Group #22611186 — subgroups

Subgroups — #22611186

Order 9344.

296 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 69 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×#11924013	C₂	1	1	yes	maximal, characteristic
2	#17988614	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988591	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14556508	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14556507	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988351	C₂	1	1	yes	maximal, characteristic, complemented
7	#418×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₂₉₂	C₂²	1	1	yes	characteristic
9	C₂×#11924013	—	2	2	no	—
10	C₂×#11924013	—	2	2	no	—
11	C₂×#11924013	—	2	2	no	—
12	C₂×#11924012	C₂²	1	1	yes	characteristic
13	C₂×#11924012	C₂²	1	1	yes	characteristic
14	C₂²×#10596831	C₂²	1	1	yes	characteristic
15	C₄⋊#10596831	—	2	2	no	—
16	#14556511	—	2	2	no	—
17	#14556508	—	2	2	no	—
18	#14556507	—	2	2	no	—
19	C₂²×D₄×C₇₃	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
21	#117×C₇₃	C₂²	1	1	yes	characteristic
22	#102×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	#11924013	—	2	8	no	—
31	#11924012	—	2	1	no	—
32	#11924012	—	2	1	no	—
33	#11924012	—	2	1	no	—
34	#11924012	—	2	1	no	—
35	C₂×#10596831	—	2	1	no	—
36	C₂×#10596831	—	2	1	no	—
37	C₂×#10596831	—	2	1	no	—
38	C₂×#10596831	—	2	1	no	—
39	C₂×#10596831	D₄	1	2	yes	—
40	C₂⁴×C₇₃	—	2	1	no	—
41	C₂×D₄×C₇₃	—	2	1	no	—
42	C₂×D₄×C₇₃	—	2	1	no	—
43	C₂×D₄×C₇₃	—	2	1	no	—
44	C₂×D₄×C₇₃	—	2	1	no	—
45	C₂×D₄×C₇₃	—	2	1	no	—
46	C₂×D₄×C₇₃	D₄	1	2	yes	—
47	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
48	M₁₆×C₇₃	—	2	1	no	—
49	M₁₆×C₇₃	—	2	1	no	—
50	C₂×C₈×C₇₃	—	2	1	no	—
51	C₄⋊C₄×C₇₃	D₄	1	2	yes	—
52	C₂²⋊C₄×C₇₃	—	2	1	no	—
53	C₄²×C₇₃	—	2	1	no	—
54	C₂³×C₇₃	—	2	2	no	—
55	C₂³×C₇₃	—	4	1	no	—
56	C₂³×C₇₃	—	4	1	no	—
57	C₂³×C₇₃	—	8	2	no	—
58	C₂³×C₇₃	#41	1	1	yes	characteristic
59	C₂²×D₇₃	—	2	2	no	—
60	C₂²×D₇₃	—	4	1	no	—
61	C₂²×D₇₃	—	4	1	no	—
62	C₂²×D₇₃	—	8	2	no	—
63	D₄×C₇₃	—	2	2	no	—
64	D₄×C₇₃	—	2	2	no	—
65	D₄×C₇₃	—	4	1	no	—
66	D₄×C₇₃	—	4	1	no	—
67	C₂×C₄×C₇₃	—	2	2	no	—
68	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
73	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
74	D₂₉₂	—	2	2	no	—
75	D₂₉₂	—	2	2	no	—
76	D₂₉₂	—	4	1	no	—
77	D₂₉₂	—	4	1	no	—
78	C₈×C₇₃	—	4	1	no	—
79	#10596831	—	2	2	no	—
80	#10596831	—	4	1	no	—
81	#10596831	—	4	1	no	—
82	C₂²×C₇₃	—	2	2	no	—
83	C₂²×C₇₃	—	4	2	no	—
84	C₂²×C₇₃	—	8	2	no	—
85	C₂²×C₇₃	—	8	2	no	—
86	C₂²×C₇₃	—	8	2	no	—
87	C₂²×C₇₃	#121	1	1	yes	characteristic
88	C₂²×C₇₃	#123	1	1	yes	characteristic
89	C₂²×C₇₃	#123	1	1	yes	characteristic
90	C₂×D₇₃	—	4	2	no	—
91	C₂×D₇₃	—	8	2	no	—
92	C₂×D₇₃	—	8	2	no	—
93	C₂×D₇₃	—	8	2	no	—
94	C₄×C₇₃	—	4	2	no	—
95	C₄×C₇₃	#120	1	1	yes	characteristic
96	C₄×C₇₃	#121	1	1	yes	characteristic
97	C₄×C₇₃	#121	1	1	yes	characteristic
98	C₄×C₇₃	#124	1	1	yes	characteristic
99	C₂×C₇₃	—	2	1	no	—
100	C₂×C₇₃	—	2	1	no	—
101	C₂×C₇₃	—	8	2	no	—
