Group #22620222 — subgroups

Subgroups — #22620222

Order 9344.

382 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 125 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	#17988911	C₂	1	2	yes	maximal, complemented
3	#17988894	C₂	1	2	yes	maximal, complemented
4	#17988787	C₂	1	2	yes	maximal, complemented
5	#17988726	C₂	1	2	yes	maximal, complemented
6	#17988685	C₂	1	2	yes	maximal, complemented
7	C₄×#11924015	C₂	1	2	yes	maximal, complemented
8	#17988648	C₂	1	1	yes	maximal, characteristic, complemented
9	#519×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11924015	C₂²	1	2	yes	—
11	C₂×#11924015	C₂²	1	2	yes	—
12	#14556588	C₂²	1	1	yes	characteristic
13	#14556583	C₂²	1	1	yes	characteristic
14	#14556576	—	2	2	no	—
15	#14556565	C₂²	1	2	yes	—
16	#14556564	—	2	2	no	—
17	C₄×D₂₉₂	C₂²	1	2	yes	—
18	#14556524	C₂²	1	2	yes	—
19	#14556522	C₂²	1	2	yes	—
20	C₄⋊#10596831	C₂²	1	2	yes	—
21	#14556515	C₂²	1	2	yes	—
22	#14556515	C₂²	1	2	yes	—
23	#14556511	C₂²	1	2	yes	—
24	#14556511	C₂²	1	2	yes	—
25	C₂²⋊#10596831	C₂²	1	2	yes	—
26	#14556502	C₂²	1	1	yes	characteristic
27	C₄×#10596831	C₂²	1	1	yes	characteristic
28	C₂²×Q₈×C₇₃	C₂²	1	1	yes	characteristic
29	C₄×Q₈×C₇₃	—	2	2	no	—
30	C₄×Q₈×C₇₃	C₂²	1	2	yes	—
31	C₄×Q₈×C₇₃	C₂²	1	2	yes	—
32	#117×C₇₃	—	2	2	no	—
33	#117×C₇₃	C₂²	1	1	yes	characteristic
34	#117×C₇₃	C₂²	1	1	yes	characteristic
35	C₂×D₂₉₂	—	2	2	no	—
36	C₂×D₂₉₂	C₂³	1	2	yes	—
37	C₂×C₂²⋊D₇₃	—	2	2	no	—
38	C₂×C₄×D₇₃	—	2	2	no	—
39	#11924015	—	2	4	no	—
40	#11924015	—	2	4	no	—
41	C₂×#10596831	C₂³	1	2	yes	—
42	C₂×#10596831	C₂³	1	2	yes	—
43	C₄⋊Dic₇₃	C₂³	1	2	yes	—
44	#11924007	—	2	2	no	—
45	#11924005	—	2	2	no	—
46	#11924005	—	2	2	no	—
47	C₂×Q₈×C₇₃	—	2	4	no	—
48	C₂×Q₈×C₇₃	—	2	1	no	—
49	C₂×Q₈×C₇₃	C₂³	1	2	yes	—
50	C₂²×C₄×C₇₃	—	2	1	no	—
51	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
52	C₄⋊C₄×C₇₃	—	2	2	no	—
53	C₄⋊C₄×C₇₃	—	2	2	no	—
54	C₄⋊C₄×C₇₃	D₄	1	2	yes	—
55	C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
56	C₂²⋊C₄×C₇₃	—	2	1	no	—
57	C₂²⋊C₄×C₇₃	D₄	1	2	yes	—
58	C₄²×C₇₃	—	2	2	no	—
59	C₄²×C₇₃	—	2	2	no	—
60	C₄²×C₇₃	—	2	1	no	—
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	Q₈×C₇₃	—	2	4	no	—
66	Q₈×C₇₃	—	2	2	no	—
67	Q₈×C₇₃	#41	1	4	yes	—
68	C₂×C₄×C₇₃	—	2	4	no	—
69	C₂×C₄×C₇₃	—	2	2	no	—
70	C₂×C₄×C₇₃	—	2	2	no	—
71	C₂×C₄×C₇₃	—	2	2	no	—
72	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
73	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
76	C₂²⋊D₇₃	—	8	2	no	—
77	C₂×Dic₇₃	—	2	2	no	—
78	D₂₉₂	—	2	2	no	—
79	D₂₉₂	—	2	2	no	—
80	D₂₉₂	—	8	2	no	—
81	C₄×D₇₃	—	8	2	no	—
82	#10596831	—	2	2	no	—
83	#10596831	—	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	4	no	—
91	C₄×C₇₃	—	2	1	no	—
92	C₄×C₇₃	—	2	2	no	—
93	C₄×C₇₃	—	2	2	no	—
94	C₄×C₇₃	—	4	2	no	—
95	C₄×C₇₃	C₂×#41	1	2	yes	—
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	D₇₃	1	1	yes	characteristic, complemented
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	—	73	1	no	—
112	C₂²×Q₈	C₂×D₇₃	1	1	yes	characteristic
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	—	2	2	no	—
121	C₄×Q₈	C₂×D₇₃	1	2	yes	—
122	C₄×Q₈	C₂×D₇₃	1	2	yes	—
123	C₄×D₄	—	73	2	no	—
124	#117	—	2	2	no	—
125	#117	C₂×D₇₃	1	1	yes	characteristic
126	#117	C₂×D₇₃	1	1	yes	characteristic
127	#107	—	73	2	no	—
128	#106	—	73	2	no	—
129	C₄⋊C₈	—	73	2	no	—
130	#103	—	73	2	no	—
131	#103	—	73	2	no	—
132	#102	—	73	2	no	—
133	#102	—	73	2	no	—
134	C₂²⋊C₈	—	73	2	no	—
135	#97	—	73	1	no	—
136	C₄×C₈	—	73	1	no	—
137	C₂×Q₈	—	2	4	no	—
138	C₂×Q₈	—	2	1	no	—
139	C₂×Q₈	C₂²×D₇₃	1	2	yes	—
140	C₂×D₄	—	73	2	no	—
141	C₂×D₄	—	146	2	no	—
142	C₂×D₄	—	146	2	no	—
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	146	2	no	—
145	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
146	SD₁₆	—	146	4	no	—
147	SD₁₆	—	146	4	no	—
148	C₂×C₈	—	73	2	no	—
149	C₂×C₈	—	73	2	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	73	2	no	—
153	C₄⋊C₄	—	146	2	no	—
154	C₄⋊C₄	C₂²⋊D₇₃	1	2	yes	—
155	C₄⋊C₄	C₂²×D₇₃	1	2	yes	—
156	C₂²⋊C₄	—	2	1	no	—
157	C₂²⋊C₄	—	146	2	no	—
158	C₂²⋊C₄	—	146	2	no	—
159	C₂²⋊C₄	C₂²⋊D₇₃	1	2	yes	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	2	1	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×C₂²⋊D₇₃	1	1	yes	characteristic
167	Q₈	—	2	4	no	—
168	Q₈	—	2	2	no	—
169	Q₈	#11924028	1	4	yes	—
170	D₄	—	584	2	no	—
171	D₄	—	584	2	no	—
172	D₄	—	146	2	no	—
173	D₄	—	146	2	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	584	2	no	—
179	C₂×C₄	—	146	2	no	—
180	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
181	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
182	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
183	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
184	C₈	—	146	2	no	—
185	C₈	—	146	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×C₂²⋊D₇₃	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	2	2	no	—
194	C₄	—	2	4	no	—
195	C₄	—	2	1	no	—
196	C₄	—	4	2	no	—
197	C₄	—	292	2	no	—
198	C₄	C₂×#11924028	1	2	yes	—
199	C₂	—	4	1	no	—
200	C₂	—	584	2	no	—
201	C₂	#17989098	1	1	yes	characteristic
202	C₂	#17989395	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17989192

