Group #22621402 — subgroups

Subgroups — #22621402

Order 9344.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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	#17989208	C₂	1	2	yes	maximal, complemented
2	#17988878	C₂	1	2	yes	maximal, complemented
3	#17988873	C₂	1	2	yes	maximal, complemented
4	#17988845	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988840	C₂	1	1	yes	maximal, characteristic
6	#17988816	C₂	1	2	yes	maximal, complemented
7	#17988810	C₂	1	2	yes	maximal, complemented
8	#17988803	C₂	1	2	yes	maximal, complemented
9	#424×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₂₉₂	C₂²	1	1	yes	characteristic
11	C₂×#11924017	C₂²	1	2	yes	—
12	C₂×D₅₈₄	C₂²	1	1	yes	characteristic
13	#14556591	C₂²	1	2	yes	—
14	#14556586	—	2	2	no	—
15	#14556579	C₂²	1	2	yes	—
16	#14556577	—	2	2	no	—
17	#14556573	—	2	2	no	—
18	#14556570	C₂²	1	2	yes	—
19	#14556569	—	2	2	no	—
20	#14556565	C₂²	1	2	yes	—
21	C₄×Dic₁₄₆	C₂²	1	2	yes	—
22	C₄×D₂₉₂	C₂²	1	2	yes	—
23	#14556523	C₂²	1	1	yes	characteristic
24	#14556523	C₂²	1	2	yes	—
25	#14556523	C₂²	1	1	yes	characteristic
26	#14556516	C₂²	1	2	yes	—
27	#14556516	C₂²	1	2	yes	—
28	#14556513	C₂²	1	2	yes	—
29	#14556513	C₂²	1	2	yes	—
30	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
31	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
32	#117×C₇₃	C₂²	1	1	yes	characteristic
33	C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
34	C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
35	C₂×Dic₁₄₆	—	2	2	no	—
36	C₂×Dic₁₄₆	C₂³	1	2	yes	—
37	C₂×D₂₉₂	C₂³	1	2	yes	—
38	C₂×C₂²⋊D₇₃	—	2	2	no	—
39	C₂×C₄×D₇₃	—	2	2	no	—
40	Dic₂₉₂	—	4	1	no	—
41	#11924017	—	4	2	no	—
42	D₅₈₄	—	4	1	no	—
43	C₄⋊Dic₇₃	—	2	2	no	—
44	C₄⋊Dic₇₃	C₂³	1	2	yes	—
45	C₄⋊Dic₇₃	C₂³	1	2	yes	—
46	#11924007	—	2	2	no	—
47	#11924007	—	2	2	no	—
48	C₂²⋊Dic₇₃	—	2	2	no	—
49	#11924005	—	2	2	no	—
50	#11924005	—	2	2	no	—
51	#11924005	—	2	2	no	—
52	#11924005	—	2	2	no	—
53	C₄×Dic₇₃	—	2	2	no	—
54	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
55	M₁₆×C₇₃	—	4	1	no	—
56	C₂×C₈×C₇₃	—	4	1	no	—
57	C₂×C₈×C₇₃	C₂³	1	2	yes	—
58	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₇₃	D₄	1	2	yes	—
61	C₂²⋊C₄×C₇₃	D₄	1	1	yes	characteristic
62	C₂²⋊C₄×C₇₃	D₄	1	1	yes	characteristic
63	C₄²×C₇₃	C₂³	1	2	yes	—
64	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×D₇₃	—	2	2	no	—
66	C₂×C₄×C₇₃	—	2	1	no	—
67	C₂×C₄×C₇₃	—	2	1	no	—
68	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
72	C₂²⋊D₇₃	—	8	2	no	—
73	C₂×Dic₇₃	—	2	2	no	—
74	C₂×Dic₇₃	—	2	2	no	—
75	C₂×Dic₇₃	—	2	2	no	—
76	D₂₉₂	—	2	2	no	—
77	D₂₉₂	—	2	2	no	—
78	C₄×D₇₃	—	8	2	no	—
79	Dic₁₄₆	—	2	2	no	—
80	Dic₁₄₆	—	2	2	no	—
81	Dic₁₄₆	—	8	2	no	—
82	C₈×C₇₃	—	2	2	no	—
83	C₈×C₇₃	—	2	1	no	—
84	C₈×C₇₃	—	2	1	no	—
85	C₂²×C₇₃	—	2	1	no	—
86	C₂²×C₇₃	—	2	2	no	—
87	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
88	C₂×D₇₃	—	4	2	no	—
89	C₂×D₇₃	—	4	2	no	—
90	C₂×D₇₃	—	4	2	no	—
91	C₄×C₇₃	—	2	1	no	—
92	C₄×C₇₃	—	2	2	no	—
93	C₄×C₇₃	—	2	2	no	—
94	C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
