Group #22611429 — subgroups

Subgroups — #22611429

Order 9344.

224 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 89 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	#17988485	C₂	1	1	yes	maximal, characteristic
3	#17988485	C₂	1	1	yes	maximal, characteristic
4	#17988350	C₂	1	1	yes	maximal, characteristic
5	#17988350	C₂	1	1	yes	maximal, characteristic
6	#17988350	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Dic₇₃	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×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
12	C₂×#11924007	—	2	1	no	—
13	C₂×#11924007	—	2	1	no	—
14	#14556523	—	2	2	no	—
15	#14556498	—	2	1	no	—
16	#14556498	—	2	1	no	—
17	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₇₃	—	2	2	no	—
21	C₂²×Dic₇₃	—	2	1	no	—
22	C₂²×Dic₇₃	—	2	1	no	—
23	C₂²×Dic₇₃	—	2	1	no	—
24	C₂²×Dic₇₃	—	2	1	no	—
25	C₄⋊Dic₇₃	—	2	1	no	—
26	C₄⋊Dic₇₃	—	2	1	no	—
27	C₄⋊Dic₇₃	—	2	1	no	—
28	C₄⋊Dic₇₃	—	2	1	no	—
29	C₄⋊Dic₇₃	—	2	1	no	—
30	C₄⋊Dic₇₃	—	2	1	no	—
31	C₄⋊Dic₇₃	Q₈	1	2	yes	—
32	C₄⋊Dic₇₃	Q₈	1	2	yes	—
33	#11924007	—	4	2	no	—
34	#11924007	—	4	2	no	—
35	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
38	C₂×C₈×C₇₃	—	2	1	no	—
39	C₂×C₈×C₇₃	—	2	1	no	—
40	C₂×C₈×C₇₃	—	2	1	no	—
41	C₂×C₈×C₇₃	—	2	1	no	—
42	C₂×C₈×C₇₃	—	2	1	no	—
43	C₂×C₈×C₇₃	D₄	1	2	yes	—
44	C₄²×C₇₃	—	2	1	no	—
45	C₄²×C₇₃	—	2	1	no	—
46	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₇₃	—	2	1	no	—
48	C₂×C₄×C₇₃	—	2	1	no	—
49	C₂×C₄×C₇₃	—	2	1	no	—
50	C₂×C₄×C₇₃	—	2	1	no	—
51	C₂×C₄×C₇₃	—	2	1	no	—
52	C₂×C₄×C₇₃	SD₁₆	1	2	yes	—
53	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
57	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
58	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
59	C₂×Dic₇₃	—	2	2	no	—
60	C₂×Dic₇₃	—	2	2	no	—
61	C₂×Dic₇₃	—	4	1	no	—
62	C₂×Dic₇₃	—	4	1	no	—
63	C₂×Dic₇₃	—	4	1	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	—	4	1	no	—
70	C₂×Dic₇₃	—	4	1	no	—
71	C₈×C₇₃	—	2	2	no	—
72	C₈×C₇₃	—	4	1	no	—
73	C₂²×C₇₃	#121	1	1	yes	characteristic
74	C₂²×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂²×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂²×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
77	C₂²×C₇₃	#135	1	1	yes	characteristic
78	C₂²×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂²×C₇₃	#137	1	1	yes	characteristic
80	C₄×C₇₃	—	2	2	no	—
81	C₄×C₇₃	—	4	1	no	—
82	C₄×C₇₃	#123	1	1	yes	characteristic
83	C₄×C₇₃	#125	1	1	yes	characteristic
84	C₄×C₇₃	#125	1	1	yes	characteristic
85	C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
86	Dic₇₃	—	4	2	no	—
87	Dic₇₃	—	4	2	no	—
88	Dic₇₃	—	8	1	no	—
89	Dic₇₃	—	8	1	no	—
90	C₂×C₇₃	#398	1	1	yes	characteristic
91	C₂×C₇₃	#465	1	1	yes	characteristic
92	C₂×C₇₃	#468	1	1	yes	characteristic
93	C₂×C₇₃	#475	1	1	yes	characteristic
