Group #34018149 — subgroups

Subgroups — #34018149

Order 18688.

702 conjugacy classes of subgroups; only the 272 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 272 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	#22626577	C₂	1	2	yes	maximal
2	#22625499	C₂	1	6	yes	maximal, complemented
3	#22625486	C₂	1	3	yes	maximal, complemented
4	#22625473	C₂	1	3	yes	maximal, complemented
5	C₈×#11924029	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556679	—	2	2	no	—
7	#17989594	—	2	6	no	—
8	#17989591	C₂²	1	6	yes	—
9	#17989588	C₂²	1	6	yes	—
10	#17989449	C₂²	1	6	yes	—
11	#17989448	C₂²	1	1	yes	characteristic
12	#17989448	C₂²	1	3	yes	—
13	#17989448	C₂²	1	6	yes	—
14	C₄×#11924029	C₂²	1	1	yes	characteristic
15	Q₈×#10596831	C₄	1	1	yes	characteristic
16	C₈×D₂₉₂	C₄	1	3	yes	—
17	#17988652	C₄	1	3	yes	—
18	#17988639	C₂²	1	3	yes	—
19	C₄×C₈×D₇₃	C₂²	1	3	yes	—
20	C₈×Q₈×C₇₃	C₄	1	1	yes	characteristic
21	#14556679	—	4	8	no	—
22	C₂×#11924038	—	2	6	no	—
23	C₂×#11924036	—	2	6	no	—
24	#14556658	—	2	6	no	—
25	C₄⋊#10596833	—	2	6	no	—
26	C₄⋊#10596833	C₂³	1	6	yes	—
27	C₄⋊#10596833	C₂³	1	2	yes	—
28	C₂×#11924029	D₄	1	1	yes	characteristic
29	C₂×C₈×D₇₃	C₂×C₄	1	3	yes	—
30	C₂×C₈×D₇₃	C₂×C₄	1	3	yes	—
31	Q₈×Dic₇₃	Q₈	1	1	yes	characteristic
32	C₄×D₂₉₂	C₂×C₄	1	3	yes	—
33	#14556553	Q₈	1	3	yes	—
34	#14556549	D₄	1	3	yes	—
35	C₄²×D₇₃	C₂³	1	3	yes	—
36	C₄⋊#10596831	C₂×C₄	1	3	yes	—
37	#14556503	C₂×C₄	1	6	yes	—
38	C₄×#10596831	C₂×C₄	1	3	yes	—
39	C₈×Dic₇₃	C₂³	1	3	yes	—
40	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
41	C₄×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₄⋊C₈×C₇₃	C₂×C₄	1	3	yes	—
43	C₄×C₈×C₇₃	C₂×C₄	1	3	yes	—
44	#11924038	—	4	12	no	—
45	#11924036	—	4	12	no	—
46	C₂×#10596833	—	2	2	no	—
47	C₂×#10596833	—	2	6	no	—
48	#11924029	SD₁₆	1	8	yes	—
49	C₂×D₂₉₂	C₄⋊C₄	1	3	yes	—
50	C₂×C₄×D₇₃	C₂²×C₄	1	3	yes	—
51	C₂×C₄×D₇₃	C₂×D₄	1	3	yes	—
52	C₂×C₄×D₇₃	C₂×Q₈	1	3	yes	—
53	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
54	C₂×#10596831	C₂²×C₄	1	3	yes	—
55	C₈×D₇₃	—	2	6	no	—
56	C₈×D₇₃	—	4	3	no	—
57	C₄⋊Dic₇₃	C₄⋊C₄	1	3	yes	—
58	#11924005	C₄⋊C₄	1	6	yes	—