95	C₄×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
96	Dic₇₃	—	4	2	no	—
97	Dic₇₃	—	4	2	no	—
98	Dic₇₃	—	4	2	no	—
99	C₂×C₇₃	—	4	1	no	—
100	C₂×C₇₃	#537	1	1	yes	characteristic
101	C₂×C₇₃	#577	1	2	yes	—
102	D₇₃	—	8	2	no	—
103	#3263	—	73	1	no	maximal, Sylow
104	C₇₃	#3263	1	1	yes	characteristic, complemented, Sylow
105	#542	—	73	2	no	—
106	#476	—	73	2	no	—
107	#475	—	73	2	no	—
108	#470	—	73	1	no	—
109	#469	—	73	1	no	—
110	#465	—	73	2	no	—
111	#464	—	73	2	no	—
112	#463	—	73	2	no	—
113	#424	D₇₃	1	1	yes	characteristic, complemented
114	C₂×Q₁₆	—	73	1	no	—
115	C₂×SD₁₆	—	73	2	no	—
116	C₂×D₈	—	73	1	no	—
117	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
118	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
119	C₄⋊Q₈	—	73	2	no	—
120	#126	—	146	2	no	—
121	#124	—	73	2	no	—
122	#124	—	146	2	no	—
123	#123	—	146	2	no	—
124	C₂²⋊Q₈	—	73	2	no	—
125	C₂²⋊Q₈	—	146	2	no	—
126	#121	—	73	2	no	—
127	C₄×Q₈	—	73	2	no	—
128	C₄×D₄	—	73	2	no	—
129	#117	C₂×D₇₃	1	1	yes	characteristic
130	#106	—	73	2	no	—
131	#106	—	73	1	no	—
132	#106	—	73	1	no	—
133	C₄⋊C₈	C₂×D₇₃	1	2	yes	—
134	C₄⋊C₈	C₂×D₇₃	1	2	yes	—
135	#103	—	73	2	no	—
136	#103	—	73	2	no	—
137	#102	—	73	2	no	—
138	#102	—	73	2	no	—
139	C₂×Q₈	—	73	2	no	—
140	C₂×Q₈	—	146	2	no	—
141	C₂×D₄	—	73	2	no	—
142	C₂×D₄	—	146	2	no	—
143	C₂²×C₄	—	146	2	no	—
144	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
145	Q₁₆	—	292	1	no	—
146	SD₁₆	—	292	2	no	—
147	D₈	—	292	1	no	—
148	M₁₆	—	4	1	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
151	C₂×C₈	C₂²×D₇₃	1	2	yes	—
152	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
153	C₄⋊C₄	—	73	2	no	—
154	C₄⋊C₄	—	73	2	no	—
155	C₄⋊C₄	—	146	2	no	—
156	C₄⋊C₄	—	146	2	no	—
157	C₄⋊C₄	—	146	2	no	—
158	C₄⋊C₄	D₂₉₂	1	2	yes	—
159	C₂²⋊C₄	—	146	2	no	—
160	C₂²⋊C₄	—	146	2	no	—
161	C₂²⋊C₄	—	146	2	no	—
162	C₂²⋊C₄	—	146	2	no	—
163	C₂²⋊C₄	—	146	2	no	—
164	C₂²⋊C₄	D₂₉₂	1	1	yes	characteristic
165	C₂²⋊C₄	D₂₉₂	1	1	yes	characteristic
166	C₄²	—	146	2	no	—
167	C₄²	C₂²×D₇₃	1	2	yes	—
168	C₂³	—	146	2	no	—
169	C₂³	C₂×D₂₉₂	1	1	yes	characteristic
170	Q₈	—	584	2	no	—
171	Q₈	—	146	2	no	—
172	Q₈	—	146	2	no	—
173	D₄	—	584	2	no	—
174	D₄	—	146	2	no	—
175	D₄	—	146	2	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	584	2	no	—
179	C₂×C₄	—	146	2	no	—
180	C₂×C₄	—	146	2	no	—
181	C₂×C₄	—	146	2	no	—
182	C₂×C₄	C₂×D₂₉₂	1	2	yes	—
183	C₂×C₄	C₂×D₂₉₂	1	2	yes	—
184	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
185	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
186	C₈	—	2	2	no	—
187	C₈	—	2	1	no	—
188	C₈	—	2	1	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	292	2	no	—
192	C₂²	—	292	2	no	—
193	C₂²	—	292	2	no	—
194	C₂²	C₂²×D₂₉₂	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	1	no	—
198	C₄	—	292	2	no	—
199	C₄	—	292	2	no	—
200	C₄	—	292	2	no	—
201	C₄	#14556636	1	1	yes	characteristic
202	C₄	#14556639	1	1	yes	characteristic
203	C₂	—	4	1	no	—
204	C₂	—	584	2	no	—
205	C₂	#17989168	1	1	yes	characteristic
206	C₂	#17989399	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17989208

