Group #30692690 — subgroups

Subgroups — C₆₇×#13439 #30692690

Order 17152.

418 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 110 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	C₂²×D₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2151	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#361×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1978	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1381	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
7	C₂×D₁₆×C₆₇	—	2	4	no	—
8	C₂×D₁₆×C₆₇	—	2	2	no	—
9	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×#108×C₆₇	C₂²	1	2	yes	—
12	#418×C₆₇	—	2	2	no	—
13	C₂×#100×C₆₇	—	2	2	no	—
14	#405×C₆₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₆₇	—	2	2	no	—
16	#361×C₆₇	—	2	4	no	—
17	C₂²×D₄×C₆₇	—	2	2	no	—
18	C₂×D₈×C₆₇	—	2	2	no	—
19	C₂×D₈×C₆₇	—	2	2	no	—
20	C₂×D₈×C₆₇	—	2	2	no	—
21	C₂×D₈×C₆₇	—	2	2	no	—
22	C₂×D₈×C₆₇	—	2	2	no	—
23	C₂×D₈×C₆₇	D₄	1	4	yes	—
24	C₂×M₁₆×C₆₇	—	2	2	no	—
25	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
27	#117×C₆₇	D₄	1	1	yes	characteristic
28	D₁₆×C₆₇	—	2	8	no	—
29	M₃₂×C₆₇	—	2	2	no	—
30	C₂×C₁₆×C₆₇	—	2	2	no	—
31	C₂×C₁₆×C₆₇	—	2	1	no	—
32	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
33	#108×C₆₇	—	2	2	no	—
34	#108×C₆₇	—	2	2	no	—
35	#102×C₆₇	—	4	4	no	—
36	#100×C₆₇	—	4	4	no	—
37	#97×C₆₇	—	2	1	no	—
38	C₄×C₈×C₆₇	—	2	1	no	—
39	C₂⁴×C₆₇	—	4	2	no	—
40	C₂×D₄×C₆₇	—	2	4	no	—
41	C₂×D₄×C₆₇	—	4	2	no	—
42	C₂×D₄×C₆₇	—	4	2	no	—
43	C₂×D₄×C₆₇	—	4	2	no	—
44	C₂×D₄×C₆₇	—	4	2	no	—
45	C₂×D₄×C₆₇	—	4	2	no	—
46	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
47	D₈×C₆₇	—	2	4	no	—
48	D₈×C₆₇	—	2	4	no	—
49	D₈×C₆₇	—	4	2	no	—
50	D₈×C₆₇	—	4	2	no	—
51	M₁₆×C₆₇	—	2	2	no	—
52	M₁₆×C₆₇	—	4	2	no	—
53	M₁₆×C₆₇	—	4	2	no	—
54	C₂×C₈×C₆₇	—	2	1	no	—
55	C₂×C₈×C₆₇	—	4	2	no	—
56	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
57	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
59	C₂×C₈×C₆₇	#41	1	2	yes	—
60	C₄⋊C₄×C₆₇	D₈	1	2	yes	—
61	C₂²⋊C₄×C₆₇	—	2	1	no	—
62	C₄²×C₆₇	—	2	1	no	—
63	C₁₆×C₆₇	—	2	2	no	—
64	C₁₆×C₆₇	—	4	1	no	—
65	C₂³×C₆₇	—	4	4	no	—
66	C₂³×C₆₇	—	8	2	no	—
67	C₂³×C₆₇	—	8	2	no	—
68	C₂³×C₆₇	—	16	4	no	—
69	C₂³×C₆₇	#135	1	1	yes	characteristic
70	D₄×C₆₇	—	4	4	no	—
71	D₄×C₆₇	—	4	4	no	—
72	D₄×C₆₇	—	8	2	no	—
73	D₄×C₆₇	—	8	2	no	—
74	C₂×C₄×C₆₇	—	2	2	no	—
75	C₂×C₄×C₆₇	#121	1	2	yes	—
76	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
79	C₈×C₆₇	—	4	1	no	—
80	C₈×C₆₇	—	8	2	no	—
81	C₈×C₆₇	#120	1	1	yes	characteristic
82	C₈×C₆₇	#123	1	2	yes	—
83	C₈×C₆₇	#124	1	1	yes	characteristic
84	C₂²×C₆₇	—	2	2	no	—
85	C₂²×C₆₇	—	8	4	no	—
86	C₂²×C₆₇	—	16	4	no	—
87	C₂²×C₆₇	—	16	4	no	—
88	C₂²×C₆₇	—	16	4	no	—
89	C₂²×C₆₇	#463	1	2	yes	—
90	C₂²×C₆₇	#466	1	1	yes	characteristic
91	C₄×C₆₇	—	4	2	no	—
92	C₄×C₆₇	#394	1	1	yes	characteristic
93	C₄×C₆₇	#459	1	2	yes	—
94	C₄×C₆₇	#469	1	1	yes	characteristic
95	#13439	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂×C₆₇	—	2	2	no	—
97	C₂×C₆₇	—	16	4	no	—
