Group #30773864 — subgroups

Subgroups — #30773864

Order 17152.

326 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 135 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×#16863224	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	#22560940	C₂	1	2	yes	maximal
4	#22558300	C₂	1	2	yes	maximal
5	C₂×#367×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14523977	C₂²	1	2	yes	—
8	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
9	#16863292	—	2	2	no	—
10	#16863224	C₄	1	4	yes	complemented
11	C₁₆×Dic₆₇	—	2	2	no	—
12	#16863158	—	2	2	no	—
13	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#107×C₆₇	C₂²	1	2	yes	—
15	#367×C₆₇	—	2	2	no	—
16	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
17	C₂×C₄⋊Dic₆₇	—	2	2	no	—
18	C₂×Dic₆₇⋊C₄	—	4	2	no	—
19	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
20	C₂×C₆₇⋊C₁₆	—	2	1	no	—
21	C₂×C₆₇⋊C₁₆	—	2	1	no	—
22	C₂×C₆₇⋊C₁₆	Q₈	1	2	yes	—
23	#14523977	—	2	2	no	—
24	#14523977	C₂×C₄	1	4	yes	—
25	C₈×Dic₆₇	—	2	1	no	—
26	C₈×Dic₆₇	—	2	1	no	—
27	C₆₇⋊#95	—	4	2	no	—
28	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
30	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
32	C₂×C₁₆×C₆₇	Q₈	1	2	yes	—
33	#107×C₆₇	—	2	2	no	—
34	#107×C₆₇	—	2	2	no	—
35	C₂²×Dic₆₇	—	4	2	no	—
36	C₂²×Dic₆₇	D₈	1	2	yes	—
37	C₂×C₆₇⋊C₈	—	2	1	no	—
38	C₂×C₆₇⋊C₈	—	2	1	no	—
39	C₂×C₆₇⋊C₈	Q₁₆	1	2	yes	—
40	C₄⋊Dic₆₇	—	2	4	no	—
41	C₄⋊Dic₆₇	—	4	2	no	—
42	Dic₆₇⋊C₄	—	8	4	no	—
43	C₄×Dic₆₇	—	2	1	no	—
44	C₄×Dic₆₇	—	2	1	no	—
45	C₆₇⋊C₁₆	—	2	2	no	—
46	C₂²×C₄×C₆₇	—	4	2	no	—
47	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
53	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
54	C₄⋊C₄×C₆₇	—	4	2	no	—
55	C₄⋊C₄×C₆₇	—	4	2	no	—
56	C₁₆×C₆₇	C₄⋊C₄	1	4	yes	complemented
57	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₆₇	—	8	2	no	—
59	C₂×C₄×C₆₇	—	8	2	no	—
60	C₂×C₄×C₆₇	—	8	2	no	—
61	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
62	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
65	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
66	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
67	C₂×Dic₆₇	—	2	2	no	—
68	C₂×Dic₆₇	—	2	2	no	—
69	C₂×Dic₆₇	—	4	4	no	—
70	C₂×Dic₆₇	—	8	2	no	—
71	C₂×Dic₆₇	—	8	2	no	—
72	C₂×Dic₆₇	SD₃₂	1	4	yes	—
73	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₈×C₆₇	C₄×Q₈	1	1	yes	characteristic
75	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
76	C₈×C₆₇	#125	1	1	yes	characteristic
77	C₆₇⋊C₈	—	2	2	no	—
78	C₂²×C₆₇	C₄×D₈	1	1	yes	characteristic
79	C₂²×C₆₇	#466	1	1	yes	characteristic
80	C₂²×C₆₇	#500	1	1	yes	characteristic
81	C₂²×C₆₇	C₂×SD₃₂	1	2	yes	—
82	C₂²×C₆₇	#508	1	2	yes	—
83	C₄×C₆₇	—	16	2	no	—
84	C₄×C₆₇	#389	1	1	yes	characteristic
85	C₄×C₆₇	C₄×Q₁₆	1	1	yes	characteristic
86	C₄×C₆₇	#467	1	1	yes	characteristic
87	C₄×C₆₇	#499	1	1	yes	characteristic
88	Dic₆₇	—	2	4	no	—
89	Dic₆₇	—	8	4	no	—
90	#13166	—	67	1	no	maximal, Sylow
91	C₂×C₆₇	#1942	1	1	yes	characteristic
92	C₂×C₆₇	C₄×SD₃₂	1	2	yes	—
93	C₂×C₆₇	#2217	1	2	yes	—
94	C₂×C₆₇	#2253	1	1	yes	characteristic
