄	—	146	1	no	—
133	C₂×D₄	—	146	1	no	—
134	C₂×D₄	—	146	1	no	—
135	C₂×D₄	—	146	1	no	—
136	C₂×D₄	—	146	1	no	—
137	C₂²×C₄	—	2	1	no	—
138	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
139	SD₁₆	—	146	8	no	—
140	M₁₆	—	2	1	no	—
141	M₁₆	—	2	1	no	—
142	M₁₆	—	2	1	no	—
143	M₁₆	—	2	1	no	—
144	M₁₆	—	146	1	no	—
145	M₁₆	—	146	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	73	2	no	—
149	C₂×C₈	—	146	1	no	—
150	C₂×C₈	—	146	1	no	—
151	C₂×C₈	—	146	1	no	—
152	C₄⋊C₄	—	73	2	no	—
153	C₂²⋊C₄	—	146	1	no	—
154	C₄²	—	146	1	no	—
155	C₂³	—	292	1	no	—
156	C₂³	—	292	1	no	—
157	C₂³	—	584	2	no	—
158	C₂³	—	146	2	no	—
159	C₂³	#11924028	1	1	yes	characteristic
160	Q₈	—	2	2	no	—
161	Q₈	—	2	2	no	—
162	Q₈	—	4	1	no	—
163	Q₈	—	4	1	no	—
164	D₄	—	292	1	no	—
165	D₄	—	292	1	no	—
166	D₄	—	146	2	no	—
167	D₄	—	146	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	146	2	no	—
172	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
173	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
174	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
175	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
176	C₂×C₄	#11924027	1	1	yes	characteristic
177	C₂×C₄	#11924028	1	1	yes	characteristic
178	C₈	—	4	1	no	—
179	C₈	—	4	1	no	—
180	C₈	—	292	1	no	—
181	C₈	—	146	2	no	—
182	C₂²	—	2	2	no	—
183	C₂²	—	292	2	no	—
184	C₂²	—	584	2	no	—
185	C₂²	—	584	2	no	—
186	C₂²	—	584	2	no	—
187	C₂²	#14556564	1	1	yes	characteristic
188	C₂²	#14556573	1	1	yes	characteristic
189	C₂²	#14556574	1	1	yes	characteristic
190	C₄	—	4	2	no	—
191	C₄	—	292	2	no	—
192	C₄	#14556559	1	1	yes	characteristic
193	C₄	#14556563	1	1	yes	characteristic
194	C₄	#14556564	1	1	yes	characteristic
195	C₄	#14556577	1	1	yes	characteristic
196	C₂	—	2	1	no	—
197	C₂	—	2	1	no	—
198	C₂	—	584	2	no	—
199	C₂	#17988502	1	1	yes	characteristic
200	C₂	#17988853	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11924015

