Group #22620389 — subgroups

Subgroups — #22620389

Order 9344.

552 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 145 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×#14556623	C₂	1	4	yes	maximal, complemented
2	#17989138	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Q₈×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988980	C₂	1	4	yes	maximal, complemented
5	#17988966	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988961	C₂	1	1	yes	maximal, characteristic, complemented
7	#17988960	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11924011	C₂	1	1	yes	maximal, characteristic, complemented
9	#494×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11924029	C₂²	1	2	yes	—
11	C₂×Q₈×D₇₃	C₂²	1	2	yes	—
12	#14556623	—	2	16	no	—
13	C₂×#11924018	C₂²	1	4	yes	—
14	C₂×#11924017	C₂²	1	4	yes	—
15	C₂×#11924015	C₂²	1	4	yes	—
16	C₂×#11924011	C₂²	1	2	yes	—
17	#14556591	C₂²	1	1	yes	characteristic
18	#14556590	—	2	2	no	—
19	#14556589	C₂²	1	2	yes	—
20	#14556588	C₂²	1	1	yes	characteristic
21	#14556580	C₂²	1	2	yes	—
22	#14556566	—	2	2	no	—
23	#14556553	—	2	2	no	—
24	C₄⋊C₄×D₇₃	—	2	2	no	—
25	C₄²×D₇₃	C₂²	1	1	yes	characteristic
26	#14556501	C₂²	1	2	yes	—
27	C₄×#10596831	C₂²	1	1	yes	characteristic
28	C₂×Q₁₆×C₇₃	C₂²	1	4	yes	—
29	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
30	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
31	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
32	#11924029	—	2	8	no	—
33	C₂×Dic₁₄₆	—	2	2	no	—
34	C₂×Dic₁₄₆	C₂³	1	2	yes	—
35	Q₈×D₇₃	—	2	8	no	—
36	C₂×D₂₉₂	—	2	2	no	—
37	C₂×D₂₉₂	—	2	2	no	—
38	C₂×D₂₉₂	C₂³	1	2	yes	—
39	C₂×C₄×D₇₃	—	2	2	no	—
40	C₂×C₄×D₇₃	—	2	2	no	—
41	C₂×C₄×D₇₃	D₄	1	2	yes	—
42	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
43	#11924018	—	2	8	no	—
44	#11924017	—	2	8	no	—
45	#11924015	—	2	8	no	—
46	#11924011	D₄	1	8	yes	complemented
47	C₂×#10596831	C₂³	1	2	yes	—
48	C₄⋊Dic₇₃	—	2	2	no	—
49	#11924007	—	2	4	no	—
50	#11924005	—	2	4	no	—
51	C₄×Dic₇₃	D₄	1	2	yes	—
52	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
53	C₂×Q₈×C₇₃	C₂³	1	2	yes	—
54	C₂×Q₈×C₇₃	C₂³	1	2	yes	—
55	Q₁₆×C₇₃	—	2	8	no	—
56	C₂×C₈×C₇₃	C₂³	1	2	yes	—
57	C₄⋊C₄×C₇₃	—	2	2	no	—
58	C₄⋊C₄×C₇₃	—	2	2	no	—
59	C₄²×C₇₃	C₂³	1	1	yes	characteristic
60	C₂²×D₇₃	—	2	2	no	—
61	C₂²×D₇₃	C₂×D₄	1	1	yes	characteristic
62	Q₈×C₇₃	—	2	4	no	—
63	Q₈×C₇₃	—	2	4	no	—
64	C₂×C₄×C₇₃	—	2	2	no	—
65	C₂×C₄×C₇₃	—	2	2	no	—
66	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
68	C₂×Dic₇₃	—	2	2	no	—
69	C₂×Dic₇₃	C₂×D₄	1	2	yes	—
70	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
71	D₂₉₂	—	2	4	no	—
72	D₂₉₂	—	4	4	no	—
73	D₂₉₂	—	4	4	no	—
74	C₄×D₇₃	—	2	4	no	—
75	C₄×D₇₃	—	4	4	no	—
76	C₄×D₇₃	—	4	4	no	—
77	C₄×D₇₃	C₂×D₄	1	4	yes	—
78	Dic₁₄₆	—	2	4	no	—
79	Dic₁₄₆	—	4	4	no	—
80	C₈×C₇₃	C₂×D₄	1	4	yes	—
81	#10596831	C₂×D₄	1	4	yes	—
82	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
83	C₂×D₇₃	—	2	2	no	—
84	C₂×D₇₃	—	4	2	no	—
