Group #30767772 — subgroups

Subgroups — #30767772

Order 17152.

210 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 75 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	C₄×#14523989	C₂	1	1	yes	maximal, characteristic
2	#22559105	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#22558282	C₂	1	1	yes	maximal, characteristic
5	#22558174	C₂	1	1	yes	maximal, characteristic
6	#22558128	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1332	C₂	1	1	yes	maximal, characteristic
8	C₂×#14523989	C₄	1	1	yes	characteristic
9	#16863523	—	2	1	no	—
10	Q₈×C₆₇⋊C₈	—	2	1	no	—
11	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
12	#16863221	C₄	1	1	yes	characteristic
13	#16863200	C₄	1	1	yes	characteristic
14	#16863199	C₄	1	1	yes	characteristic
15	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	#16863149	C₂²	1	1	yes	characteristic
17	Q₈⋊(C₆₇⋊C₈)	—	2	1	no	—
18	#16863126	—	2	1	no	—
19	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
21	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
22	#335×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₁₃₄	—	2	1	no	—
25	#14523989	—	2	2	no	—
26	C₂×C₆₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₆₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
28	#14523977	C₂×C₄	1	1	yes	characteristic
29	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
30	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
31	Dic₆₇⋊C₈	—	4	1	no	—
32	#14523968	—	2	1	no	—
33	#14523953	—	4	1	no	—
34	C₄×C₆₇⋊C₈	—	4	1	no	—
35	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
36	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
37	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₄×Q₈×C₆₇	—	2	1	no	—
39	C₂×C₁₆×C₆₇	—	2	1	no	—
40	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₄⋊C₈×C₆₇	—	2	1	no	—
42	#103×C₆₇	—	2	1	no	—
43	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
44	C₂×Dic₁₃₄	—	2	1	no	—
45	Dic₂₆₈	—	2	1	no	—
46	Dic₂₆₈	—	2	1	no	—
47	C₂×C₆₇⋊C₈	—	2	1	no	—
48	C₂×C₆₇⋊C₈	—	4	1	no	—
49	C₂×C₆₇⋊C₈	D₈	1	1	yes	characteristic
50	C₂×C₆₇⋊C₈	D₈	1	1	yes	characteristic
51	C₄⋊Dic₆₇	—	2	1	no	—
52	Dic₆₇⋊C₄	—	4	1	no	—
53	C₄×Dic₆₇	—	4	1	no	—
54	C₆₇⋊C₁₆	—	2	1	no	—
55	C₆₇⋊C₁₆	—	2	1	no	—
56	C₂×Q₈×C₆₇	—	2	1	no	—
57	Q₁₆×C₆₇	—	2	1	no	—
58	Q₁₆×C₆₇	M₁₆	1	2	yes	—
59	C₂×C₈×C₆₇	—	4	1	no	—
60	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
62	C₄⋊C₄×C₆₇	—	2	1	no	—
63	C₄⋊C₄×C₆₇	—	4	1	no	—
64	C₄²×C₆₇	—	4	1	no	—
65	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₁₆×C₆₇	—	4	1	no	—
67	Q₈×C₆₇	—	2	2	no	—
68	Q₈×C₆₇	—	4	1	no	—
69	C₂×C₄×C₆₇	—	4	1	no	—
70	C₂×C₄×C₆₇	—	4	1	no	—
71	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
74	C₂×Dic₆₇	—	4	1	no	—
75	C₂×Dic₆₇	—	4	1	no	—
76	Dic₁₃₄	—	4	1	no	—
77	Dic₁₃₄	—	4	1	no	—
78	C₈×C₆₇	—	2	1	no	—
79	C₈×C₆₇	—	8	1	no	—
80	C₈×C₆₇	C₂×M₁₆	1	1	yes	characteristic
81	C₈×C₆₇	#131	1	1	yes	characteristic
82	C₆₇⋊C₈	—	2	2	no	—
83	C₆₇⋊C₈	—	2	1	no	—
84	C₆₇⋊C₈	—	2	1	no	—
85	C₆₇⋊C₈	—	8	1	no	—
86	C₂²×C₆₇	C₄×D₈	1	1	yes	characteristic
87	C₄×C₆₇	—	2	1	no	—
88	C₄×C₆₇	—	4	2	no	—
89	C₄×C₆₇	—	8	1	no	—
90	C₄×C₆₇	#435	1	1	yes	characteristic
