Group #30760935 — subgroups

Subgroups — #30760935

Order 17152.

526 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 153 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	#22561015	C₂	1	1	yes	maximal, characteristic
2	#22561014	C₂	1	2	yes	maximal
3	#22560365	C₂	1	2	yes	maximal, complemented
4	C₆₇⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1844	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14524040	C₂²	1	1	yes	characteristic
7	C₂×#14523974	—	2	4	no	—
8	C₂×#14523965	—	2	2	no	—
9	C₂×#14523963	—	2	4	no	—
10	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
11	#16863266	C₂²	1	1	yes	characteristic
12	C₂×#106×C₆₇	—	2	2	no	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	#384×C₆₇	C₂²	1	2	yes	—
15	C₂²×D₂₆₈	—	2	2	no	—
16	C₂²×D₂₆₈	—	2	1	no	—
17	C₂²×C₆₇⋊C₈	D₄	1	4	yes	—
18	#14524040	—	2	2	no	—
19	#14524040	—	2	1	no	—
20	#14524040	Q₈	1	2	yes	—
21	C₂×C₄⋊Dic₆₇	—	2	1	no	—
22	C₂×C₄⋊Dic₆₇	—	2	2	no	—
23	#14523974	—	4	8	no	—
24	#14523965	—	4	4	no	—
25	#14523963	—	4	8	no	—
26	C₄×C₆₇⋊C₈	—	2	2	no	—
27	C₄×C₆₇⋊C₈	—	2	2	no	—
28	C₂²×C₈×C₆₇	D₄	1	2	yes	—
29	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
32	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
33	#106×C₆₇	—	4	4	no	—
34	C₄×C₈×C₆₇	—	2	2	no	—
35	C₂³×D₆₇	—	4	1	no	—
36	C₂×D₂₆₈	—	2	2	no	—
37	C₂×D₂₆₈	—	2	4	no	—
38	C₂×D₂₆₈	—	4	4	no	—
39	C₂×D₂₆₈	—	4	2	no	—
40	C₂×D₂₆₈	—	4	2	no	—
41	C₂×D₂₆₈	—	4	4	no	—
42	C₂×D₂₆₈	—	4	1	no	—
43	C₂×D₂₆₈	—	4	2	no	—
44	C₂²×Dic₆₇	—	4	1	no	—
45	C₂×C₆₇⋊C₈	—	2	4	no	—
46	C₂×C₆₇⋊C₈	—	2	4	no	—
47	C₂×C₆₇⋊C₈	SD₁₆	1	8	yes	—
48	C₄⋊Dic₆₇	—	4	2	no	—
49	C₄⋊Dic₆₇	—	4	4	no	—
50	C₂²×C₄×C₆₇	—	4	2	no	—
51	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₆₇	—	2	2	no	—
54	C₂×C₈×C₆₇	—	2	2	no	—
55	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
56	C₄⋊C₄×C₆₇	—	4	4	no	—
57	C₄⋊C₄×C₆₇	—	4	4	no	—
58	C₄⋊C₄×C₆₇	—	4	4	no	—
59	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
60	C₄²×C₆₇	#41	1	2	yes	—
61	C₄²×C₆₇	#41	1	1	yes	characteristic
62	C₂³×C₆₇	#127	1	1	yes	characteristic
63	C₂²×D₆₇	—	4	2	no	—
64	C₂²×D₆₇	—	8	2	no	—
65	C₂²×D₆₇	—	8	1	no	—
66	C₂²×D₆₇	—	8	2	no	—
67	C₂²×D₆₇	—	8	1	no	—
68	C₂×C₄×C₆₇	—	8	2	no	—
69	C₂×C₄×C₆₇	—	8	2	no	—
70	C₂×C₄×C₆₇	—	8	2	no	—
71	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
73	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
74	C₂×C₄×C₆₇	#123	1	2	yes	—
75	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
76	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
77	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
78	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
79	C₂×Dic₆₇	—	8	1	no	—
80	C₂×Dic₆₇	—	8	2	no	—
81	D₂₆₈	—	4	4	no	—
82	D₂₆₈	—	4	8	no	—
83	D₂₆₈	—	8	2	no	—
84	D₂₆₈	—	8	4	no	—
85	C₈×C₆₇	—	2	4	no	—
86	C₆₇⋊C₈	—	2	8	no	—
87	C₂²×C₆₇	#399	1	1	yes	characteristic
88	C₂²×C₆₇	#492	1	2	yes	—
89	C₂²×C₆₇	#492	1	1	yes	characteristic
90	C₂²×C₆₇	#496	1	2	yes	—
91	C₂²×C₆₇	#496	1	1	yes	characteristic
92	C₂×D₆₇	—	8	4	no	—
93	C₂×D₆₇	—	8	2	no	—
94	C₂×D₆₇	—	16	1	no	—
95	C₂×D₆₇	—	16	2	no	—
96	C₂×D₆₇	—	16	1	no	—
97	C₄×C₆₇	—	16	2	no	—
98	C₄×C₆₇	#476	1	2	yes	—
99	C₄×C₆₇	#476	1	4	yes	—
100	C₄×C₆₇	#481	1	2	yes	—
101	C₄×C₆₇	#481	1	4	yes	—
102	Dic₆₇	—	16	1	no	—
103	#10683	—	67	1	no	maximal, Sylow
