Group #30772311 — subgroups

Subgroups — #30772311

Order 17152.

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

normal subgroups only

This group has 370 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	#22561974	C₂	1	2	yes	maximal, complemented
2	#22561823	C₂	1	1	yes	maximal, characteristic
3	#22560009	C₂	1	1	yes	maximal, characteristic
4	#22558299	C₂	1	2	yes	maximal, complemented
5	C₆₇×#3407	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₂²	1	1	yes	characteristic
7	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	#16863380	C₄	1	1	yes	characteristic
9	#16863370	C₄	1	1	yes	characteristic
10	#16863351	C₂²	1	2	yes	—
11	#16863216	—	2	4	no	—
12	C₂²⋊(C₆₇⋊C₁₆)	—	2	2	no	—
13	#16863167	C₄	1	2	yes	—
14	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
15	#504×C₆₇	—	2	4	no	—
16	C₂×M₃₂×C₆₇	—	2	2	no	—
17	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
18	C₂×#11761612	C₂×C₄	1	2	yes	—
19	#14524002	C₂×C₄	1	2	yes	—
20	#14523983	—	2	2	no	—
21	C₂×C₆₇⋊C₁₆	—	2	1	no	—
22	C₂×C₆₇⋊C₁₆	—	2	1	no	—
23	C₂×C₆₇⋊C₁₆	—	2	2	no	—
24	#14523962	C₈	1	2	yes	—
25	#14523960	C₈	1	2	yes	—
26	#14523958	—	2	2	no	—
27	#14523953	—	2	2	no	—
28	C₈×Dic₆₇	—	2	2	no	—
29	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
30	#131×C₆₇	—	2	2	no	—
31	#131×C₆₇	—	2	2	no	—
32	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂×M₁₆×C₆₇	Q₈	1	2	yes	—
34	C₂²×C₈×C₆₇	D₄	1	2	yes	—
35	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
36	M₃₂×C₆₇	—	2	4	no	—
37	M₃₂×C₆₇	—	2	2	no	—
38	M₃₂×C₆₇	D₄	1	4	yes	—
39	C₂×C₁₆×C₆₇	—	2	4	no	—
40	C₂×C₁₆×C₆₇	—	2	2	no	—
41	#11761612	—	2	2	no	—
42	#11761612	C₂×C₈	1	4	yes	—
43	C₂×C₆₇⋊C₈	—	2	2	no	—
44	C₄⋊Dic₆₇	—	2	2	no	—
45	C₂²⋊Dic₆₇	—	2	2	no	—
46	C₄×Dic₆₇	—	2	2	no	—
47	C₆₇⋊C₁₆	—	4	1	no	—
48	C₆₇⋊C₁₆	—	4	1	no	—
49	#41×C₆₇	—	2	2	no	—
50	#41×C₆₇	—	2	2	no	—
51	C₂×Q₈×C₆₇	C₂×C₈	1	1	yes	characteristic
52	C₂×D₄×C₆₇	C₂×C₈	1	1	yes	characteristic
53	C₂×D₄×C₆₇	M₁₆	1	2	yes	—
54	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
56	M₁₆×C₆₇	—	2	2	no	—
57	M₁₆×C₆₇	—	2	4	no	—
58	C₂×C₈×C₆₇	—	2	2	no	—
59	C₂×C₈×C₆₇	—	2	2	no	—
60	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
61	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₈×C₆₇	#41	1	2	yes	—
65	C₁₆×C₆₇	—	2	4	no	—
66	C₂³×C₆₇	C₄⋊C₈	1	2	yes	—
67	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
68	Q₈×C₆₇	—	2	2	no	—
69	D₄×C₆₇	—	2	4	no	—
70	D₄×C₆₇	—	2	2	no	—
71	C₂×C₄×C₆₇	—	2	2	no	—
72	C₂×C₄×C₆₇	—	2	2	no	—
73	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
74	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
76	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₆₇	#131	1	2	yes	—
78	C₂×Dic₆₇	—	4	2	no	—
79	C₈×C₆₇	—	2	2	no	—
80	C₈×C₆₇	#120	1	1	yes	characteristic
81	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
82	C₈×C₆₇	#123	1	1	yes	characteristic
83	C₆₇⋊C₈	—	2	4	no	—
84	C₂²×C₆₇	—	2	2	no	—
85	C₂²×C₆₇	—	4	1	no	—
86	C₂²×C₆₇	—	4	2	no	—
87	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
88	C₂²×C₆₇	#435	1	2	yes	—
89	C₄×C₆₇	—	2	2	no	—
90	C₄×C₆₇	#385	1	1	yes	characteristic
91	C₄×C₆₇	#424	1	1	yes	characteristic
92	C₄×C₆₇	C₈×D₄	1	2	yes	—
93	Dic₆₇	—	8	2	no	—
