Group #22623012 — subgroups

Subgroups — #22623012

Order 9344.

356 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 121 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	#17989310	C₂	1	2	yes	maximal, complemented
2	#17988981	C₂	1	2	yes	maximal, complemented
3	#17988971	C₂	1	1	yes	maximal, characteristic
4	#17988810	C₂	1	2	yes	maximal, complemented
5	#17988803	C₂	1	2	yes	maximal
6	#17988798	C₂	1	2	yes	maximal, complemented
7	#17988689	C₂	1	2	yes	maximal, complemented
8	C₄×Dic₂₉₂	C₂	1	1	yes	maximal, characteristic, complemented
9	#446×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₂₉₂	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₉₂	C₂²	1	1	yes	characteristic
12	C₂×Dic₂₉₂	C₂²	1	2	yes	—
13	C₂×#11924017	C₂²	1	2	yes	—
14	C₂×#11924017	C₂²	1	2	yes	—
15	C₂×D₅₈₄	C₂²	1	1	yes	characteristic
16	#14556591	C₂²	1	2	yes	—
17	#14556586	—	2	2	no	—
18	#14556586	—	2	2	no	—
19	#14556579	C₂²	1	2	yes	—
20	#14556579	C₂²	1	2	yes	—
21	#14556567	—	2	2	no	—
22	C₄×Dic₁₄₆	C₂²	1	2	yes	—
23	C₄×Dic₁₄₆	C₂²	1	2	yes	—
24	C₄×D₂₉₂	C₂²	1	2	yes	—
25	#14556549	—	2	2	no	—
26	#14556525	C₂²	1	1	yes	characteristic
27	#14556516	C₂²	1	2	yes	—
28	#14556516	C₂²	1	2	yes	—
29	#14556513	C₂²	1	2	yes	—
30	C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
31	C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
32	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
33	#97×C₇₃	C₂²	1	2	yes	—
34	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
35	C₂×Dic₁₄₆	C₂³	1	2	yes	—
36	C₂×Dic₁₄₆	C₂³	1	2	yes	—
37	C₂×D₂₉₂	C₂³	1	2	yes	—
38	C₂×C₄×D₇₃	—	2	2	no	—
39	Dic₂₉₂	—	2	2	no	—
40	Dic₂₉₂	—	4	1	no	—
41	Dic₂₉₂	—	4	2	no	—
42	#11924017	—	2	4	no	—
43	#11924017	—	4	2	no	—
44	D₅₈₄	—	4	1	no	—
45	C₄⋊Dic₇₃	—	2	2	no	—
46	C₄⋊Dic₇₃	—	2	2	no	—
47	C₄⋊Dic₇₃	C₂³	1	2	yes	—
48	#11924007	—	2	2	no	—
49	#11924007	—	2	2	no	—
50	#11924005	—	2	2	no	—
51	#11924005	—	2	2	no	—
52	#11924005	—	2	2	no	—
53	#11924005	—	2	2	no	—
54	#11924005	—	2	2	no	—
55	C₄×Dic₇₃	—	2	2	no	—
56	C₄×Dic₇₃	—	2	2	no	—
57	C₂×Q₈×C₇₃	D₄	1	1	yes	characteristic
58	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₇₃	C₂³	1	2	yes	—
60	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₇₃	D₄	1	2	yes	—
62	C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
63	C₄²×C₇₃	C₂³	1	1	yes	characteristic
64	C₄²×C₇₃	C₂³	1	2	yes	—
65	C₂²×D₇₃	—	2	2	no	—
66	Q₈×C₇₃	—	4	1	no	—
67	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
72	C₂×Dic₇₃	—	2	2	no	—
73	C₂×Dic₇₃	—	2	2	no	—
74	C₂×Dic₇₃	—	2	2	no	—
75	D₂₉₂	—	2	2	no	—
76	D₂₉₂	—	2	2	no	—
77	C₄×D₇₃	—	4	4	no	—
78	Dic₁₄₆	—	2	2	no	—
79	Dic₁₄₆	—	2	2	no	—
80	Dic₁₄₆	—	2	2	no	—
81	Dic₁₄₆	—	2	2	no	—
82	C₈×C₇₃	—	2	1	no	—
83	C₈×C₇₃	—	2	2	no	—
84	C₈×C₇₃	#41	1	2	yes	—
85	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
86	C₂×D₇₃	—	4	2	no	—
87	C₂×D₇₃	—	4	2	no	—
88	C₂×D₇₃	—	4	2	no	—
89	C₄×C₇₃	—	2	2	no	—
90	C₄×C₇₃	—	2	1	no	—
91	C₄×C₇₃	—	2	1	no	—
92	C₄×C₇₃	—	2	2	no	—
93	C₄×C₇₃	C₂×#41	1	1	yes	characteristic
94	C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