95	C₂×C₆₇	#2254	1	1	yes	characteristic
96	C₂×#367	—	67	1	no	—
97	C₂×#367	D₆₇	1	1	yes	characteristic
98	C₂×C₄×C₁₆	—	67	1	no	—
99	#1942	—	67	2	no	—
100	#1380	—	67	2	no	—
101	C₆₇	#13166	1	1	yes	characteristic, complemented, Sylow
102	C₂²×C₁₆	—	67	1	no	—
103	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
104	C₂×#107	—	67	2	no	—
105	C₂×#107	C₂×D₆₇	1	2	yes	—
106	C₂×C₄×C₈	—	67	1	no	—
107	#389	—	134	2	no	—
108	#367	—	2	2	no	—
109	#367	—	67	4	no	—
110	C₄×C₁₆	—	134	2	no	—
111	#340	—	134	2	no	—
112	C₂²×C₈	—	67	1	no	—
113	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	268	2	no	—
116	C₂×C₄⋊C₄	—	134	2	no	—
117	C₂×C₄²	—	67	1	no	—
118	C₂×C₁₆	—	67	2	no	—
119	C₂×C₁₆	—	134	1	no	—
120	C₂×C₁₆	—	134	1	no	—
121	C₂×C₁₆	Dic₁₃₄	1	2	yes	—
122	C₂×C₁₆	C₄×D₆₇	1	2	yes	—
123	C₂×C₁₆	C₂²⋊D₆₇	1	2	yes	—
124	#107	—	2	2	no	—
125	#107	—	2	2	no	—
126	#107	—	67	4	no	—
127	#107	—	134	2	no	—
128	C₄×C₈	—	134	1	no	—
129	C₄×C₈	—	134	1	no	—
130	#95	—	268	2	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	268	2	no	—
133	C₂²×C₄	—	67	2	no	—
134	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
135	C₂×C₈	—	67	2	no	—
136	C₂×C₈	—	134	1	no	—
137	C₂×C₈	—	134	1	no	—
138	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
139	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
140	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
141	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
142	C₂×C₈	#11761627	1	1	yes	characteristic
143	C₂×C₈	#11761629	1	1	yes	characteristic
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	268	2	no	—
147	C₄⋊C₄	—	536	4	no	—
148	C₄⋊C₄	—	134	4	no	—
149	C₄²	—	134	1	no	—
150	C₄²	—	134	1	no	—
151	C₁₆	—	134	2	no	—
152	C₁₆	Dic₆₇⋊C₄	1	4	yes	complemented
153	C₂³	D₈×D₆₇	1	1	yes	characteristic
154	C₂×C₄	—	8	2	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	268	4	no	—
158	C₂×C₄	—	536	2	no	—
159	C₂×C₄	—	536	2	no	—
160	C₂×C₄	—	67	4	no	—
161	C₂×C₄	—	134	2	no	—
162	C₂×C₄	—	134	2	no	—
163	C₂×C₄	#14524005	1	1	yes	characteristic
164	C₂×C₄	#14524014	1	1	yes	characteristic
165	C₂×C₄	#14524042	1	1	yes	characteristic
166	C₂×C₄	Q₁₆×D₆₇	1	1	yes	characteristic
167	C₂×C₄	#14524062	1	1	yes	characteristic
168	C₂×C₄	#14524066	1	1	yes	characteristic
169	C₈	—	134	2	no	—
170	C₈	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
171	C₈	#14524008	1	1	yes	characteristic
172	C₈	#14524019	1	1	yes	characteristic
173	C₈	#14524034	1	1	yes	characteristic
174	C₂²	#16863478	1	1	yes	characteristic
175	C₂²	#16863629	1	1	yes	characteristic
176	C₂²	#16863800	1	1	yes	characteristic
177	C₂²	SD₃₂×D₆₇	1	2	yes	—
178	C₂²	#16863827	1	2	yes	—
179	C₄	—	16	2	no	—
180	C₄	—	536	4	no	—
181	C₄	—	134	4	no	—
182	C₄	#16863292	1	1	yes	characteristic
183	C₄	#16863490	1	1	yes	characteristic
184	C₄	#16863634	1	1	yes	characteristic
185	C₄	#16863797	1	1	yes	characteristic
186	C₂	#22560940	1	1	yes	characteristic
187	C₂	#22562060	1	2	yes	—
188	C₂	#22562302	1	2	yes	—
189	C₂	#22562487	1	1	yes	characteristic
190	C₂	#22562490	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863224

