Group #34034358 — subgroups

Subgroups — #34034358

Order 18688.

398 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 145 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×C₄⋊#11924040	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11924040	C₂	1	1	yes	maximal, characteristic
3	#22627001	C₂	1	4	yes	maximal
4	C₂×Q₈×#10596833	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924040	C₂²	1	2	yes	—
6	C₂²×#11924040	C₂²	1	2	yes	—
7	C₄⋊#11924040	—	2	2	no	—
8	C₄×#11924040	—	2	2	no	—
9	Q₈×#10596833	—	2	2	no	—
10	Q₈×#10596833	—	2	2	no	—
11	C₂×C₄⋊#10596833	C₄	1	1	yes	characteristic
12	C₂×C₄⋊#10596833	C₄	1	2	yes	—
13	C₂×C₄×#10596833	C₂²	1	2	yes	—
14	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
15	C₂×Q₈×Dic₇₃	C₄	1	1	yes	characteristic
16	C₂×#11924040	—	2	2	no	—
17	C₂×#11924040	—	2	2	no	—
18	C₂×#11924040	—	2	2	no	—
19	C₂×#11924040	—	2	4	no	—
20	C₂×#11924040	Q₈	1	4	yes	complemented
21	C₂²×#10596833	C₂×C₄	1	2	yes	—
22	C₂²×#10596833	C₂×C₄	1	1	yes	characteristic
23	C₂²×#10596833	C₂³	1	1	yes	characteristic
24	C₄⋊#10596833	—	2	4	no	—
25	C₄⋊#10596833	—	2	2	no	—
26	C₄×#10596833	—	2	2	no	—
27	C₄×#10596833	—	2	2	no	—
28	C₄×#10596833	D₄	1	2	yes	—
29	C₄×#10596833	D₄	1	2	yes	—
30	Q₈×Dic₇₃	—	2	2	no	—
31	Q₈×Dic₇₃	—	2	2	no	—
32	C₂×C₄⋊Dic₇₃	C₈	1	2	yes	—
33	C₂×C₄⋊Dic₇₃	C₈	1	1	yes	characteristic
34	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
35	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×Q₈×C₇₃	C₈	1	1	yes	characteristic
37	#11924040	—	2	4	no	—
38	#11924040	—	4	2	no	—
39	C₂×#10596833	—	2	2	no	—
40	C₂×#10596833	—	2	2	no	—
41	C₂×#10596833	—	2	4	no	—
42	C₂×#10596833	C₂²⋊C₄	1	2	yes	—
43	C₂×#10596833	C₂×D₄	1	1	yes	characteristic
44	C₂×#10596833	C₂×D₄	1	1	yes	characteristic
45	C₂×#10596833	C₂×Q₈	1	2	yes	—
46	C₂×#10596833	#41	1	2	yes	—
47	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
48	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
49	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
50	C₄⋊Dic₇₃	—	2	2	no	—
51	C₄⋊Dic₇₃	—	2	4	no	—
52	C₄×Dic₇₃	—	2	2	no	—
53	C₄×Dic₇₃	—	2	2	no	—
54	C₄×Dic₇₃	C₂²⋊C₄	1	2	yes	—
55	C₄×Dic₇₃	M₁₆	1	2	yes	—
56	C₂×Q₈×C₇₃	—	2	2	no	—
57	C₂×Q₈×C₇₃	—	2	2	no	—
58	C₂×Q₈×C₇₃	C₁₆	1	4	yes	complemented
59	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
61	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
62	Q₈×C₇₃	—	2	8	no	—
63	C₂×C₄×C₇₃	—	2	2	no	—
64	C₂×C₄×C₇₃	—	2	2	no	—
65	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
66	C₂×C₄×C₇₃	C₂×C₁₆	1	4	yes	—
67	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
68	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
69	C₂×Dic₇₃	—	2	2	no	—
70	C₂×Dic₇₃	—	2	2	no	—
71	C₂×Dic₇₃	—	2	4	no	—
72	C₂×Dic₇₃	C₂²⋊C₈	1	2	yes	—
73	C₂×Dic₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×Dic₇₃	C₄×Q₈	1	2	yes	—
75	C₂×Dic₇₃	C₂×M₁₆	1	1	yes	characteristic
76	C₂×Dic₇₃	#131	1	2	yes	—
77	#10596833	—	2	2	no	—
78	#10596833	—	4	2	no	—
79	#10596833	C₂²⋊Q₈	1	2	yes	—
80	#10596833	#124	1	1	yes	characteristic
81	#10596833	C₄⋊Q₈	1	1	yes	characteristic
82	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
83	C₂²×C₇₃	C₈×Q₈	1	2	yes	—
84	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
