Group #22620223 — subgroups

Subgroups — #22620223

Order 9344.

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

normal subgroups only

This group has 382 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	#17989192	C₂	1	1	yes	maximal, characteristic
2	#17989064	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988912	C₂	1	2	yes	maximal, complemented
4	#17988895	C₂	1	2	yes	maximal, complemented
5	#17988788	C₂	1	2	yes	maximal, complemented
6	#17988728	C₂	1	2	yes	maximal, complemented
7	#17988689	C₂	1	2	yes	maximal, complemented
8	C₄×#11924017	C₂	1	2	yes	maximal, complemented
9	#433×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Dic₁₄₆	C₂²	1	1	yes	characteristic
11	C₂×#11924017	C₂²	1	2	yes	—
12	C₂×#11924017	C₂²	1	2	yes	—
13	#14556588	C₂²	1	1	yes	characteristic
14	#14556583	C₂²	1	1	yes	characteristic
15	#14556576	—	2	2	no	—
16	#14556565	C₂²	1	2	yes	—
17	#14556564	—	2	2	no	—
18	C₄×Dic₁₄₆	C₂²	1	2	yes	—
19	C₄×Dic₁₄₆	C₂²	1	2	yes	—
20	#14556556	—	2	2	no	—
21	C₄×D₂₉₂	C₂²	1	2	yes	—
22	#14556550	C₂²	1	1	yes	characteristic
23	#14556547	—	2	2	no	—
24	#14556525	C₂²	1	2	yes	—
25	#14556523	C₂²	1	2	yes	—
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	#117×C₇₃	C₂²	1	1	yes	characteristic
31	C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
32	C₂²⋊C₈×C₇₃	C₂²	1	2	yes	—
33	#97×C₇₃	C₂²	1	1	yes	characteristic
34	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
35	C₂×Dic₁₄₆	—	2	1	no	—
36	C₂×Dic₁₄₆	—	2	4	no	—
37	C₂×Dic₁₄₆	C₂³	1	2	yes	—
38	C₂×D₂₉₂	—	2	2	no	—
39	C₂×D₂₉₂	C₂³	1	2	yes	—
40	C₂×C₂²⋊D₇₃	—	2	2	no	—
41	C₂²×Dic₇₃	—	2	1	no	—
42	C₂×C₄×D₇₃	—	2	2	no	—
43	#11924017	—	2	4	no	—
44	#11924017	—	2	4	no	—
45	C₄⋊Dic₇₃	C₂³	1	2	yes	—
46	C₄⋊Dic₇₃	C₂³	1	2	yes	—
47	#11924007	—	2	2	no	—
48	#11924007	—	2	2	no	—
49	#11924007	—	2	2	no	—
50	C₂²⋊Dic₇₃	—	2	1	no	—
51	#11924005	—	2	2	no	—
52	#11924005	—	2	2	no	—
53	C₄×Dic₇₃	—	2	2	no	—
54	C₄×Dic₇₃	—	2	2	no	—
55	C₄×Dic₇₃	—	2	1	no	—
56	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₇₃	C₂³	1	2	yes	—
58	C₂×C₈×C₇₃	C₂³	1	2	yes	—
59	C₄⋊C₄×C₇₃	D₄	1	2	yes	—
60	C₂²⋊C₄×C₇₃	D₄	1	2	yes	—
61	C₄²×C₇₃	C₂³	1	1	yes	characteristic
62	C₄²×C₇₃	C₂³	1	1	yes	characteristic
63	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×D₇₃	—	2	2	no	—
65	C₂×C₄×C₇₃	—	2	2	no	—
66	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
70	C₂²⋊D₇₃	—	8	2	no	—
71	C₂×Dic₇₃	—	2	4	no	—
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	D₂₉₂	—	8	2	no	—
78	C₄×D₇₃	—	8	2	no	—
79	Dic₁₄₆	—	2	4	no	—
80	Dic₁₄₆	—	2	2	no	—
81	Dic₁₄₆	#41	1	4	yes	—
82	C₈×C₇₃	—	2	2	no	—
83	C₈×C₇₃	—	2	2	no	—
84	C₂²×C₇₃	—	2	1	no	—
85	C₂²×C₇₃	—	2	2	no	—
86	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
87	C₂×D₇₃	—	4	2	no	—
88	C₂×D₇₃	—	4	2	no	—
89	C₂×D₇₃	—	4	2	no	—
90	C₄×C₇₃	—	2	1	no	—
91	C₄×C₇₃	—	2	2	no	—
92	C₄×C₇₃	—	2	2	no	—