102	C₂×C₇₃	#394	1	1	yes	characteristic
103	C₂×C₇₃	#472	1	2	yes	—
104	D₇₃	—	8	2	no	—
105	#2045	—	73	1	no	maximal, Sylow
106	C₇₃	#2045	1	1	yes	characteristic, complemented, Sylow
107	C₂²×D₈	—	73	1	no	—
108	#424	—	73	1	no	—
109	#418	—	73	1	no	—
110	#418	D₇₃	1	1	yes	characteristic, complemented
111	C₂×#100	—	73	1	no	—
112	C₂×#100	—	73	1	no	—
113	#341	—	73	1	no	—
114	C₂²×D₄	—	73	1	no	—
115	C₂²×D₄	C₂×D₇₃	1	1	yes	characteristic
116	C₂×D₈	—	146	2	no	—
117	C₂×D₈	—	146	2	no	—
118	C₂×D₈	—	146	2	no	—
119	C₂×M₁₆	—	73	1	no	—
120	C₂×M₁₆	—	73	1	no	—
121	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
122	C₂²×C₈	—	73	1	no	—
123	#117	C₂×D₇₃	1	1	yes	characteristic
124	C₄⋊C₈	—	146	2	no	—
125	#102	—	2	2	no	—
126	#102	—	146	2	no	—
127	#100	—	146	2	no	—
128	#100	—	146	2	no	—
129	C₂⁴	—	2	1	no	—
130	C₂⁴	—	146	1	no	—
131	C₂×D₄	—	2	1	no	—
132	C₂×D₄	—	2	1	no	—
133	C₂×D₄	—	2	1	no	—
134	C₂×D₄	—	2	1	no	—
135	C₂×D₄	—	2	1	no	—
136	C₂×D₄	—	73	2	no	—
137	C₂×D₄	—	146	1	no	—
138	C₂×D₄	—	146	1	no	—
139	C₂×D₄	—	146	1	no	—
140	C₂×D₄	—	146	1	no	—
141	C₂×D₄	—	146	1	no	—
142	C₂×D₄	D₂₉₂	1	2	yes	—
143	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
144	D₈	—	146	8	no	—
145	M₁₆	—	2	1	no	—
146	M₁₆	—	2	1	no	—
147	M₁₆	—	146	1	no	—
148	M₁₆	—	146	1	no	—
149	M₁₆	—	146	1	no	—
150	M₁₆	—	146	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	73	2	no	—
153	C₂×C₈	—	146	1	no	—
154	C₂×C₈	—	146	1	no	—
155	C₂×C₈	—	146	1	no	—
156	C₂×C₈	—	146	1	no	—
157	C₄⋊C₄	C₂²⋊D₇₃	1	2	yes	—
158	C₂²⋊C₄	—	2	1	no	—
159	C₄²	—	2	1	no	—
160	C₂³	—	2	2	no	—
161	C₂³	—	4	1	no	—
162	C₂³	—	4	1	no	—
163	C₂³	—	8	2	no	—
164	C₂³	—	292	1	no	—
165	C₂³	—	292	1	no	—
166	C₂³	—	584	2	no	—
167	C₂³	—	146	2	no	—
168	C₂³	#11924027	1	1	yes	characteristic
169	D₄	—	2	2	no	—
170	D₄	—	2	2	no	—
171	D₄	—	4	1	no	—
172	D₄	—	4	1	no	—
173	D₄	—	292	1	no	—
174	D₄	—	292	1	no	—
175	D₄	—	146	2	no	—
176	D₄	—	146	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
179	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
180	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
181	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
182	C₂×C₄	#11924027	1	1	yes	characteristic
183	C₂×C₄	#11924028	1	1	yes	characteristic
184	C₈	—	4	1	no	—
185	C₈	—	292	1	no	—
186	C₈	—	292	1	no	—
187	C₈	—	146	2	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	4	2	no	—
190	C₂²	—	8	2	no	—
191	C₂²	—	8	2	no	—
192	C₂²	—	8	2	no	—
193	C₂²	—	292	2	no	—
194	C₂²	—	584	2	no	—
195	C₂²	—	584	2	no	—
196	C₂²	—	584	2	no	—
197	C₂²	#14556563	1	1	yes	characteristic
198	C₂²	#14556573	1	1	yes	characteristic
199	C₂²	#14556575	1	1	yes	characteristic
200	C₄	—	4	2	no	—
201	C₄	#14556561	1	1	yes	characteristic
202	C₄	#14556562	1	1	yes	characteristic
203	C₄	#14556565	1	1	yes	characteristic
204	C₄	#14556577	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	8	2	no	—
208	C₂	—	584	2	no	—
209	C₂	#17988503	1	1	yes	characteristic
210	C₂	#17988855	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11924013

