Group #22611451 — subgroups

Subgroups — #22611451

Order 9344.

274 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 87 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	C₂×#14556525	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14556523	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14556513	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556513	C₂	1	1	yes	maximal, characteristic
5	#17988489	C₂	1	1	yes	maximal, characteristic
6	#17988453	C₂	1	1	yes	maximal, characteristic, complemented
7	#336×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₂₉₂	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
12	C₂×#11924005	—	2	2	no	—
13	#14556525	—	2	2	no	—
14	#14556523	—	2	2	no	—
15	#14556513	—	2	2	no	—
16	#14556513	—	2	2	no	—
17	#14556498	—	2	2	no	—
18	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
21	C₂³×D₇₃	—	2	1	no	—
22	C₂×D₂₉₂	—	2	1	no	—
23	C₂×D₂₉₂	—	2	1	no	—
24	C₂×D₂₉₂	—	2	1	no	—
25	C₂×D₂₉₂	—	2	1	no	—
26	C₂×D₂₉₂	—	2	1	no	—
27	C₂×D₂₉₂	Q₈	1	2	yes	—
28	C₂²×Dic₇₃	—	2	1	no	—
29	C₂²×Dic₇₃	—	2	1	no	—
30	C₂²×Dic₇₃	—	2	1	no	—
31	C₄⋊Dic₇₃	—	2	1	no	—
32	C₄⋊Dic₇₃	—	2	1	no	—
33	C₄⋊Dic₇₃	—	2	1	no	—
34	C₄⋊Dic₇₃	—	2	1	no	—
35	C₄⋊Dic₇₃	—	2	1	no	—
36	C₄⋊Dic₇₃	—	2	1	no	—
37	#11924005	—	4	4	no	—
38	C₂²×C₄×C₇₃	D₄	1	2	yes	—
39	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₇₃	—	2	1	no	—
41	C₂×C₈×C₇₃	—	2	1	no	—
42	C₂×C₈×C₇₃	—	2	1	no	—
43	C₂×C₈×C₇₃	—	2	1	no	—
44	C₂×C₈×C₇₃	D₄	1	2	yes	—
45	C₂×C₈×C₇₃	D₄	1	2	yes	—
46	C₄²×C₇₃	—	2	1	no	—
47	C₄²×C₇₃	—	2	1	no	—
48	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×D₇₃	—	2	2	no	—
50	C₂²×D₇₃	—	4	1	no	—
51	C₂²×D₇₃	—	4	1	no	—
52	C₂²×D₇₃	—	4	2	no	—
53	C₂²×D₇₃	—	4	2	no	—
54	C₂×C₄×C₇₃	—	2	2	no	—
55	C₂×C₄×C₇₃	—	2	2	no	—
56	C₂×C₄×C₇₃	—	2	2	no	—
57	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
61	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
62	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
63	C₂×Dic₇₃	—	4	1	no	—
64	C₂×Dic₇₃	—	4	1	no	—
65	C₂×Dic₇₃	—	4	1	no	—
66	C₂×Dic₇₃	—	4	2	no	—
67	C₂×Dic₇₃	—	4	2	no	—
68	C₂×Dic₇₃	—	4	2	no	—
69	D₂₉₂	—	2	2	no	—
70	D₂₉₂	—	2	2	no	—
71	D₂₉₂	—	4	1	no	—
72	D₂₉₂	—	4	1	no	—
73	C₈×C₇₃	—	2	2	no	—
74	C₈×C₇₃	—	2	2	no	—
75	C₂²×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂²×C₇₃	#123	1	1	yes	characteristic
77	C₂²×C₇₃	#127	1	1	yes	characteristic
78	C₂²×C₇₃	#135	1	2	yes	—
79	C₂²×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×D₇₃	—	4	2	no	—
81	C₂×D₇₃	—	4	4	no	—
82	C₂×D₇₃	—	8	2	no	—
83	C₂×D₇₃	—	8	1	no	—
84	C₂×D₇₃	—	8	1	no	—
85	C₄×C₇₃	—	4	2	no	—
86	C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
87	C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
88	C₄×C₇₃	#123	1	1	yes	characteristic
89	C₄×C₇₃	#123	1	1	yes	characteristic
90	Dic₇₃	—	8	1	no	—
91	Dic₇₃	—	8	1	no	—
92	Dic₇₃	—	8	1	no	—
93	C₂×C₇₃	#399	1	1	yes	characteristic
94	C₂×C₇₃	#478	1	2	yes	—