C₂

yes

maximal, complemented

#17988878

C₂

yes

maximal, complemented

#17988873

C₂

yes

maximal, complemented

#17988845

C₂

yes

maximal, characteristic, complemented

#17988840

C₂

yes

maximal, characteristic

#17988816

C₂

yes

maximal, complemented

#17988810

C₂

yes

maximal, complemented

#17988803

C₂

yes

maximal, complemented

#424×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₉₂

C₂²

yes

characteristic

C₂×#11924017

C₂²

yes

—

C₂×D₅₈₄

C₂²

yes

characteristic

#14556591

C₂²

yes

—

#14556586

—

#14556579

C₂²

yes

—

#14556577

—

#14556573

—

#14556570

C₂²

yes

—

#14556569

—

#14556565

C₂²

yes

—

C₄×Dic₁₄₆

C₂²

yes

—

C₄×D₂₉₂

C₂²

yes

—

#14556523

C₂²

yes

characteristic

#14556523

C₂²

yes

—

#14556523

C₂²

yes

characteristic

#14556516

C₂²

yes

—

#14556516

C₂²

yes

—

#14556513

C₂²

yes

—

#14556513

C₂²

yes

—

C₂×M₁₆×C₇₃

C₂²

yes

characteristic

C₂²×C₈×C₇₃

C₂²

yes

characteristic

#117×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

—

C₄⋊C₈×C₇₃

C₂²

yes

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

C₂³

yes

—

C₂×D₂₉₂

C₂³

yes

—

C₂×C₂²⋊D₇₃

—

C₂×C₄×D₇₃

—

Dic₂₉₂

—

#11924017

—

D₅₈₄

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂³

yes

—

C₄⋊Dic₇₃

C₂³

yes

—

#11924007

—

#11924007

—

C₂²⋊Dic₇₃

—

—

—

—

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂³

yes

—

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

D₄

yes

—

C₂²⋊C₄×C₇₃

D₄

yes

characteristic

C₂²⋊C₄×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

C₂³

yes

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂²×D₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂⁴

yes

characteristic

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

H(5,𝔽₂)

