Group #30754916 — subgroups

Subgroups — #30754916

Order 17152.

538 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 171 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	#22568607	C₂	1	1	yes	maximal, characteristic
2	#22560326	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863141	C₂	1	4	yes	maximal
4	C₆₇×#1767	C₂	1	1	yes	maximal, characteristic
5	C₂×#14524043	—	2	2	no	—
6	C₂×#14524043	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
8	#16863266	—	2	1	no	—
9	#16863266	—	2	1	no	—
10	#16863266	C₄	1	4	yes	—
11	#16863151	—	2	2	no	—
12	#16863141	—	2	8	no	—
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
14	C₂×#97×C₆₇	—	2	1	no	—
15	C₄³×C₆₇	C₂²	1	1	yes	characteristic
16	#336×C₆₇	—	2	1	no	—
17	C₂²×Dic₁₃₄	—	2	2	no	—
18	C₂²×C₆₇⋊C₈	—	2	2	no	—
19	#14524043	—	2	4	no	—
20	#14524043	—	2	4	no	—
21	#14524043	—	4	8	no	—
22	C₂×C₄⋊Dic₆₇	—	2	4	no	—
23	C₂×C₄⋊Dic₆₇	—	2	4	no	—
24	C₂×C₄⋊Dic₆₇	—	2	2	no	—
25	C₂×C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
26	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
27	C₂²×C₈×C₆₇	—	2	2	no	—
28	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₆₇	D₄	1	2	yes	—
30	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₆₇	—	2	2	no	—
34	C₄⋊C₈×C₆₇	D₄	1	4	yes	—
35	#97×C₆₇	—	4	2	no	—
36	C₂×Dic₁₃₄	—	4	2	no	—
37	C₂×Dic₁₃₄	—	4	2	no	—
38	C₂×Dic₁₃₄	—	4	8	no	—
39	C₂²×Dic₆₇	—	2	4	no	—
40	C₂×C₆₇⋊C₈	—	4	2	no	—
41	C₂×C₆₇⋊C₈	—	4	4	no	—
42	C₄⋊Dic₆₇	—	2	8	no	—
43	C₄⋊Dic₆₇	—	4	8	no	—
44	C₄⋊Dic₆₇	—	4	8	no	—
45	C₄⋊Dic₆₇	—	4	4	no	—
46	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₆₇	—	2	4	no	—
52	C₂×C₈×C₆₇	—	4	2	no	—
53	C₂×C₈×C₆₇	—	4	2	no	—
54	C₄²×C₆₇	—	2	4	no	—
55	C₄²×C₆₇	—	2	4	no	—
56	C₄²×C₆₇	—	2	2	no	—
57	C₄²×C₆₇	SD₁₆	1	2	yes	—
58	C₄²×C₆₇	Q₁₆	1	2	yes	—
59	C₄²×C₆₇	C₂×D₄	1	2	yes	—
60	C₄²×C₆₇	#41	1	2	yes	—
61	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₇	—	2	2	no	—
63	C₂×C₄×C₆₇	—	2	2	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	#101	1	4	yes	—
68	C₂×C₄×C₆₇	#103	1	4	yes	—
69	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
70	C₂×C₄×C₆₇	#121	1	2	yes	—
71	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
72	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
76	C₂×C₄×C₆₇	#137	1	2	yes	—
77	C₂×Dic₆₇	—	4	4	no	—
78	C₂×Dic₆₇	—	4	8	no	—
79	Dic₁₃₄	—	8	4	no	—
80	Dic₁₃₄	—	8	4	no	—
81	C₈×C₆₇	—	4	4	no	—
82	C₆₇⋊C₈	—	8	2	no	—
83	C₂²×C₆₇	#390	1	1	yes	characteristic
84	C₂²×C₆₇	C₂×#101	1	2	yes	—
85	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
86	C₂²×C₆₇	#418	1	2	yes	—
87	C₂²×C₆₇	#419	1	1	yes	characteristic
88	C₄×C₆₇	—	2	4	no	—
89	C₄×C₆₇	—	2	4	no	—
90	C₄×C₆₇	—	2	2	no	—
91	C₄×C₆₇	#461	1	2	yes	—
92	C₄×C₆₇	#462	1	2	yes	—
93	C₄×C₆₇	#490	1	2	yes	—
94	C₄×C₆₇	#496	1	2	yes	—
95	Dic₆₇	—	8	4	no	—
96	#9703	—	67	1	no	maximal, Sylow
97	C₂×C₆₇	#1549	1	2	yes	—
98	C₂×C₆₇	#1556	1	2	yes	—
99	C₂×C₆₇	#1961	1	1	yes	characteristic
100	C₂×C₆₇	#1966	1	1	yes	characteristic
101	C₂×C₆₇	#1981	1	1	yes	characteristic
102	#2868	—	67	1	no	—
103	#1831	—	67	1	no	—
104	#1767	D₆₇	1	1	yes	characteristic
105	C₂×#332	—	67	4	no	—
106	C₆₇	#9703	1	1	yes	characteristic, complemented, Sylow
107	C₂×C₄⋊Q₈	—	67	2	no	—
108	C₂×C₄⋊Q₈	—	134	2	no	—
109	C₂×C₄⋊C₈	—	67	2	no	—
110	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