85	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
86	C₂²×C₇₃	#504	1	2	yes	—
87	C₄×C₇₃	—	2	4	no	—
88	C₄×C₇₃	C₂²⋊C₁₆	1	4	yes	—
89	Dic₇₃	—	2	2	no	—
90	Dic₇₃	—	4	2	no	—
91	Dic₇₃	#391	1	1	yes	characteristic
92	Dic₇₃	#408	1	1	yes	characteristic
93	Dic₇₃	#446	1	2	yes	—
94	#13263	—	73	1	no	maximal, Sylow
95	C₂×C₇₃	#1984	1	1	yes	characteristic
96	C₂×C₇₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
97	C₂×C₇₃	#2113	1	1	yes	characteristic
98	C₂×C₇₃	Q₈×C₁₆	1	2	yes	—
99	C₂×C₇₃	#2183	1	2	yes	—
100	C₂×C₈×Q₈	—	73	1	no	—
101	C₂×C₄⋊C₁₆	—	73	1	no	—
102	C₂×C₄×C₁₆	—	73	1	no	—
103	#1374	—	73	4	no	—
104	C₇₃	#13263	1	1	yes	characteristic, complemented, Sylow
105	C₂×C₄×Q₈	—	73	1	no	—
106	C₂²×C₁₆	—	73	2	no	—
107	C₂²×C₁₆	—	73	2	no	—
108	C₈×Q₈	—	146	2	no	—
109	C₈×Q₈	—	146	2	no	—
110	C₂×C₄⋊C₈	—	73	1	no	—
111	C₂×C₄⋊C₈	—	73	2	no	—
112	C₂×C₄×C₈	—	73	1	no	—
113	C₂×C₄×C₈	—	73	2	no	—
114	C₄⋊C₁₆	—	146	2	no	—
115	C₄×C₁₆	—	146	2	no	—
116	C₂²×Q₈	#10596842	1	1	yes	characteristic
117	C₂²×C₈	—	73	1	no	—
118	C₂²×C₈	—	73	1	no	—
119	C₂²×C₈	—	73	2	no	—
120	C₄×Q₈	—	146	2	no	—
121	C₄×Q₈	—	146	2	no	—
122	C₂×C₄⋊C₄	—	73	2	no	—
123	C₂×C₄⋊C₄	—	73	1	no	—
124	C₂×C₄²	—	73	1	no	—
125	C₂×C₄²	—	73	2	no	—
126	C₂×C₁₆	—	73	4	no	—
127	C₂×C₁₆	—	146	2	no	—
128	C₂×C₁₆	—	146	4	no	—
129	C₂×C₁₆	—	146	2	no	—
130	C₂×C₁₆	—	146	2	no	—
131	C₄⋊C₈	—	146	2	no	—
132	C₄⋊C₈	—	146	4	no	—
133	C₄×C₈	—	73	2	no	—
134	C₄×C₈	—	73	2	no	—
135	C₄×C₈	—	146	2	no	—
136	C₄×C₈	—	146	2	no	—
137	C₂×Q₈	—	2	2	no	—
138	C₂×Q₈	—	2	2	no	—
139	C₂×Q₈	#11924040	1	4	yes	complemented
140	C₂²×C₄	—	73	1	no	—
141	C₂²×C₄	—	73	1	no	—
142	C₂²×C₄	—	73	2	no	—
143	C₂²×C₄	C₂×#10596842	1	2	yes	—
144	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
145	C₂×C₈	—	73	1	no	—
146	C₂×C₈	—	73	1	no	—
147	C₂×C₈	—	73	2	no	—
148	C₂×C₈	—	73	2	no	—
149	C₂×C₈	—	73	2	no	—
150	C₂×C₈	—	146	2	no	—
151	C₂×C₈	—	146	4	no	—
152	C₂×C₈	—	146	2	no	—
153	C₄⋊C₄	—	146	4	no	—
154	C₄⋊C₄	—	146	2	no	—
155	C₄²	—	73	2	no	—
156	C₄²	—	73	2	no	—
157	C₄²	—	146	2	no	—
158	C₄²	—	146	2	no	—
159	C₁₆	—	292	2	no	—
160	C₁₆	—	146	4	no	—
161	C₂³	C₂²×#10596842	1	1	yes	characteristic
162	Q₈	—	2	8	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	73	1	no	—
166	C₂×C₄	—	73	1	no	—
167	C₂×C₄	—	73	2	no	—
168	C₂×C₄	—	73	2	no	—
169	C₂×C₄	—	73	2	no	—
170	C₂×C₄	—	146	2	no	—
171	C₂×C₄	—	146	4	no	—
172	C₂×C₄	—	146	2	no	—
173	C₂×C₄	C₂²⋊#10596842	1	2	yes	—
174	C₂×C₄	C₂×#11924040	1	4	yes	—
175	C₂×C₄	C₂×#11924040	1	2	yes	—
176	C₂×C₄	#14556687	1	2	yes	—
177	C₈	—	292	2	no	—
178	C₈	—	73	1	no	—
179	C₈	—	73	2	no	—
180	C₈	—	73	1	no	—
181	C₈	—	146	2	no	—
182	C₂²	C₂×C₂²⋊#10596842	1	1	yes	characteristic
183	C₂²	Q₈×#10596842	1	2	yes	—
184	C₂²	C₂²×#11924040	1	1	yes	characteristic
185	C₂²	C₂×#14556687	1	1	yes	characteristic
186	C₂²	#17989709	1	2	yes	—
187	C₄	—	2	4	no	—
188	C₄	—	292	2	no	—
189	C₄	—	73	1	no	—
190	C₄	—	73	1	no	—
191	C₄	—	73	2	no	—
192	C₄	—	146	2	no	—
193	C₄	C₂²⋊#11924040	1	4	yes	—
194	C₂	(C₂×#10596842)⋊Q₈	1	1	yes	characteristic
195	C₂	C₂×C₂²⋊#11924040	1	1	yes	characteristic
196	C₂	#22627103	1	1	yes	characteristic
197	C₂	Q₈×#11924040	1	2	yes	—
198	C₂	#22627157	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11924040

