Group #30746757 — subgroups

Subgroups — #30746757

Order 17152.

444 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 151 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	#22559958	C₂	1	2	yes	maximal
2	C₂×#16863138	C₂	1	2	yes	maximal
3	#22558090	C₂	1	2	yes	maximal
4	C₄×#125×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523954	C₂²	1	4	yes	—
6	#16863151	—	2	2	no	—
7	#16863151	—	2	2	no	—
8	#16863138	C₄	1	8	yes	complemented
9	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
10	#389×C₆₇	C₄	1	2	yes	—
11	#387×C₆₇	—	2	2	no	—
12	#382×C₆₇	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
14	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
15	C₄³×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₆₇⋊C₈	—	2	4	no	—
17	#14523954	—	2	4	no	—
18	#14523954	C₂×C₄	1	8	yes	—
19	#125×C₆₇	—	2	2	no	—
20	#125×C₆₇	C₂×C₄	1	4	yes	—
21	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
22	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
23	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
24	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₆₇	—	2	2	no	—
26	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₆₇	D₄	1	2	yes	—
29	C₂×C₄²×C₆₇	D₄	1	2	yes	—
30	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
31	#95×C₆₇	—	2	2	no	—
32	#95×C₆₇	—	2	2	no	—
33	#95×C₆₇	—	2	2	no	—
34	C₂×C₆₇⋊C₈	—	2	8	no	—
35	C₂×C₆₇⋊C₈	—	4	4	no	—
36	C₂×C₆₇⋊C₈	—	4	4	no	—
37	C₂²×C₄×C₆₇	—	2	2	no	—
38	C₂²×C₄×C₆₇	—	2	2	no	—
39	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
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	2	no	—
45	C₄⋊C₄×C₆₇	—	2	8	no	—
46	C₄⋊C₄×C₆₇	—	4	2	no	—
47	C₄⋊C₄×C₆₇	—	4	4	no	—
48	C₄⋊C₄×C₆₇	C₄²	1	4	yes	—
49	C₄²×C₆₇	—	2	2	no	—
50	C₄²×C₆₇	—	4	1	no	—
51	C₄²×C₆₇	—	4	2	no	—
52	C₄²×C₆₇	—	4	2	no	—
53	C₄²×C₆₇	—	4	4	no	—
54	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
55	C₄²×C₆₇	#41	1	2	yes	—
56	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₆₇	—	2	1	no	—
58	C₂×C₄×C₆₇	—	2	1	no	—
59	C₂×C₄×C₆₇	—	2	2	no	—
60	C₂×C₄×C₆₇	—	2	2	no	—
61	C₂×C₄×C₆₇	—	2	2	no	—
62	C₂×C₄×C₆₇	—	2	4	no	—
63	C₂×C₄×C₆₇	—	2	1	no	—
64	C₂×C₄×C₆₇	—	2	2	no	—
65	C₂×C₄×C₆₇	—	2	2	no	—
66	C₂×C₄×C₆₇	—	2	2	no	—
67	C₂×C₄×C₆₇	—	2	2	no	—
68	C₂×C₄×C₆₇	—	4	2	no	—
69	C₂×C₄×C₆₇	—	4	2	no	—
70	C₂×C₄×C₆₇	—	4	2	no	—
71	C₂×C₄×C₆₇	—	4	2	no	—
72	C₂×C₄×C₆₇	—	4	2	no	—
73	C₂×C₄×C₆₇	#101	1	2	yes	—
74	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
77	C₆₇⋊C₈	—	4	8	no	—
78	C₂²×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₆₇	C₂×#101	1	1	yes	characteristic
80	C₂²×C₆₇	#413	1	1	yes	characteristic
81	C₂²×C₆₇	#421	1	2	yes	—
82	C₂²×C₆₇	#421	1	2	yes	—