C₂

yes

maximal, characteristic

#17988911

C₂

yes

maximal, complemented

#17988894

C₂

yes

maximal, complemented

#17988787

C₂

yes

maximal, complemented

#17988726

C₂

yes

maximal, complemented

#17988685

C₂

yes

maximal, complemented

C₄×#11924015

C₂

yes

maximal, complemented

#17988648

C₂

yes

maximal, characteristic, complemented

#519×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924015

C₂²

yes

—

C₂×#11924015

C₂²

yes

—

#14556588

C₂²

yes

characteristic

#14556583

C₂²

yes

characteristic

#14556576

—

#14556565

C₂²

yes

—

#14556564

—

C₄×D₂₉₂

C₂²

yes

—

#14556524

C₂²

yes

—

#14556522

C₂²

yes

—

C₄⋊#10596831

C₂²

yes

—

#14556515

C₂²

yes

—

#14556515

C₂²

yes

—

#14556511

C₂²

yes

—

#14556511

C₂²

yes

—

C₂²⋊#10596831

C₂²

yes

—

#14556502

C₂²

yes

characteristic

C₄×#10596831

C₂²

yes

characteristic

C₂²×Q₈×C₇₃

C₂²

yes

characteristic

C₄×Q₈×C₇₃

—

C₄×Q₈×C₇₃

C₂²

yes

—

C₄×Q₈×C₇₃

C₂²

yes

—

#117×C₇₃

—

#117×C₇₃

C₂²

yes

characteristic

#117×C₇₃

C₂²

yes

characteristic

C₂×D₂₉₂

—

C₂×D₂₉₂

C₂³

yes

—

C₂×C₂²⋊D₇₃

—

C₂×C₄×D₇₃

—

#11924015

—

#11924015

—

C₂×#10596831

C₂³

yes

—

C₂×#10596831

C₂³

yes

—

C₄⋊Dic₇₃

C₂³

yes

—

#11924007

—

#11924005

—

#11924005

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

C₂³

yes

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

D₄

yes

—

C₄⋊C₄×C₇₃

C₂³

yes

—

C₂²⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

D₄

yes

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂²×D₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