95	C₂×C₇₃	#482	1	2	yes	—
96	C₂×C₇₃	#495	1	1	yes	characteristic
97	C₂×C₇₃	#496	1	1	yes	characteristic
98	D₇₃	—	8	2	no	—
99	#2092	—	73	1	no	maximal, Sylow
100	C₇₃	#2092	1	1	yes	characteristic, complemented, Sylow
101	C₂×#107	—	73	1	no	—
102	C₂×#106	—	73	1	no	—
103	C₂×#102	—	73	1	no	—
104	C₂×#102	—	73	1	no	—
105	#390	—	73	1	no	—
106	#382	—	73	1	no	—
107	#336	D₇₃	1	1	yes	characteristic, complemented
108	C₂²×D₄	—	73	1	no	—
109	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
110	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
111	C₂×C₄⋊C₄	—	73	1	no	—
112	C₂×C₄⋊C₄	—	73	1	no	—
113	C₂×C₄⋊C₄	—	73	1	no	—
114	C₂×C₂²⋊C₄	—	146	2	no	—
115	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
116	#107	—	146	2	no	—
117	#106	—	146	2	no	—
118	#102	—	146	2	no	—
119	#102	—	146	2	no	—
120	#95	—	146	2	no	—
121	C₂⁴	—	146	1	no	—
122	C₂×D₄	—	73	2	no	—
123	C₂×D₄	—	146	1	no	—
124	C₂×D₄	—	146	1	no	—
125	C₂×D₄	—	146	1	no	—
126	C₂×D₄	—	146	1	no	—
127	C₂×D₄	—	146	1	no	—
128	C₂²×C₄	—	146	1	no	—
129	C₂²×C₄	—	146	1	no	—
130	C₂²×C₄	—	146	1	no	—
131	C₂²×C₄	D₂₉₂	1	2	yes	—
132	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	D₂₉₂	1	2	yes	—
138	C₂×C₈	D₂₉₂	1	2	yes	—
139	C₄⋊C₄	—	146	1	no	—
140	C₄⋊C₄	—	146	1	no	—
141	C₄⋊C₄	—	146	1	no	—
142	C₄⋊C₄	—	146	1	no	—
143	C₄⋊C₄	—	146	1	no	—
144	C₄⋊C₄	—	146	1	no	—
145	C₂²⋊C₄	—	292	4	no	—
146	C₄²	—	2	1	no	—
147	C₄²	—	2	1	no	—
148	C₂³	—	292	1	no	—
149	C₂³	—	292	1	no	—
150	C₂³	—	292	2	no	—
151	C₂³	—	292	2	no	—
152	C₂³	—	146	2	no	—
153	C₂³	C₂×D₂₉₂	1	1	yes	characteristic
154	D₄	—	292	1	no	—
155	D₄	—	292	1	no	—
156	D₄	—	146	2	no	—
157	D₄	—	146	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	292	1	no	—
162	C₂×C₄	—	292	1	no	—
163	C₂×C₄	—	292	1	no	—
164	C₂×C₄	—	292	2	no	—
165	C₂×C₄	—	292	2	no	—
166	C₂×C₄	—	292	2	no	—
167	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
168	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
169	C₂×C₄	Q₈×D₇₃	1	1	yes	characteristic
170	C₂×C₄	#11924027	1	1	yes	characteristic
171	C₂×C₄	#11924027	1	1	yes	characteristic
172	C₂×C₄	#11924027	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	2	2	no	—
175	C₂²	—	292	4	no	—
176	C₂²	—	292	2	no	—
177	C₂²	—	584	2	no	—
178	C₂²	—	584	1	no	—
179	C₂²	—	584	1	no	—
180	C₂²	#14556571	1	1	yes	characteristic
181	C₂²	#14556575	1	1	yes	characteristic
182	C₂²	#14556588	1	1	yes	characteristic
183	C₂²	#14556613	1	2	yes	—
184	C₂²	#14556619	1	2	yes	—
185	C₄	—	4	2	no	—
186	C₄	—	584	1	no	—
187	C₄	—	584	1	no	—
188	C₄	—	584	1	no	—
189	C₄	#14556571	1	1	yes	characteristic
190	C₄	#14556571	1	1	yes	characteristic
191	C₄	#14556575	1	1	yes	characteristic
192	C₄	#14556575	1	1	yes	characteristic
193	C₂	—	584	2	no	—
194	C₂	#17988524	1	1	yes	characteristic
195	C₂	#17988888	1	2	yes	—
196	C₂	#17988912	1	2	yes	—
197	C₂	#17988971	1	1	yes	characteristic
198	C₂	#17988976	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556525

