Group #30749503 — subgroups

Subgroups — #30749503

Order 17152.

322 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 131 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	#22560325	C₂	1	2	yes	maximal
2	#22559954	C₂	1	1	yes	maximal, characteristic
3	#22558044	C₂	1	1	yes	maximal, characteristic
4	#22558043	C₂	1	2	yes	maximal
5	C₆₇×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
9	#16863150	—	2	2	no	—
10	#16863150	—	2	2	no	—
11	#16863150	C₄	1	2	yes	—
12	#16863150	C₄	1	2	yes	—
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₆₇⋊C₈	—	2	2	no	—
16	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
20	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
21	C₂×C₄×Dic₆₇	Q₈	1	2	yes	—
22	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₄×C₆₇⋊C₈	—	2	2	no	—
25	C₂²×C₈×C₆₇	—	2	2	no	—
26	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₆₇	—	2	2	no	—
29	C₄⋊C₈×C₆₇	—	2	2	no	—
30	C₄×C₈×C₆₇	—	2	2	no	—
31	C₂²×Dic₆₇	C₄⋊C₄	1	2	yes	—
32	C₂²×Dic₆₇	C₄⋊C₄	1	2	yes	—
33	C₂×C₆₇⋊C₈	—	2	1	no	—
34	C₂×C₆₇⋊C₈	—	2	1	no	—
35	C₂×C₆₇⋊C₈	—	2	2	no	—
36	C₂×C₆₇⋊C₈	—	2	2	no	—
37	C₂×C₆₇⋊C₈	—	4	2	no	—
38	C₂×C₆₇⋊C₈	—	4	2	no	—
39	C₂×C₆₇⋊C₈	—	4	2	no	—
40	C₄×Dic₆₇	—	2	2	no	—
41	C₄×Dic₆₇	—	2	2	no	—
42	C₄×Dic₆₇	—	2	1	no	—
43	C₄×Dic₆₇	—	2	2	no	—
44	C₄×Dic₆₇	—	2	1	no	—
45	C₄×Dic₆₇	—	4	2	no	—
46	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₆₇	—	2	2	no	—
50	C₂×C₈×C₆₇	—	2	2	no	—
51	C₂×C₈×C₆₇	—	2	2	no	—
52	C₂×C₈×C₆₇	—	4	2	no	—
53	C₂×C₈×C₆₇	—	4	2	no	—
54	C₂×C₈×C₆₇	—	4	2	no	—
55	C₄²×C₆₇	#41	1	2	yes	—
56	C₄²×C₆₇	#41	1	2	yes	—
57	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₆₇	—	2	2	no	—
59	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
60	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
61	C₂×C₄×C₆₇	#124	1	2	yes	—
62	C₂×C₄×C₆₇	#125	1	2	yes	—
63	C₂×C₄×C₆₇	#131	1	2	yes	—
64	C₂×C₄×C₆₇	#131	1	2	yes	—
65	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
66	C₂×C₄×C₆₇	#135	1	2	yes	—
67	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
68	C₂×Dic₆₇	—	2	2	no	—
69	C₂×Dic₆₇	—	2	2	no	—
70	C₂×Dic₆₇	—	2	2	no	—
71	C₂×Dic₆₇	—	2	2	no	—
72	C₂×Dic₆₇	—	2	2	no	—
73	C₂×Dic₆₇	—	2	2	no	—
74	C₈×C₆₇	—	4	2	no	—
75	C₈×C₆₇	—	8	2	no	—
76	C₆₇⋊C₈	—	4	1	no	—
77	C₆₇⋊C₈	—	4	1	no	—
78	C₆₇⋊C₈	—	8	2	no	—
79	C₂²×C₆₇	#382	1	1	yes	characteristic
80	C₂²×C₆₇	#424	1	2	yes	—
81	C₂²×C₆₇	#427	1	1	yes	characteristic
82	C₂²×C₆₇	#427	1	1	yes	characteristic
83	C₂²×C₆₇	#430	1	2	yes	—
84	C₄×C₆₇	—	2	2	no	—
85	C₄×C₆₇	#433	1	2	yes	—
86	C₄×C₆₇	#433	1	1	yes	characteristic
87	C₄×C₆₇	#433	1	1	yes	characteristic
88	C₄×C₆₇	#488	1	2	yes	—
89	C₄×C₆₇	#488	1	2	yes	—
90	Dic₆₇	—	4	2	no	—
91	Dic₆₇	—	4	2	no	—
92	#8788	—	67	1	no	maximal, Sylow
93	C₂×C₆₇	#1572	1	2	yes	—
94	C₂×C₆₇	#1572	1	2	yes	—