59	C₄×Dic₇₃	C₂×D₄	1	3	yes	—
60	C₄×Dic₇₃	C₂×Q₈	1	3	yes	—
61	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
62	C₂×Q₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₇₃	C₂²×C₄	1	3	yes	—
64	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
65	C₄⋊C₄×C₇₃	C₄⋊C₄	1	3	yes	—
66	C₄²×C₇₃	C₂²×C₄	1	3	yes	—
67	C₂²×D₇₃	C₂×C₄⋊C₄	1	3	yes	—
68	Q₈×C₇₃	#106	1	4	yes	—
69	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	3	yes	—
70	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	3	yes	—
71	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
72	C₂×Dic₇₃	C₂×C₄⋊C₄	1	3	yes	—
73	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
74	C₂×Dic₇₃	C₂²×Q₈	1	1	yes	characteristic
75	D₂₉₂	#106	1	12	yes	—
76	C₄×D₇₃	—	2	3	no	—
77	C₄×D₇₃	—	2	3	no	—
78	C₄×D₇₃	—	4	3	no	—
79	C₄×D₇₃	C₂×SD₁₆	1	12	yes	—
80	#10596833	—	4	2	no	—
81	#10596833	—	4	6	no	—
82	C₈×C₇₃	—	2	3	no	—
83	C₈×C₇₃	#131	1	2	yes	complemented
84	#10596831	—	2	3	no	—
85	#10596831	—	2	1	no	—
86	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
87	C₂×D₇₃	—	2	3	no	—
88	C₂×D₇₃	—	2	3	no	—
89	C₂×D₇₃	C₂×#106	1	6	yes	—
90	C₄×C₇₃	—	2	3	no	—
91	C₄×C₇₃	C₂×#106	1	6	yes	—
92	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
93	C₄×C₇₃	#568	1	1	yes	characteristic
94	Dic₇₃	—	2	3	no	—
95	Dic₇₃	—	2	1	no	—
96	Dic₇₃	C₂²×SD₁₆	1	2	yes	—
97	#17826	—	73	1	no	maximal, Sylow
98	C₂×C₇₃	#2906	1	1	yes	characteristic
99	C₂×C₇₃	C₂²×#106	1	1	yes	characteristic
100	C₂×C₇₃	#2916	1	1	yes	characteristic
101	D₇₃	—	2	6	no	—
102	C₈×#41	—	73	1	no	—
103	#2906	—	73	2	no	—
104	#1570	—	73	6	no	—
105	#1566	—	73	3	no	—
106	C₂×#334	—	73	3	no	—
107	C₇₃	#17826	1	1	yes	characteristic, complemented, Sylow
108	C₂×#131	—	146	2	no	—
109	C₄×#41	—	73	1	no	—
110	C₈×Q₈	—	73	1	no	—
111	C₈×Q₈	#64408	1	1	yes	characteristic
112	C₈×D₄	—	73	3	no	—
113	C₈×D₄	—	73	3	no	—
114	#431	—	73	3	no	—
115	#424	—	146	6	no	—
116	#423	—	73	6	no	—
117	C₂×C₄⋊C₈	—	73	6	no	—
118	C₂×C₄×C₈	—	73	3	no	—
119	#334	—	73	1	no	—
120	#334	—	73	3	no	—
121	#334	—	73	6	no	—
122	#334	—	73	6	no	—
123	C₂×#41	—	73	1	no	—
124	#131	—	292	8	no	—