104	C₂×C₆₇	#1705	1	1	yes	characteristic
105	C₂×C₆₇	#1715	1	2	yes	—
106	C₂×C₆₇	#1715	1	1	yes	characteristic
107	C₂×C₆₇	#2094	1	2	yes	—
108	C₂×C₆₇	#2094	1	1	yes	characteristic
109	D₆₇	—	16	2	no	—
110	#1957	—	67	1	no	—
111	#1957	—	67	2	no	—
112	#1844	—	67	2	no	—
113	#1844	D₆₇	1	1	yes	characteristic, complemented
114	#1273	—	67	1	no	—
115	C₆₇	#10683	1	1	yes	characteristic, complemented, Sylow
116	C₂×#127	—	67	1	no	—
117	C₂×#106	—	2	2	no	—
118	C₂×#106	—	134	4	no	—
119	C₂×#102	—	134	4	no	—
120	C₂×#102	—	134	2	no	—
121	C₂×C₄×C₈	—	67	2	no	—
122	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
123	#384	—	67	1	no	—
124	#384	C₂×D₆₇	1	2	yes	—
125	C₂²×D₄	—	134	2	no	—
126	C₂²×D₄	—	134	1	no	—
127	C₂²×C₈	—	67	4	no	—
128	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
129	#127	—	67	2	no	—
130	#127	—	134	2	no	—
131	#127	—	134	1	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	134	2	no	—
136	C₂×C₄⋊C₄	—	134	1	no	—
137	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
138	#106	—	4	4	no	—
139	#106	—	268	8	no	—
140	#102	—	268	8	no	—
141	#102	—	268	4	no	—
142	C₄×C₈	—	2	2	no	—
143	C₄×C₈	—	134	2	no	—
144	C₄×C₈	—	134	2	no	—
145	C₂⁴	—	268	1	no	—
146	C₂×D₄	—	268	1	no	—
147	C₂×D₄	—	268	2	no	—
148	C₂×D₄	—	268	4	no	—
149	C₂×D₄	—	268	2	no	—
150	C₂×D₄	—	268	2	no	—
151	C₂×D₄	—	268	4	no	—
152	C₂×D₄	—	134	2	no	—
153	C₂×D₄	—	134	4	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	—	268	1	no	—
156	C₂²×C₄	D₄×D₆₇	1	2	yes	—
157	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	67	8	no	—
161	C₂×C₈	—	134	4	no	—
162	C₂×C₈	—	134	4	no	—
163	C₂×C₈	#11761615	1	4	yes	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄⋊C₄	—	4	4	no	—
166	C₄⋊C₄	—	4	4	no	—
167	C₄⋊C₄	—	268	2	no	—
168	C₄⋊C₄	—	268	4	no	—
169	C₄²	Q₈×D₆₇	1	1	yes	characteristic
170	C₄²	#11761628	1	2	yes	—
171	C₄²	#11761629	1	1	yes	characteristic
172	C₂³	—	268	2	no	—
173	C₂³	—	536	2	no	—
174	C₂³	—	536	1	no	—
175	C₂³	—	536	2	no	—
176	C₂³	—	536	1	no	—
177	C₂³	#14524039	1	1	yes	characteristic
178	D₄	—	268	4	no	—
179	D₄	—	268	8	no	—
180	D₄	—	536	2	no	—
181	D₄	—	536	4	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	—	8	2	no	—
184	C₂×C₄	—	8	2	no	—
185	C₂×C₄	—	536	1	no	—
186	C₂×C₄	—	536	2	no	—
187	C₂×C₄	#14524019	1	1	yes	characteristic
188	C₂×C₄	#14524021	1	2	yes	—
189	C₂×C₄	#14524026	1	1	yes	characteristic
190	C₂×C₄	#14524028	1	2	yes	—
191	C₂×C₄	SD₁₆×D₆₇	1	4	yes	—
192	C₂×C₄	C₂×#11761615	1	2	yes	—
193	C₂×C₄	#14524069	1	2	yes	—
194	C₂×C₄	#14524070	1	4	yes	—
195	C₈	—	2	4	no	—
196	C₈	—	134	8	no	—
197	C₂²	—	536	4	no	—
198	C₂²	—	536	2	no	—
199	C₂²	—	1072	1	no	—
200	C₂²	—	1072	2	no	—
201	C₂²	—	1072	1	no	—
202	C₂²	#16863331	1	1	yes	characteristic
203	C₂²	#16863767	1	2	yes	—
204	C₂²	#16863770	1	1	yes	characteristic
205	C₂²	#16863782	1	2	yes	—
206	C₂²	#16863783	1	1	yes	characteristic
207	C₄	—	16	2	no	—
208	C₄	—	1072	1	no	—
209	C₄	#16863683	1	2	yes	—
210	C₄	#16863685	1	4	yes	—
211	C₄	#16863714	1	2	yes	—
212	C₄	#16863716	1	4	yes	—
213	C₂	—	1072	2	no	—
214	C₂	#22559797	1	1	yes	characteristic
215	C₂	#22559836	1	2	yes	—
216	C₂	#22559838	1	1	yes	characteristic
217	C₂	#22561739	1	2	yes	—
218	C₂	#22561743	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22561015