94	C₂×C₇₃	#476	1	1	yes	characteristic
95	C₂×C₇₃	#478	1	1	yes	characteristic
96	C₂×C₇₃	#479	1	1	yes	characteristic
97	#2088	—	73	1	no	maximal, Sylow
98	C₇₃	#2088	1	1	yes	characteristic, complemented, Sylow
99	C₂×#106	—	73	1	no	—
100	C₂×C₄⋊C₈	D₇₃	1	1	yes	characteristic
101	#389	—	73	1	no	—
102	#389	—	73	1	no	—
103	#340	—	73	1	no	—
104	#340	—	73	1	no	—
105	#340	—	73	1	no	—
106	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
107	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
108	C₂×C₄⋊C₄	—	73	1	no	—
109	C₂×C₄⋊C₄	—	73	1	no	—
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×C₄²	C₂×D₇₃	1	1	yes	characteristic
115	#106	—	146	2	no	—
116	C₄⋊C₈	—	2	2	no	—
117	#95	—	146	1	no	—
118	#95	—	146	1	no	—
119	C₂²×C₄	—	146	1	no	—
120	C₂²×C₄	—	146	1	no	—
121	C₂²×C₄	—	146	1	no	—
122	C₂²×C₄	—	146	1	no	—
123	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
124	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	C₂²⋊D₇₃	1	2	yes	—
132	C₄⋊C₄	—	292	2	no	—
133	C₄⋊C₄	—	292	2	no	—
134	C₄⋊C₄	—	73	2	no	—
135	C₄⋊C₄	—	73	2	no	—
136	C₄⋊C₄	—	146	1	no	—
137	C₄⋊C₄	—	146	1	no	—
138	C₄⋊C₄	—	146	1	no	—
139	C₄⋊C₄	—	146	1	no	—
140	C₄⋊C₄	—	146	1	no	—
141	C₄⋊C₄	—	146	1	no	—
142	C₄²	—	2	1	no	—
143	C₄²	—	2	1	no	—
144	C₂³	C₂×D₂₉₂	1	1	yes	characteristic
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	292	1	no	—
151	C₂×C₄	—	292	1	no	—
152	C₂×C₄	—	292	1	no	—
153	C₂×C₄	—	292	1	no	—
154	C₂×C₄	—	292	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	—	146	2	no	—
161	C₂×C₄	—	146	2	no	—
162	C₂×C₄	#11924017	1	2	yes	—
163	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
164	C₂×C₄	Q₈×D₇₃	1	1	yes	characteristic
165	C₂×C₄	Q₈×D₇₃	1	1	yes	characteristic
166	C₂×C₄	#11924027	1	1	yes	characteristic
167	C₂×C₄	#11924027	1	1	yes	characteristic
168	C₂×C₄	#11924028	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	—	4	1	no	—
171	C₂²	#14556565	1	1	yes	characteristic
172	C₂²	#14556571	1	1	yes	characteristic
173	C₂²	#14556571	1	1	yes	characteristic
174	C₂²	C₂×#11924017	1	1	yes	characteristic
175	C₂²	#14556613	1	1	yes	characteristic
176	C₂²	#14556619	1	1	yes	characteristic
177	C₂²	#14556624	1	1	yes	characteristic
178	C₄	—	2	2	no	—
179	C₄	—	4	1	no	—
180	C₄	—	292	2	no	—
181	C₄	—	292	2	no	—
182	C₄	—	584	1	no	—
183	C₄	—	584	1	no	—
184	C₄	#14556572	1	1	yes	characteristic
185	C₄	#14556581	1	1	yes	characteristic
186	C₄	#14556581	1	1	yes	characteristic
187	C₄	#14556589	1	1	yes	characteristic
188	C₂	#17988517	1	1	yes	characteristic
189	C₂	#17988816	1	1	yes	characteristic
190	C₂	#17988833	1	1	yes	characteristic
191	C₂	#17988873	1	1	yes	characteristic
192	C₂	#17988878	1	1	yes	characteristic
193	C₂	#17988888	1	1	yes	characteristic
194	C₂	#17988895	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556523