94	#12916	—	67	1	no	maximal, Sylow
95	C₂×C₆₇	—	2	2	no	—
96	C₂×C₆₇	—	2	1	no	—
97	C₂×C₆₇	—	4	2	no	—
98	C₂×C₆₇	#1850	1	1	yes	characteristic
99	#3407	D₆₇	1	1	yes	characteristic
100	#2153	—	67	2	no	—
101	#2113	—	67	1	no	—
102	#1757	—	67	1	no	—
103	#1379	—	67	2	no	—
104	C₆₇	#12916	1	1	yes	characteristic, complemented, Sylow
105	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
106	#504	—	2	4	no	—
107	C₂×M₃₂	—	2	2	no	—
108	C₂×M₃₂	—	67	1	no	—
109	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
110	C₂²×C₁₆	—	67	1	no	—
111	#413	—	67	1	no	—
112	#410	—	67	1	no	—
113	#405	—	67	2	no	—
114	#364	—	134	4	no	—
115	C₂²⋊C₁₆	—	134	2	no	—
116	#344	—	67	2	no	—
117	C₂×#41	C₄×D₆₇	1	1	yes	characteristic
118	#131	—	2	2	no	—
119	#131	—	2	2	no	—
120	C₂×M₁₆	—	67	2	no	—
121	C₂×M₁₆	Dic₁₃₄	1	2	yes	—
122	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
123	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
124	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
125	#117	—	67	2	no	—
126	M₃₂	—	2	4	no	—
127	M₃₂	—	2	2	no	—
128	M₃₂	—	134	2	no	—
129	M₃₂	C₂²⋊D₆₇	1	4	yes	—
130	C₂×C₁₆	—	2	4	no	—
131	C₂×C₁₆	—	2	2	no	—
132	C₂×C₁₆	—	134	1	no	—
133	C₂×C₁₆	—	134	1	no	—
134	C₂×C₁₆	—	134	2	no	—
135	#101	—	67	2	no	—
136	#100	—	67	2	no	—
137	#99	—	134	2	no	—
138	#97	—	134	2	no	—
139	C₄×C₈	—	134	2	no	—
140	#41	—	2	2	no	—
141	#41	—	2	2	no	—
142	C₂×Q₈	C₈×D₆₇	1	1	yes	characteristic
143	C₂×D₄	C₈×D₆₇	1	1	yes	characteristic
144	C₂×D₄	#11761611	1	2	yes	—
145	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
146	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
147	M₁₆	—	2	2	no	—
148	M₁₆	—	2	4	no	—
149	M₁₆	—	67	4	no	—
150	M₁₆	—	134	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	134	2	no	—
154	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
155	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
156	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
157	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
158	C₂×C₈	#11761628	1	2	yes	—
159	C₄⋊C₄	—	134	2	no	—
160	C₂²⋊C₄	—	134	2	no	—
161	C₄²	—	134	2	no	—
162	C₁₆	—	2	4	no	—
163	C₁₆	—	268	1	no	—
164	C₁₆	—	268	1	no	—
165	C₂³	Dic₆₇⋊C₈	1	2	yes	—
166	C₂³	C₂×#11761611	1	1	yes	characteristic
167	Q₈	—	2	2	no	—
168	D₄	—	2	4	no	—
169	D₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	268	2	no	—
173	C₂×C₄	Dic₆₇⋊C₈	1	2	yes	—
174	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
175	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
176	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
177	C₂×C₄	#14524049	1	2	yes	—
178	C₈	—	2	2	no	—
179	C₈	—	134	4	no	—
180	C₈	#14524012	1	1	yes	characteristic
181	C₈	#14524022	1	2	yes	—
182	C₈	#14524026	1	1	yes	characteristic
183	C₂²	—	2	2	no	—
184	C₂²	—	4	1	no	—
185	C₂²	—	4	2	no	—
186	C₂²	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
187	C₂²	#16863467	1	2	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	536	2	no	—
190	C₄	#16863269	1	1	yes	characteristic
191	C₄	#16863422	1	1	yes	characteristic
192	C₄	C₈×C₂²⋊D₆₇	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	—	2	1	no	—
195	C₂	—	4	2	no	—
196	C₂	#22560387	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22561974