95	C₂×C₆₇	#1832	1	1	yes	characteristic
96	C₂×C₆₇	#1835	1	1	yes	characteristic
97	C₂×C₆₇	#1838	1	1	yes	characteristic
98	#1831	—	67	2	no	—
99	#1744	—	67	1	no	—
100	#1277	—	67	2	no	—
101	#1277	—	67	1	no	—
102	#1277	D₆₇	1	1	yes	characteristic
103	C₆₇	#8788	1	1	yes	characteristic, complemented, Sylow
104	C₂×C₄⋊C₈	—	67	2	no	—
105	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
106	C₂×C₄×C₈	—	67	1	no	—
107	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
108	C₄³	—	67	1	no	—
109	#336	—	67	2	no	—
110	#336	—	67	2	no	—
111	#336	—	134	2	no	—
112	#336	—	134	2	no	—
113	C₂²×C₈	—	2	2	no	—
114	C₂²×C₈	—	67	1	no	—
115	C₂²×C₈	—	67	1	no	—
116	C₂²×C₈	—	134	2	no	—
117	C₂²×C₈	C₄×D₆₇	1	2	yes	—
118	C₂×C₄²	—	67	2	no	—
119	C₂×C₄²	—	67	2	no	—
120	C₂×C₄²	—	67	1	no	—
121	C₂×C₄²	—	67	1	no	—
122	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	134	2	no	—
126	C₄⋊C₈	—	134	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	134	2	no	—
129	C₂²×C₄	—	67	2	no	—
130	C₂²×C₄	—	67	2	no	—
131	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
132	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
133	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	268	2	no	—
141	C₂×C₈	—	268	2	no	—
142	C₂×C₈	—	268	2	no	—
143	C₂×C₈	—	134	1	no	—
144	C₂×C₈	—	134	1	no	—
145	C₂×C₈	—	134	2	no	—
146	C₂×C₈	—	134	2	no	—
147	C₄²	—	268	2	no	—
148	C₄²	—	134	2	no	—
149	C₄²	—	134	2	no	—
150	C₄²	—	134	1	no	—
151	C₄²	—	134	2	no	—
152	C₄²	—	134	1	no	—
153	C₄²	#11761627	1	2	yes	—
154	C₄²	#11761628	1	2	yes	—
155	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	134	2	no	—
158	C₂×C₄	—	134	2	no	—
159	C₂×C₄	—	134	2	no	—
160	C₂×C₄	—	134	2	no	—
161	C₂×C₄	—	134	2	no	—
162	C₂×C₄	—	134	2	no	—
163	C₂×C₄	#14523999	1	1	yes	characteristic
164	C₂×C₄	#14524000	1	1	yes	characteristic
165	C₂×C₄	#14524029	1	2	yes	—
166	C₂×C₄	#14524033	1	2	yes	—
167	C₂×C₄	#14524049	1	2	yes	—
168	C₂×C₄	#14524051	1	2	yes	—
169	C₂×C₄	#14524062	1	1	yes	characteristic
170	C₂×C₄	#14524064	1	2	yes	—
171	C₂×C₄	#14524066	1	1	yes	characteristic
172	C₈	—	4	2	no	—
173	C₈	—	8	2	no	—
174	C₈	—	268	1	no	—
175	C₈	—	268	1	no	—
176	C₈	—	536	2	no	—
177	C₂²	#16863260	1	1	yes	characteristic
178	C₂²	#16863421	1	2	yes	—
179	C₂²	#16863431	1	1	yes	characteristic
180	C₂²	#16863433	1	1	yes	characteristic
181	C₂²	#16863443	1	2	yes	—
182	C₄	—	2	2	no	—
183	C₄	—	268	2	no	—
184	C₄	—	268	2	no	—
185	C₄	#16863454	1	2	yes	—
186	C₄	#16863455	1	1	yes	characteristic
187	C₄	#16863458	1	1	yes	characteristic
188	C₄	#16863749	1	2	yes	—
189	C₄	#16863752	1	2	yes	—
190	C₂	#22558986	1	2	yes	—
191	C₂	#22558988	1	2	yes	—
192	C₂	#22560327	1	1	yes	characteristic
193	C₂	#22560338	1	1	yes	characteristic
194	C₂	#22560348	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22560325