C₂

yes

maximal, characteristic, complemented

C₂×#14556523

C₂

yes

maximal, characteristic, complemented

C₂×#14556513

C₂

yes

maximal, characteristic

C₂×#14556513

C₂

yes

maximal, characteristic

#17988489

C₂

yes

maximal, characteristic

#17988453

C₂

yes

maximal, characteristic, complemented

#336×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₉₂

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

—

—

—

—

—

—

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₂³×D₇₃

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

Q₈

yes

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924005

—

C₂²×C₄×C₇₃

D₄

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

—

C₂×C₈×C₇₃

D₄

yes

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₈

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

—

C₈×C₇₃

—

C₂²×C₇₃

C₂²⋊Q₈

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₂²×C₇₃

AGL(1,ℤ/8)

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₄×C₇₃

C₂²⋊Q₈

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

#399

yes

characteristic

C₂×C₇₃

#478

yes

—

C₂×C₇₃

#482

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#2092

—

maximal, Sylow

100

C₇₃

#2092

yes

characteristic, complemented, Sylow

101

C₂×#107

—

102

C₂×#106

—

103

C₂×#102

—

104

C₂×#102

—

105

#390

—

106

#382

—

107

#336

D₇₃

yes

characteristic, complemented

108

C₂²×D₄

—

109

C₂²×C₈

C₂×D₇₃

yes

characteristic

110

C₂²×C₈

C₂×D₇₃

yes

characteristic

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

114

C₂×C₂²⋊C₄

—

146

—

115

C₂×C₄²

C₂×D₇₃

yes

characteristic

116

#107

—

146

—

117

#106

—

146

—

118

#102

—

146

—

119

#102

—

146

—

120

#95

—

146

—

121

C₂⁴

—

146

—

122

C₂×D₄

—

123

C₂×D₄

—

146

—

124

C₂×D₄

—

146

—

125

C₂×D₄

—

146

—

126

C₂×D₄

—

146

—

127

C₂×D₄

—

146

—

128

C₂²×C₄

—

146

—

129

C₂²×C₄

—

146

—

130

C₂²×C₄

—

146

—

131

C₂²×C₄

D₂₉₂

yes

—

132

C₂²×C₄

C₂²×D₇₃

yes

characteristic

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

D₂₉₂

yes

—

138

C₂×C₈

D₂₉₂

yes

—

139

C₄⋊C₄

—

146

—

140

C₄⋊C₄

—

146

—

141

C₄⋊C₄

—

146

—

142

C₄⋊C₄

—

146

—

143

C₄⋊C₄

—

146

—

144

C₄⋊C₄

—

146

—

145

C₂²⋊C₄

—

292

—

146

C₄²

—

147

C₄²

—

148

C₂³

—

292

—

149

C₂³

—

292

—

150

C₂³

—

292

—

151

C₂³

—

292

—

152

C₂³

—

146

—

153

C₂³

C₂×D₂₉₂

yes

characteristic

154

D₄

—

292

—

155

D₄

—

292

—

156

D₄

—

146

—

157

D₄

—

146

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

292

—

162

C₂×C₄

—

292

—

163

C₂×C₄

—

292

—

164

C₂×C₄

—

292

—

165

C₂×C₄

—

292

—

166

C₂×C₄

—

292

—

167

C₂×C₄

C₂×D₂₉₂

yes

characteristic

168

C₂×C₄

C₂×D₂₉₂

yes

characteristic

169

C₂×C₄

Q₈×D₇₃

yes

characteristic

170

C₂×C₄

#11924027

yes

characteristic

171

C₂×C₄

#11924027

yes

characteristic

172

C₂×C₄

#11924027

yes

characteristic

173

C₈

—

174

C₈

—

175

C₂²

—

292

—

176

C₂²

—

292

—

177

C₂²

—

584

—

178

C₂²

—

584

—

179

C₂²

—

584

—

180

C₂²

#14556571

yes

characteristic

181

C₂²

#14556575

yes

characteristic

182

C₂²

#14556588

yes

characteristic

183

C₂²

#14556613

yes

—

184

C₂²

#14556619

yes

—

185

C₄

—

186

C₄

—

584

—

187

C₄

—

584

—

188

C₄

—

584

—

189

C₄

#14556571

yes

characteristic

190

C₄

#14556571

yes

characteristic

191

C₄

#14556575

yes

characteristic

192

C₄

#14556575

yes

characteristic

193

C₂

—

584

—

194

C₂

#17988524

yes

characteristic

195

C₂

#17988888

yes

—

196

C₂

#17988912

yes

—

197

C₂

#17988971

yes

characteristic

198

C₂

#17988976

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₉₂
commutator	C₂²×C₄×C₇₃	C₂³
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	#336×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₄
hypercenter	#336	D₇₃
hypocenter	C₇₃	#2092
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₂₉₂

commutator

C₂²×C₄×C₇₃

C₂³

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

#336×C₇₃

C₂

socle

C₂³×C₇₃

C₂×D₄

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
1. C₂×#14556525	C₂	semidirect
2. C₂×#14556523	C₂	semidirect
6. #17988453	C₂	semidirect
7. #336×C₇₃	C₂	semidirect
100. C₇₃	#2092	semidirect
107. #336	D₇₃	semidirect

H ≅ G/N

type

1. C₂×#14556525

C₂

semidirect

2. C₂×#14556523

C₂

semidirect

6. #17988453

C₂

semidirect

7. #336×C₇₃

C₂

semidirect

100. C₇₃

#2092

semidirect

107. #336

D₇₃

semidirect

Subgroups — #22611451

Conjugacy classes of subgroups

Complements