85	C₂×D₇₃	—	4	4	no	—
86	C₂×D₇₃	#127	1	2	yes	—
87	C₄×C₇₃	—	4	2	no	—
88	C₄×C₇₃	—	4	2	no	—
89	C₄×C₇₃	#137	1	4	yes	complemented
90	C₄×C₇₃	C₂²×D₄	1	2	yes	—
91	Dic₇₃	—	2	2	no	—
92	Dic₇₃	—	4	2	no	—
93	Dic₇₃	#127	1	2	yes	—
94	C₂×C₇₃	C₂×#127	1	1	yes	characteristic
95	C₂×C₇₃	C₂×#137	1	2	yes	—
96	D₇₃	—	2	2	no	—
97	D₇₃	—	8	2	no	—
98	#3152	—	73	1	no	maximal, Sylow
99	C₇₃	#3152	1	1	yes	characteristic, complemented, Sylow
100	C₂×#137	—	73	4	no	—
101	#532	—	73	1	no	—
102	C₂×C₄⋊Q₈	—	73	1	no	—
103	#497	—	73	4	no	—
104	#494	—	73	1	no	—
105	#494	D₇₃	1	1	yes	characteristic, complemented
106	#492	—	73	1	no	—
107	#492	—	73	1	no	—
108	C₄×M₁₆	—	73	1	no	—
109	C₂×#41	—	73	2	no	—
110	C₂²×Q₈	—	73	2	no	—
111	#137	—	146	16	no	—
112	C₂×Q₁₆	—	73	4	no	—
113	C₂×Q₁₆	C₂×D₇₃	1	4	yes	—
114	C₂×SD₁₆	—	73	4	no	—
115	C₂×SD₁₆	—	73	4	no	—
116	C₂×M₁₆	—	73	2	no	—
117	C₄⋊Q₈	—	73	2	no	—
118	C₄⋊Q₈	—	73	1	no	—
119	C₄⋊Q₈	—	146	2	no	—
120	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
121	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
122	#127	—	73	1	no	—
123	#124	—	73	2	no	—
124	#121	—	146	2	no	—
125	C₄×D₄	—	146	2	no	—
126	C₂×C₄⋊C₄	—	146	2	no	—
127	C₂×C₄²	—	73	1	no	—
128	#97	—	73	2	no	—
129	C₄×C₈	—	73	1	no	—
130	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
131	#41	—	146	8	no	—
132	C₂×Q₈	—	73	2	no	—
133	C₂×Q₈	—	146	2	no	—
134	C₂×Q₈	—	146	8	no	—
135	C₂×Q₈	C₂²×D₇₃	1	2	yes	—
136	C₂×Q₈	C₂²×D₇₃	1	2	yes	—
137	C₂×D₄	—	73	2	no	—
138	C₂×D₄	—	146	2	no	—
139	C₂×D₄	—	146	2	no	—
140	C₂²×C₄	—	73	1	no	—
141	C₂²×C₄	—	73	2	no	—
142	C₂²×C₄	—	146	2	no	—
143	C₂²×C₄	—	146	2	no	—
144	Q₁₆	—	2	8	no	—
145	Q₁₆	—	146	8	no	—
146	SD₁₆	—	146	8	no	—
147	SD₁₆	—	146	8	no	—
148	M₁₆	—	73	8	no	—
149	C₂×C₈	—	73	2	no	—
150	C₂×C₈	C₂²×D₇₃	1	2	yes	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	146	4	no	—
154	C₄⋊C₄	—	146	2	no	—
155	C₂²⋊C₄	—	146	4	no	—
156	C₄²	—	73	2	no	—
157	C₄²	—	73	1	no	—
158	C₄²	C₂²×D₇₃	1	1	yes	characteristic
159	C₂³	—	73	1	no	—
160	C₂³	—	146	2	no	—
161	Q₈	—	2	4	no	—
162	Q₈	—	2	4	no	—
163	Q₈	—	292	4	no	—
164	Q₈	—	146	4	no	—
165	D₄	—	292	4	no	—
166	D₄	—	292	4	no	—
167	D₄	—	146	4	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	292	4	no	—
171	C₂×C₄	—	292	4	no	—
172	C₂×C₄	—	73	1	no	—
173	C₂×C₄	—	73	4	no	—
174	C₂×C₄	—	73	2	no	—
175	C₂×C₄	—	146	2	no	—
176	C₂×C₄	—	146	4	no	—
177	C₂×C₄	D₄×D₇₃	1	2	yes	—
178	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
179	C₈	—	73	4	no	—
180	C₈	D₄×D₇₃	1	4	yes	—
181	C₂²	—	292	2	no	—
182	C₂²	—	292	4	no	—
183	C₂²	—	73	2	no	—
184	C₂²	—	146	2	no	—
185	C₂²	C₂×D₄×D₇₃	1	1	yes	characteristic
186	C₄	—	4	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	292	2	no	—
189	C₄	—	73	2	no	—
190	C₄	—	146	2	no	—
191	C₄	#14556623	1	4	yes	complemented
192	C₄	C₂×D₄×D₇₃	1	2	yes	—
193	C₂	—	584	2	no	—
194	C₂	—	146	2	no	—
195	C₂	#127×D₇₃	1	1	yes	characteristic
196	C₂	C₂×#14556623	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556623