125	C₂×M₁₆	—	146	6	no	—
126	C₂²×C₈	—	73	3	no	—
127	C₂²×C₈	—	73	3	no	—
128	C₂²×C₈	—	146	6	no	—
129	C₄×Q₈	—	73	1	no	—
130	C₄×Q₈	C₂×#64408	1	1	yes	characteristic
131	C₄×D₄	—	73	3	no	—
132	C₄×D₄	—	73	3	no	—
133	#117	—	73	3	no	—
134	C₂×C₄²	—	73	3	no	—
135	C₄⋊C₈	—	73	2	no	—
136	C₄⋊C₈	—	73	6	no	—
137	C₄⋊C₈	—	73	3	no	—
138	C₄⋊C₈	—	146	6	no	—
139	C₄⋊C₈	—	146	6	no	—
140	C₄⋊C₈	C₂×#64408	1	3	yes	—
141	C₂²⋊C₈	—	73	6	no	—
142	C₄×C₈	—	73	1	no	—
143	C₄×C₈	—	73	3	no	—
144	C₄×C₈	—	73	3	no	—
145	C₄×C₈	C₂×#64408	1	3	yes	—
146	#41	—	73	8	no	—
147	C₂×Q₈	C₄⋊#64408	1	1	yes	characteristic
148	C₂×D₄	—	73	3	no	—
149	C₂²×C₄	—	73	3	no	—
150	C₂²×C₄	—	73	3	no	—
151	C₂²×C₄	—	73	3	no	—
152	M₁₆	—	292	12	no	—
153	C₂×C₈	—	292	3	no	—
154	C₂×C₈	—	292	12	no	—
155	C₂×C₈	—	73	1	no	—
156	C₂×C₈	—	73	3	no	—
157	C₂×C₈	—	146	2	no	—
158	C₂×C₈	—	146	6	no	—
159	C₂×C₈	—	146	6	no	—
160	C₂×C₈	C₂²×#64408	1	3	yes	—
161	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
162	C₄⋊C₄	—	73	3	no	—
163	C₄⋊C₄	C₄⋊#64408	1	3	yes	—
164	C₂²⋊C₄	—	73	6	no	—
165	C₄²	—	73	1	no	—
166	C₄²	—	73	3	no	—
167	C₄²	—	73	3	no	—
168	C₄²	C₂²×#64408	1	3	yes	—
169	C₂³	—	73	3	no	—
170	Q₈	#14556659	1	4	yes	—
171	D₄	—	73	12	no	—
172	C₂×C₄	—	292	3	no	—
173	C₂×C₄	—	73	1	no	—
174	C₂×C₄	—	73	12	no	—
175	C₂×C₄	—	73	1	no	—
176	C₂×C₄	—	73	3	no	—
177	C₂×C₄	—	146	3	no	—
178	C₂×C₄	—	146	3	no	—
179	C₂×C₄	C₂×C₄⋊#64408	1	3	yes	—
180	C₂×C₄	C₂×C₄⋊#64408	1	3	yes	—
181	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
182	C₈	—	2	3	no	—
183	C₈	—	292	2	no	—
184	C₈	—	292	6	no	—
185	C₈	—	146	3	no	—
186	C₈	—	146	1	no	—
187	C₈	#14556679	1	2	yes	complemented
188	C₂²	—	73	6	no	—
189	C₂²	—	146	3	no	—
190	C₂²	—	146	3	no	—
191	C₂²	C₂²×C₄⋊#64408	1	1	yes	characteristic
192	C₄	—	2	3	no	—
193	C₄	—	73	2	no	—
194	C₄	—	146	3	no	—
195	C₄	—	146	1	no	—
196	C₄	C₂×#14556659	1	6	yes	—
197	C₄	C₂×#14556679	1	1	yes	characteristic
198	C₄	#17989666	1	1	yes	characteristic
199	C₂	—	146	6	no	—
200	C₂	#22626577	1	1	yes	characteristic
201	C₂	C₂²×#14556659	1	1	yes	characteristic
202	C₂	#22626592	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626577