C₂

yes

maximal, characteristic

#22561014

C₂

yes

maximal

#22560365

C₂

yes

maximal, complemented

C₆₇⋊#1273

C₂

yes

maximal, characteristic, complemented

C₆₇×#1844

C₂

yes

maximal, characteristic, complemented

C₂×#14524040

C₂²

yes

—

—

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

#16863266

C₂²

yes

characteristic

C₂×#106×C₆₇

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#384×C₆₇

C₂²

yes

—

C₂²×D₂₆₈

—

C₂²×D₂₆₈

—

C₂²×C₆₇⋊C₈

D₄

yes

—

#14524040

—

#14524040

—

#14524040

Q₈

yes

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

#14523974

—

#14523965

—

#14523963

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#106×C₆₇

—

C₄×C₈×C₆₇

—

C₂³×D₆₇

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

SD₁₆

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

SD₁₆

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₄²×C₆₇

#41

yes

—

C₄²×C₆₇

yes

C₂³×C₆₇

yes

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₇

#123

yes

characteristic

C₂×C₄×C₆₇

#123

yes

—

C₂×C₄×C₆₇

C₂×SD₁₆

yes

—

C₂×C₄×C₆₇

C₂×SD₁₆

yes

—

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

#399

yes

characteristic

C₂²×C₆₇

#492

yes

—

C₂²×C₆₇

#492

yes

characteristic

C₂²×C₆₇

#496

yes

—

C₂²×C₆₇

#496

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#476

yes

—

C₄×C₆₇

#476

yes

—

100

C₄×C₆₇

#481

yes

—

101

C₄×C₆₇

#481

yes

—

102

Dic₆₇

—

103

#10683

—

maximal, Sylow

104

C₂×C₆₇

#1705

yes

characteristic

105

C₂×C₆₇

#1715

yes

—

106

C₂×C₆₇

#1715

yes

characteristic

107

C₂×C₆₇

#2094

yes

—

108

C₂×C₆₇

#2094

yes

characteristic

109

D₆₇

—

110

#1957

—

111

#1957

—

112

#1844

—

113

#1844

D₆₇

yes

characteristic, complemented

114

#1273

—

115

C₆₇

#10683

yes

characteristic, complemented, Sylow

116

C₂×#127

—

117

C₂×#106

—

118

C₂×#106

—

134

—

119

C₂×#102

—

134

—

120

C₂×#102

—

134

—

121

C₂×C₄×C₈

—

122

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

123

#384

—

124

#384

C₂×D₆₇

yes

—

125

C₂²×D₄

—

134

—

126

C₂²×D₄

—

134

—

127

C₂²×C₈

—

128

C₂²×C₈

C₂²⋊D₆₇

yes

—

129

#127

—

130

#127

—

134

—

131

#127

—

134

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

134

C₂×C₄⋊C₄

—

135

C₂×C₄⋊C₄

—

134

—

136

C₂×C₄⋊C₄

—

134

—

137

C₂×C₄²

C₂²×D₆₇

yes

characteristic

138

#106

—

139

#106

—

268

—

140

#102

—

268

—

141

#102

—

268

—

142

C₄×C₈

—

143

C₄×C₈

—

134

—

144

C₄×C₈

—

134

—

145

C₂⁴

—

268

—

146

C₂×D₄

—

268

—

147

C₂×D₄

—

268

—

148

C₂×D₄

—

268

—

149

C₂×D₄

—

268

—

150

C₂×D₄

—

268

—

151

C₂×D₄

—

268

—

152

C₂×D₄

—

134

—

153

C₂×D₄

—

134

—

154

C₂²×C₄

—

