Group #22625269 — subgroups

Subgroups — #22625269

Order 9344.

266 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 70 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	#17989591	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#11924038	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈⋊#10596833	C₂	1	2	yes	maximal, complemented
4	#17989435	C₂	1	2	yes	maximal, complemented
5	C₄×#11924029	C₂	1	1	yes	maximal, characteristic
6	C₂×#11924038	—	2	1	no	—
7	C₂×#11924038	—	2	1	no	—
8	C₄⋊#10596833	—	2	1	no	—
9	C₄⋊#10596833	C₂²	1	2	yes	—
10	#14556646	—	2	1	no	—
11	C₄×#10596833	C₂²	1	2	yes	—
12	C₂×#11924029	C₄	1	1	yes	characteristic
13	Q₈×Dic₇₃	C₂²	1	1	yes	characteristic
14	C₄×D₂₉₂	—	2	1	no	—
15	C₄×D₂₉₂	C₄	1	1	yes	characteristic
16	#14556553	—	2	1	no	—
17	#14556553	C₂²	1	1	yes	characteristic
18	#14556549	—	2	1	no	—
19	#14556549	C₄	1	1	yes	characteristic
20	C₄²×D₇₃	—	2	1	no	—
21	C₄²×D₇₃	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₇₃	C₄	1	1	yes	characteristic
23	#11924038	—	2	4	no	—
24	#11924038	—	4	2	no	—
25	C₂×#10596833	—	2	2	no	—
26	C₂×#10596833	—	2	2	no	—
27	#11924029	—	2	2	no	—
28	#11924029	—	2	2	no	—
29	C₂×D₂₉₂	—	2	1	no	—
30	C₂×D₂₉₂	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₇₃	—	2	1	no	—
32	C₂×C₄×D₇₃	—	2	1	no	—
33	C₂×C₄×D₇₃	—	2	1	no	—
34	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
36	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
37	C₄⋊Dic₇₃	—	2	1	no	—
38	C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
39	#11924005	—	2	1	no	—
40	#11924005	—	2	1	no	—
41	#11924005	—	2	1	no	—
42	C₄×Dic₇₃	—	2	1	no	—
43	C₄×Dic₇₃	—	2	1	no	—
44	C₄×Dic₇₃	D₄	1	1	yes	characteristic
45	C₄×Dic₇₃	D₄	1	1	yes	characteristic
46	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
47	C₂×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
48	C₄⋊C₄×C₇₃	—	2	1	no	—
49	C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
50	C₄²×C₇₃	—	2	1	no	—
51	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
52	C₂²×D₇₃	—	2	1	no	—
53	C₂²×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
54	Q₈×C₇₃	—	2	1	no	—
55	Q₈×C₇₃	M₁₆	1	2	yes	complemented
56	C₂×C₄×C₇₃	—	2	1	no	—
57	C₂×C₄×C₇₃	—	2	1	no	—
58	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×Dic₇₃	—	2	1	no	—
62	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
64	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
65	D₂₉₂	—	2	4	no	—
66	D₂₉₂	—	2	1	no	—
67	D₂₉₂	M₁₆	1	2	yes	complemented
68	C₄×D₇₃	—	2	4	no	—
69	C₄×D₇₃	—	2	2	no	—
70	C₄×D₇₃	—	2	1	no	—
71	C₄×D₇₃	—	2	1	no	—
72	C₄×D₇₃	—	2	1	no	—
73	C₄×D₇₃	—	2	1	no	—
74	C₄×D₇₃	—	4	2	no	—
75	C₄×D₇₃	—	4	1	no	—
76	C₄×D₇₃	—	4	1	no	—
77	#10596833	—	2	4	no	—
78	#10596833	—	4	2	no	—
79	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×D₇₃	—	2	1	no	—
81	C₂×D₇₃	—	2	2	no	—
82	C₂×D₇₃	—	2	1	no	—
83	C₂×D₇₃	—	2	1	no	—
84	C₂×D₇₃	—	4	1	no	—
85	C₂×D₇₃	—	4	1	no	—
86	C₄×C₇₃	—	2	2	no	—
87	C₄×C₇₃	—	2	1	no	—
88	C₄×C₇₃	—	4	1	no	—
89	C₄×C₇₃	#104	1	2	yes	—
90	C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
91	C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
