Group #22608633 — subgroups

Subgroups — #22608633

Order 9344.

242 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 65 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	#17989135	C₂	1	1	yes	maximal, characteristic
2	#17988643	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988639	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988334	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988332	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988331	C₂	1	1	yes	maximal, characteristic, complemented
7	#331×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11924027	—	2	1	no	—
9	#14556591	C₂²	1	1	yes	characteristic
10	#14556578	—	2	1	no	—
11	#14556562	—	2	1	no	—
12	#14556562	C₄	1	2	yes	—
13	D₄×Dic₇₃	—	2	1	no	—
14	D₄×Dic₇₃	C₄	1	2	yes	—
15	C₄²×D₇₃	C₂²	1	1	yes	characteristic
16	C₄⋊#10596831	C₂²	1	1	yes	characteristic
17	C₄⋊#10596831	C₂²	1	1	yes	characteristic
18	#14556503	—	2	1	no	—
19	#14556503	—	2	1	no	—
20	#14556501	—	2	1	no	—
21	C₈×Dic₇₃	—	2	1	no	—
22	#127×C₇₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
25	#11924027	—	4	2	no	—
26	#11924027	—	4	2	no	—
27	C₂×Dic₁₄₆	—	2	1	no	—
28	C₂×C₂²⋊D₇₃	—	2	2	no	—
29	C₂²×Dic₇₃	—	2	2	no	—
30	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
32	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
33	C₂×#10596831	—	2	1	no	—
34	C₂×#10596831	—	2	1	no	—
35	C₄⋊Dic₇₃	—	2	1	no	—
36	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
37	C₂²⋊Dic₇₃	—	2	2	no	—
38	C₂²⋊Dic₇₃	—	2	2	no	—
39	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
40	C₄×Dic₇₃	D₄	1	1	yes	characteristic
41	C₄×Dic₇₃	D₄	1	1	yes	characteristic
42	C₂×D₄×C₇₃	—	2	1	no	—
43	C₂×D₄×C₇₃	—	2	1	no	—
44	C₂×D₄×C₇₃	C₂×C₄	1	2	yes	—
45	C₂×C₈×C₇₃	—	2	1	no	—
46	C₂×C₈×C₇₃	—	2	1	no	—
47	C₄²×C₇₃	C₂³	1	1	yes	characteristic
48	C₂³×C₇₃	—	2	2	no	—
49	C₂²×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
50	D₄×C₇₃	—	4	2	no	—
51	D₄×C₇₃	—	4	1	no	—
52	D₄×C₇₃	—	4	1	no	—
53	D₄×C₇₃	—	4	1	no	—
54	D₄×C₇₃	—	4	1	no	—
55	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
58	C₂²⋊D₇₃	—	4	4	no	—
59	C₂×Dic₇₃	—	2	2	no	—
60	C₂×Dic₇₃	—	4	4	no	—
61	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₄×D₇₃	—	2	1	no	—
65	C₄×D₇₃	—	2	1	no	—
66	C₄×D₇₃	—	2	1	no	—
67	C₄×D₇₃	—	2	1	no	—
68	C₄×D₇₃	—	4	1	no	—
69	Dic₁₄₆	—	4	1	no	—
70	Dic₁₄₆	—	4	1	no	—
71	C₈×C₇₃	—	4	1	no	—
72	C₈×C₇₃	—	4	1	no	—
73	#10596831	—	4	1	no	—
74	#10596831	—	4	1	no	—
75	C₂²×C₇₃	—	4	2	no	—
76	C₂²×C₇₃	—	4	2	no	—
77	C₂²×C₇₃	—	4	2	no	—
78	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×D₇₃	—	2	1	no	—
80	C₂×D₇₃	—	2	1	no	—
81	C₂×D₇₃	—	2	1	no	—
82	C₄×C₇₃	—	2	1	no	—
83	C₄×C₇₃	#135	1	1	yes	characteristic
84	C₄×C₇₃	#135	1	1	yes	characteristic
85	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
86	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
87	Dic₇₃	—	2	1	no	—