91	C₄×C₆₇	#444	1	1	yes	characteristic
92	C₄×C₆₇	#510	1	1	yes	characteristic
93	C₄×C₆₇	#510	1	1	yes	characteristic
94	Dic₆₇	—	8	1	no	—
95	Dic₆₇	—	8	1	no	—
96	#12076	—	67	1	no	maximal, Sylow
97	C₂×C₆₇	#1581	1	1	yes	characteristic
98	C₂×C₆₇	#2176	1	1	yes	characteristic
99	C₂×C₆₇	#2178	1	1	yes	characteristic
100	C₄×Q₃₂	—	67	1	no	—
101	#1591	—	67	1	no	—
102	C₈×Q₁₆	—	67	1	no	—
103	#1372	—	67	1	no	—
104	#1332	—	67	1	no	—
105	#1332	D₆₇	1	1	yes	characteristic
106	#1312	—	67	1	no	—
107	C₆₇	#12076	1	1	yes	characteristic, complemented, Sylow
108	C₂×Q₃₂	—	67	1	no	—
109	#446	—	134	1	no	—
110	C₈×Q₈	—	134	1	no	—
111	C₄×Q₁₆	—	67	1	no	—
112	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
113	#366	—	67	1	no	—
114	#358	—	67	1	no	—
115	#358	—	67	1	no	—
116	C₄×C₁₆	—	67	1	no	—
117	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
118	#335	—	67	1	no	—
119	#335	C₂×D₆₇	1	1	yes	characteristic
120	Q₈⋊C₈	—	134	1	no	—
121	Q₈⋊C₈	—	134	1	no	—
122	C₈²	—	67	1	no	—
123	C₂×Q₁₆	—	67	1	no	—
124	C₂×Q₁₆	C₄×D₆₇	1	1	yes	characteristic
125	C₄×Q₈	—	2	1	no	—
126	C₄×Q₈	—	134	1	no	—
127	Q₃₂	—	134	2	no	—
128	C₂×C₁₆	—	2	1	no	—
129	C₂×C₁₆	—	67	1	no	—
130	C₂×C₁₆	—	67	1	no	—
131	#107	—	67	1	no	—
132	#107	C₄×D₆₇	1	1	yes	characteristic
133	C₄⋊C₈	—	2	1	no	—
134	C₄⋊C₈	—	268	1	no	—
135	C₄⋊C₈	—	268	1	no	—
136	C₄⋊C₈	—	134	1	no	—
137	#103	—	2	1	no	—
138	#103	—	134	1	no	—
139	#97	—	268	1	no	—
140	C₄×C₈	—	268	1	no	—
141	C₄×C₈	—	67	1	no	—
142	C₄×C₈	—	67	1	no	—
143	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
144	C₂×Q₈	—	2	1	no	—
145	C₂×Q₈	—	134	1	no	—
146	Q₁₆	—	2	1	no	—
147	Q₁₆	—	134	1	no	—
148	Q₁₆	—	134	1	no	—
149	Q₁₆	#11761611	1	2	yes	—
150	C₂×C₈	—	4	1	no	—
151	C₂×C₈	—	268	1	no	—
152	C₂×C₈	—	67	1	no	—
153	C₂×C₈	—	67	1	no	—
154	C₂×C₈	—	134	1	no	—
155	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
156	C₂×C₈	#11761627	1	1	yes	characteristic
157	C₄⋊C₄	—	2	1	no	—
158	C₄⋊C₄	—	4	1	no	—
159	C₄⋊C₄	—	268	1	no	—
160	C₄⋊C₄	—	134	1	no	—
161	C₄²	—	4	1	no	—
162	C₄²	—	268	1	no	—
163	C₄²	D₄×D₆₇	1	1	yes	characteristic
164	C₁₆	—	4	1	no	—
165	C₁₆	—	134	1	no	—
166	C₁₆	—	134	1	no	—
167	Q₈	—	2	2	no	—
168	Q₈	—	4	1	no	—
169	Q₈	—	268	1	no	—
170	Q₈	—	268	1	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	268	1	no	—
174	C₂×C₄	—	268	1	no	—
175	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
176	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
177	C₂×C₄	#14524064	1	1	yes	characteristic
178	C₈	—	2	1	no	—
179	C₈	—	8	1	no	—
180	C₈	—	536	1	no	—
181	C₈	—	134	2	no	—
182	C₈	—	134	1	no	—
183	C₈	—	134	1	no	—
184	C₈	C₂×#11761611	1	1	yes	characteristic
185	C₈	#14524051	1	1	yes	characteristic
186	C₂²	#16863476	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	4	2	no	—
189	C₄	—	8	1	no	—
190	C₄	—	536	1	no	—
191	C₄	—	536	1	no	—
192	C₄	#16863464	1	1	yes	characteristic
193	C₄	#16863515	1	1	yes	characteristic
194	C₄	#16863835	1	1	yes	characteristic
195	C₄	#16863838	1	1	yes	characteristic
196	C₂	#22559035	1	1	yes	characteristic
197	C₂	#22562077	1	1	yes	characteristic
198	C₂	#22562087	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14523989

