Group #30744159 — subgroups

Subgroups — #30744159

Order 17152.

456 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 159 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	#22560326	C₂	1	2	yes	maximal
2	#22558035	C₂	1	4	yes	maximal
3	C₄×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
5	#16863151	—	2	2	no	—
6	#16863151	C₂²	1	4	yes	—
7	C₂×C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₆₇	C₄	1	2	yes	—
9	#391×C₆₇	C₂²	1	2	yes	—
10	#389×C₆₇	—	2	2	no	—
11	#384×C₆₇	C₄	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
13	C₄³×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×C₆₇⋊C₈	—	2	4	no	—
15	C₄⋊(C₆₇⋊C₈)	—	4	4	no	—
16	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
17	C₄⋊Q₈×C₆₇	—	2	4	no	—
18	C₄×Q₈×C₆₇	—	4	4	no	—
19	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
20	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
21	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
22	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₆₇	—	2	2	no	—
24	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₆₇	D₄	1	2	yes	—
27	C₂×C₄²×C₆₇	D₄	1	2	yes	—
28	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
29	#95×C₆₇	—	2	4	no	—
30	C₂×C₆₇⋊C₈	—	4	4	no	—
31	C₂×C₆₇⋊C₈	—	4	4	no	—
32	C₂×C₆₇⋊C₈	—	4	4	no	—
33	C₂×Q₈×C₆₇	—	2	2	no	—
34	C₂×Q₈×C₆₇	—	4	2	no	—
35	C₂×Q₈×C₆₇	—	4	2	no	—
36	C₂×Q₈×C₆₇	M₁₆	1	4	yes	—
37	C₂²×C₄×C₆₇	—	2	2	no	—
38	C₂²×C₄×C₆₇	—	2	2	no	—
39	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
44	C₄⋊C₄×C₆₇	—	2	8	no	—
45	C₄⋊C₄×C₆₇	—	4	2	no	—
46	C₄⋊C₄×C₆₇	—	4	4	no	—
47	C₄⋊C₄×C₆₇	—	4	4	no	—
48	C₄²×C₆₇	—	2	2	no	—
49	C₄²×C₆₇	—	2	2	no	—
50	C₄²×C₆₇	—	4	2	no	—
51	C₄²×C₆₇	—	4	1	no	—
52	C₄²×C₆₇	—	4	4	no	—
53	C₄²×C₆₇	SD₁₆	1	4	yes	—
54	C₄²×C₆₇	Q₁₆	1	4	yes	—
55	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	Q₈×C₆₇	—	4	4	no	—
57	Q₈×C₆₇	—	4	4	no	—
58	C₂×C₄×C₆₇	—	2	2	no	—
59	C₂×C₄×C₆₇	—	2	1	no	—
60	C₂×C₄×C₆₇	—	2	2	no	—
61	C₂×C₄×C₆₇	—	2	1	no	—
62	C₂×C₄×C₆₇	—	2	2	no	—
63	C₂×C₄×C₆₇	—	2	2	no	—
64	C₂×C₄×C₆₇	—	2	4	no	—
65	C₂×C₄×C₆₇	—	2	1	no	—
66	C₂×C₄×C₆₇	—	2	2	no	—
67	C₂×C₄×C₆₇	—	4	2	no	—
68	C₂×C₄×C₆₇	—	4	2	no	—
69	C₂×C₄×C₆₇	—	4	2	no	—
70	C₂×C₄×C₆₇	—	4	4	no	—
71	C₂×C₄×C₆₇	#100	1	2	yes	—
72	C₂×C₄×C₆₇	#103	1	4	yes	—
73	C₂×C₄×C₆₇	#103	1	4	yes	—
74	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
75	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
76	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
77	C₆₇⋊C₈	—	8	4	no	—
78	C₂²×C₆₇	#407	1	1	yes	characteristic
79	C₂²×C₆₇	#410	1	1	yes	characteristic
80	C₂²×C₆₇	C₂×#100	1	1	yes	characteristic
81	C₂²×C₆₇	C₂×#103	1	2	yes	—
82	C₂²×C₆₇	#421	1	2	yes	—
83	C₄×C₆₇	—	2	2	no	—
84	C₄×C₆₇	—	2	4	no	—
85	C₄×C₆₇	—	2	2	no	—
86	C₄×C₆₇	—	2	2	no	—
87	C₄×C₆₇	—	4	1	no	—
88	C₄×C₆₇	—	4	4	no	—
89	C₄×C₆₇	—	8	2	no	—
90	C₄×C₆₇	#333	1	4	yes	—
91	#7653	—	67	1	no	maximal, Sylow