#41

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

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

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×#41

yes

—

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

D₇₃

yes

characteristic, complemented

105

#482

—

106

#479

—

107

#460

—

108

#448

—

109

#440

—

110

C₄×SD₁₆

—

111

#433

—

112

C₂²×Q₈

C₂×D₇₃

yes

characteristic

113

C₂×SD₁₆

—

114

C₂×SD₁₆

—

115

#127

—

116

#125

—

117

#123

—

146

—

118

#121

—

119

#121

—

146

—

120

C₄×Q₈

—

121

C₄×Q₈

C₂×D₇₃

yes

—

122

C₄×Q₈

C₂×D₇₃

yes

—

123

C₄×D₄

—

124

#117

—

125

#117

C₂×D₇₃

yes

characteristic

126

#117

C₂×D₇₃

yes

characteristic

127

#107

—

128

#106

—

129

C₄⋊C₈

—

130

#103

—

131

#103

—

132

#102

—

133

#102

—

134

C₂²⋊C₈

—

135

#97

—

136

C₄×C₈

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

C₂²×D₇₃

yes

—

140

C₂×D₄

—

141

C₂×D₄

—

146

—

142

C₂×D₄

—

146

—

143

C₂²×C₄

—

144

C₂²×C₄

—

146

—

145

C₂²×C₄

C₂²×D₇₃

yes

characteristic

146

SD₁₆

—

146

—

147

SD₁₆

—

146

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

146

—

154

C₄⋊C₄

C₂²⋊D₇₃

yes

—

155

C₄⋊C₄

C₂²×D₇₃

yes

—

156

C₂²⋊C₄

—

157

C₂²⋊C₄

—

146

—

158

C₂²⋊C₄

—

146

—

159

C₂²⋊C₄

C₂²⋊D₇₃

yes

—

160

C₄²

—

161

C₄²

—

162

C₄²

—

163

C₄²

C₂²×D₇₃

yes

characteristic

164

C₄²

C₂²×D₇₃

yes

characteristic

165

C₂³

—

146

—

166

C₂³

C₂×C₂²⋊D₇₃

yes

characteristic

167

Q₈

—

168

Q₈

—

169

Q₈

#11924028

yes

—

170

D₄

—

584

—

171

D₄

—

584

—

172

D₄

—

146

—

173

D₄

—

146

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

584

—

179

C₂×C₄

—

146

—

180

C₂×C₄

C₂×C₂²⋊D₇₃

yes

—

181

C₂×C₄

C₂×C₂²⋊D₇₃

yes

—

182

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

183

C₂×C₄

C₂³×D₇₃

yes

characteristic

184

C₈

—

146

—

185

C₈

—

146

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

292

—

189

C₂²

—

292

—

190

C₂²

—

292

—

191

C₂²

C₂²×C₂²⋊D₇₃

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

—

292

—

198

C₄

C₂×#11924028

yes

—

199

C₂

—

200

C₂

—

584

—

201

C₂

#17989098

yes

characteristic

202

C₂

#17989395

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₃
Fitting	#519×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₄
hypercenter	#519	D₇₃
hypocenter	C₇₃	#3126
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #519

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₇₃

commutator

C₂×C₄×C₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₃

Fitting

#519×C₇₃

C₂

socle

C₂²×C₇₃

C₂²×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ #117 ◁ #519

N	H ≅ G/N	type
2. #17988911	C₂	semidirect
3. #17988894	C₂	semidirect
4. #17988787	C₂	semidirect
5. #17988726	C₂	semidirect
6. #17988685	C₂	semidirect
7. C₄×#11924015	C₂	semidirect
8. #17988648	C₂	semidirect
9. #519×C₇₃	C₂	semidirect
102. C₇₃	#3126	semidirect
104. #519	D₇₃	semidirect

H ≅ G/N

type

2. #17988911

C₂

semidirect

3. #17988894

C₂

semidirect

4. #17988787

C₂

semidirect

5. #17988726

C₂

semidirect

6. #17988685

C₂

semidirect

7. C₄×#11924015

C₂

semidirect

8. #17988648

C₂

semidirect

9. #519×C₇₃

C₂

semidirect

102. C₇₃

#3126

semidirect

104. #519

D₇₃

semidirect

Subgroups — #22620222

Conjugacy classes of subgroups

Complements