83	C₄×C₆₇	—	2	1	no	—
84	C₄×C₆₇	—	2	1	no	—
85	C₄×C₆₇	—	2	4	no	—
86	C₄×C₆₇	—	2	2	no	—
87	C₄×C₆₇	—	4	1	no	—
88	C₄×C₆₇	—	4	4	no	—
89	C₄×C₆₇	—	4	2	no	—
90	C₄×C₆₇	—	8	2	no	—
91	#8225	—	67	1	no	maximal, Sylow
92	C₂×C₆₇	#1534	1	1	yes	characteristic
93	C₂×C₆₇	#1535	1	2	yes	—
94	C₂×C₆₇	#1537	1	1	yes	characteristic
95	C₂×C₆₇	C₄×#101	1	1	yes	characteristic
96	C₂×C₆₇	#1759	1	2	yes	—
97	C₄×#125	D₆₇	1	1	yes	characteristic
98	#1745	—	67	2	no	—
99	C₂×#330	—	67	2	no	—
100	#1295	—	67	2	no	—
101	C₆₇	#8225	1	1	yes	characteristic, complemented, Sylow
102	C₂×#125	C₂×D₆₇	1	1	yes	characteristic
103	C₂×#97	—	67	4	no	—
104	#389	Dic₆₇	1	2	yes	—
105	#387	—	2	2	no	—
106	#382	C₂×D₆₇	1	1	yes	characteristic
107	C₄×C₄⋊C₄	—	2	2	no	—
108	C₄×C₄⋊C₄	Dic₆₇	1	2	yes	—
109	C₄³	C₂×D₆₇	1	1	yes	characteristic
110	#336	—	134	2	no	—
111	#336	—	134	2	no	—
112	#330	—	67	8	no	—
113	C₂²×C₈	—	134	4	no	—
114	#125	—	2	2	no	—
115	#125	C₄×D₆₇	1	4	yes	—
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×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
120	C₂×C₄²	—	2	2	no	—
121	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
122	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
123	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
124	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
125	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
126	#97	—	67	8	no	—
127	#97	—	134	4	no	—
128	#95	—	2	2	no	—
129	#95	—	2	2	no	—
130	#95	—	2	2	no	—
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	—	134	8	no	—
141	C₄⋊C₄	—	2	2	no	—
142	C₄⋊C₄	—	2	8	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	4	4	no	—
145	C₄⋊C₄	C₄×Dic₆₇	1	4	yes	—
146	C₄²	—	2	2	no	—
147	C₄²	—	4	1	no	—
148	C₄²	—	4	2	no	—
149	C₄²	—	4	2	no	—
150	C₄²	—	4	4	no	—
151	C₄²	C₂×C₄×D₆₇	1	2	yes	—
152	C₄²	#11761628	1	2	yes	—
153	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	#14523962	1	2	yes	—
171	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
172	C₂×C₄	#14524002	1	1	yes	characteristic
173	C₂×C₄	C₄×C₂²⋊D₆₇	1	4	yes	—
174	C₈	—	268	8	no	—
175	C₂²	C₄×C₂²⋊Dic₆₇	1	1	yes	characteristic
176	C₂²	C₂×#14523962	1	1	yes	characteristic
177	C₂²	#16863380	1	1	yes	characteristic
178	C₂²	#16863413	1	2	yes	—
179	C₂²	#16863413	1	2	yes	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	2	4	no	—
183	C₄	—	2	2	no	—
184	C₄	—	4	4	no	—
185	C₄	—	4	2	no	—
186	C₄	—	4	1	no	—
187	C₄	—	8	2	no	—
188	C₂	#22558836	1	1	yes	characteristic
189	C₂	#22558840	1	2	yes	—
190	C₂	#22558847	1	1	yes	characteristic
191	C₂	C₄×#14523962	1	1	yes	characteristic
192	C₂	#22560019	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22559958