93	C₄×C₇₃	C₂×#41	1	2	yes	—
94	Dic₇₃	—	2	4	no	—
95	Dic₇₃	—	4	2	no	—
96	Dic₇₃	—	4	2	no	—
97	C₂×C₇₃	—	4	1	no	—
98	C₂×C₇₃	#525	1	1	yes	characteristic
99	C₂×C₇₃	#577	1	2	yes	—
100	D₇₃	—	8	2	no	—
101	#3126	—	73	1	no	maximal, Sylow
102	C₇₃	#3126	1	1	yes	characteristic, complemented, Sylow
103	#540	—	73	1	no	—
104	#519	—	73	1	no	—
105	#482	—	73	2	no	—
106	#479	—	73	2	no	—
107	#460	—	73	2	no	—
108	#448	—	73	2	no	—
109	#440	—	73	2	no	—
110	C₄×SD₁₆	—	73	2	no	—
111	#433	D₇₃	1	1	yes	characteristic, complemented
112	C₂²×Q₈	—	73	1	no	—
113	C₂×SD₁₆	—	73	2	no	—
114	C₂×SD₁₆	—	73	2	no	—
115	#127	—	73	1	no	—
116	#125	—	73	1	no	—
117	#123	—	146	2	no	—
118	#121	—	73	2	no	—
119	#121	—	146	2	no	—
120	C₄×Q₈	—	73	2	no	—
121	C₄×Q₈	—	73	2	no	—
122	C₄×Q₈	—	146	2	no	—
123	C₄×D₄	—	73	2	no	—
124	#117	—	73	1	no	—
125	#117	—	146	2	no	—
126	#117	C₂×D₇₃	1	1	yes	characteristic
127	#107	—	73	2	no	—
128	#106	—	73	2	no	—
129	C₄⋊C₈	C₂×D₇₃	1	2	yes	—
130	#103	—	73	2	no	—
131	#103	—	73	2	no	—
132	#102	—	73	2	no	—
133	#102	—	73	2	no	—
134	C₂²⋊C₈	C₂×D₇₃	1	2	yes	—
135	#97	C₂×D₇₃	1	1	yes	characteristic
136	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
137	C₂×Q₈	—	73	2	no	—
138	C₂×Q₈	—	146	4	no	—
139	C₂×Q₈	—	146	1	no	—
140	C₂×D₄	—	73	2	no	—
141	C₂×D₄	—	146	2	no	—
142	C₂×D₄	—	146	2	no	—
143	C₂²×C₄	—	146	2	no	—
144	C₂²×C₄	—	146	1	no	—
145	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
146	SD₁₆	—	146	4	no	—
147	SD₁₆	—	146	4	no	—
148	C₂×C₈	C₂²×D₇₃	1	2	yes	—
149	C₂×C₈	C₂²×D₇₃	1	2	yes	—
150	C₄⋊C₄	—	73	2	no	—
151	C₄⋊C₄	—	73	2	no	—
152	C₄⋊C₄	—	146	2	no	—
153	C₄⋊C₄	—	146	2	no	—
154	C₄⋊C₄	—	146	2	no	—
155	C₄⋊C₄	D₂₉₂	1	2	yes	—
156	C₂²⋊C₄	—	146	1	no	—
157	C₂²⋊C₄	—	146	2	no	—
158	C₂²⋊C₄	—	146	2	no	—
159	C₂²⋊C₄	D₂₉₂	1	2	yes	—
160	C₄²	—	146	1	no	—
161	C₄²	—	146	2	no	—
162	C₄²	—	146	2	no	—
163	C₄²	C₂²×D₇₃	1	1	yes	characteristic
164	C₄²	C₂²×D₇₃	1	1	yes	characteristic
165	C₂³	—	146	2	no	—
166	C₂³	C₂×D₂₉₂	1	1	yes	characteristic
167	Q₈	—	73	4	no	—
168	Q₈	—	146	4	no	—
169	Q₈	—	146	2	no	—
170	D₄	—	584	2	no	—
171	D₄	—	584	2	no	—
172	D₄	—	146	2	no	—
173	D₄	—	146	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	584	2	no	—
176	C₂×C₄	—	146	2	no	—
177	C₂×C₄	—	146	2	no	—
178	C₂×C₄	—	146	4	no	—
179	C₂×C₄	—	146	2	no	—
180	C₂×C₄	C₂×D₂₉₂	1	2	yes	—
181	C₂×C₄	C₂×D₂₉₂	1	2	yes	—
182	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
183	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
184	C₈	—	2	2	no	—
185	C₈	—	2	2	no	—
186	C₂²	—	2	2	no	—
187	C₂²	—	2	1	no	—
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	2	no	—
194	C₄	—	2	1	no	—
195	C₄	—	292	2	no	—
196	C₄	—	292	2	no	—
197	C₄	—	146	4	no	—
198	C₄	#41×D₇₃	1	2	yes	—
199	C₂	—	4	1	no	—
200	C₂	—	584	2	no	—
201	C₂	#17989099	1	1	yes	characteristic
202	C₂	#17989399	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17989192