C₂

yes

maximal, characteristic

C₂×C₁₆×Dic₆₇

C₂

yes

maximal, characteristic

#22560940

C₂

yes

maximal

#22558300

C₂

yes

maximal

C₂×#367×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₂×#14523977

C₂²

yes

—

C₂×C₈×Dic₆₇

C₂²

yes

characteristic

#16863292

—

#16863224

C₄

yes

complemented

C₁₆×Dic₆₇

—

#16863158

—

C₂²×C₁₆×C₆₇

C₂²

yes

characteristic

C₂×#107×C₆₇

C₂²

yes

—

#367×C₆₇

—

C₂²×C₆₇⋊C₈

D₄

yes

characteristic

C₂×C₄⋊Dic₆₇

—

C₂×Dic₆₇⋊C₄

—

C₂×C₄×Dic₆₇

D₄

yes

characteristic

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

Q₈

yes

—

#14523977

—

#14523977

C₂×C₄

yes

—

C₈×Dic₆₇

—

C₈×Dic₆₇

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₆₇

—

C₂×C₁₆×C₆₇

C₂×C₄

yes

—

C₂×C₁₆×C₆₇

D₄

yes

—

C₂×C₁₆×C₆₇

Q₈

yes

—

#107×C₆₇

—

#107×C₆₇

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

D₈

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

Q₁₆

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

C₂×C₈×C₆₇

yes

C₂×C₈×C₆₇

yes

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₁₆×C₆₇

C₄⋊C₄

yes

complemented

C₂³×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄×D₄

yes

characteristic

C₂×C₄×C₆₇

#121

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₁₆

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

SD₃₂

yes

—

C₈×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₈×C₆₇

C₄×Q₈

yes

characteristic

C₈×C₆₇

C₂²⋊Q₈

yes

characteristic

C₈×C₆₇

#125

yes

characteristic

C₆₇⋊C₈

—

C₂²×C₆₇

C₄×D₈

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

C₂×SD₃₂

yes

—

C₂²×C₆₇

#508

yes

—

C₄×C₆₇

—

C₄×C₆₇

#389

yes

characteristic

C₄×C₆₇

C₄×Q₁₆

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

C₄×SD₃₂

yes

—

C₂×C₆₇

#2217

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

D₆₇

yes

characteristic

C₂×C₄×C₁₆

—

#1942

—

100

#1380

—

101

C₆₇

#13166

yes

characteristic, complemented, Sylow

102

C₂²×C₁₆

—

103

C₂²×C₁₆

C₂×D₆₇

yes

characteristic

104

C₂×#107

—

105

C₂×#107

C₂×D₆₇

yes

—

106

C₂×C₄×C₈

—

107

#389

—

134

—

108

#367

—

109

#367

—

110

C₄×C₁₆

—

134

—

111

#340

—

134

—

112

C₂²×C₈

—

113

C₂²×C₈

C₂²×D₆₇

yes

characteristic

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

268

—

116

C₂×C₄⋊C₄

—

134

—

117

C₂×C₄²

—

118

C₂×C₁₆

—

119

C₂×C₁₆

—

134

—

120

C₂×C₁₆

—

134

—

121

C₂×C₁₆

Dic₁₃₄

yes

—

122

C₂×C₁₆

C₄×D₆₇

yes

—

123

C₂×C₁₆

C₂²⋊D₆₇

yes

—

124

#107

—

125

#107

—

126

#107

—

127

#107

—

134

—

128

C₄×C₈

—

134

—

129

C₄×C₈

—

134

—

130

#95

—

268

—

131

C₂²×C₄

—

132

C₂²×C₄

—

268

—

133

C₂²×C₄

—

134

C₂²×C₄

D₄×D₆₇

yes

characteristic

135

C₂×C₈

—

136

C₂×C₈

—

134