C₂

yes

maximal, characteristic

#17988485

C₂

yes

maximal, characteristic

#17988485

C₂

yes

maximal, characteristic

#17988350

C₂

yes

maximal, characteristic

#17988350

C₂

yes

maximal, characteristic

#17988350

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₇₃

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

—

—

—

—

—

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

Q₈

yes

—

C₄⋊Dic₇₃

Q₈

yes

—

#11924007

—

#11924007

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

SD₁₆

yes

—

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×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

—

C₈×C₇₃

—

C₈×C₇₃

—

C₂²×C₇₃

#121

yes

characteristic

C₂²×C₇₃

C₂²⋊Q₈

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

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

C₄⋊Q₈

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×#106

—

100

C₂×C₄⋊C₈

D₇₃

yes

characteristic

101

#389

—

102

#389

—

103

#340

—

104

#340

—

105

#340

—

106

C₂²×C₈

C₂×D₇₃

yes

characteristic

107

C₂²×C₈

C₂×D₇₃

yes

characteristic

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

146

—

113

C₂×C₄⋊C₄

—

146

—

114

C₂×C₄²

C₂×D₇₃

yes

characteristic

115

#106

—

146

—

116

C₄⋊C₈

—

117

#95

—

146

—

118

#95

—

146

—

119

C₂²×C₄

—

146

—

120

C₂²×C₄

—

146

—

121

C₂²×C₄

—

146

—

122

C₂²×C₄

—

146

—

123

C₂²×C₄

D₂₉₂

yes

characteristic

124

C₂²×C₄

D₂₉₂

yes

characteristic

125

C₂²×C₄

C₂²×D₇₃

yes

characteristic

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

C₂²⋊D₇₃

yes

—

132

C₄⋊C₄

—

292

—

133

C₄⋊C₄

—

292

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

146

—

137

C₄⋊C₄

—

146

—

138

C₄⋊C₄

—

146

—

139

C₄⋊C₄

—

146

—

140

C₄⋊C₄

—

146

—

141

C₄⋊C₄

—

146

—

142

C₄²

—

143

C₄²

—

144

C₂³

C₂×D₂₉₂

yes

characteristic

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

292

—

151

C₂×C₄

—

292

—

152

C₂×C₄

—

292

—

153

C₂×C₄

—

292

—

154

C₂×C₄

—

292

—

155

C₂×C₄

—

292

—

156

C₂×C₄

—

292

—

157

C₂×C₄

—

292

—

158

C₂×C₄

—

292

—

159

C₂×C₄

—

292

—

160

C₂×C₄

—

146

—

161

C₂×C₄

—

146

—

162

C₂×C₄

#11924017

yes

—

163

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

164

C₂×C₄

Q₈×D₇₃

yes

characteristic

165

C₂×C₄

Q₈×D₇₃

yes

characteristic

166

C₂×C₄

#11924027

yes

characteristic

167

C₂×C₄

#11924027

yes

characteristic

168

C₂×C₄

#11924028

yes

characteristic

169

C₈

—

170

C₈

—

171

C₂²

#14556565

yes

characteristic

172

C₂²

#14556571

yes

characteristic

173

C₂²

#14556571

yes

characteristic

174

C₂²

C₂×#11924017

yes

characteristic

175

C₂²

#14556613

yes

characteristic

176

C₂²

#14556619

yes

characteristic

177

C₂²

#14556624

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

292

—

181

C₄

—

292

—

182

C₄

—

584

—

183

C₄

—

584

—

184

C₄

#14556572

yes

characteristic

185

C₄

#14556581

yes

characteristic

186

C₄

#14556581

yes

characteristic

187

C₄

#14556589

yes

characteristic

188

C₂

#17988517

yes

characteristic

189

C₂

#17988816

yes

characteristic

190

C₂

#17988833

yes

characteristic

191

C₂

#17988873

yes

characteristic

192

C₂

#17988878

yes

characteristic

193

C₂

#17988888

yes

characteristic

194

C₂

#17988895

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₉₂
commutator	C₂²×C₄×C₇₃	C₂³
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	C₂×C₄⋊C₈×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₇₃
hypocenter	C₇₃	#2088
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₂₉₂

commutator

C₂²×C₄×C₇₃

C₂³

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

C₂×C₄⋊C₈×C₇₃

C₂

socle

C₂³×C₇₃

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₇₃

hypocenter

C₇₃

#2088

LCS

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
98. C₇₃	#2088	semidirect

H ≅ G/N

type

98. C₇₃

#2088

semidirect

Subgroups — #22611429

Conjugacy classes of subgroups

Complements