98	C₂×C₆₇	#2054	1	1	yes	characteristic
99	C₂×C₆₇	#2221	1	2	yes	—
100	C₂²×D₁₆	C₂×C₆₇	1	1	yes	characteristic
101	#2151	C₂×C₆₇	1	1	yes	characteristic, complemented
102	C₂×#361	C₂×C₆₇	1	2	yes	complemented
103	#1978	C₂×C₆₇	1	2	yes	complemented
104	#1381	C₂×C₆₇	1	1	yes	characteristic, complemented
105	C₆₇	#13439	1	1	yes	characteristic, direct product factor, Sylow
106	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
107	C₂×D₁₆	—	2	4	no	—
108	C₂×D₁₆	—	2	2	no	—
109	C₂×M₃₂	C₂²×C₆₇	1	1	yes	characteristic
110	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
111	C₂×#108	C₂²×C₆₇	1	2	yes	—
112	#418	—	2	2	no	—
113	C₂×#100	—	2	2	no	—
114	#405	C₂²×C₆₇	1	1	yes	characteristic
115	C₄⋊C₁₆	—	2	2	no	—
116	#361	—	2	4	no	—
117	C₂²×D₄	—	2	2	no	—
118	C₂×D₈	—	2	2	no	—
119	C₂×D₈	—	2	2	no	—
120	C₂×D₈	—	2	2	no	—
121	C₂×D₈	—	2	2	no	—
122	C₂×D₈	—	2	2	no	—
123	C₂×D₈	D₄×C₆₇	1	4	yes	—
124	C₂×M₁₆	—	2	2	no	—
125	C₂×M₁₆	D₄×C₆₇	1	1	yes	characteristic
126	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
127	#117	D₄×C₆₇	1	1	yes	characteristic
128	D₁₆	—	2	8	no	—
129	M₃₂	—	2	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	2	1	no	—
132	C₂×C₁₆	D₄×C₆₇	1	2	yes	—
133	#108	—	2	2	no	—
134	#108	—	2	2	no	—
135	#102	—	4	4	no	—
136	#100	—	4	4	no	—
137	#97	—	2	1	no	—
138	C₄×C₈	—	2	1	no	—
139	C₂⁴	—	4	2	no	—
140	C₂×D₄	—	2	4	no	—
141	C₂×D₄	—	4	2	no	—
142	C₂×D₄	—	4	2	no	—
143	C₂×D₄	—	4	2	no	—
144	C₂×D₄	—	4	2	no	—
145	C₂×D₄	—	4	2	no	—
146	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
147	D₈	—	2	4	no	—
148	D₈	—	2	4	no	—
149	D₈	—	4	2	no	—
150	D₈	—	4	2	no	—
151	M₁₆	—	2	2	no	—
152	M₁₆	—	4	2	no	—
153	M₁₆	—	4	2	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
157	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
158	C₂×C₈	#41×C₆₇	1	1	yes	characteristic
159	C₂×C₈	#41×C₆₇	1	2	yes	—
160	C₄⋊C₄	D₈×C₆₇	1	2	yes	—
161	C₂²⋊C₄	—	2	1	no	—
162	C₄²	—	2	1	no	—
163	C₁₆	—	2	2	no	—
164	C₁₆	—	4	1	no	—
165	C₂³	—	4	4	no	—
166	C₂³	—	8	2	no	—
167	C₂³	—	8	2	no	—
168	C₂³	—	16	4	no	—
169	C₂³	#135×C₆₇	1	1	yes	characteristic
170	D₄	—	4	4	no	—
171	D₄	—	4	4	no	—
172	D₄	—	8	2	no	—
173	D₄	—	8	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	#121×C₆₇	1	2	yes	—
176	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
177	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
178	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
179	C₈	—	4	1	no	—
180	C₈	—	8	2	no	—
181	C₈	#120×C₆₇	1	1	yes	characteristic
182	C₈	#123×C₆₇	1	2	yes	—
183	C₈	#124×C₆₇	1	1	yes	characteristic
184	C₂²	—	2	2	no	—
185	C₂²	—	8	4	no	—
186	C₂²	—	16	4	no	—
187	C₂²	—	16	4	no	—
188	C₂²	—	16	4	no	—
189	C₂²	#463×C₆₇	1	2	yes	—
190	C₂²	#466×C₆₇	1	1	yes	characteristic
191	C₄	—	4	2	no	—
192	C₄	#394×C₆₇	1	1	yes	characteristic
193	C₄	#459×C₆₇	1	2	yes	—
194	C₄	#469×C₆₇	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	16	4	no	—
197	C₂	C₆₇×#2054	1	1	yes	characteristic
198	C₂	C₆₇×#2221	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₁₆×C₆₇