95	Dic₇₃	—	4	2	no	—
96	Dic₇₃	—	4	2	no	—
97	Dic₇₃	—	4	2	no	—
98	C₂×C₇₃	#550	1	1	yes	characteristic
99	C₂×C₇₃	#575	1	1	yes	characteristic
100	C₂×C₇₃	#578	1	1	yes	characteristic
101	D₇₃	—	8	2	no	—
102	#3402	—	73	1	no	maximal, Sylow
103	C₇₃	#3402	1	1	yes	characteristic, complemented, Sylow
104	#556	—	73	2	no	—
105	#497	—	73	2	no	—
106	#495	—	73	1	no	—
107	#464	—	73	2	no	—
108	#463	—	73	2	no	—
109	#462	—	73	2	no	—
110	#446	D₇₃	1	1	yes	characteristic, complemented
111	#440	—	73	2	no	—
112	C₄×Q₁₆	—	73	1	no	—
113	C₂×Q₁₆	—	73	1	no	—
114	C₂×Q₁₆	—	73	2	no	—
115	C₂×Q₁₆	—	73	1	no	—
116	C₂×SD₁₆	—	73	2	no	—
117	C₂×SD₁₆	—	73	2	no	—
118	C₂×D₈	—	73	1	no	—
119	C₄⋊Q₈	—	73	2	no	—
120	#126	—	146	2	no	—
121	#126	—	146	2	no	—
122	#124	—	73	2	no	—
123	#124	—	73	2	no	—
124	C₂²⋊Q₈	—	146	2	no	—
125	C₄×Q₈	—	73	2	no	—
126	C₄×Q₈	—	73	2	no	—
127	C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
128	C₄×D₄	—	73	2	no	—
129	#117	—	146	2	no	—
130	#107	—	73	1	no	—
131	C₄⋊C₈	C₂×D₇₃	1	2	yes	—
132	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
133	#103	—	73	2	no	—
134	#103	—	73	2	no	—
135	#102	—	73	2	no	—
136	#97	C₂×D₇₃	1	2	yes	—
137	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
138	C₂×Q₈	—	73	2	no	—
139	C₂×Q₈	—	73	2	no	—
140	C₂×Q₈	D₂₉₂	1	1	yes	characteristic
141	C₂×D₄	—	73	2	no	—
142	C₂²×C₄	—	146	2	no	—
143	Q₁₆	—	292	1	no	—
144	Q₁₆	—	292	2	no	—
145	Q₁₆	—	146	2	no	—
146	SD₁₆	—	292	2	no	—
147	SD₁₆	—	146	4	no	—
148	D₈	—	292	1	no	—
149	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
150	C₂×C₈	C₂²×D₇₃	1	2	yes	—
151	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
152	C₄⋊C₄	—	73	2	no	—
153	C₄⋊C₄	—	146	2	no	—
154	C₄⋊C₄	—	146	2	no	—
155	C₄⋊C₄	—	146	2	no	—
156	C₄⋊C₄	—	146	2	no	—
157	C₄⋊C₄	D₂₉₂	1	2	yes	—
158	C₄⋊C₄	D₂₉₂	1	1	yes	characteristic
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	C₄²	—	146	2	no	—
165	C₄²	—	146	2	no	—
166	C₄²	C₂²×D₇₃	1	1	yes	characteristic
167	C₄²	C₂²×D₇₃	1	2	yes	—
168	C₂³	—	146	2	no	—
169	Q₈	—	4	1	no	—
170	Q₈	—	146	2	no	—
171	Q₈	—	146	2	no	—
172	Q₈	—	146	2	no	—
173	Q₈	—	146	2	no	—
174	D₄	—	146	2	no	—
175	D₄	—	146	2	no	—
176	C₂×C₄	—	292	4	no	—
177	C₂×C₄	—	146	2	no	—
178	C₂×C₄	—	146	2	no	—
179	C₂×C₄	—	146	2	no	—
180	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
181	C₂×C₄	C₂×D₂₉₂	1	2	yes	—
182	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
183	C₂×C₄	C₂×D₂₉₂	1	2	yes	—
184	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
185	C₈	—	2	1	no	—
186	C₈	—	2	2	no	—
187	C₈	#11924029	1	2	yes	—
188	C₂²	—	292	2	no	—
189	C₂²	—	292	2	no	—
190	C₂²	—	292	2	no	—
191	C₂²	C₂²×D₂₉₂	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	2	2	no	—
196	C₄	—	292	2	no	—
197	C₄	—	292	2	no	—
198	C₄	—	292	2	no	—
199	C₄	C₂×#11924029	1	1	yes	characteristic
200	C₄	#14556637	1	1	yes	characteristic
201	C₂	—	584	2	no	—
202	C₂	#17989259	1	1	yes	characteristic
203	C₂	#17989387	1	1	yes	characteristic
204	C₂	#17989404	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989310