C₂

yes

maximal, characteristic

#17989064

C₂

yes

maximal, characteristic, complemented

#17988912

C₂

yes

maximal, complemented

#17988895

C₂

yes

maximal, complemented

#17988788

C₂

yes

maximal, complemented

#17988728

C₂

yes

maximal, complemented

#17988689

C₂

yes

maximal, complemented

C₄×#11924017

C₂

yes

maximal, complemented

#433×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₁₄₆

C₂²

yes

characteristic

C₂×#11924017

C₂²

yes

—

C₂×#11924017

C₂²

yes

—

#14556588

C₂²

yes

characteristic

#14556583

C₂²

yes

characteristic

#14556576

—

#14556565

C₂²

yes

—

#14556564

—

C₄×Dic₁₄₆

C₂²

yes

—

C₄×Dic₁₄₆

C₂²

yes

—

#14556556

—

C₄×D₂₉₂

C₂²

yes

—

#14556550

C₂²

yes

characteristic

#14556547

—

#14556525

C₂²

yes

—

#14556523

C₂²

yes

—

#14556516

C₂²

yes

—

#14556516

C₂²

yes

—

#14556513

C₂²

yes

—

#14556513

C₂²

yes

—

#117×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

—

C₂²⋊C₈×C₇₃

C₂²

yes

—

#97×C₇₃

C₂²

yes

characteristic

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

C₂³

yes

—

C₂×D₂₉₂

—

C₂×D₂₉₂

C₂³

yes

—

C₂×C₂²⋊D₇₃

—

C₂²×Dic₇₃

—

C₂×C₄×D₇₃

—

#11924017

—

#11924017

—

C₄⋊Dic₇₃

C₂³

yes

—

C₄⋊Dic₇₃

C₂³

yes

—

#11924007

—

#11924007

—

#11924007

—

C₂²⋊Dic₇₃

—

#11924005

—

#11924005

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

C₂³

yes

—

C₂×C₈×C₇₃

C₂³

yes

—

C₄⋊C₄×C₇₃

D₄

yes

—

C₂²⋊C₄×C₇₃

D₄

yes

—

C₄²×C₇₃

C₂³

yes

characteristic

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂²×D₇₃

—

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

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

#41

yes

—

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

C₂×#41

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#525

yes

characteristic

C₂×C₇₃

#577

yes

—

100

D₇₃

—

101

#3126

—

maximal, Sylow

102

C₇₃

#3126

yes

characteristic, complemented, Sylow

103

#540

—

104

#519

—

105

#482

—

106

#479

—

107

#460

—

108

#448

—

109

#440

—

110

C₄×SD₁₆

—

111

#433

D₇₃

yes

characteristic, complemented

112

C₂²×Q₈

—

113

C₂×SD₁₆

—

114

C₂×SD₁₆

—

115

#127

—

116

#125

—

117

#123

—

146

—

118

#121

—

119

#121

—

146

—

120

C₄×Q₈

—

121

C₄×Q₈

—

122

C₄×Q₈

—

146

—

123

C₄×D₄

—

124

#117

—

125

#117

—

146

—

126

#117

C₂×D₇₃

yes

characteristic

127

#107

—

128

#106

—

129

C₄⋊C₈

C₂×D₇₃

yes

—

130

#103

—

131

#103

—

132

#102

—

133

#102

—

134

C₂²⋊C₈

C₂×D₇₃

yes

—

135

#97

C₂×D₇₃

yes

characteristic

136

C₄×C₈

C₂×D₇₃

yes

characteristic

137

