Group #22620426 — subgroups

Subgroups — #22620426

Order 9344.

532 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14556621	C₂	1	2	yes	maximal, complemented
2	C₂²×Dic₂₉₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#17989158	C₂	1	2	yes	maximal, complemented
4	#17988978	C₂	1	4	yes	maximal, complemented
5	#17988970	C₂	1	2	yes	maximal, complemented
6	#17988965	C₂	1	2	yes	maximal, complemented
7	#17988542	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₄₆	C₂²	1	2	yes	—
10	#14556621	—	2	8	no	—
11	C₂×Dic₂₉₂	—	2	4	no	—
12	C₂×Dic₂₉₂	C₂²	1	2	yes	—
13	C₂×Dic₂₉₂	C₂²	1	2	yes	—
14	C₂×#11924018	C₂²	1	4	yes	—
15	C₂×#11924016	C₂²	1	4	yes	—
16	C₂×#11924012	C₂²	1	1	yes	characteristic
17	#14556589	C₂²	1	4	yes	—
18	#14556578	C₂²	1	4	yes	—
19	#14556572	—	2	4	no	—
20	#14556570	—	2	4	no	—
21	#14556550	C₂²	1	1	yes	characteristic
22	#14556501	C₂²	1	2	yes	—
23	C₈×Dic₇₃	C₂²	1	2	yes	—
24	C₂×#41×C₇₃	C₂²	1	2	yes	—
25	#135×C₇₃	—	2	4	no	—
26	C₂×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
27	C₂×SD₁₆×C₇₃	C₂²	1	2	yes	—
28	C₂×D₈×C₇₃	C₂²	1	1	yes	characteristic
29	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
30	C₂×Dic₁₄₆	—	2	4	no	—
31	C₂×Dic₁₄₆	—	2	4	no	—
32	C₂×Dic₁₄₆	—	2	2	no	—
33	C₂×Dic₁₄₆	C₂³	1	4	yes	—
34	C₂²×Dic₇₃	—	2	2	no	—
35	Dic₂₉₂	—	2	4	no	—
36	Dic₂₉₂	—	2	4	no	—
37	#11924018	—	2	8	no	—
38	#11924016	—	2	8	no	—
39	#11924012	D₄	1	4	yes	—
40	C₂×#10596831	C₂³	1	2	yes	—
41	C₄⋊Dic₇₃	D₄	1	2	yes	—
42	#11924007	—	2	8	no	—
43	C₂²⋊Dic₇₃	—	2	8	no	—
44	C₂²⋊Dic₇₃	D₄	1	2	yes	—
45	C₄×Dic₇₃	C₂³	1	2	yes	—
46	#41×C₇₃	—	2	4	no	—
47	#41×C₇₃	—	2	4	no	—
48	C₂×Q₈×C₇₃	C₂³	1	2	yes	—
49	C₂×D₄×C₇₃	—	2	2	no	—
50	C₂×D₄×C₇₃	C₂³	1	2	yes	—
51	C₂²×C₄×C₇₃	—	2	2	no	—
52	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
53	Q₁₆×C₇₃	—	2	2	no	—
54	SD₁₆×C₇₃	—	2	4	no	—
55	D₈×C₇₃	—	2	2	no	—
56	C₂×C₈×C₇₃	D₄	1	4	yes	—
57	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
59	C₂³×C₇₃	—	2	2	no	—
60	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
61	Q₈×C₇₃	—	2	2	no	—
62	Q₈×C₇₃	—	2	2	no	—
63	D₄×C₇₃	—	2	2	no	—
64	D₄×C₇₃	—	2	2	no	—
65	D₄×C₇₃	—	4	4	no	—
66	C₂×C₄×C₇₃	—	2	2	no	—
67	C₂×C₄×C₇₃	—	4	4	no	—
68	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
72	C₂×Dic₇₃	—	2	4	no	—
73	C₂×Dic₇₃	—	4	4	no	—
74	C₂×Dic₇₃	C₂×D₄	1	4	yes	—
75	Dic₁₄₆	—	2	4	no	—
76	Dic₁₄₆	—	2	4	no	—
77	Dic₁₄₆	—	4	4	no	—
78	C₈×C₇₃	C₂×D₄	1	2	yes	—
79	C₈×C₇₃	C₂×D₄	1	2	yes	—
80	#10596831	C₂×D₄	1	4	yes	—
81	C₂²×C₇₃	—	2	2	no	—
82	C₂²×C₇₃	—	4	2	no	—
83	C₂²×C₇₃	—	4	4	no	—
84	C₂²×C₇₃	#127	1	2	yes	—
85	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
86	C₄×C₇₃	—	4	2	no	—
87	C₄×C₇₃	#127	1	2	yes	—
88	C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
89	C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
90	Dic₇₃	—	2	4	no	—
91	Dic₇₃	—	4	4	no	—
92	C₂×C₇₃	—	2	2	no	—
93	C₂×C₇₃	—	8	2	no	—
94	C₂×C₇₃	C₂×#127	1	1	yes	characteristic