C₂

yes

maximal, characteristic

C₂×C₄×#11924040

C₂

yes

maximal, characteristic

#22627001

C₂

yes

maximal

C₂×Q₈×#10596833

C₂

yes

maximal, characteristic

C₂²×#11924040

C₂²

yes

—

C₂²×#11924040

C₂²

yes

—

—

—

—

—

C₄

yes

characteristic

C₂×C₄⋊#10596833

C₄

yes

—

C₂×C₄×#10596833

C₂²

yes

—

C₂×C₄×#10596833

C₂²

yes

characteristic

C₂×Q₈×Dic₇₃

C₄

yes

—

—

—

—

Q₈

yes

complemented

C₂²×#10596833

C₂×C₄

yes

—

C₂²×#10596833

C₂×C₄

yes

characteristic

C₂²×#10596833

C₂³

yes

—

—

—

—

D₄

yes

—

C₄×#10596833

D₄

yes

—

Q₈×Dic₇₃

—

Q₈×Dic₇₃

—

C₂×C₄⋊Dic₇₃

C₈

yes

—

C₂×C₄⋊Dic₇₃

C₈

yes

characteristic

C₂×C₄×Dic₇₃

C₂×C₄

yes

—

C₂×C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₂²×Q₈×C₇₃

C₈

yes

—

—

—

—

—

C₂²⋊C₄

yes

—

C₂×#10596833

C₂×D₄

yes

characteristic

C₂×#10596833

C₂×D₄

yes

characteristic

C₂×#10596833

C₂×Q₈

yes

—

C₂×#10596833

#41

yes

—

C₂²×Dic₇₃

C₂×C₈

yes

—

C₂²×Dic₇₃

C₂×C₈

yes

characteristic

C₂²×Dic₇₃

C₂²×C₄

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂²⋊C₄

yes

—

C₄×Dic₇₃

M₁₆

yes

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

C₁₆

yes

complemented

C₂²×C₄×C₇₃

C₂×C₈

yes

—

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂³×C₇₃

C₂²×C₈

yes

characteristic

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

—

C₂×C₄×C₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

M₃₂

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂²⋊C₈

yes

—

C₂×Dic₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₄×Q₈

yes

—

C₂×Dic₇₃

C₂×M₁₆

yes

characteristic

C₂×Dic₇₃

#131

yes

—

#10596833

—

#10596833

—

#10596833

C₂²⋊Q₈

yes

—

yes

C₄⋊Q₈

yes

characteristic

C₂²×C₇₃

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₇₃

C₈×Q₈

yes

—

C₂²×C₇₃

C₂²×C₁₆

yes

characteristic

C₂²×C₇₃

C₂×M₃₂

yes

characteristic

C₂²×C₇₃

#504

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₂²⋊C₁₆

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

yes

Dic₇₃

yes

Dic₇₃

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

C₂×C₂²⋊C₁₆

yes

characteristic

C₂×C₇₃

#2113

yes

characteristic

C₂×C₇₃

Q₈×C₁₆

yes

—

C₂×C₇₃

#2183

yes

—

100

C₂×C₈×Q₈

—

101

C₂×C₄⋊C₁₆

—

102

C₂×C₄×C₁₆

—

103

#1374

—

104

C₇₃

#13263

yes

characteristic, complemented, Sylow

105

C₂×C₄×Q₈

—

106

C₂²×C₁₆

—

107

C₂²×C₁₆

—

108

C₈×Q₈

—

146

—

109

C₈×Q₈

—

146

—

110

C₂×C₄⋊C₈

—

111

C₂×C₄⋊C₈

—

112

C₂×C₄×C₈

—

113

C₂×C₄×C₈

—

114

C₄⋊C₁₆

—

146

—

115

C₄×C₁₆

—

146

—

116

C₂²×Q₈

#10596842

yes

characteristic

117

C₂²×C₈

—

118

C₂²×C₈

—

119

C₂²×C₈

—

120

C₄×Q₈

—

146

—

121

