Group #22608451 — subgroups

Subgroups — #22608451

Order 9344.

294 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 85 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	#17988551	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊Dic₇₃	C₂	1	1	yes	maximal, characteristic
3	#17988324	C₂	1	4	yes	maximal, complemented
4	C₂×C₂²⋊Q₈×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11924012	—	2	1	no	—
6	C₂×C₂²⋊Dic₇₃	—	2	1	no	—
7	C₂×C₄×Dic₇₃	—	2	1	no	—
8	C₂²⋊#10596831	—	2	1	no	—
9	C₂²⋊#10596831	C₂²	1	2	yes	—
10	#14556498	C₂²	1	2	yes	—
11	C₂²×Q₈×C₇₃	C₄	1	1	yes	characteristic, complemented
12	C₂³×C₄×C₇₃	C₂²	1	1	yes	characteristic
13	C₂²⋊Q₈×C₇₃	—	2	1	no	—
14	C₂²⋊Q₈×C₇₃	—	2	1	no	—
15	C₂²⋊Q₈×C₇₃	C₄	1	2	yes	complemented
16	C₂²⋊Q₈×C₇₃	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
18	C₂×C₄⋊C₄×C₇₃	C₄	1	1	yes	characteristic, complemented
19	C₂×C₂²⋊C₄×C₇₃	—	2	1	no	—
20	C₂²×Dic₇₃	—	2	2	no	—
21	#11924012	—	4	2	no	—
22	C₂×#10596831	—	2	2	no	—
23	C₂²⋊Dic₇₃	—	2	4	no	—
24	C₄×Dic₇₃	—	4	2	no	—
25	C₂⁴×C₇₃	D₄	1	1	yes	characteristic
26	C₂×Q₈×C₇₃	—	2	1	no	—
27	C₂×Q₈×C₇₃	—	4	2	no	—
28	C₂×Q₈×C₇₃	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₇₃	—	2	1	no	—
30	C₂²×C₄×C₇₃	—	2	1	no	—
31	C₂²×C₄×C₇₃	—	4	2	no	—
32	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₇₃	—	2	2	no	—
39	C₄⋊C₄×C₇₃	—	2	1	no	—
40	C₄⋊C₄×C₇₃	—	2	2	no	—
41	C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
42	C₂²⋊C₄×C₇₃	—	2	2	no	—
43	C₂²⋊C₄×C₇₃	—	2	2	no	—
44	C₂³×C₇₃	—	2	2	no	—
45	C₂³×C₇₃	—	2	2	no	—
46	C₂³×C₇₃	—	4	1	no	—
47	C₂³×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂³×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
50	Q₈×C₇₃	—	4	2	no	—
51	Q₈×C₇₃	—	4	2	no	—
52	C₂×C₄×C₇₃	—	2	2	no	—
53	C₂×C₄×C₇₃	—	2	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	—	4	1	no	—
58	C₂×C₄×C₇₃	—	4	1	no	—
59	C₂×C₄×C₇₃	—	4	1	no	—
60	C₂×C₄×C₇₃	—	4	2	no	—
61	C₂×C₄×C₇₃	—	4	2	no	—
62	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
66	C₂×Dic₇₃	—	2	4	no	—
67	C₂×Dic₇₃	—	4	2	no	—
68	C₂×Dic₇₃	—	4	2	no	—
69	#10596831	—	4	2	no	—
70	C₂²×C₇₃	—	2	1	no	—
71	C₂²×C₇₃	—	2	1	no	—
72	C₂²×C₇₃	—	2	1	no	—
73	C₂²×C₇₃	—	2	1	no	—
74	C₂²×C₇₃	—	2	1	no	—
75	C₂²×C₇₃	—	2	1	no	—
76	C₂²×C₇₃	—	2	1	no	—
77	C₂²×C₇₃	—	4	4	no	—
78	C₂²×C₇₃	#104	1	2	yes	—
79	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₇₃	#137	1	2	yes	—
81	C₄×C₇₃	—	2	1	no	—
82	C₄×C₇₃	—	2	1	no	—
83	C₄×C₇₃	—	2	1	no	—
84	C₄×C₇₃	—	4	2	no	—
85	C₄×C₇₃	—	4	2	no	—
86	C₄×C₇₃	#99	1	2	yes	—
87	Dic₇₃	—	4	4	no	—
88	C₂×C₇₃	—	2	2	no	—
89	C₂×C₇₃	—	2	2	no	—
90	C₂×C₇₃	—	4	1	no	—
91	C₂×C₇₃	C₂×#99	1	1	yes	characteristic
92	C₂×C₇₃	#419	1	1	yes	characteristic
93	C₂×C₇₃	C₂×#104	1	1	yes	characteristic