95	C₂×C₇₃	#578	1	2	yes	—
96	#3156	—	73	1	no	maximal, Sylow
97	C₇₃	#3156	1	1	yes	characteristic, complemented, Sylow
98	C₂×#137	—	73	2	no	—
99	C₂×#135	D₇₃	1	1	yes	characteristic
100	C₂²×Q₁₆	—	73	1	no	—
101	#536	—	73	2	no	—
102	#497	—	73	4	no	—
103	#495	—	73	2	no	—
104	#494	—	73	2	no	—
105	#405	—	73	1	no	—
106	C₂×#41	C₂×D₇₃	1	2	yes	—
107	C₂²×Q₈	—	73	2	no	—
108	#137	—	146	8	no	—
109	#135	—	2	4	no	—
110	C₂×Q₁₆	—	73	4	no	—
111	C₂×Q₁₆	—	73	2	no	—
112	C₂×Q₁₆	—	73	2	no	—
113	C₂×Q₁₆	—	146	4	no	—
114	C₂×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
115	C₂×SD₁₆	—	73	4	no	—
116	C₂×SD₁₆	C₂×D₇₃	1	2	yes	—
117	C₂×D₈	C₂×D₇₃	1	1	yes	characteristic
118	C₂×M₁₆	—	73	1	no	—
119	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
120	C₄⋊Q₈	—	73	4	no	—
121	#124	—	73	4	no	—
122	#123	—	146	4	no	—
123	C₂²⋊Q₈	—	146	4	no	—
124	#117	—	73	1	no	—
125	#97	—	73	2	no	—
126	C₄×C₈	—	73	2	no	—
127	#41	—	2	4	no	—
128	#41	—	2	4	no	—
129	C₂×Q₈	—	73	4	no	—
130	C₂×Q₈	—	146	4	no	—
131	C₂×Q₈	—	146	4	no	—
132	C₂×Q₈	—	146	2	no	—
133	C₂×Q₈	C₂²×D₇₃	1	2	yes	—
134	C₂×D₄	—	2	2	no	—
135	C₂×D₄	C₂²×D₇₃	1	2	yes	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	—	146	2	no	—
138	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
139	Q₁₆	—	2	2	no	—
140	Q₁₆	—	146	8	no	—
141	Q₁₆	—	146	4	no	—
142	Q₁₆	—	146	4	no	—
143	SD₁₆	—	2	4	no	—
144	SD₁₆	—	146	8	no	—
145	D₈	—	2	2	no	—
146	M₁₆	—	73	4	no	—
147	C₂×C₈	—	73	2	no	—
148	C₂×C₈	C₂²⋊D₇₃	1	4	yes	—
149	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
150	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
151	C₄⋊C₄	—	73	2	no	—
152	C₄⋊C₄	—	146	8	no	—
153	C₂²⋊C₄	—	73	2	no	—
154	C₂²⋊C₄	—	146	8	no	—
155	C₄²	—	73	2	no	—
156	C₂³	—	2	2	no	—
157	C₂³	D₄×D₇₃	1	1	yes	characteristic
158	Q₈	—	2	2	no	—
159	Q₈	—	2	2	no	—
160	Q₈	—	292	4	no	—
161	Q₈	—	146	4	no	—
162	Q₈	—	146	4	no	—
163	D₄	—	2	2	no	—
164	D₄	—	2	2	no	—
165	D₄	—	4	4	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	—	292	4	no	—
169	C₂×C₄	—	73	4	no	—
170	C₂×C₄	—	146	4	no	—
171	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
172	C₂×C₄	D₄×D₇₃	1	2	yes	—
173	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
174	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
175	C₈	—	73	4	no	—
176	C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
177	C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
178	C₂²	—	2	2	no	—
179	C₂²	—	4	2	no	—
180	C₂²	—	4	4	no	—
181	C₂²	#14556587	1	2	yes	—
182	C₂²	C₂×D₄×D₇₃	1	1	yes	characteristic
183	C₄	—	4	2	no	—
184	C₄	—	292	4	no	—
185	C₄	—	146	4	no	—
186	C₄	#14556587	1	2	yes	—
187	C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
188	C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
189	C₂	—	2	2	no	—
190	C₂	—	8	2	no	—
191	C₂	C₂×#14556587	1	1	yes	characteristic
192	C₂	#17989401	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556621