C₂×Q₈

—

138

C₂×Q₈

—

146

—

139

C₂×Q₈

—

146

—

140

C₂×D₄

—

141

C₂×D₄

—

146

—

142

C₂×D₄

—

146

—

143

C₂²×C₄

—

146

—

144

C₂²×C₄

—

146

—

145

C₂²×C₄

C₂²×D₇₃

yes

characteristic

146

SD₁₆

—

146

—

147

SD₁₆

—

146

—

148

C₂×C₈

C₂²×D₇₃

yes

—

149

C₂×C₈

C₂²×D₇₃

yes

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

146

—

153

C₄⋊C₄

—

146

—

154

C₄⋊C₄

—

146

—

155

C₄⋊C₄

D₂₉₂

yes

—

156

C₂²⋊C₄

—

146

—

157

C₂²⋊C₄

—

146

—

158

C₂²⋊C₄

—

146

—

159

C₂²⋊C₄

D₂₉₂

yes

—

160

C₄²

—

146

—

161

C₄²

—

146

—

162

C₄²

—

146

—

163

C₄²

C₂²×D₇₃

yes

characteristic

164

C₄²

C₂²×D₇₃

yes

characteristic

165

C₂³

—

146

—

166

C₂³

C₂×D₂₉₂

yes

characteristic

167

Q₈

—

168

Q₈

—

146

—

169

Q₈

—

146

—

170

D₄

—

584

—

171

D₄

—

584

—

172

D₄

—

146

—

173

D₄

—

146

—

174

C₂×C₄

—

175

C₂×C₄

—

584

—

176

C₂×C₄

—

146

—

177

C₂×C₄

—

146

—

178

C₂×C₄

—

146

—

179

C₂×C₄

—

146

—

180

C₂×C₄

C₂×D₂₉₂

yes

—

181

C₂×C₄

C₂×D₂₉₂

yes

—

182

C₂×C₄

C₂×D₂₉₂

yes

characteristic

183

C₂×C₄

C₂³×D₇₃

yes

characteristic

184

C₈

—

185

C₈

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

292

—

189

C₂²

—

292

—

190

C₂²

—

292

—

191

C₂²

C₂²×D₂₉₂

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

292

—

196

C₄

—

292

—

197

C₄

—

146

—

198

C₄

#41×D₇₃

yes

—

199

C₂

—

200

C₂

—

584

—

201

C₂

#17989099

yes

characteristic

202

C₂

#17989399

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₃
Fitting	#433×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₄
hypercenter	#433	D₇₃
hypocenter	C₇₃	#3126
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #433

subgroup

G/H

center

C₂²

C₂²×D₂₉₂

commutator

C₂×C₄×C₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₃

Fitting

#433×C₇₃

C₂

socle

C₂²×C₇₃

C₂²×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ #117 ◁ #433

N	H ≅ G/N	type
2. #17989064	C₂	semidirect
3. #17988912	C₂	semidirect
4. #17988895	C₂	semidirect
5. #17988788	C₂	semidirect
6. #17988728	C₂	semidirect
7. #17988689	C₂	semidirect
8. C₄×#11924017	C₂	semidirect
9. #433×C₇₃	C₂	semidirect
102. C₇₃	#3126	semidirect
111. #433	D₇₃	semidirect

H ≅ G/N

type

2. #17989064

C₂

semidirect

3. #17988912

C₂

semidirect

4. #17988895

C₂

semidirect

5. #17988788

C₂

semidirect

6. #17988728

C₂

semidirect

7. #17988689

C₂

semidirect

8. C₄×#11924017

C₂

semidirect

9. #433×C₇₃

C₂

semidirect

102. C₇₃

#3126

semidirect

111. #433

D₇₃

semidirect

Subgroups — #22620223

Conjugacy classes of subgroups

Complements