155

C₂²×C₄

—

268

—

156

C₂²×C₄

D₄×D₆₇

yes

—

157

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

134

—

162

C₂×C₈

—

134

—

163

C₂×C₈

#11761615

yes

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

268

—

168

C₄⋊C₄

—

268

—

169

C₄²

Q₈×D₆₇

yes

characteristic

170

C₄²

#11761628

yes

—

171

C₄²

#11761629

yes

characteristic

172

C₂³

—

268

—

173

C₂³

—

536

—

174

C₂³

—

536

—

175

C₂³

—

536

—

176

C₂³

—

536

—

177

C₂³

#14524039

yes

characteristic

178

D₄

—

268

—

179

D₄

—

268

—

180

D₄

—

536

—

181

D₄

—

536

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

536

—

186

C₂×C₄

—

536

—

187

C₂×C₄

#14524019

yes

characteristic

188

C₂×C₄

#14524021

yes

—

189

C₂×C₄

#14524026

yes

characteristic

190

C₂×C₄

#14524028

yes

—

191

C₂×C₄

SD₁₆×D₆₇

yes

—

192

C₂×C₄

C₂×#11761615

yes

—

193

C₂×C₄

#14524069

yes

—

194

C₂×C₄

#14524070

yes

—

195

C₈

—

196

C₈

—

134

—

197

C₂²

—

536

—

198

C₂²

—

536

—

199

C₂²

—

1072

—

200

C₂²

—

1072

—

201

C₂²

—

1072

—

202

C₂²

#16863331

yes

characteristic

203

C₂²

#16863767

yes

—

204

C₂²

#16863770

yes

characteristic

205

C₂²

#16863782

yes

—

206

C₂²

#16863783

yes

characteristic

207

C₄

—

208

C₄

—

1072

—

209

C₄

#16863683

yes

—

210

C₄

#16863685

yes

—

211

C₄

#16863714

yes

—

212

C₄

#16863716

yes

—

213

C₂

—

1072

—

214

C₂

#22559797

yes

characteristic

215

C₂

#22559836

yes

—

216

C₂

#22559838

yes

characteristic

217

C₂

#22561739

yes

—

218

C₂

#22561743

yes

characteristic

	subgroup	G/H
center	C₂³	#14524039
commutator, cosocle	C₂×C₄²×C₆₇	C₂³
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1844	C₂
socle	C₂³×C₆₇	#127
prime radical	C₂×#14524040	C₂²
hypercenter	#1844	D₆₇
hypocenter	C₆₇	#10683
LCS	G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

subgroup

G/H

center

C₂³

#14524039

commutator, cosocle

C₂×C₄²×C₆₇

C₂³

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1844

C₂

C₂³×C₆₇

C₂²

D₆₇

C₆₇

G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

N	H ≅ G/N	type
3. #22560365	C₂	semidirect
4. C₆₇⋊#1273	C₂	semidirect
5. C₆₇×#1844	C₂	semidirect
113. #1844	D₆₇	semidirect
115. C₆₇	#10683	semidirect

H ≅ G/N

type

3. #22560365

C₂

semidirect

4. C₆₇⋊#1273

C₂

semidirect

5. C₆₇×#1844

C₂

semidirect

113. #1844

D₆₇

semidirect

115. C₆₇

#10683

semidirect

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

subgroup	index
#22561015	2
#22561014	2
#22560365	2
C₆₇⋊#1273	2
C₆₇×#1844	2
#10683	67

subgroup

index

#22561015

#22561014

#22560365

C₆₇⋊#1273

C₆₇×#1844

#10683

Subgroups — #30760935

Conjugacy classes of subgroups

Complements

Maximal subgroups