yes

characteristic

C₄×C₇₃

2₋¹⁺⁴

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

100

C₂×C₇₃

#537

yes

characteristic

101

C₂×C₇₃

#577

yes

—

102

D₇₃

—

103

#3263

—

maximal, Sylow

104

C₇₃

#3263

yes

characteristic, complemented, Sylow

105

#542

—

106

#476

—

107

#475

—

108

#470

—

109

#469

—

110

#465

—

111

#464

—

112

#463

—

113

#424

D₇₃

yes

characteristic, complemented

114

C₂×Q₁₆

—

115

C₂×SD₁₆

—

116

C₂×D₈

—

117

C₂×M₁₆

C₂×D₇₃

yes

characteristic

118

C₂²×C₈

C₂×D₇₃

yes

characteristic

119

C₄⋊Q₈

—

120

#126

—

146

—

121

#124

—

122

#124

—

146

—

123

#123

—

146

—

124

C₂²⋊Q₈

—

125

C₂²⋊Q₈

—

146

—

126

#121

—

127

C₄×Q₈

—

128

C₄×D₄

—

129

#117

C₂×D₇₃

yes

characteristic

130

#106

—

131

#106

—

132

#106

—

133

C₄⋊C₈

C₂×D₇₃

yes

—

134

C₄⋊C₈

C₂×D₇₃

yes

—

135

#103

—

136

#103

—

137

#102

—

138

#102

—

139

C₂×Q₈

—

140

C₂×Q₈

—

146

—

141

C₂×D₄

—

142

C₂×D₄

—

146

—

143

C₂²×C₄

—

146

—

144

C₂²×C₄

C₂²×D₇₃

yes

characteristic

145

Q₁₆

—

292

—

146

SD₁₆

—

292

—

147

D₈

—

292

—

148

M₁₆

—

149

C₂×C₈

—

150

C₂×C₈

C₂²×D₇₃

yes

characteristic

151

C₂×C₈

C₂²×D₇₃

yes

—

152

C₂×C₈

C₂²×D₇₃

yes

characteristic

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

146

—

156

C₄⋊C₄

—

146

—

157

C₄⋊C₄

—

146

—

158

C₄⋊C₄

D₂₉₂

yes

—

159

C₂²⋊C₄

—

146

—

160

C₂²⋊C₄

—

146

—

161

C₂²⋊C₄

—

146

—

162

C₂²⋊C₄

—

146

—

163

C₂²⋊C₄

—

146

—

164

C₂²⋊C₄

D₂₉₂

yes

characteristic

165

C₂²⋊C₄

D₂₉₂

yes

characteristic

166

C₄²

—

146

—

167

C₄²

C₂²×D₇₃

yes

—

168

C₂³

—

146

—

169

C₂³

C₂×D₂₉₂

yes

characteristic

170

Q₈

—

584

—

171

Q₈

—

146

—

172

Q₈

—

146

—

173

D₄

—

584

—

174

D₄

—

146

—

175

D₄

—

146

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

584

—

179

C₂×C₄

—

146

—

180

C₂×C₄

—

146

—

181

C₂×C₄

—

146

—

182

C₂×C₄

C₂×D₂₉₂

yes

—

183

C₂×C₄

C₂×D₂₉₂

yes

—

184

C₂×C₄

C₂×D₂₉₂

yes

characteristic

185

C₂×C₄

C₂³×D₇₃

yes

characteristic

186

C₈

—

187

C₈

—

188

C₈

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

292

—

192

C₂²

—

292

—

193

C₂²

—

292

—

194

C₂²

C₂²×D₂₉₂

yes

characteristic

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

292

—

199

C₄

—

292

—

200

C₄

—

292

—

201

C₄

#14556636

yes

characteristic

202

C₄

#14556639

yes

characteristic

203

C₂

—

204

C₂

—

584

—

205

C₂

#17989168

yes

characteristic

206

C₂

#17989399

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₃
Fitting	#424×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₄
hypercenter	#424	D₇₃
hypocenter	C₇₃	#3263
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #424

subgroup

G/H

center

C₂²

C₂²×D₂₉₂

commutator

C₂×C₄×C₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₃

Fitting

#424×C₇₃

C₂

socle

C₂²×C₇₃

C₂²×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ #117 ◁ #424

N	H ≅ G/N	type
1. #17989208	C₂	semidirect
2. #17988878	C₂	semidirect
3. #17988873	C₂	semidirect
4. #17988845	C₂	semidirect
6. #17988816	C₂	semidirect
7. #17988810	C₂	semidirect
8. #17988803	C₂	semidirect
9. #424×C₇₃	C₂	semidirect
104. C₇₃	#3263	semidirect
113. #424	D₇₃	semidirect

H ≅ G/N

type

1. #17989208

C₂

semidirect

2. #17988878

C₂

semidirect

3. #17988873

C₂

semidirect

4. #17988845

C₂

semidirect

6. #17988816

C₂

semidirect

7. #17988810

C₂

semidirect

8. #17988803

C₂

semidirect

9. #424×C₇₃

C₂

semidirect

104. C₇₃

#3263

semidirect

113. #424

D₇₃

semidirect

Subgroups — #22621402

Conjugacy classes of subgroups

Complements