C₂

yes

maximal

#22625499

C₂

yes

maximal, complemented

#22625486

C₂

yes

maximal, complemented

#22625473

C₂

yes

maximal, complemented

C₈×#11924029

C₂

yes

maximal, characteristic

C₂×#14556679

—

#17989594

—

#17989591

C₂²

yes

—

#17989588

C₂²

yes

—

#17989449

C₂²

yes

—

#17989448

C₂²

yes

characteristic

#17989448

C₂²

yes

—

#17989448

C₂²

yes

—

C₄×#11924029

C₂²

yes

characteristic

Q₈×#10596831

C₄

yes

characteristic

C₈×D₂₉₂

C₄

yes

—

#17988652

C₄

yes

—

#17988639

C₂²

yes

—

C₄×C₈×D₇₃

C₂²

yes

—

C₈×Q₈×C₇₃

C₄

yes

—

—

—

—

—

C₂³

yes

—

C₄⋊#10596833

C₂³

yes

—

C₂×#11924029

D₄

yes

characteristic

C₂×C₈×D₇₃

C₂×C₄

yes

—

C₂×C₈×D₇₃

C₂×C₄

yes

—

Q₈×Dic₇₃

Q₈

yes

characteristic

C₄×D₂₉₂

C₂×C₄

yes

—

#14556553

Q₈

yes

—

#14556549

D₄

yes

—

C₄²×D₇₃

C₂³

yes

—

C₄⋊#10596831

C₂×C₄

yes

—

#14556503

C₂×C₄

yes

—

C₄×#10596831

C₂×C₄

yes

—

C₈×Dic₇₃

C₂³

yes

—

C₈×Dic₇₃

C₂³

yes

characteristic

C₄×Q₈×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

—

—

—

—

SD₁₆

yes

—

C₂×D₂₉₂

C₄⋊C₄

yes

—

C₂×C₄×D₇₃

C₂²×C₄

yes

—

C₂×C₄×D₇₃

C₂×D₄

yes

—

C₂×C₄×D₇₃

C₂×Q₈

yes

—

C₂×#10596831

C₂²×C₄

yes

characteristic

C₂×#10596831

C₂²×C₄

yes

—

C₈×D₇₃

—

C₈×D₇₃

—

C₄⋊Dic₇₃

C₄⋊C₄

yes

—

#11924005

C₄⋊C₄

yes

—

C₄×Dic₇₃

C₂×D₄

yes

—

C₄×Dic₇₃

C₂×Q₈

yes

—

C₄×Dic₇₃

C₂⁴

yes

characteristic

C₂×Q₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₇₃

C₄⋊C₄

yes

—

C₄²×C₇₃

C₂²×C₄

yes

—

C₂²×D₇₃

C₂×C₄⋊C₄

yes

—

Q₈×C₇₃

#106

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₃

C₂³×C₄

yes

characteristic

C₂×Dic₇₃

C₂×C₄⋊C₄

yes

—

C₂×Dic₇₃

C₂²×D₄

yes

characteristic

C₂×Dic₇₃

C₂²×Q₈

yes

characteristic

D₂₉₂

#106

yes

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂×SD₁₆

yes

—

#10596833

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

yes

—

—

C₂²×C₇₃

C₂²×C₄⋊C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

C₂×#106

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₂×#106

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

C₂²×SD₁₆

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

C₂²×#106

yes

characteristic

100

C₂×C₇₃

#2916

yes

characteristic

101

D₇₃

—

102

C₈×#41

—

103

#2906

—

104

#1570

—

105

#1566

—

106

C₂×#334

—

107

C₇₃

#17826

yes

characteristic, complemented, Sylow

108

C₂×#131

—

146

—

109

C₄×#41

—

110

C₈×Q₈

—

111

C₈×Q₈

#64408

yes

characteristic

112

C₈×D₄

—

113

C₈×D₄

—

114

#431

—

115

#424

—

146

—

116

#423

—

117

C₂×C₄⋊C₈

—

118

C₂×C₄×C₈

—

119

#334

—

120

#334

—

121

#334

—

122

#334

—

123

C₂×#41

—

124

#131

—

292

—

125

C₂×M₁₆

—

146

—

126

C₂²×C₈

—

127

C₂²×C₈

—

128

C₂²×C₈

—

146

—

129

C₄×Q₈

—

130