C₂

yes

maximal, complemented

#17988981

C₂

yes

maximal, complemented

#17988971

C₂

yes

maximal, characteristic

#17988810

C₂

yes

maximal, complemented

#17988803

C₂

yes

maximal

#17988798

C₂

yes

maximal, complemented

#17988689

C₂

yes

maximal, complemented

C₄×Dic₂₉₂

C₂

yes

maximal, characteristic, complemented

#446×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₉₂

C₂²

yes

characteristic

C₂×Dic₂₉₂

C₂²

yes

characteristic

C₂×Dic₂₉₂

C₂²

yes

—

C₂×#11924017

C₂²

yes

—

C₂×#11924017

C₂²

yes

—

C₂×D₅₈₄

C₂²

yes

characteristic

#14556591

C₂²

yes

—

#14556586

—

#14556586

—

#14556579

C₂²

yes

—

#14556579

C₂²

yes

—

#14556567

—

C₄×Dic₁₄₆

C₂²

yes

—

C₄×Dic₁₄₆

C₂²

yes

—

C₄×D₂₉₂

C₂²

yes

—

#14556549

—

#14556525

C₂²

yes

characteristic

#14556516

C₂²

yes

—

#14556516

C₂²

yes

—

#14556513

C₂²

yes

—

C₄×Q₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

—

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

#97×C₇₃

C₂²

yes

—

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

C₂³

yes

—

C₂×Dic₁₄₆

C₂³

yes

—

C₂×D₂₉₂

C₂³

yes

—

C₂×C₄×D₇₃

—

Dic₂₉₂

—

Dic₂₉₂

—

Dic₂₉₂

—

#11924017

—

#11924017

—

D₅₈₄

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂³

yes

—

—

—

—

—

—

—

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂×Q₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

C₂³

yes

—

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

D₄

yes

—

C₄⋊C₄×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

C₂³

yes

characteristic

C₄²×C₇₃

C₂³

yes

—

C₂²×D₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂⁴

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

#41

yes

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×#41

yes

characteristic

C₄×C₇₃

H(5,𝔽₂)