C₂

yes

maximal

#22559954

C₂

yes

maximal, characteristic

#22558044

C₂

yes

maximal, characteristic

#22558043

C₂

yes

maximal

C₆₇×#1277

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₄²×Dic₆₇

C₂²

yes

—

—

C₄

yes

—

#16863150

C₄

yes

—

C₂×C₄⋊C₈×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

—

C₂²×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄×Dic₆₇

C₂×C₄

yes

—

C₂×C₄×Dic₆₇

D₄

yes

characteristic

C₂×C₄×Dic₆₇

D₄

yes

characteristic

C₂×C₄×Dic₆₇

Q₈

yes

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

—

C₂²×C₈×C₆₇

—

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄×C₈×C₆₇

—

C₂²×Dic₆₇

C₄⋊C₄

yes

—

C₂²×Dic₆₇

C₄⋊C₄

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

#41

yes

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

—

C₂×C₄×C₆₇

#125

yes

—

C₂×C₄×C₆₇

#131

yes

—

C₂×C₄×C₆₇

#131

yes

—

C₂×C₄×C₆₇

#135

yes

characteristic

C₂×C₄×C₆₇

#135

yes

—

C₂×C₄×C₆₇

#135

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

#382

yes

characteristic

C₂²×C₆₇

#424

yes

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₄×C₆₇

—

C₄×C₆₇

#433

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₄×C₆₇

#488

yes

—

Dic₆₇

—

Dic₆₇

—

#8788

—

maximal, Sylow

C₂×C₆₇

#1572

yes

—

C₂×C₆₇

#1572

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

—

100

#1277

—

101

#1277

—

102

#1277

D₆₇

yes

characteristic

103

C₆₇

#8788

yes

characteristic, complemented, Sylow

104

C₂×C₄⋊C₈

—

105

C₂×C₄⋊C₈

C₂×D₆₇

yes

—

106

C₂×C₄×C₈

—

107

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

108

C₄³

—

109

#336

—

110

#336

—

111

#336

—

134

—

112

#336

—

134

—

113

C₂²×C₈

—

114

C₂²×C₈

—

115

C₂²×C₈

—

116

C₂²×C₈

—

134

—

117

C₂²×C₈

C₄×D₆₇

yes

—

118

C₂×C₄²

—

119

C₂×C₄²

—

120

C₂×C₄²

—

121

C₂×C₄²

—

122

C₂×C₄²

C₂²×D₆₇

yes

characteristic

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

C₄⋊C₈

—

134

—

126

C₄⋊C₈

—

134

—

127

C₄×C₈

—

128

C₄×C₈

—

134

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

132

C₂²×C₄

D₄×D₆₇

yes

characteristic

133

C₂²×C₄

Q₈×D₆₇

yes

characteristic

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

268

—

141

C₂×C₈

—

268

—

142

C₂×C₈

—

268

—

143

C₂×C₈

—

134

—

144

C₂×C₈

—

134

—

145

C₂×C₈

—

134

—

146

C₂×C₈

—

134

—

147

C₄²

—

268

—

148

C₄²

—

134

—

149

C₄²

—

134

—

150

C₄²

—

134

—

151

C₄²

—

134

—

152

C₄²

—

134

—

153

C₄²

#11761627

yes

—

154

C₄²

#11761628

yes

—

155

C₂³

C₄⋊C₄×D₆₇

yes

characteristic

156

C₂×C₄

—

157

C₂×C₄

—

134

—

158

C₂×C₄

—

134

—

159

C₂×C₄

—

134

—

160

C₂×C₄

—

134

—

161

C₂×C₄

—

134

—

162

C₂×C₄

—

134

—

163

C₂×C₄

#14523999

yes

characteristic

164

C₂×C₄

#14524000

yes

characteristic

165

C₂×C₄

#14524029

yes

—

166

C₂×C₄

#14524033

yes

—

167

C₂×C₄

#14524049

yes

—

168

C₂×C₄

#14524051

yes

—

169

C₂×C₄

#14524062

yes

characteristic

170

C₂×C₄

#14524064

yes

—

171

C₂×C₄

#14524066

yes

characteristic

172

C₈

—

173

C₈

—

174

C₈

—

268

—

175

C₈

—

268

—

176

C₈

—

536

—

177

C₂²

#16863260

yes

characteristic

178

C₂²

#16863421

yes

—

179

C₂²

#16863431

yes

characteristic

180

C₂²

#16863433

yes

characteristic

181

C₂²

#16863443

yes

—

182

C₄

—

183

C₄

—

268

—

184

C₄

—

268

—

185

C₄

#16863454

yes

—

186

C₄

#16863455

yes

characteristic

187

C₄

#16863458

yes

characteristic

188

C₄

#16863749

yes

—

189

C₄

#16863752

yes

—

190

C₂

#22558986

yes

—

191

C₂

#22558988

yes

—

192

C₂

#22560327

yes

characteristic

193

C₂

#22560338

yes

characteristic

194

C₂

#22560348

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1277	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1277	D₆₇
hypocenter	C₆₇	#8788
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

subgroup

G/H

center

C₂²×C₄

D₄×D₆₇

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1277

C₂

socle, prime radical

C₂³×C₆₇

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

N	H ≅ G/N	type
103. C₆₇	#8788	semidirect

H ≅ G/N

type

103. C₆₇

#8788

semidirect

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

subgroup	index
#22560325	2
#22559954	2
#22558044	2
#22558043	2
C₆₇×#1277	2
#8788	67

subgroup

index

C₆₇×#1277

Subgroups — #30749503

Conjugacy classes of subgroups

Complements

Maximal subgroups