C₄×Q₈

C₂×#64408

yes

characteristic

131

C₄×D₄

—

132

C₄×D₄

—

133

#117

—

134

C₂×C₄²

—

135

C₄⋊C₈

—

136

C₄⋊C₈

—

137

C₄⋊C₈

—

138

C₄⋊C₈

—

146

—

139

C₄⋊C₈

—

146

—

140

C₄⋊C₈

C₂×#64408

yes

—

141

C₂²⋊C₈

—

142

C₄×C₈

—

143

C₄×C₈

—

144

C₄×C₈

—

145

C₄×C₈

C₂×#64408

yes

—

146

#41

—

147

C₂×Q₈

C₄⋊#64408

yes

characteristic

148

C₂×D₄

—

149

C₂²×C₄

—

150

C₂²×C₄

—

151

C₂²×C₄

—

152

M₁₆

—

292

—

153

C₂×C₈

—

292

—

154

C₂×C₈

—

292

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

146

—

158

C₂×C₈

—

146

—

159

C₂×C₈

—

146

—

160

C₂×C₈

C₂²×#64408

yes

—

161

C₂×C₈

C₂²×#64408

yes

characteristic

162

C₄⋊C₄

—

163

C₄⋊C₄

C₄⋊#64408

yes

—

164

C₂²⋊C₄

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₄²

C₂²×#64408

yes

—

169

C₂³

—

170

Q₈

#14556659

yes

—

171

D₄

—

172

C₂×C₄

—

292

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

146

—

178

C₂×C₄

—

146

—

179

C₂×C₄

C₂×C₄⋊#64408

yes

—

180

C₂×C₄

C₂×C₄⋊#64408

yes

—

181

C₂×C₄

C₂³×#64408

yes

characteristic

182

C₈

—

183

C₈

—

292

—

184

C₈

—

292

—

185

C₈

—

146

—

186

C₈

—

146

—

187

C₈

#14556679

yes

complemented

188

C₂²

—

189

C₂²

—

146

—

190

C₂²

—

146

—

191

C₂²

C₂²×C₄⋊#64408

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

146

—

195

C₄

—

146

—

196

C₄

C₂×#14556659

yes

—

197

C₄

C₂×#14556679

yes

characteristic

198

C₄

#17989666

yes

characteristic

199

C₂

—

146

—

200

C₂

#22626577

yes

characteristic

201

C₂

C₂²×#14556659

yes

characteristic

202

C₂

#22626592

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#64408
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₈×Q₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂²×C₄⋊C₄
prime radical	C₂×#11924029	D₄
cosocle	C₄×Dic₇₃	C₂⁴
hypercenter	C₈×Q₈	#64408
hypocenter	C₇₃	#17826
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×C₄⋊#64408

commutator

C₂×C₄×C₇₃

C₂³×C₄

Frattini

C₂×C₄

C₂³×#64408

Fitting

C₈×Q₈×C₇₃

C₄

socle

C₂²×C₇₃

C₂²×C₄⋊C₄

prime radical

C₂×#11924029

D₄

cosocle

C₄×Dic₇₃

C₂⁴

C₈×Q₈

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
2. #22625499	C₂	semidirect
3. #22625486	C₂	semidirect
4. #22625473	C₂	semidirect
83. C₈×C₇₃	#131	semidirect
107. C₇₃	#17826	semidirect
187. C₈	#14556679	semidirect

H ≅ G/N

type

2. #22625499

C₂

semidirect

3. #22625486

C₂

semidirect

4. #22625473

C₂

semidirect

83. C₈×C₇₃

#131

semidirect

107. C₇₃

#17826

semidirect

187. C₈

#14556679

semidirect

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

subgroup	index
#22626577	2
#22625499	2
#22625486	2
#22625473	2
C₈×#11924029	2
#17826	73

subgroup

index

Subgroups — #34018149

Conjugacy classes of subgroups

Complements

Maximal subgroups