C₂

yes

maximal, complemented

#17989138

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₇₃

C₂

yes

maximal, characteristic, complemented

#17988980

C₂

yes

maximal, complemented

#17988966

C₂

yes

maximal, characteristic, complemented

#17988961

C₂

yes

maximal, characteristic, complemented

#17988960

C₂

yes

maximal, characteristic, complemented

C₄×#11924011

C₂

yes

maximal, characteristic, complemented

#494×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924029

C₂²

yes

—

C₂×Q₈×D₇₃

C₂²

yes

—

#14556623

—

C₂×#11924018

C₂²

yes

—

C₂×#11924017

C₂²

yes

—

C₂×#11924015

C₂²

yes

—

C₂×#11924011

C₂²

yes

—

#14556591

C₂²

yes

characteristic

#14556590

—

#14556589

C₂²

yes

—

#14556588

C₂²

yes

characteristic

#14556580

C₂²

yes

—

#14556566

—

#14556553

—

C₄⋊C₄×D₇₃

—

C₄²×D₇₃

C₂²

yes

characteristic

#14556501

C₂²

yes

—

C₄×#10596831

C₂²

yes

characteristic

C₂×Q₁₆×C₇₃

C₂²

yes

—

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

C₄×C₈×C₇₃

C₂²

yes

characteristic

#11924029

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

C₂³

yes

—

Q₈×D₇₃

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

C₂³

yes

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

D₄

yes

—

C₂×C₄×D₇₃

C₂³

yes

—

—

—

D₄

yes

complemented

C₂×#10596831

C₂³

yes

—

C₄⋊Dic₇₃

—

#11924007

—

#11924005

—

C₄×Dic₇₃

D₄

yes

—

C₄×Dic₇₃

C₂³

yes

characteristic

C₂×Q₈×C₇₃

C₂³

yes

—

C₂×Q₈×C₇₃

C₂³

yes

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

C₂³

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

C₂×D₄

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂⁴

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×D₄

yes

—

C₂×Dic₇₃

C₂×D₄

yes

characteristic

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂×D₄

yes

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

C₂×D₄

yes

—

#10596831

C₂×D₄

yes

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

#127

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#137

yes

complemented

C₄×C₇₃

C₂²×D₄

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#127

yes

—

C₂×C₇₃

C₂×#127

yes

characteristic

C₂×C₇₃

C₂×#137

yes

—

D₇₃

—

D₇₃

—

#3152

—

maximal, Sylow

C₇₃

#3152

yes

characteristic, complemented, Sylow

100

C₂×#137

—

101

#532

—

102

C₂×C₄⋊Q₈

—

103

#497

—

104

#494

—

105

#494

D₇₃

yes

characteristic, complemented

106

#492

—

107

#492

—

108

C₄×M₁₆

—

109

C₂×#41

—

110

C₂²×Q₈

—

111

#137

—

146

—

112

C₂×Q₁₆

—

113

C₂×Q₁₆

C₂×D₇₃

yes

—

114

C₂×SD₁₆

—

115

C₂×SD₁₆

—

116

C₂×M₁₆

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

—

119

C₄⋊Q₈

—

146

—

120

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

121

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