C₂

yes

maximal, characteristic

#22559105

C₂

yes

maximal, characteristic

Q₁₆×C₆₇⋊C₈

C₂

yes

maximal, characteristic

#22558282

C₂

yes

maximal, characteristic

#22558174

C₂

yes

maximal, characteristic

#22558128

C₂

yes

maximal, characteristic

C₆₇×#1332

C₂

yes

maximal, characteristic

C₂×#14523989

C₄

yes

characteristic

#16863523

—

Q₈×C₆₇⋊C₈

—

C₄×Dic₂₆₈

C₂²

yes

characteristic

#16863221

C₄

yes

characteristic

#16863200

C₄

yes

characteristic

#16863199

C₄

yes

characteristic

C₄×C₆₇⋊C₁₆

C₂²

yes

characteristic

#16863149

C₂²

yes

characteristic

Q₈⋊(C₆₇⋊C₈)

—

#16863126

—

C₈×C₆₇⋊C₈

C₂²

yes

characteristic

C₄×Q₁₆×C₆₇

C₂²

yes

characteristic

C₄×C₁₆×C₆₇

C₂²

yes

characteristic

#335×C₆₇

C₂²

yes

characteristic

C₂×Dic₂₆₈

C₂×C₄

yes

characteristic

C₄×Dic₁₃₄

—

#14523989

—

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

characteristic

#14523977

C₂×C₄

yes

characteristic

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

Dic₆₇⋊C₈

—

#14523968

—

#14523953

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

D₄

yes

characteristic

C₄×C₆₇⋊C₈

D₄

yes

characteristic

C₂×Q₁₆×C₆₇

C₂×C₄

yes

characteristic

C₄×Q₈×C₆₇

—

C₂×C₁₆×C₆₇

—

#107×C₆₇

C₂×C₄

yes

characteristic

C₄⋊C₈×C₆₇

—

#103×C₆₇

—

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×Dic₁₃₄

—

Dic₂₆₈

—

Dic₂₆₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

D₈

yes

characteristic

C₂×C₆₇⋊C₈

D₈

yes

characteristic

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

—

C₆₇⋊C₁₆

—

C₂×Q₈×C₆₇

—

Q₁₆×C₆₇

—

Q₁₆×C₆₇

M₁₆

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

#41

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₁₆×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