—

137

C₂×C₈

—

134

—

138

C₂×C₈

C₂×C₄×D₆₇

yes

characteristic

139

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

140

C₂×C₈

Q₈×D₆₇

yes

characteristic

141

C₂×C₈

C₂×Dic₁₃₄

yes

characteristic

142

C₂×C₈

#11761627

yes

characteristic

143

C₂×C₈

#11761629

yes

characteristic

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

268

—

147

C₄⋊C₄

—

536

—

148

C₄⋊C₄

—

134

—

149

C₄²

—

134

—

150

C₄²

—

134

—

151

C₁₆

—

134

—

152

C₁₆

Dic₆₇⋊C₄

yes

complemented

153

C₂³

D₈×D₆₇

yes

characteristic

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

268

—

158

C₂×C₄

—

536

—

159

C₂×C₄

—

536

—

160

C₂×C₄

—

161

C₂×C₄

—

134

—

162

C₂×C₄

—

134

—

163

C₂×C₄

#14524005

yes

characteristic

164

C₂×C₄

#14524014

yes

characteristic

165

C₂×C₄

#14524042

yes

characteristic

166

C₂×C₄

Q₁₆×D₆₇

yes

characteristic

167

C₂×C₄

#14524062

yes

characteristic

168

C₂×C₄

#14524066

yes

characteristic

169

C₈

—

134

—

170

C₈

C₂×Dic₆₇⋊C₄

yes

characteristic

171

C₈

#14524008

yes

characteristic

172

C₈

#14524019

yes

characteristic

173

C₈

#14524034

yes

characteristic

174

C₂²

#16863478

yes

characteristic

175

C₂²

#16863629

yes

characteristic

176

C₂²

#16863800

yes

characteristic

177

C₂²

SD₃₂×D₆₇

yes

—

178

C₂²

#16863827

yes

—

179

C₄

—

180

C₄

—

536

—

181

C₄

—

134

—

182

C₄

#16863292

yes

characteristic

183

C₄

#16863490

yes

characteristic

184

C₄

#16863634

yes

characteristic

185

C₄

#16863797

yes

characteristic

186

C₂

#22560940

yes

characteristic

187

C₂

#22562060

yes

—

188

C₂

#22562302

yes

—

189

C₂

#22562487

yes

characteristic

190

C₂

#22562490

yes

characteristic

	subgroup	G/H
center	C₂³	D₈×D₆₇
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#367×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#367	D₆₇
hypocenter	C₆₇	#13166
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

subgroup

G/H

center

C₂³

D₈×D₆₇

commutator

C₂×C₈×C₆₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₂×#367×C₆₇

C₂

socle, prime radical

C₂³×C₆₇

C₂×D₈

cosocle

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

N	H ≅ G/N	type
10. #16863224	C₄	semidirect
56. C₁₆×C₆₇	C₄⋊C₄	semidirect
101. C₆₇	#13166	semidirect
152. C₁₆	Dic₆₇⋊C₄	semidirect

H ≅ G/N

type

10. #16863224

C₄

semidirect

56. C₁₆×C₆₇

C₄⋊C₄

semidirect

101. C₆₇

#13166

semidirect

152. C₁₆

Dic₆₇⋊C₄

semidirect

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

subgroup	index
C₂×#16863224	2
C₂×C₁₆×Dic₆₇	2
#22560940	2
#22558300	2
C₂×#367×C₆₇	2
#13166	67

subgroup

index

C₂×C₁₆×Dic₆₇

C₂×#367×C₆₇

Subgroups — #30773864

Conjugacy classes of subgroups

Complements

Maximal subgroups