92	Dic₇₃	—	2	1	no	—
93	Dic₇₃	—	2	1	no	—
94	Dic₇₃	—	4	1	no	—
95	Dic₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
96	Dic₇₃	#137	1	1	yes	characteristic
97	C₂×C₇₃	#407	1	1	yes	characteristic
98	C₂×C₇₃	#417	1	1	yes	characteristic
99	C₂×C₇₃	C₂×#104	1	1	yes	characteristic
100	D₇₃	—	4	1	no	—
101	D₇₃	—	4	2	no	—
102	#1489	—	73	1	no	maximal, Sylow
103	C₇₃	#1489	1	1	yes	characteristic, complemented, Sylow
104	C₄×#41	—	73	1	no	—
105	#423	—	73	1	no	—
106	C₄×M₁₆	—	73	1	no	—
107	Q₈⋊C₈	—	73	2	no	—
108	D₄⋊C₈	—	73	2	no	—
109	C₂×#41	—	73	1	no	—
110	C₂×M₁₆	—	146	1	no	—
111	C₂×M₁₆	—	146	1	no	—
112	C₄×Q₈	—	73	1	no	—
113	C₄×Q₈	#64408	1	1	yes	characteristic
114	C₄×D₄	—	73	1	no	—
115	C₄×D₄	—	73	1	no	—
116	C₄×D₄	—	146	1	no	—
117	C₄×D₄	—	146	1	no	—
118	#117	—	73	1	no	—
119	#117	—	146	1	no	—
120	C₂×C₄²	—	73	1	no	—
121	C₂×C₄²	—	146	1	no	—
122	C₄⋊C₈	—	73	2	no	—
123	C₄⋊C₈	—	146	1	no	—
124	#97	—	146	1	no	—
125	C₄×C₈	—	73	2	no	—
126	#41	—	146	2	no	—
127	#41	—	146	2	no	—
128	C₂×Q₈	C₂×#64408	1	1	yes	characteristic
129	C₂×D₄	—	73	1	no	—
130	C₂×D₄	—	146	1	no	—
131	C₂²×C₄	—	73	1	no	—
132	C₂²×C₄	—	73	1	no	—
133	C₂²×C₄	—	73	1	no	—
134	C₂²×C₄	—	146	1	no	—
135	C₂²×C₄	—	146	1	no	—
136	C₂²×C₄	—	146	1	no	—
137	M₁₆	—	292	2	no	—
138	M₁₆	—	146	4	no	—
139	C₂×C₈	—	146	2	no	—
140	C₂×C₈	—	146	2	no	—
141	C₄⋊C₄	—	2	1	no	—
142	C₄⋊C₄	—	73	1	no	—
143	C₄⋊C₄	—	146	1	no	—
144	C₄⋊C₄	C₂×#64408	1	1	yes	characteristic
145	C₂²⋊C₄	—	146	1	no	—
146	C₂²⋊C₄	—	146	1	no	—
147	C₂²⋊C₄	—	146	1	no	—
148	C₄²	—	2	1	no	—
149	C₄²	—	73	1	no	—
150	C₄²	—	73	1	no	—
151	C₄²	—	73	1	no	—
152	C₄²	—	146	1	no	—
153	C₄²	—	146	1	no	—
154	C₄²	C₂×#64408	1	1	yes	characteristic
155	C₂³	—	73	1	no	—
156	C₂³	—	146	1	no	—
157	Q₈	—	2	1	no	—
158	Q₈	#11924038	1	2	yes	complemented
159	D₄	—	73	2	no	—
160	D₄	—	146	4	no	—
161	D₄	—	146	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	292	1	no	—
165	C₂×C₄	—	292	1	no	—
166	C₂×C₄	—	292	2	no	—
167	C₂×C₄	—	73	1	no	—
168	C₂×C₄	—	73	1	no	—
169	C₂×C₄	—	73	1	no	—
170	C₂×C₄	—	146	1	no	—
171	C₂×C₄	—	146	1	no	—
172	C₂×C₄	—	146	4	no	—
173	C₂×C₄	—	146	1	no	—
174	C₂×C₄	—	146	1	no	—
175	C₂×C₄	—	146	1	no	—
176	C₂×C₄	—	146	2	no	—
177	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
178	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
179	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
180	C₈	—	292	2	no	—
181	C₈	—	146	4	no	—
182	C₂²	—	292	1	no	—
183	C₂²	—	292	1	no	—
184	C₂²	—	146	2	no	—
185	C₂²	—	146	1	no	—
186	C₂²	—	146	1	no	—
187	C₂²	—	146	1	no	—
188	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	—	2	1	no	—
191	C₄	—	4	1	no	—
192	C₄	—	292	1	no	—
193	C₄	—	73	1	no	—
194	C₄	—	73	1	no	—
195	C₄	—	146	1	no	—
196	C₄	—	146	1	no	—
197	C₄	#14556656	1	2	yes	—
198	C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
199	C₄	C₂×#11924038	1	1	yes	characteristic
200	C₂	—	292	2	no	—
201	C₂	—	292	1	no	—
202	C₂	#17989558	1	1	yes	characteristic
203	C₂	#17989578	1	1	yes	characteristic
204	C₂	C₂×#14556656	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989591