#135

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₂×M₁₆

yes

characteristic

C₈×C₆₇

#131

yes

characteristic

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

C₄×D₈

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

100

C₄×Q₃₂

—

101

#1591

—

102

C₈×Q₁₆

—

103

#1372

—

104

#1332

—

105

#1332

D₆₇

yes

characteristic

106

#1312

—

107

C₆₇

#12076

yes

characteristic, complemented, Sylow

108

C₂×Q₃₂

—

109

#446

—

134

—

110

C₈×Q₈

—

134

—

111

C₄×Q₁₆

—

112

C₄×Q₁₆

C₂×D₆₇

yes

characteristic

113

#366

—

114

#358

—

115

#358

—

116

C₄×C₁₆

—

117

C₄×C₁₆

C₂×D₆₇

yes

characteristic

118

#335

—

119

#335

C₂×D₆₇

yes

characteristic

120

Q₈⋊C₈

—

134

—

121

Q₈⋊C₈

—

134

—

122

C₈²

—

123

C₂×Q₁₆

—

124

C₂×Q₁₆

C₄×D₆₇

yes

characteristic

125

C₄×Q₈

—

126

C₄×Q₈

—

134

—

127

Q₃₂

—

134

—

128

C₂×C₁₆

—

129

C₂×C₁₆

—

130

C₂×C₁₆

—

131

#107

—

132

#107

C₄×D₆₇

yes

characteristic

133

C₄⋊C₈

—

134

C₄⋊C₈

—

268

—

135

C₄⋊C₈

—

268

—

136

C₄⋊C₈

—

134

—

137

#103

—

138

#103

—

134

—

139

#97

—

268

—

140

C₄×C₈

—

268

—

141

C₄×C₈

—

142

C₄×C₈

—

143

C₄×C₈

C₂²×D₆₇

yes

characteristic

144

C₂×Q₈

—

145

C₂×Q₈

—

134

—

146

Q₁₆

—

147

Q₁₆

—

134

—

148

Q₁₆

—

134

—

149

Q₁₆

#11761611

yes

—

150

C₂×C₈

—

151

C₂×C₈

—

268

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

134

—

155

C₂×C₈

C₂×C₄×D₆₇

yes

characteristic

156

C₂×C₈

#11761627

yes

characteristic

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

268

—

160

C₄⋊C₄

—

134

—

161

C₄²

—

162

C₄²

—

268

—

163

C₄²

D₄×D₆₇

yes

characteristic

164

C₁₆

—

165

C₁₆

—

134

—

166

C₁₆

—

134

—

167

Q₈

—

168

Q₈

—

169

Q₈

—

268

—

170

Q₈

—

268

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

268

—

174

C₂×C₄

—

268

—

175

C₂×C₄

(C₂²⋊D₆₇)⋊C₄

yes

characteristic

176

C₂×C₄

D₈×D₆₇

yes

characteristic

177

C₂×C₄

#14524064

yes

characteristic

178

C₈

—

179

C₈

—

180

C₈

—

536

—

181

C₈

—

134

—

182

C₈

—

134

—

183

C₈

—

134

—

184

C₈

C₂×#11761611

yes

characteristic

185

C₈

#14524051

yes

characteristic

186

C₂²

#16863476

yes

characteristic

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

536

—

191

C₄

—

536

—

192

C₄

#16863464

yes

characteristic

193

C₄

#16863515

yes

characteristic

194

C₄

#16863835

yes

characteristic

195

C₄

#16863838

yes

characteristic

196

C₂

#22559035

yes

characteristic

197

C₂

#22562077

yes

characteristic

198

C₂

#22562087

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₆₇
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#1332	C₂
socle, prime radical	C₂²×C₆₇	C₄×D₈
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	#1332	D₆₇
hypocenter	C₆₇	#12076
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1332

subgroup

G/H

center

C₂×C₄

D₈×D₆₇

commutator

C₂×C₈×C₆₇

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#1332

C₂

socle, prime radical

C₂²×C₆₇

C₄×D₈

cosocle

C₄×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1332

N	H ≅ G/N	type
107. C₆₇	#12076	semidirect

H ≅ G/N

type

107. C₆₇

#12076

semidirect

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

subgroup	index
C₄×#14523989	2
#22559105	2
Q₁₆×C₆₇⋊C₈	2
#22558282	2
#22558174	2
#22558128	2
C₆₇×#1332	2
#12076	67

subgroup

index

Q₁₆×C₆₇⋊C₈

C₆₇×#1332

Subgroups — #30767772

Conjugacy classes of subgroups

Complements

Maximal subgroups