C₂

yes

maximal, complemented

#22561823

C₂

yes

maximal, characteristic

#22560009

C₂

yes

maximal, characteristic

#22558299

C₂

yes

maximal, complemented

C₆₇×#3407

C₂

yes

maximal, characteristic

C₂×#14523983

C₂²

yes

characteristic

C₂²×C₆₇⋊C₁₆

C₂²

yes

characteristic

#16863380

C₄

yes

characteristic

#16863370

C₄

yes

characteristic

#16863351

C₂²

yes

—

#16863216

—

C₂²⋊(C₆₇⋊C₁₆)

—

#16863167

C₄

yes

—

C₂×#131×C₆₇

C₂²

yes

characteristic

#504×C₆₇

—

C₂×M₃₂×C₆₇

—

C₂×M₃₂×C₆₇

C₂²

yes

—

C₂×#11761612

C₂×C₄

yes

—

#14524002

C₂×C₄

yes

—

#14523983

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

—

#14523962

C₈

yes

—

#14523960

C₈

yes

—

#14523958

—

#14523953

—

C₈×Dic₆₇

—

C₂×#41×C₆₇

C₂×C₄

yes

characteristic

#131×C₆₇

—

#131×C₆₇

—

C₂×M₁₆×C₆₇

C₂×C₄

yes

characteristic

C₂×M₁₆×C₆₇

Q₈

yes

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

M₃₂×C₆₇

—

M₃₂×C₆₇

—

M₃₂×C₆₇

D₄

yes

—

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

—

#11761612

—

#11761612

C₂×C₈

yes

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

—

C₆₇⋊C₁₆

—

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

C₂×C₈

yes

characteristic

C₂×D₄×C₆₇

C₂×C₈

yes

characteristic

C₂×D₄×C₆₇

M₁₆

yes

—

C₂²×C₄×C₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₁₆×C₆₇

—

C₂³×C₆₇

C₄⋊C₈

yes

—

C₂³×C₆₇

C₂×M₁₆

yes

characteristic

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄⋊C₈

yes

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₇

#131

yes

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#120

yes

characteristic

C₈×C₆₇

C₂²⋊Q₈

yes

—

C₈×C₆₇

#123

yes

characteristic

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₆₇

#435

yes

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

C₈×D₄

yes

—

Dic₆₇

—

#12916

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#1850

yes

characteristic

#3407

D₆₇

yes

characteristic

100

#2153

—

101

#2113

—

102

#1757

—

103

#1379

—

104

C₆₇

#12916

yes

characteristic, complemented, Sylow

105

C₂×#131

C₂×D₆₇

yes

characteristic

106

#504

—

107

C₂×M₃₂

—

108

C₂×M₃₂

—

109

C₂×M₃₂

C₂×D₆₇

yes

—

110

C₂²×C₁₆

—

111

#413

—

112

#410

—

113

#405

—

114

#364

—

134

—

115

C₂²⋊C₁₆

—

134

—

116

#344

—

117

C₂×#41

C₄×D₆₇

yes

characteristic

118

#131

—

119

#131

—

120

C₂×M₁₆

—

121

C₂×M₁₆

Dic₁₃₄

yes

—

122

C₂×M₁₆

C₄×D₆₇

yes

characteristic

123

C₂²×C₈

C₂²⋊D₆₇

yes

—

124

C₂²×C₈

C₂²×D₆₇

yes

characteristic

125

#117

—

126

M₃₂

—

127

M₃₂

—

128

M₃₂

—

134

—

129

M₃₂

C₂²⋊D₆₇

yes

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