92	C₂×C₆₇	#1466	1	1	yes	characteristic
93	C₂×C₆₇	#1487	1	2	yes	—
94	C₂×C₆₇	#1518	1	2	yes	—
95	C₂×C₆₇	#1536	1	1	yes	characteristic
96	C₂×C₆₇	C₂×#333	1	1	yes	characteristic
97	C₄×C₄⋊Q₈	D₆₇	1	1	yes	characteristic
98	#1831	—	67	2	no	—
99	#1272	—	67	4	no	—
100	C₆₇	#7653	1	1	yes	characteristic, complemented, Sylow
101	C₂×C₄⋊Q₈	Dic₆₇	1	1	yes	characteristic
102	C₂×C₄×Q₈	Dic₆₇	1	2	yes	—
103	C₂×C₄⋊C₈	—	134	2	no	—
104	#391	C₂×D₆₇	1	2	yes	—
105	#389	—	2	2	no	—
106	#384	Dic₆₇	1	1	yes	characteristic
107	C₄×C₄⋊C₄	—	2	2	no	—
108	C₄³	C₂×D₆₇	1	1	yes	characteristic
109	#336	—	67	4	no	—
110	#336	—	134	2	no	—
111	C₂²×Q₈	C₂×Dic₆₇	1	2	yes	—
112	C₂²×C₈	—	134	4	no	—
113	C₄⋊Q₈	—	2	4	no	—
114	C₄×Q₈	—	4	4	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
119	C₂×C₄²	—	2	2	no	—
120	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
121	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
122	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
123	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
124	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
125	C₄⋊C₈	—	268	4	no	—
126	#95	—	2	4	no	—
127	C₂×Q₈	—	2	2	no	—
128	C₂×Q₈	—	4	2	no	—
129	C₂×Q₈	—	4	2	no	—
130	C₂×Q₈	#11761612	1	4	yes	—
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	—	2	2	no	—
133	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
134	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
135	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
136	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
137	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
138	C₂×C₈	—	268	4	no	—
139	C₂×C₈	—	268	4	no	—
140	C₂×C₈	—	268	4	no	—
141	C₄⋊C₄	—	2	8	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	4	4	no	—
144	C₄⋊C₄	—	4	4	no	—
145	C₄²	—	2	2	no	—
146	C₄²	—	2	2	no	—
147	C₄²	—	4	2	no	—
148	C₄²	—	4	1	no	—
149	C₄²	—	4	4	no	—
150	C₄²	#11761615	1	4	yes	—
151	C₄²	#11761618	1	4	yes	—
152	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
153	Q₈	—	4	4	no	—
154	Q₈	—	4	4	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	#14523960	1	2	yes	—
169	C₂×C₄	#14523967	1	4	yes	—
170	C₂×C₄	#14523967	1	4	yes	—
171	C₂×C₄	C₂×#11761612	1	2	yes	—
172	C₂×C₄	C₂×#11761615	1	2	yes	—
173	C₂×C₄	C₂×#11761618	1	2	yes	—
174	C₈	—	536	4	no	—
175	C₂²	#16863360	1	1	yes	characteristic
176	C₂²	#16863370	1	1	yes	characteristic
177	C₂²	C₂×#14523960	1	1	yes	characteristic
178	C₂²	C₂×#14523967	1	2	yes	—
179	C₂²	#16863413	1	2	yes	—
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	2	2	no	—
183	C₄	—	2	4	no	—
184	C₄	—	4	4	no	—
185	C₄	—	4	1	no	—
186	C₄	—	8	2	no	—
187	C₄	#16863145	1	4	yes	—
188	C₂	#22558550	1	1	yes	characteristic
189	C₂	#22558638	1	2	yes	—
190	C₂	#22558773	1	2	yes	—
191	C₂	#22558844	1	1	yes	characteristic
192	C₂	C₂×#16863145	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22560326