C₄×Q₈

—

146

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄²

—

125

C₂×C₄²

—

126

C₂×C₁₆

—

127

C₂×C₁₆

—

146

—

128

C₂×C₁₆

—

146

—

129

C₂×C₁₆

—

146

—

130

C₂×C₁₆

—

146

—

131

C₄⋊C₈

—

146

—

132

C₄⋊C₈

—

146

—

133

C₄×C₈

—

134

C₄×C₈

—

135

C₄×C₈

—

146

—

136

C₄×C₈

—

146

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

#11924040

yes

complemented

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

C₂×#10596842

yes

—

144

C₂²×C₄

C₂×#10596842

yes

characteristic

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

146

—

151

C₂×C₈

—

146

—

152

C₂×C₈

—

146

—

153

C₄⋊C₄

—

146

—

154

C₄⋊C₄

—

146

—

155

C₄²

—

156

C₄²

—

157

C₄²

—

146

—

158

C₄²

—

146

—

159

C₁₆

—

292

—

160

C₁₆

—

146

—

161

C₂³

C₂²×#10596842

yes

characteristic

162

Q₈

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

146

—

171

C₂×C₄

—

146

—

172

C₂×C₄

—

146

—

173

C₂×C₄

C₂²⋊#10596842

yes

—

174

C₂×C₄

C₂×#11924040

yes

—

175

C₂×C₄

C₂×#11924040

yes

—

176

C₂×C₄

#14556687

yes

—

177

C₈

—

292

—

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

146

—

182

C₂²

C₂×C₂²⋊#10596842

yes

characteristic

183

C₂²

Q₈×#10596842

yes

—

184

C₂²

C₂²×#11924040

yes

characteristic

185

C₂²

C₂×#14556687

yes

characteristic

186

C₂²

#17989709

yes

—

187

C₄

—

188

C₄

—

292

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

146

—

193

C₄

C₂²⋊#11924040

yes

—

194

C₂

(C₂×#10596842)⋊Q₈

yes

characteristic

195

C₂

C₂×C₂²⋊#11924040

yes

characteristic

196

C₂

#22627103

yes

characteristic

197

C₂

Q₈×#11924040

yes

—

198

C₂

#22627157

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#10596842
commutator	C₂²×C₇₃	C₂²×C₁₆
Fitting	C₂²×Q₈×C₇₃	C₈
socle, prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂²×#10596833	C₂³
hypercenter	C₂²×Q₈	#10596842
hypocenter	C₇₃	#13263
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂²×#10596842

commutator

C₂²×C₇₃

C₂²×C₁₆

Fitting

C₂²×Q₈×C₇₃

C₈

socle, prime radical

C₂³×C₇₃

C₂²×C₈

C₂³

C₂²×Q₈

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂³ ◁ C₂²×Q₈

N	H ≅ G/N	type
20. C₂×#11924040	Q₈	semidirect
58. C₂×Q₈×C₇₃	C₁₆	semidirect
104. C₇₃	#13263	semidirect
139. C₂×Q₈	#11924040	semidirect

H ≅ G/N

type

20. C₂×#11924040

Q₈

semidirect

58. C₂×Q₈×C₇₃

C₁₆

semidirect

104. C₇₃

#13263

semidirect

139. C₂×Q₈

#11924040

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄⋊#11924040	2
C₂×C₄×#11924040	2
#22627001	2
C₂×Q₈×#10596833	2
#13263	73

subgroup

index

Subgroups — #34034358

Conjugacy classes of subgroups

Complements

Maximal subgroups