C₂

yes

maximal, characteristic

#17988614

C₂

yes

maximal, characteristic, complemented

#17988591

C₂

yes

maximal, characteristic, complemented

C₂×#14556508

C₂

yes

maximal, characteristic, complemented

C₂×#14556507

C₂

yes

maximal, characteristic, complemented

#17988351

C₂

yes

maximal, characteristic, complemented

#418×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₉₂

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₂×#11924012

C₂²

yes

characteristic

C₂²×#10596831

C₂²

yes

—

—

—

—

C₂²×D₄×C₇₃

C₂²

yes

characteristic

C₂×M₁₆×C₇₃

C₂²

yes

characteristic

#117×C₇₃

C₂²

yes

characteristic

#102×C₇₃

—

C₂³×D₇₃

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

D₄

yes

—

—

—

—

—

—

—

—

—

—

D₄

yes

—

C₂⁴×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

D₄

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

D₄

yes

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

#41

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×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

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

—

#10596831

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

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

C₂×C₇₃

—

100

C₂×C₇₃

—

101

C₂×C₇₃

—

102

C₂×C₇₃

#394

yes

characteristic

103

C₂×C₇₃

#472

yes

—

104

D₇₃

—

105

#2045

—

maximal, Sylow

106

C₇₃

#2045

yes

characteristic, complemented, Sylow

107

C₂²×D₈

—

108

#424

—

109

#418

—

110

#418

D₇₃

yes

characteristic, complemented

111

C₂×#100

—

112

C₂×#100

—

113

#341

—

114

C₂²×D₄

—

115

C₂²×D₄

C₂×D₇₃

yes

characteristic

116

C₂×D₈

—

146

—

117

C₂×D₈

—

146

—

118

C₂×D₈

—

146

—

119

C₂×M₁₆

—

120

C₂×M₁₆

—

121

C₂×M₁₆

C₂×D₇₃

yes

characteristic

122

C₂²×C₈

—

123

#117

C₂×D₇₃

yes

characteristic

124

C₄⋊C₈

—

146

—

125

#102

—

126

#102

—

146

—

127

#100

—

146

—

128

#100

—

146

—

129

C₂⁴

—

130

C₂⁴

—

146

—

131

C₂×D₄

—

132

C₂×D₄

—

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

—

136

C₂×D₄

—

137

C₂×D₄

—

146

—

138

C₂×D₄

—

146

—

139

C₂×D₄

—

146

—

140

C₂×D₄

—

146

—

141

C₂×D₄

—

146

—

142

C₂×D₄

D₂₉₂

yes

—

143

C₂²×C₄

C₂²×D₇₃

yes

characteristic

144

D₈

—

146

—

145

M₁₆

—

146

M₁₆

—

147

M₁₆

—

146

—

148

M₁₆

—

146

—

149

M₁₆

—

146

—

150

M₁₆

—