C₂

yes

maximal, complemented

C₂²×Dic₂₉₂

C₂

yes

maximal, characteristic, complemented

#17989158

C₂

yes

maximal, complemented

#17988978

C₂

yes

maximal, complemented

#17988970

C₂

yes

maximal, complemented

#17988965

C₂

yes

maximal, complemented

#17988542

C₂

yes

maximal, characteristic, complemented

C₂×#135×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₆

C₂²

yes

—

#14556621

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

C₂²

yes

—

C₂×Dic₂₉₂

C₂²

yes

—

C₂×#11924018

C₂²

yes

—

C₂×#11924016

C₂²

yes

—

C₂×#11924012

C₂²

yes

characteristic

#14556589

C₂²

yes

—

#14556578

C₂²

yes

—

#14556572

—

#14556570

—

#14556550

C₂²

yes

characteristic

#14556501

C₂²

yes

—

C₈×Dic₇₃

C₂²

yes

—

C₂×#41×C₇₃

C₂²

yes

—

#135×C₇₃

—

C₂×Q₁₆×C₇₃

C₂²

yes

characteristic

C₂×SD₁₆×C₇₃

C₂²

yes

—

C₂×D₈×C₇₃

C₂²

yes

characteristic

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

C₂³

yes

—

C₂²×Dic₇₃

—

Dic₂₉₂

—

Dic₂₉₂

—

#11924018

—

#11924016

—

#11924012

D₄

yes

—

C₂×#10596831

C₂³

yes

—

C₄⋊Dic₇₃

D₄

yes

—

#11924007

—

C₂²⋊Dic₇₃

—

C₂²⋊Dic₇₃

D₄

yes

—

C₄×Dic₇₃

C₂³

yes

—

#41×C₇₃

—

#41×C₇₃

—

C₂×Q₈×C₇₃

C₂³

yes

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

C₂³

yes

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

Q₁₆×C₇₃

—

SD₁₆×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

—

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂⁴

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×D₄

yes

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

C₂×D₄

yes

—

C₈×C₇₃

C₂×D₄

yes

—

#10596831

C₂×D₄

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#127

yes

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

#127

yes

—

C₄×C₇₃

C₂²×D₄

yes

characteristic

C₄×C₇₃

C₂²×D₄

yes

characteristic

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

C₂×#127

yes

characteristic

C₂×C₇₃

#578

yes

—

#3156

—

maximal, Sylow

C₇₃

#3156

yes

characteristic, complemented, Sylow

C₂×#137

—

C₂×#135

D₇₃

yes

characteristic

100

C₂²×Q₁₆

—

101

#536

—

102

#497

—

103

#495

—

104

#494

—

105

#405

—

106

C₂×#41

C₂×D₇₃

yes

—

107

C₂²×Q₈

—

108

#137

—

146

—

109

#135

—

110

C₂×Q₁₆

—

111

C₂×Q₁₆

—

112

C₂×Q₁₆

—

113

C₂×Q₁₆

—

146

—

114

C₂×Q₁₆

C₂×D₇₃

yes

characteristic

115

C₂×SD₁₆

—

116

C₂×SD₁₆

C₂×D₇₃

yes

—

117

C₂×D₈

C₂×D₇₃

yes

characteristic

118

C₂×M₁₆

—

119

C₂²×C₈

C₂×D₇₃

yes

characteristic

120

C₄⋊Q₈

—

121

#124

—

122

#123