C₂

yes

maximal, characteristic, complemented

C₄×#11924038

C₂

yes

maximal, characteristic, complemented

Q₈⋊#10596833

C₂

yes

maximal, complemented

#17989435

C₂

yes

maximal, complemented

C₄×#11924029

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

#14556646

—

C₄×#10596833

C₂²

yes

—

C₂×#11924029

C₄

yes

characteristic

Q₈×Dic₇₃

C₂²

yes

characteristic

C₄×D₂₉₂

—

C₄×D₂₉₂

C₄

yes

characteristic

#14556553

—

#14556553

C₂²

yes

characteristic

#14556549

—

#14556549

C₄

yes

characteristic

C₄²×D₇₃

—

C₄²×D₇₃

C₂²

yes

characteristic

C₄×Q₈×C₇₃

C₄

yes

—

—

—

—

—

—

C₂×D₂₉₂

—

C₂×D₂₉₂

C₂×C₄

yes

characteristic

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

C₂×C₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

—

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

C₂³

yes

characteristic

C₂×Q₈×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂×C₄

yes

characteristic

C₄²×C₇₃

—

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

C₂²⋊C₄

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

M₁₆

yes

complemented

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

D₂₉₂

—

D₂₉₂

—

D₂₉₂

M₁₆

yes

complemented

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

#10596833

—

#10596833

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#104

yes

—

C₄×C₇₃

C₂×M₁₆

yes

characteristic

C₄×C₇₃

C₂×M₁₆

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

yes

Dic₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

100

D₇₃

—

101

D₇₃

—

102

#1489

—

maximal, Sylow

103

C₇₃

#1489

yes

characteristic, complemented, Sylow

104

C₄×#41

—

105

#423

—

106

C₄×M₁₆

—

107

Q₈⋊C₈

—

108

D₄⋊C₈

—

109

C₂×#41

—

110

C₂×M₁₆

—

146

—

111

C₂×M₁₆

—

146

—

112

C₄×Q₈

—

113

C₄×Q₈

#64408

yes

characteristic

114

C₄×D₄

—

115

C₄×D₄

—

116

C₄×D₄

—

146

—

117

C₄×D₄

—

146

—

118

#117

—

119

#117

—

146

—

120

C₂×C₄²

—

121

C₂×C₄²

—

146

—

122

C₄⋊C₈

—

123

C₄⋊C₈

—

146

—

124

#97

—

146

—

125

C₄×C₈

—

126

#41

—

146

—

127

#41

—

146

—

128

C₂×Q₈

C₂×#64408

yes

characteristic

129

C₂×D₄

—

130

C₂×D₄

—

146

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

—

146

—

135

C₂²×C₄

—

146

—

136

C₂²×C₄

—

146

—

137

M₁₆

—

292

—

138

M₁₆

—

146

—

139

C₂×C₈

—

146

—

140

C₂×C₈

—

146

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

146

—

144

C₄⋊C₄