94	#1508	—	73	1	no	maximal, Sylow
95	C₇₃	#1508	1	1	yes	characteristic, complemented, Sylow
96	C₂×C₂²⋊Q₈	D₇₃	1	1	yes	characteristic
97	#407	—	73	1	no	—
98	C₄×C₂²⋊C₄	—	73	1	no	—
99	#328	—	73	4	no	—
100	C₂²×Q₈	Dic₇₃	1	1	yes	characteristic, complemented
101	C₂³×C₄	C₂×D₇₃	1	1	yes	characteristic
102	C₂×M₁₆	—	146	1	no	—
103	C₂²⋊Q₈	—	2	1	no	—
104	C₂²⋊Q₈	—	2	1	no	—
105	C₂²⋊Q₈	Dic₇₃	1	2	yes	complemented
106	C₂²⋊Q₈	C₂×D₇₃	1	2	yes	—
107	C₂×C₄⋊C₄	—	2	1	no	—
108	C₂×C₄⋊C₄	Dic₇₃	1	1	yes	characteristic, complemented
109	C₂×C₂²⋊C₄	—	2	1	no	—
110	C₂×C₂²⋊C₄	—	146	1	no	—
111	C₂×C₄²	—	146	1	no	—
112	C₂²⋊C₈	—	73	2	no	—
113	C₂²⋊C₈	—	146	1	no	—
114	#95	—	73	2	no	—
115	C₂⁴	C₂²⋊D₇₃	1	1	yes	characteristic
116	C₂×Q₈	—	2	1	no	—
117	C₂×Q₈	—	4	2	no	—
118	C₂×Q₈	C₂×Dic₇₃	1	2	yes	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	—	4	2	no	—
122	C₂²×C₄	—	146	2	no	—
123	C₂²×C₄	C₂×Dic₇₃	1	1	yes	characteristic
124	C₂²×C₄	C₂×Dic₇₃	1	1	yes	characteristic
125	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
126	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
127	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
128	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
129	M₁₆	—	292	2	no	—
130	C₂×C₈	—	146	2	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	C₂×Dic₇₃	1	2	yes	—
135	C₂²⋊C₄	—	2	2	no	—
136	C₂²⋊C₄	—	2	2	no	—
137	C₂²⋊C₄	—	146	4	no	—
138	C₄²	—	292	2	no	—
139	C₂³	—	2	2	no	—
140	C₂³	—	2	2	no	—
141	C₂³	—	4	1	no	—
142	C₂³	C₂²⋊Dic₇₃	1	1	yes	characteristic
143	C₂³	C₂²⋊Dic₇₃	1	1	yes	characteristic
144	C₂³	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
145	Q₈	—	4	2	no	—
146	Q₈	—	4	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	292	2	no	—
158	C₂×C₄	—	292	2	no	—
159	C₂×C₄	—	146	4	no	—
160	C₂×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
161	C₂×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
162	C₂×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
163	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
164	C₈	—	292	2	no	—
165	C₂²	—	2	1	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	1	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	—	2	1	no	—
172	C₂²	—	4	4	no	—
173	C₂²	#14556518	1	2	yes	—
174	C₂²	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
175	C₂²	#14556622	1	2	yes	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	4	2	no	—
180	C₄	—	4	2	no	—
181	C₄	—	292	4	no	—
182	C₄	#14556506	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	—	2	2	no	—
185	C₂	—	4	1	no	—
186	C₂	C₂×#14556506	1	1	yes	characteristic
187	C₂	#17988596	1	1	yes	characteristic
188	C₂	C₂×#14556518	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988551