C₂

yes

maximal, characteristic

#17988615

C₂

yes

maximal, characteristic, complemented

#17988596

C₂

yes

maximal, characteristic, complemented

#17988593

C₂

yes

maximal, characteristic

C₂×#14556507

C₂

yes

maximal, characteristic

#17988351

C₂

yes

maximal, characteristic, complemented

C₂×#101×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₉₂

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₂²×#10596831

C₂²

yes

characteristic

#14556550

C₂²

yes

—

—

—

—

C₂²×Q₈×C₇₃

C₂²

yes

characteristic

C₂×M₁₆×C₇₃

C₂²

yes

characteristic

C₂×M₁₆×C₇₃

C₂²

yes

characteristic

#101×C₇₃

—

C₂³×D₇₃

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

D₄

yes

—

—

—

—

—

—

—

D₄

yes

—

C₄⋊Dic₇₃

D₄

yes

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

D₄

yes

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂³×C₇₃

#41

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

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

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#394

yes

characteristic

C₂×C₇₃

#471

yes

—

D₇₃

—

100

#2046

—

maximal, Sylow

101

C₇₃

#2046

yes

characteristic, complemented, Sylow

102

C₂²×SD₁₆

—

103

#424

—

104

#419

—

105

#418

—

106

C₂×#101

D₇₃

yes

characteristic, complemented

107

C₂×#100

—

108

#341

—

109

C₂²×Q₈

C₂×D₇₃

yes

characteristic

110

C₂²×D₄

—

111

C₂×SD₁₆

—

146

—

112

C₂×SD₁₆

—

146

—

113

C₂×SD₁₆

—

146

—

114

C₂×M₁₆

—

115

C₂×M₁₆

C₂×D₇₃

yes

characteristic

116

C₂×M₁₆

C₂×D₇₃

yes

characteristic

117

C₂²×C₈

—

118

#117

—

119

C₄⋊C₈

—

146

—

120

#103

—

146

—

121

#102

—

146

—

122

#101

—

123

#100

—

146

—

124

C₂⁴

—

146

—

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

C₂²⋊D₇₃

yes

—

131

C₂×D₄

—

132

C₂×D₄

—

146

—

133

C₂×D₄

—

146

—

134

C₂×D₄

—

146

—

135

C₂×D₄

—

146

—

136

C₂×D₄

—

146

—

137

C₂²×C₄

—

138

C₂²×C₄

C₂²×D₇₃

yes

characteristic

139

SD₁₆

—

146

—

140

M₁₆

—

141

M₁₆

—

142

M₁₆

—

143

M₁₆

—

144

M₁₆

—

146

—

145

M₁₆

—

146

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

146

—

150

C₂×C₈

—

146

—

151

C₂×C₈

—

146

—

152

C₄⋊C₄

—

153

C₂²⋊C₄

—

146

—

154

C₄²

—

146

—

155

C₂³

—

292

—

156

C₂³

—

292

—

157

C₂³

—

584

—

158

C₂³

—

146

—

159

C₂³

#11924028

yes

characteristic

160

Q₈

—

161

Q₈

—

162

Q₈

—

163

Q₈

—

164

D₄

—

292

—

165

D₄

—

292

—

166

D₄

—

146

—

167

D₄

—

146

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

146

—

172

C₂×C₄

D₄×D₇₃

yes

characteristic

173

C₂×C₄

D₄×D₇₃

yes

characteristic

174

C₂×C₄

D₄×D₇₃

yes

characteristic

175

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

176

C₂×C₄

#11924027

yes

characteristic

177

C₂×C₄

#11924028

yes

characteristic

178

C₈

—

179

C₈

—

180

C₈

—

292

—

181

C₈

—

146

—

182

C₂²

—

183

C₂²

—

292

—

184

C₂²

—

584

—

185

C₂²

—

584

—

186

C₂²

—

584

—

187

C₂²

#14556564

yes

characteristic

188

C₂²

#14556573

yes

characteristic

189

C₂²

#14556574

yes

characteristic

190

C₄

—

191

C₄

—

292

—

192

C₄

#14556559

yes

characteristic

193

C₄

#14556563

yes

characteristic

194

C₄

#14556564

yes

characteristic

195

C₄

#14556577

yes

characteristic

196

C₂

—

197

C₂

—

198

C₂

—

584

—

199

C₂

#17988502

yes

characteristic

200

C₂

#17988853

yes

—

	subgroup	G/H
center	C₂²	#14556564
commutator	C₂²×C₄×C₇₃	C₂³
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	C₂×#101×C₇₃	C₂
socle	C₂²×C₇₃	#121
hypercenter	C₂×#101	D₇₃
hypocenter	C₇₃	#2046
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂²

#14556564

commutator

C₂²×C₄×C₇₃

C₂³

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

C₂×#101×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
2. #17988615	C₂	semidirect
3. #17988596	C₂	semidirect
6. #17988351	C₂	semidirect
7. C₂×#101×C₇₃	C₂	semidirect
101. C₇₃	#2046	semidirect
106. C₂×#101	D₇₃	semidirect

H ≅ G/N

type

2. #17988615

C₂

semidirect

3. #17988596

C₂

semidirect

6. #17988351

C₂

semidirect

7. C₂×#101×C₇₃

C₂

semidirect

101. C₇₃

#2046

semidirect

106. C₂×#101

D₇₃

semidirect

Subgroups — #22611190

Conjugacy classes of subgroups

Complements