—

146

—

123

C₂²⋊Q₈

—

146

—

124

#117

—

125

#97

—

126

C₄×C₈

—

127

#41

—

128

#41

—

129

C₂×Q₈

—

130

C₂×Q₈

—

146

—

131

C₂×Q₈

—

146

—

132

C₂×Q₈

—

146

—

133

C₂×Q₈

C₂²×D₇₃

yes

—

134

C₂×D₄

—

135

C₂×D₄

C₂²×D₇₃

yes

—

136

C₂²×C₄

—

137

C₂²×C₄

—

146

—

138

C₂²×C₄

C₂²×D₇₃

yes

characteristic

139

Q₁₆

—

140

Q₁₆

—

146

—

141

Q₁₆

—

146

—

142

Q₁₆

—

146

—

143

SD₁₆

—

144

SD₁₆

—

146

—

145

D₈

—

146

M₁₆

—

147

C₂×C₈

—

148

C₂×C₈

C₂²⋊D₇₃

yes

—

149

C₂×C₈

C₂²×D₇₃

yes

characteristic

150

C₂×C₈

C₂²×D₇₃

yes

characteristic

151

C₄⋊C₄

—

152

C₄⋊C₄

—

146

—

153

C₂²⋊C₄

—

154

C₂²⋊C₄

—

146

—

155

C₄²

—

156

C₂³

—

157

C₂³

D₄×D₇₃

yes

characteristic

158

Q₈

—

159

Q₈

—

160

Q₈

—

292

—

161

Q₈

—

146

—

162

Q₈

—

146

—

163

D₄

—

164

D₄

—

165

D₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

292

—

169

C₂×C₄

—

170

C₂×C₄

—

146

—

171

C₂×C₄

D₄×D₇₃

yes

characteristic

172

C₂×C₄

D₄×D₇₃

yes

—

173

C₂×C₄

C₂×C₂²⋊D₇₃

yes

—

174

C₂×C₄

C₂³×D₇₃

yes

characteristic

175

C₈

—

176

C₈

C₂×C₂²⋊D₇₃

yes

—

177

C₈

C₂×C₂²⋊D₇₃

yes

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

#14556587

yes

—

182

C₂²

C₂×D₄×D₇₃

yes

characteristic

183

C₄

—

184

C₄

—

292

—

185

C₄

—

146

—

186

C₄

#14556587

yes

—

187

C₄

C₂×D₄×D₇₃

yes

characteristic

188

C₄

C₂²×C₂²⋊D₇₃

yes

characteristic

189

C₂

—

190

C₂

—

191

C₂

C₂×#14556587

yes

characteristic

192

C₂

#17989401

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₃
Fitting	C₂×#135×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₄
hypercenter	C₂×#135	D₇₃
hypocenter	C₇₃	#3156
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#135

subgroup

G/H

center

C₂²

C₂×D₄×D₇₃

commutator

C₂×C₄×C₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₃

Fitting

C₂×#135×C₇₃

C₂

socle

C₂²×C₇₃

C₂²×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#135

N	H ≅ G/N	type
1. C₂×#14556621	C₂	semidirect
2. C₂²×Dic₂₉₂	C₂	semidirect
3. #17989158	C₂	semidirect
4. #17988978	C₂	semidirect
5. #17988970	C₂	semidirect
6. #17988965	C₂	semidirect
7. #17988542	C₂	semidirect
97. C₇₃	#3156	semidirect

H ≅ G/N

type

1. C₂×#14556621

C₂

semidirect

2. C₂²×Dic₂₉₂

C₂

semidirect

3. #17989158

C₂

semidirect

4. #17988978

C₂

semidirect

5. #17988970

C₂

semidirect

6. #17988965

C₂

semidirect

7. #17988542

C₂

semidirect

97. C₇₃

#3156

semidirect

Subgroups — #22620426

Conjugacy classes of subgroups

Complements