122

#127

—

123

#124

—

124

#121

—

146

—

125

C₄×D₄

—

146

—

126

C₂×C₄⋊C₄

—

146

—

127

C₂×C₄²

—

128

#97

—

129

C₄×C₈

—

130

C₄×C₈

C₂×D₇₃

yes

characteristic

131

#41

—

146

—

132

C₂×Q₈

—

133

C₂×Q₈

—

146

—

134

C₂×Q₈

—

146

—

135

C₂×Q₈

C₂²×D₇₃

yes

—

136

C₂×Q₈

C₂²×D₇₃

yes

—

137

C₂×D₄

—

138

C₂×D₄

—

146

—

139

C₂×D₄

—

146

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

146

—

143

C₂²×C₄

—

146

—

144

Q₁₆

—

145

Q₁₆

—

146

—

146

SD₁₆

—

146

—

147

SD₁₆

—

146

—

148

M₁₆

—

149

C₂×C₈

—

150

C₂×C₈

C₂²×D₇₃

yes

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

146

—

154

C₄⋊C₄

—

146

—

155

C₂²⋊C₄

—

146

—

156

C₄²

—

157

C₄²

—

158

C₄²

C₂²×D₇₃

yes

characteristic

159

C₂³

—

160

C₂³

—

146

—

161

Q₈

—

162

Q₈

—

163

Q₈

—

292

—

164

Q₈

—

146

—

165

D₄

—

292

—

166

D₄

—

292

—

167

D₄

—

146

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

292

—

171

C₂×C₄

—

292

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

146

—

176

C₂×C₄

—

146

—

177

C₂×C₄

D₄×D₇₃

yes

—

178

C₂×C₄

C₂³×D₇₃

yes

characteristic

179

C₈

—

180

C₈

D₄×D₇₃

yes

—

181

C₂²

—

292

—

182

C₂²

—

292

—

183

C₂²

—

184

C₂²

—

146

—

185

C₂²

C₂×D₄×D₇₃

yes

characteristic

186

C₄

—

187

C₄

—

188

C₄

—

292

—

189

C₄

—

190

C₄

—

146

—

191

C₄

#14556623

yes

complemented

192

C₄

C₂×D₄×D₇₃

yes

—

193

C₂

—

584

—

194

C₂

—

146

—

195

C₂

#127×D₇₃

yes

characteristic

196

C₂

C₂×#14556623

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₃
Fitting	#494×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₄
hypercenter	#494	D₇₃
hypocenter	C₇₃	#3152
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

C₂×D₄×D₇₃

commutator

C₂×C₄×C₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₃

Fitting

#494×C₇₃

C₂

socle

C₂²×C₇₃

C₂²×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
1. C₂×#14556623	C₂	semidirect
3. C₄⋊Q₈×D₇₃	C₂	semidirect
4. #17988980	C₂	semidirect
5. #17988966	C₂	semidirect
6. #17988961	C₂	semidirect
7. #17988960	C₂	semidirect
8. C₄×#11924011	C₂	semidirect
9. #494×C₇₃	C₂	semidirect
46. #11924011	D₄	semidirect
89. C₄×C₇₃	#137	semidirect
99. C₇₃	#3152	semidirect
105. #494	D₇₃	semidirect
191. C₄	#14556623	semidirect

H ≅ G/N

type

1. C₂×#14556623

C₂

semidirect

3. C₄⋊Q₈×D₇₃

C₂

semidirect

4. #17988980

C₂

semidirect

5. #17988966

C₂

semidirect

6. #17988961

C₂

semidirect

7. #17988960

C₂

semidirect

8. C₄×#11924011

C₂

semidirect

9. #494×C₇₃

C₂

semidirect

46. #11924011

D₄

semidirect

89. C₄×C₇₃

#137

semidirect

99. C₇₃

#3152

semidirect

105. #494

D₇₃

semidirect

191. C₄

#14556623

semidirect

Subgroups — #22620389

Conjugacy classes of subgroups

Complements