88	Dic₇₃	—	2	1	no	—
89	Dic₇₃	—	2	1	no	—
90	Dic₇₃	—	4	2	no	—
91	C₂×C₇₃	—	8	2	no	—
92	C₂×C₇₃	#413	1	1	yes	characteristic
93	C₂×C₇₃	#416	1	1	yes	characteristic
94	C₂×C₇₃	#418	1	1	yes	characteristic
95	D₇₃	—	4	1	no	—
96	#1553	—	73	1	no	maximal, Sylow
97	C₇₃	#1553	1	1	yes	characteristic, complemented, Sylow
98	#532	—	73	1	no	—
99	#432	—	73	1	no	—
100	#431	—	73	1	no	—
101	#332	—	73	1	no	—
102	#332	—	73	1	no	—
103	#331	—	73	1	no	—
104	#331	D₇₃	1	1	yes	characteristic, complemented
105	C₂×#41	—	146	1	no	—
106	C₄⋊Q₈	—	73	1	no	—
107	#127	C₂×D₇₃	1	1	yes	characteristic
108	#124	—	146	1	no	—
109	#121	—	73	2	no	—
110	#121	—	146	1	no	—
111	C₄×D₄	—	73	2	no	—
112	C₄×D₄	—	146	1	no	—
113	C₂×C₄²	—	73	1	no	—
114	C₄⋊C₈	—	73	1	no	—
115	C₄⋊C₈	—	73	1	no	—
116	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
117	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
118	C₂²⋊C₈	—	146	1	no	—
119	C₂²⋊C₈	—	146	1	no	—
120	#97	—	146	1	no	—
121	C₄×C₈	—	146	1	no	—
122	#41	—	292	2	no	—
123	#41	—	292	2	no	—
124	C₂×Q₈	—	146	1	no	—
125	C₂×D₄	—	2	1	no	—
126	C₂×D₄	—	2	1	no	—
127	C₂×D₄	—	146	2	no	—
128	C₂×D₄	C₄×D₇₃	1	2	yes	—
129	C₂²×C₄	—	73	1	no	—
130	C₂²×C₄	—	73	1	no	—
131	C₂²×C₄	—	73	1	no	—
132	C₂²×C₄	—	146	2	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	146	1	no	—
136	C₂×C₈	—	146	1	no	—
137	C₄⋊C₄	—	73	2	no	—
138	C₄⋊C₄	—	146	1	no	—
139	C₂²⋊C₄	—	146	2	no	—
140	C₂²⋊C₄	—	146	2	no	—
141	C₄²	—	73	1	no	—
142	C₄²	—	73	1	no	—
143	C₄²	—	73	1	no	—
144	C₄²	C₂²×D₇₃	1	1	yes	characteristic
145	C₂³	—	2	2	no	—
146	C₂³	—	73	1	no	—
147	Q₈	—	292	1	no	—
148	Q₈	—	292	1	no	—
149	D₄	—	4	2	no	—
150	D₄	—	4	1	no	—
151	D₄	—	4	1	no	—
152	D₄	—	4	1	no	—
153	D₄	—	4	1	no	—
154	D₄	—	292	4	no	—
155	C₂×C₄	—	292	1	no	—
156	C₂×C₄	—	292	4	no	—
157	C₂×C₄	—	73	1	no	—
158	C₂×C₄	—	73	1	no	—
159	C₂×C₄	—	73	1	no	—
160	C₂×C₄	—	146	2	no	—
161	C₂×C₄	—	146	1	no	—
162	C₂×C₄	—	146	1	no	—
163	C₂×C₄	—	146	1	no	—
164	C₂×C₄	—	146	1	no	—
165	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
166	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
167	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
168	C₈	—	4	1	no	—
169	C₈	—	4	1	no	—
170	C₈	—	292	1	no	—
171	C₈	—	292	1	no	—
172	C₂²	—	4	2	no	—
173	C₂²	—	4	2	no	—
174	C₂²	—	4	2	no	—
175	C₂²	—	146	1	no	—
176	C₂²	—	146	1	no	—
177	C₂²	—	146	1	no	—
178	C₂²	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
179	C₄	—	2	1	no	—
180	C₄	—	292	2	no	—
181	C₄	—	146	1	no	—
182	C₄	—	146	1	no	—
183	C₄	—	146	1	no	—
184	C₄	#14556610	1	1	yes	characteristic
185	C₄	#14556612	1	1	yes	characteristic
186	C₄	#14556615	1	1	yes	characteristic
187	C₄	#14556618	1	1	yes	characteristic
188	C₂	—	8	2	no	—
189	C₂	—	292	1	no	—
190	C₂	#17988568	1	1	yes	characteristic
191	C₂	#17988580	1	1	yes	characteristic
192	C₂	#17988590	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989135