C₂

yes

maximal, characteristic

C₆₇×#2151

C₂

yes

maximal, characteristic, complemented

C₂×#361×C₆₇

C₂

yes

maximal, complemented

C₆₇×#1978

C₂

yes

maximal, complemented

C₆₇×#1381

C₂

yes

maximal, characteristic, complemented

C₂²×D₈×C₆₇

C₂²

yes

—

C₂×D₁₆×C₆₇

—

C₂×D₁₆×C₆₇

—

C₂×M₃₂×C₆₇

C₂²

yes

characteristic

C₂²×C₁₆×C₆₇

C₂²

yes

characteristic

C₂×#108×C₆₇

C₂²

yes

—

#418×C₆₇

—

C₂×#100×C₆₇

—

#405×C₆₇

C₂²

yes

characteristic

C₄⋊C₁₆×C₆₇

—

#361×C₆₇

—

C₂²×D₄×C₆₇

—

C₂×D₈×C₆₇

—

C₂×D₈×C₆₇

—

C₂×D₈×C₆₇

—

C₂×D₈×C₆₇

—

C₂×D₈×C₆₇

—

C₂×D₈×C₆₇

D₄

yes

—

C₂×M₁₆×C₆₇

—

C₂×M₁₆×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

#117×C₆₇

D₄

yes

characteristic

D₁₆×C₆₇

—

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

D₄

yes

—

#108×C₆₇

—

#108×C₆₇

—

#102×C₆₇

—

#100×C₆₇

—

#97×C₆₇

—

C₄×C₈×C₆₇

—

C₂⁴×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

D₈×C₆₇

—

D₈×C₆₇

—

D₈×C₆₇

—

D₈×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

C₂×C₈×C₆₇

yes

C₂×C₈×C₆₇

yes

—

C₄⋊C₄×C₆₇

D₈

yes

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₁₆×C₆₇

—

C₁₆×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

#135

yes

characteristic

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

#121

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#120

yes

characteristic

C₈×C₆₇

#123

yes

—

C₈×C₆₇

#124

yes

characteristic

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#463

yes

—

C₂²×C₆₇

#466

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

#394

yes

characteristic

C₄×C₆₇

#459

yes

—

C₄×C₆₇

#469

yes

characteristic

#13439

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#2054

yes

characteristic

C₂×C₆₇

#2221

yes

—

100

C₂²×D₁₆

C₂×C₆₇

yes

characteristic

101

#2151

C₂×C₆₇

yes

characteristic, complemented

102

C₂×#361

C₂×C₆₇

yes

complemented

103

#1978

C₂×C₆₇

yes

complemented

104

#1381

C₂×C₆₇

yes

characteristic, complemented

105

C₆₇

#13439

yes

characteristic, direct product factor, Sylow

106

C₂²×D₈

C₂²×C₆₇

yes

—

107

C₂×D₁₆

—

108

C₂×D₁₆

—

109

C₂×M₃₂

C₂²×C₆₇

yes

characteristic

110

C₂²×C₁₆

C₂²×C₆₇

yes

characteristic

111

C₂×#108

C₂²×C₆₇

yes

—

112

#418

—

113

C₂×#100

—

114

#405

C₂²×C₆₇

yes

characteristic

115

C₄⋊C₁₆

—

116

#361

—

117

C₂²×D₄

—

118

C₂×D₈

—

119

C₂×D₈

—

120

C₂×D₈

—

121

C₂×D₈

—

122

C₂×D₈

—

123

C₂×D₈

D₄×C₆₇

yes

—

124

C₂×M₁₆

—

125

C₂×M₁₆

D₄×C₆₇

yes

characteristic

126

C₂²×C₈

C₂³×C₆₇

yes

characteristic

127

#117

D₄×C₆₇

yes

characteristic

128

D₁₆

—

129

M₃₂

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

132

C₂×C₁₆

D₄×C₆₇

yes

—

133

#108

—

134

#108

—

135

#102