C₂

yes

maximal, characteristic

C₄×C₂²⋊Dic₇₃

C₂

yes

maximal, characteristic

#17988324

C₂

yes

maximal, complemented

C₂×C₂²⋊Q₈×C₇₃

C₂

yes

maximal, characteristic

C₂×#11924012

—

C₂×C₂²⋊Dic₇₃

—

C₂×C₄×Dic₇₃

—

C₂²⋊#10596831

—

C₂²⋊#10596831

C₂²

yes

—

#14556498

C₂²

yes

—

C₂²×Q₈×C₇₃

C₄

yes

characteristic, complemented

C₂³×C₄×C₇₃

C₂²

yes

characteristic

C₂²⋊Q₈×C₇₃

—

C₂²⋊Q₈×C₇₃

—

C₂²⋊Q₈×C₇₃

C₄

yes

complemented

C₂²⋊Q₈×C₇₃

C₂²

yes

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

C₄

yes

characteristic, complemented

C₂×C₂²⋊C₄×C₇₃

—

C₂²×Dic₇₃

—

#11924012

—

C₂×#10596831

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₂⁴×C₇₃

D₄

yes

characteristic

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

C₂×C₄

yes

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₂²⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

C₂²⋊C₄

yes

characteristic

C₂³×C₇₃

C₂²⋊C₄

yes

characteristic

C₂³×C₇₃

C₂×D₄

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#104

yes

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₃

#137

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#99

yes

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

C₂×#99

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×C₂²⋊Q₈

D₇₃

yes

characteristic

#407

—

C₄×C₂²⋊C₄

—

#328

—

100

C₂²×Q₈

Dic₇₃

yes

characteristic, complemented

101

C₂³×C₄

C₂×D₇₃

yes

characteristic

102

C₂×M₁₆

—

146

—

103

C₂²⋊Q₈

—

104

C₂²⋊Q₈

—

105

C₂²⋊Q₈

Dic₇₃

yes

complemented

106

C₂²⋊Q₈

C₂×D₇₃

yes

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

Dic₇₃

yes

characteristic, complemented

109

C₂×C₂²⋊C₄

—

110

C₂×C₂²⋊C₄

—

146

—

111

C₂×C₄²

—

146

—

112

C₂²⋊C₈

—

113

C₂²⋊C₈

—

146

—

114

#95

—

115

C₂⁴

C₂²⋊D₇₃

yes

characteristic

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

C₂×Dic₇₃

yes

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

146

—

123

C₂²×C₄

C₂×Dic₇₃

yes

characteristic

124

C₂²×C₄

C₂×Dic₇₃

yes

characteristic

125

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

126

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

127

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

128

C₂²×C₄

C₂²×D₇₃

yes

characteristic

129

M₁₆

—

292

—

130

C₂×C₈

—

146

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

C₂×Dic₇₃

yes

—

135

C₂²⋊C₄

—

136

C₂²⋊C₄

—

137

C₂²⋊C₄

—

146

—

138

C₄²

—

292

—

139

C₂³

—

140

C₂³

—

141

C₂³

—

142

C₂³

C₂²⋊Dic₇₃

yes

characteristic

143

C₂³

C₂²⋊Dic₇₃

yes

characteristic

144

C₂³

C₂×C₂²⋊D₇₃

yes

characteristic

145

Q₈

—

146

Q₈

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

292

—

158

C₂×C₄

—

292

—

159

C₂×C₄

—

146

—

160

C₂×C₄

C₂²⋊Dic₇₃

yes

characteristic

161

C₂×C₄

C₂²⋊Dic₇₃

yes

characteristic

162

C₂×C₄

C₂²×Dic₇₃

yes

characteristic

163

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

164

C₈

—

292

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

#14556518

yes

—

174

C₂²

C₂×C₂²⋊Dic₇₃

yes

characteristic

175

C₂²

#14556622

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

292

—

182

C₄

#14556506

yes

—

183

C₂

—

184

C₂

—

185

C₂

—

186

C₂

C₂×#14556506

yes

characteristic

187

C₂

#17988596

yes

characteristic

188

C₂

C₂×#14556518

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₇₃
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	C₂×C₂²⋊Q₈×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	C₂×C₂²⋊Q₈	D₇₃
hypocenter	C₇₃	#1508
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×C₂²⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₇₃

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

C₂×C₂²⋊Q₈×C₇₃

C₂

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

hypercenter

C₂×C₂²⋊Q₈

D₇₃

hypocenter

C₇₃

#1508

LCS

G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×C₂²⋊Q₈

N	H ≅ G/N	type
3. #17988324	C₂	semidirect
11. C₂²×Q₈×C₇₃	C₄	semidirect
15. C₂²⋊Q₈×C₇₃	C₄	semidirect
18. C₂×C₄⋊C₄×C₇₃	C₄	semidirect
95. C₇₃	#1508	semidirect
100. C₂²×Q₈	Dic₇₃	semidirect
105. C₂²⋊Q₈	Dic₇₃	semidirect
108. C₂×C₄⋊C₄	Dic₇₃	semidirect

H ≅ G/N

type

3. #17988324

C₂

semidirect

11. C₂²×Q₈×C₇₃

C₄

semidirect

15. C₂²⋊Q₈×C₇₃

C₄

semidirect

18. C₂×C₄⋊C₄×C₇₃

C₄

semidirect

95. C₇₃

#1508

semidirect

100. C₂²×Q₈

Dic₇₃

semidirect

105. C₂²⋊Q₈

Dic₇₃

semidirect

108. C₂×C₄⋊C₄

Dic₇₃

semidirect

Subgroups — #22608451

Conjugacy classes of subgroups

Complements