132

C₂×C₁₆

—

134

—

133

C₂×C₁₆

—

134

—

134

C₂×C₁₆

—

134

—

135

#101

—

136

#100

—

137

#99

—

134

—

138

#97

—

134

—

139

C₄×C₈

—

134

—

140

#41

—

141

#41

—

142

C₂×Q₈

C₈×D₆₇

yes

characteristic

143

C₂×D₄

C₈×D₆₇

yes

characteristic

144

C₂×D₄

#11761611

yes

—

145

C₂²×C₄

Dic₆₇⋊C₄

yes

—

146

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

147

M₁₆

—

148

M₁₆

—

149

M₁₆

—

150

M₁₆

—

134

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

134

—

154

C₂×C₈

Dic₆₇⋊C₄

yes

—

155

C₂×C₈

C₂×C₂²⋊D₆₇

yes

—

156

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

157

C₂×C₈

C₂×Dic₁₃₄

yes

characteristic

158

C₂×C₈

#11761628

yes

—

159

C₄⋊C₄

—

134

—

160

C₂²⋊C₄

—

134

—

161

C₄²

—

134

—

162

C₁₆

—

163

C₁₆

—

268

—

164

C₁₆

—

268

—

165

C₂³

Dic₆₇⋊C₈

yes

—

166

C₂³

C₂×#11761611

yes

characteristic

167

Q₈

—

168

D₄

—

169

D₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

268

—

173

C₂×C₄

Dic₆₇⋊C₈

yes

—

174

C₂×C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

175

C₂×C₄

C₄×C₂²⋊D₆₇

yes

—

176

C₂×C₄

C₂×C₈×D₆₇

yes

characteristic

177

C₂×C₄

#14524049

yes

—

178

C₈

—

179

C₈

—

134

—

180

C₈

#14524012

yes

characteristic

181

C₈

#14524022

yes

—

182

C₈

#14524026

yes

characteristic

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

C₂×Dic₆₇⋊C₈

yes

characteristic

187

C₂²

#16863467

yes

—

188

C₄

—

189

C₄

—

536

—

190

C₄

#16863269

yes

characteristic

191

C₄

#16863422

yes

characteristic

192

C₄

C₈×C₂²⋊D₆₇

yes

—

193

C₂

—

194

C₂

—

195

C₂

—

196

C₂

#22560387

yes

characteristic

	subgroup	G/H
center	C₈	#14524012
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#3407	C₂
socle	C₂×C₆₇	#1850
prime radical	C₂×#41×C₆₇	C₂×C₄
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#3407	D₆₇
hypocenter	C₆₇	#12916
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #3407

subgroup

G/H

center

C₈

#14524012

commutator

C₂×C₄×C₆₇

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#3407

C₂

socle

C₂×C₆₇

#1850

prime radical

C₂×#41×C₆₇

C₂×C₄

cosocle

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #3407

N	H ≅ G/N	type
1. #22561974	C₂	semidirect
4. #22558299	C₂	semidirect
104. C₆₇	#12916	semidirect

H ≅ G/N

type

1. #22561974

C₂

semidirect

4. #22558299

C₂

semidirect

104. C₆₇

#12916

semidirect

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

subgroup	index
#22561974	2
#22561823	2
#22560009	2
#22558299	2
C₆₇×#3407	2
#12916	67

subgroup

index

C₆₇×#3407

Subgroups — #30772311

Conjugacy classes of subgroups

Complements

Maximal subgroups