C₂

yes

maximal

#22558035

C₂

yes

maximal

C₄×C₄⋊Q₈×C₆₇

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₆₇⋊C₈)

—

#16863151

—

#16863151

C₂²

yes

—

C₂×C₄⋊Q₈×C₆₇

C₄

yes

characteristic

C₂×C₄×Q₈×C₆₇

C₄

yes

—

#391×C₆₇

C₂²

yes

—

#389×C₆₇

—

#384×C₆₇

C₄

yes

characteristic

C₄×C₄⋊C₄×C₆₇

—

C₄³×C₆₇

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

—

C₄⋊(C₆₇⋊C₈)

—

C₂²×Q₈×C₆₇

C₂×C₄

yes

—

C₄⋊Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#95×C₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

M₁₆

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

SD₁₆

yes

—

C₄²×C₆₇

Q₁₆

yes

—

C₂³×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#100

yes

—

C₂×C₄×C₆₇

#103

yes

—

C₂×C₄×C₆₇

#103

yes

—

C₂×C₄×C₆₇

C₂×M₁₆

yes

—

C₂×C₄×C₆₇

C₂×SD₁₆

yes

—

C₂×C₄×C₆₇

C₂×Q₁₆

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₂²×C₆₇

#421

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#333

yes

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×C₆₇

#1518

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₄×C₄⋊Q₈

D₆₇

yes

characteristic

#1831

—

#1272

—

100

C₆₇

#7653

yes

characteristic, complemented, Sylow

101

C₂×C₄⋊Q₈

Dic₆₇

yes

characteristic

102

C₂×C₄×Q₈

Dic₆₇

yes

—

103

C₂×C₄⋊C₈

—

134

—

104

#391

C₂×D₆₇

yes

—

105

#389

—

106

#384

Dic₆₇

yes

characteristic

107

C₄×C₄⋊C₄

—

108

C₄³

C₂×D₆₇

yes

characteristic

109

#336

—

110

#336

—

134

—

111

C₂²×Q₈

C₂×Dic₆₇

yes

—

112

C₂²×C₈

—

134

—

113

C₄⋊Q₈

—

114

C₄×Q₈

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

C₂×Dic₆₇

yes

—

119

C₂×C₄²

—

120

C₂×C₄²

C₂×Dic₆₇

yes

characteristic

121

C₂×C₄²

C₂×Dic₆₇

yes

characteristic

122

C₂×C₄²

C₂²⋊D₆₇

yes

—

123

C₂×C₄²

C₂²⋊D₆₇

yes

—

124

C₂×C₄²

C₂²×D₆₇

yes

characteristic

125

C₄⋊C₈

—

268

—

126

#95

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

#11761612

yes

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

C₂²⋊Dic₆₇

yes

characteristic

134

C₂²×C₄

C₂²⋊Dic₆₇

yes

—

135

C₂²×C₄

C₂²⋊Dic₆₇

yes

characteristic

136

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

137

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

—

138

C₂×C₈

—

268

—

139

C₂×C₈

—

268

—

140

C₂×C₈

—

268

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄²

—

146

C₄²

—

147

C₄²

—

148

C₄²

—

149

C₄²

—

150

C₄²

#11761615

yes

—

151

C₄²

#11761618

yes

—

152

C₂³

C₂×C₂²⋊Dic₆₇

yes

characteristic

153

Q₈

—

154

Q₈

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

#14523960

yes

—

169

C₂×C₄

#14523967

yes

—

170

C₂×C₄

#14523967

yes

—

171

C₂×C₄

C₂×#11761612

yes

—

172

C₂×C₄

C₂×#11761615

yes

—

173

C₂×C₄

C₂×#11761618

yes

—

174

C₈

—

536

—

175

C₂²

#16863360

yes

characteristic

176

C₂²

#16863370

yes

characteristic

177

C₂²

C₂×#14523960

yes

characteristic

178

C₂²

C₂×#14523967

yes

—

179

C₂²

#16863413

yes

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

#16863145

yes

—

188

C₂

#22558550

yes

characteristic

189

C₂

#22558638

yes

—

190

C₂

#22558773

yes

—

191

C₂

#22558844

yes

characteristic

192

C₂

C₂×#16863145

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×C₄⋊Q₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×C₄⋊Q₈	D₆₇
hypocenter	C₆₇	#7653
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₆₇

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₄×C₄⋊Q₈×C₆₇

C₂

socle, prime radical

C₂³×C₆₇

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₆₇

C₂³

hypercenter

C₄×C₄⋊Q₈

D₆₇

hypocenter

C₆₇

#7653

LCS

G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

N	H ≅ G/N	type
100. C₆₇	#7653	semidirect

H ≅ G/N

type

100. C₆₇

#7653

semidirect

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

subgroup	index
#22560326	2
#22558035	2
C₄×C₄⋊Q₈×C₆₇	2
#7653	67

subgroup

index

#22560326

#22558035

C₄×C₄⋊Q₈×C₆₇

#7653

Subgroups — #30744159

Conjugacy classes of subgroups

Complements

Maximal subgroups