—

136

#100

—

137

#97

—

138

C₄×C₈

—

139

C₂⁴

—

140

C₂×D₄

—

141

C₂×D₄

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂²×C₄

C₂×D₄×C₆₇

yes

characteristic

147

D₈

—

148

D₈

—

149

D₈

—

150

D₈

—

151

M₁₆

—

152

M₁₆

—

153

M₁₆

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

C₂×D₄×C₆₇

yes

—

157

C₂×C₈

C₂×D₄×C₆₇

yes

characteristic

158

C₂×C₈

#41×C₆₇

yes

characteristic

159

C₂×C₈

#41×C₆₇

yes

—

160

C₄⋊C₄

D₈×C₆₇

yes

—

161

C₂²⋊C₄

—

162

C₄²

—

163

C₁₆

—

164

C₁₆

—

165

C₂³

—

166

C₂³

—

167

C₂³

—

168

C₂³

—

169

C₂³

#135×C₆₇

yes

characteristic

170

D₄

—

171

D₄

—

172

D₄

—

173

D₄

—

174

C₂×C₄

—

175

C₂×C₄

#121×C₆₇

yes

—

176

C₂×C₄

#121×C₆₇

yes

characteristic

177

C₂×C₄

C₂×D₈×C₆₇

yes

characteristic

178

C₂×C₄

AGL(1,ℤ/8)×C₆₇

yes

—

179

C₈

—

180

C₈

—

181

C₈

#120×C₆₇

yes

characteristic

182

C₈

#123×C₆₇

yes

—

183

C₈

#124×C₆₇

yes

characteristic

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

#463×C₆₇

yes

—

190

C₂²

#466×C₆₇

yes

characteristic

191

C₄

—

192

C₄

#394×C₆₇

yes

characteristic

193

C₄

#459×C₆₇

yes

—

194

C₄

#469×C₆₇

yes

characteristic

195

C₂

—

196

C₂

—

197

C₂

C₆₇×#2054

yes

characteristic

198

C₂

C₆₇×#2221

yes

—

	subgroup	G/H
center, socle	C₂²×C₆₇	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₆₇
prime radical	C₂²×D₁₆×C₆₇	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ #405×C₆₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₆₇

#466

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₆₇

prime radical

C₂²×D₁₆×C₆₇

C₂

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ #405×C₆₇ ◁ G

N	H ≅ G/N	type
2. C₆₇×#2151	C₂	semidirect
3. C₂×#361×C₆₇	C₂	semidirect
4. C₆₇×#1978	C₂	semidirect
5. C₆₇×#1381	C₂	semidirect
95. #13439	C₆₇	direct (N × H)
101. #2151	C₂×C₆₇	semidirect
102. C₂×#361	C₂×C₆₇	semidirect
103. #1978	C₂×C₆₇	semidirect
104. #1381	C₂×C₆₇	semidirect
105. C₆₇	#13439	direct (N × H)

H ≅ G/N

type

2. C₆₇×#2151

C₂

semidirect

3. C₂×#361×C₆₇

C₂

semidirect

4. C₆₇×#1978

C₂

semidirect

5. C₆₇×#1381

C₂

semidirect

95. #13439

C₆₇

direct (N × H)

101. #2151

C₂×C₆₇

semidirect

102. C₂×#361

C₂×C₆₇

semidirect

103. #1978

C₂×C₆₇

semidirect

104. #1381

C₂×C₆₇

semidirect

105. C₆₇

#13439

direct (N × H)

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

subgroup	index
C₂²×D₁₆×C₆₇	2
C₆₇×#2151	2
C₂×#361×C₆₇	2
C₆₇×#1978	2
C₆₇×#1381	2
#13439	67

subgroup

index

C₂²×D₁₆×C₆₇

C₆₇×#2151

C₂×#361×C₆₇

C₆₇×#1978

C₆₇×#1381

#13439

Subgroups — C₆₇×#13439 #30692690

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#13439 #30692690

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#13439 #30692690