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

100

C₂×C₇₃

#578

yes

characteristic

101

D₇₃

—

102

#3402

—

maximal, Sylow

103

C₇₃

#3402

yes

characteristic, complemented, Sylow

104

#556

—

105

#497

—

106

#495

—

107

#464

—

108

#463

—

109

#462

—

110

#446

D₇₃

yes

characteristic, complemented

111

#440

—

112

C₄×Q₁₆

—

113

C₂×Q₁₆

—

114

C₂×Q₁₆

—

115

C₂×Q₁₆

—

116

C₂×SD₁₆

—

117

C₂×SD₁₆

—

118

C₂×D₈

—

119

C₄⋊Q₈

—

120

#126

—

146

—

121

#126

—

146

—

122

#124

—

123

#124

—

124

C₂²⋊Q₈

—

146

—

125

C₄×Q₈

—

126

C₄×Q₈

—

127

C₄×Q₈

C₂×D₇₃

yes

characteristic

128

C₄×D₄

—

129

#117

—

146

—

130

#107

—

131

C₄⋊C₈

C₂×D₇₃

yes

—

132

C₄⋊C₈

C₂×D₇₃

yes

characteristic

133

#103

—

134

#103

—

135

#102

—

136

#97

C₂×D₇₃

yes

—

137

C₄×C₈

C₂×D₇₃

yes

characteristic

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

D₂₉₂

yes

characteristic

141

C₂×D₄

—

142

C₂²×C₄

—

146

—

143

Q₁₆

—

292

—

144

Q₁₆

—

292

—

145

Q₁₆

—

146

—

146

SD₁₆

—

292

—

147

SD₁₆

—

146

—

148

D₈

—

292

—

149

C₂×C₈

C₂²×D₇₃

yes

characteristic

150

C₂×C₈

C₂²×D₇₃

yes

—

151

C₂×C₈

C₂²×D₇₃

yes

characteristic

152

C₄⋊C₄

—

153

C₄⋊C₄

—

146

—

154

C₄⋊C₄

—

146

—

155

C₄⋊C₄

—

146

—

156

C₄⋊C₄

—

146

—

157

C₄⋊C₄

D₂₉₂

yes

—

158

C₄⋊C₄

D₂₉₂

yes

characteristic

159

C₂²⋊C₄

—

146

—

160

C₂²⋊C₄

—

146

—

161

C₂²⋊C₄

—

146

—

162

C₂²⋊C₄

—

146

—

163

C₂²⋊C₄

—

146

—

164

C₄²

—

146

—

165

C₄²

—

146

—

166

C₄²

C₂²×D₇₃

yes

characteristic

167

C₄²

C₂²×D₇₃

yes

—

168

C₂³

—

146

—

169

Q₈

—

170

Q₈

—

146

—

171

Q₈

—

146

—

172

Q₈

—

146

—

173

Q₈

—

146

—

174

D₄

—

146

—

175

D₄

—

146

—

176

C₂×C₄

—

292

—

177

C₂×C₄

—

146

—

178

C₂×C₄

—

146

—

179

C₂×C₄

—

146

—

180

C₂×C₄

C₂×D₂₉₂

yes

characteristic

181

C₂×C₄

C₂×D₂₉₂

yes

—

182

C₂×C₄

C₂×D₂₉₂

yes

characteristic

183

C₂×C₄

C₂×D₂₉₂

yes

—

184

C₂×C₄

C₂³×D₇₃

yes

characteristic

185

C₈

—

186

C₈

—

187

C₈

#11924029

yes

—

188

C₂²

—

292

—

189

C₂²

—

292

—

190

C₂²

—

292

—

191

C₂²

C₂²×D₂₉₂

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

292

—

197

C₄

—

292

—

198

C₄

—

292

—

199

C₄

C₂×#11924029

yes

characteristic

200

C₄

#14556637

yes

characteristic

201

C₂

—

584

—

202

C₂

#17989259

yes

characteristic

203

C₂

#17989387

yes

characteristic

204

C₂

#17989404

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₃
Fitting	#446×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₄
hypercenter	#446	D₇₃
hypocenter	C₇₃	#3402
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

subgroup

G/H

center

C₂²

C₂²×D₂₉₂

commutator

C₂×C₄×C₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₃

Fitting

#446×C₇₃

C₂

socle

C₂²×C₇₃

C₂²×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

N	H ≅ G/N	type
1. #17989310	C₂	semidirect
2. #17988981	C₂	semidirect
4. #17988810	C₂	semidirect
6. #17988798	C₂	semidirect
7. #17988689	C₂	semidirect
8. C₄×Dic₂₉₂	C₂	semidirect
9. #446×C₇₃	C₂	semidirect
103. C₇₃	#3402	semidirect
110. #446	D₇₃	semidirect

H ≅ G/N

type

1. #17989310

C₂

semidirect

2. #17988981

C₂

semidirect

4. #17988810

C₂

semidirect

6. #17988798

C₂

semidirect

7. #17988689

C₂

semidirect

8. C₄×Dic₂₉₂

C₂

semidirect

9. #446×C₇₃

C₂

semidirect

103. C₇₃

#3402

semidirect

110. #446

D₇₃

semidirect

Subgroups — #22623012

Conjugacy classes of subgroups

Complements