C₂×#64408

yes

characteristic

145

C₂²⋊C₄

—

146

—

146

C₂²⋊C₄

—

146

—

147

C₂²⋊C₄

—

146

—

148

C₄²

—

149

C₄²

—

150

C₄²

—

151

C₄²

—

152

C₄²

—

146

—

153

C₄²

—

146

—

154

C₄²

C₂×#64408

yes

characteristic

155

C₂³

—

156

C₂³

—

146

—

157

Q₈

—

158

Q₈

#11924038

yes

complemented

159

D₄

—

160

D₄

—

146

—

161

D₄

—

146

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

292

—

165

C₂×C₄

—

292

—

166

C₂×C₄

—

292

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

146

—

171

C₂×C₄

—

146

—

172

C₂×C₄

—

146

—

173

C₂×C₄

—

146

—

174

C₂×C₄

—

146

—

175

C₂×C₄

—

146

—

176

C₂×C₄

—

146

—

177

C₂×C₄

C₂²⋊#64408

yes

characteristic

178

C₂×C₄

C₂²⋊#64408

yes

characteristic

179

C₂×C₄

C₂²×#64408

yes

characteristic

180

C₈

—

292

—

181

C₈

—

146

—

182

C₂²

—

292

—

183

C₂²

—

292

—

184

C₂²

—

146

—

185

C₂²

—

146

—

186

C₂²

—

146

—

187

C₂²

—

146

—

188

C₂²

C₂×C₂²⋊#64408

yes

characteristic

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

292

—

193

C₄

—

194

C₄

—

195

C₄

—

146

—

196

C₄

—

146

—

197

C₄

#14556656

yes

—

198

C₄

D₇₃⋊M₁₆

yes

characteristic

199

C₄

C₂×#11924038

yes

characteristic

200

C₂

—

292

—

201

C₂

—

292

—

202

C₂

#17989558

yes

characteristic

203

C₂

#17989578

yes

characteristic

204

C₂

C₂×#14556656

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#64408
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#64408
Fitting	C₄×Q₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	C₄×Q₈	#64408
hypocenter	C₇₃	#1489
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#64408

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#64408

Fitting

C₄×Q₈×C₇₃

C₄

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

C₄×Q₈

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

N	H ≅ G/N	type
1. #17989591	C₂	semidirect
2. C₄×#11924038	C₂	semidirect
3. Q₈⋊#10596833	C₂	semidirect
4. #17989435	C₂	semidirect
55. Q₈×C₇₃	M₁₆	semidirect
67. D₂₉₂	M₁₆	semidirect
103. C₇₃	#1489	semidirect
158. Q₈	#11924038	semidirect

H ≅ G/N

type

1. #17989591

C₂

semidirect

2. C₄×#11924038

C₂

semidirect

3. Q₈⋊#10596833

C₂

semidirect

4. #17989435

C₂

semidirect

55. Q₈×C₇₃

M₁₆

semidirect

67. D₂₉₂

M₁₆

semidirect

103. C₇₃

#1489

semidirect

158. Q₈

#11924038

semidirect

Subgroups — #22625269

Conjugacy classes of subgroups

Complements