C₂

yes

maximal

C₂×#16863138

C₂

yes

maximal

#22558090

C₂

yes

maximal

C₄×#125×C₆₇

C₂

yes

maximal, characteristic

C₂×#14523954

C₂²

yes

—

#16863151

—

#16863151

—

#16863138

C₄

yes

complemented

C₂×#125×C₆₇

C₂²

yes

characteristic

#389×C₆₇

C₄

yes

—

#387×C₆₇

—

#382×C₆₇

C₂²

yes

characteristic

C₄×C₄⋊C₄×C₆₇

—

C₄×C₄⋊C₄×C₆₇

C₄

yes

—

C₄³×C₆₇

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

—

#14523954

—

#14523954

C₂×C₄

yes

—

#125×C₆₇

—

#125×C₆₇

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂×C₄

yes

—

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

—

#95×C₆₇

—

#95×C₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

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

—

C₄⋊C₄×C₆₇

C₄²

yes

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂²×C₄

yes

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

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

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#101

yes

—

C₂×C₄×C₆₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₇

#117

yes

characteristic

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

C₄×C₂²⋊C₄

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

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₄×#125

D₆₇

yes

characteristic

#1745

—

C₂×#330

—

100

#1295

—

101

C₆₇

#8225

yes

characteristic, complemented, Sylow

102

C₂×#125

C₂×D₆₇

yes

characteristic

103

C₂×#97

—

104

#389

Dic₆₇

yes

—

105

#387

—

106

#382

C₂×D₆₇

yes

characteristic

107

C₄×C₄⋊C₄

—

108

C₄×C₄⋊C₄

Dic₆₇

yes

—

109

C₄³

C₂×D₆₇

yes

characteristic

110

#336

—

134

—

111

#336

—

134

—

112

#330

—

113

C₂²×C₈

—

134

—

114

#125

—

115

#125

C₄×D₆₇

yes

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

C₂×Dic₆₇

yes

—

119

C₂×C₄⋊C₄

C₂×Dic₆₇

yes

—

120

C₂×C₄²

—

121

C₂×C₄²

C₂×Dic₆₇

yes

characteristic

122

C₂×C₄²

C₂×Dic₆₇

yes

characteristic

123

C₂×C₄²

C₂²⋊D₆₇

yes

—

124

C₂×C₄²

C₂²⋊D₆₇

yes

—

125

C₂×C₄²

C₂²×D₆₇

yes

characteristic

126

#97

—

127

#97

—

134

—

128

#95

—

129

#95

—

130

#95

—

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

—

134

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

C₄×Dic₆₇

yes

—

146

C₄²

—

147

C₄²

—

148

C₄²

—

149

C₄²

—

150

C₄²

—

151

C₄²

C₂×C₄×D₆₇

yes

—

152

C₄²

#11761628

yes

—

153

C₂³

C₂×C₂²⋊Dic₆₇

yes

characteristic

154

C₂×C₄

—

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

—

169

C₂×C₄

—

170

C₂×C₄

#14523962

yes

—

171

C₂×C₄

C₂×C₄×Dic₆₇

yes

characteristic

172

C₂×C₄

#14524002

yes

characteristic

173

C₂×C₄

C₄×C₂²⋊D₆₇

yes

—

174

C₈

—

268

—

175

C₂²

C₄×C₂²⋊Dic₆₇

yes

characteristic

176

C₂²

C₂×#14523962

yes

characteristic

177

C₂²

#16863380

yes

characteristic

178

C₂²

#16863413

yes

—

179

C₂²

#16863413

yes

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₂

#22558836

yes

characteristic

189

C₂

#22558840

yes

—

190

C₂

#22558847

yes

characteristic

191

C₂

C₄×#14523962

yes

characteristic

192

C₂

#22560019

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×#125×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×#125	D₆₇
hypocenter	C₆₇	#8225
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₆₇

commutator

C₂×C₄×C₆₇

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₄×#125×C₆₇

C₂

socle, prime radical

C₂³×C₆₇

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

N	H ≅ G/N	type
8. #16863138	C₄	semidirect
101. C₆₇	#8225	semidirect

H ≅ G/N

type

8. #16863138

C₄

semidirect

101. C₆₇

#8225

semidirect

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

subgroup	index
#22559958	2
C₂×#16863138	2
#22558090	2
C₄×#125×C₆₇	2
#8225	67

subgroup

index

C₄×#125×C₆₇

Subgroups — #30746757

Conjugacy classes of subgroups

Complements

Maximal subgroups