C₂

yes

maximal, characteristic

#17988643

C₂

yes

maximal, characteristic, complemented

#17988639

C₂

yes

maximal, characteristic, complemented

#17988334

C₂

yes

maximal, characteristic, complemented

#17988332

C₂

yes

maximal, characteristic, complemented

#17988331

C₂

yes

maximal, characteristic, complemented

#331×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924027

—

#14556591

C₂²

yes

—

—

C₄

yes

—

D₄×Dic₇₃

—

D₄×Dic₇₃

C₄

yes

—

C₄²×D₇₃

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

—

—

—

C₈×Dic₇₃

—

#127×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

#11924027

—

#11924027

—

C₂×Dic₁₄₆

—

C₂×C₂²⋊D₇₃

—

C₂²×Dic₇₃

—

C₂×C₄×D₇₃

C₂×C₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×#10596831

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

C₂²⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

C₂×C₄

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

—

C₂²×D₇₃

C₂²⋊C₄

yes

characteristic

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1553

—

maximal, Sylow

C₇₃

#1553

yes

characteristic, complemented, Sylow

#532

—

#432

—

100

#431

—

101

#332

—

102

#332

—

103

#331

—

104

#331

D₇₃

yes

characteristic, complemented

105

C₂×#41

—

146

—

106

C₄⋊Q₈

—

107

#127

C₂×D₇₃

yes

characteristic

108

#124

—

146

—

109

#121

—

110

#121

—

146

—

111

C₄×D₄

—

112

C₄×D₄

—

146

—

113

C₂×C₄²

—

114

C₄⋊C₈

—

115

C₄⋊C₈

—

116

C₄⋊C₈

C₂×D₇₃

yes

characteristic

117

C₄⋊C₈

C₂×D₇₃

yes

characteristic

118

C₂²⋊C₈

—

146

—

119

C₂²⋊C₈

—

146

—

120

#97

—

146

—

121

C₄×C₈

—

146

—

122

#41

—

292

—

123

#41

—

292

—

124

C₂×Q₈

—

146

—

125

C₂×D₄

—

126

C₂×D₄

—

127

C₂×D₄

—

146

—

128

C₂×D₄

C₄×D₇₃

yes

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

146

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

146

—

136

C₂×C₈

—

146

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

146

—

139

C₂²⋊C₄

—

146

—

140

C₂²⋊C₄

—

146

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

C₂²×D₇₃

yes

characteristic

145

C₂³

—

146

C₂³

—

147

Q₈

—

292

—

148

Q₈

—

292

—

149

D₄

—

150

D₄

—

151

D₄

—

152

D₄

—

153

D₄

—

154

D₄

—

292

—

155

C₂×C₄

—

292

—

156

C₂×C₄

—

292

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

146

—

161

C₂×C₄

—

146

—

162

C₂×C₄

—

146

—

163

C₂×C₄

—

146

—

164

C₂×C₄

—

146

—

165

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

166

C₂×C₄

D₄×D₇₃

yes

characteristic

167

C₂×C₄

D₄×D₇₃

yes

characteristic

168

C₈

—

169

C₈

—

170

C₈

—

292

—

171

C₈

—

292

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

—

146

—

176

C₂²

—

146

—

177

C₂²

—

146

—

178

C₂²

C₂²⋊C₄×D₇₃

yes

characteristic

179

C₄

—

180

C₄

—

292

—

181

C₄

—

146

—

182

C₄

—

146

—

183

C₄

—

146

—

184

C₄

#14556610

yes

characteristic

185

C₄

#14556612

yes

characteristic

186

C₄

#14556615

yes

characteristic

187

C₄

#14556618

yes

characteristic

188

C₂

—

189

C₂

—

292

—

190

C₂

#17988568

yes

characteristic

191

C₂

#17988580

yes

characteristic

192

C₂

#17988590

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₄²	C₂²×D₇₃
Fitting	#331×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	#331	D₇₃
hypocenter	C₇₃	#1553
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₇₃

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₄²

C₂²×D₇₃

Fitting

#331×C₇₃

C₂

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
2. #17988643	C₂	semidirect
3. #17988639	C₂	semidirect
4. #17988334	C₂	semidirect
5. #17988332	C₂	semidirect
6. #17988331	C₂	semidirect
7. #331×C₇₃	C₂	semidirect
97. C₇₃	#1553	semidirect
104. #331	D₇₃	semidirect

H ≅ G/N

type

2. #17988643

C₂

semidirect

3. #17988639

C₂

semidirect

4. #17988334

C₂

semidirect

5. #17988332

C₂

semidirect

6. #17988331

C₂

semidirect

7. #331×C₇₃

C₂

semidirect

97. C₇₃

#1553

semidirect

104. #331

D₇₃

semidirect

Subgroups — #22608633

Conjugacy classes of subgroups

Complements