146

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

146

—

154

C₂×C₈

—

146

—

155

C₂×C₈

—

146

—

156

C₂×C₈

—

146

—

157

C₄⋊C₄

C₂²⋊D₇₃

yes

—

158

C₂²⋊C₄

—

159

C₄²

—

160

C₂³

—

161

C₂³

—

162

C₂³

—

163

C₂³

—

164

C₂³

—

292

—

165

C₂³

—

292

—

166

C₂³

—

584

—

167

C₂³

—

146

—

168

C₂³

#11924027

yes

characteristic

169

D₄

—

170

D₄

—

171

D₄

—

172

D₄

—

173

D₄

—

292

—

174

D₄

—

292

—

175

D₄

—

146

—

176

D₄

—

146

—

177

C₂×C₄

—

178

C₂×C₄

D₄×D₇₃

yes

characteristic

179

C₂×C₄

D₄×D₇₃

yes

characteristic

180

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

181

C₂×C₄

C₂×D₂₉₂

yes

characteristic

182

C₂×C₄

#11924027

yes

characteristic

183

C₂×C₄

#11924028

yes

characteristic

184

C₈

—

185

C₈

—

292

—

186

C₈

—

292

—

187

C₈

—

146

—

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

292

—

194

C₂²

—

584

—

195

C₂²

—

584

—

196

C₂²

—

584

—

197

C₂²

#14556563

yes

characteristic

198

C₂²

#14556573

yes

characteristic

199

C₂²

#14556575

yes

characteristic

200

C₄

—

201

C₄

#14556561

yes

characteristic

202

C₄

#14556562

yes

characteristic

203

C₄

#14556565

yes

characteristic

204

C₄

#14556577

yes

characteristic

205

C₂

—

206

C₂

—

207

C₂

—

208

C₂

—

584

—

209

C₂

#17988503

yes

characteristic

210

C₂

#17988855

yes

—

	subgroup	G/H
center	C₂²	#14556563
commutator	C₂²×C₄×C₇₃	C₂³
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	#418×C₇₃	C₂
socle	C₂²×C₇₃	#121
hypercenter	#418	D₇₃
hypocenter	C₇₃	#2045
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #418

subgroup

G/H

center

C₂²

#14556563

commutator

C₂²×C₄×C₇₃

C₂³

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

#418×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #418

N	H ≅ G/N	type
2. #17988614	C₂	semidirect
3. #17988591	C₂	semidirect
4. C₂×#14556508	C₂	semidirect
5. C₂×#14556507	C₂	semidirect
6. #17988351	C₂	semidirect
7. #418×C₇₃	C₂	semidirect
106. C₇₃	#2045	semidirect
110. #418	D₇₃	semidirect

H ≅ G/N

type

2. #17988614

C₂

semidirect

3. #17988591

C₂

semidirect

4. C₂×#14556508

C₂

semidirect

5. C₂×#14556507

C₂

semidirect

6. #17988351

C₂

semidirect

7. #418×C₇₃

C₂

semidirect

106. C₇₃

#2045

semidirect

110. #418

D₇₃

semidirect

Subgroups — #22611186

Conjugacy classes of subgroups

Complements