111	C₂×#97	—	2	1	no	—
112	#384	—	67	4	no	—
113	#384	—	134	1	no	—
114	#384	—	134	1	no	—
115	C₄³	C₂×D₆₇	1	1	yes	characteristic
116	#336	—	2	1	no	—
117	#336	—	134	2	no	—
118	#332	—	134	8	no	—
119	C₂²×Q₈	—	134	2	no	—
120	C₂²×C₈	—	2	2	no	—
121	C₂²×C₈	—	134	2	no	—
122	C₄⋊Q₈	—	268	8	no	—
123	C₄⋊Q₈	—	134	4	no	—
124	C₄⋊Q₈	—	134	4	no	—
125	C₂×C₄⋊C₄	—	67	4	no	—
126	C₂×C₄⋊C₄	—	134	4	no	—
127	C₂×C₄⋊C₄	—	134	4	no	—
128	C₂×C₄⋊C₄	—	134	2	no	—
129	C₂×C₄²	C₄×D₆₇	1	2	yes	—
130	C₂×C₄²	D₂₆₈	1	2	yes	—
131	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
132	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
133	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	134	4	no	—
136	C₄⋊C₈	D₂₆₈	1	4	yes	—
137	#97	—	4	2	no	—
138	C₂×Q₈	—	268	2	no	—
139	C₂×Q₈	—	268	2	no	—
140	C₂×Q₈	—	268	8	no	—
141	C₂²×C₄	—	134	4	no	—
142	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
143	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
144	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
146	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
147	C₂×C₈	—	2	4	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	268	2	no	—
151	C₂×C₈	—	268	4	no	—
152	C₄⋊C₄	—	268	8	no	—
153	C₄⋊C₄	—	268	8	no	—
154	C₄⋊C₄	—	268	4	no	—
155	C₄⋊C₄	—	134	8	no	—
156	C₄²	—	2	4	no	—
157	C₄²	—	2	4	no	—
158	C₄²	—	2	2	no	—
159	C₄²	#11761616	1	2	yes	—
160	C₄²	#11761618	1	2	yes	—
161	C₄²	C₂×D₂₆₈	1	2	yes	—
162	C₄²	#11761628	1	2	yes	—
163	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
164	Q₈	—	536	4	no	—
165	Q₈	—	536	4	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	268	4	no	—
172	C₂×C₄	—	268	8	no	—
173	C₂×C₄	#14523961	1	4	yes	—
174	C₂×C₄	#14523966	1	4	yes	—
175	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
176	C₂×C₄	#14524017	1	2	yes	—
177	C₂×C₄	#14524031	1	1	yes	characteristic
178	C₂×C₄	#14524040	1	1	yes	characteristic
179	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
180	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
181	C₂×C₄	#14524071	1	4	yes	—
182	C₂×C₄	#14524073	1	2	yes	—
183	C₈	—	4	4	no	—
184	C₈	—	536	2	no	—
185	C₂²	#16863298	1	1	yes	characteristic
186	C₂²	C₂×#14523961	1	2	yes	—
187	C₂²	C₂×#14523966	1	1	yes	characteristic
188	C₂²	#16863402	1	2	yes	—
189	C₂²	#16863404	1	1	yes	characteristic
190	C₄	—	2	2	no	—
191	C₄	—	2	4	no	—
192	C₄	—	2	4	no	—
193	C₄	—	536	4	no	—
194	C₄	#16863601	1	2	yes	—
195	C₄	#16863606	1	2	yes	—
196	C₄	#16863761	1	2	yes	—
197	C₄	#16863785	1	2	yes	—
198	C₂	#22558895	1	2	yes	—
199	C₂	#22558925	1	2	yes	—
200	C₂	#22561033	1	1	yes	characteristic
201	C₂	#22561053	1	1	yes	characteristic
202	C₂	#22561132	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22568607

C₂

yes

maximal, characteristic

#22560326

C₂

yes

maximal, characteristic

C₂×#16863141

C₂

yes

maximal

C₆₇×#1767

C₂

yes

maximal, characteristic

C₂×#14524043

—

C₂×#14524043

C₂²

yes

—

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

#16863266

—

#16863266

—

#16863266

C₄

yes

—

#16863151

—

#16863141

—

C₂×C₄⋊C₈×C₆₇

C₂²

yes

—

C₂×#97×C₆₇

—

C₄³×C₆₇

C₂²

yes

characteristic

#336×C₆₇

—

C₂²×Dic₁₃₄

—

C₂²×C₆₇⋊C₈

—

#14524043

—

#14524043

—

#14524043

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

C₂×C₄

yes

—

C₄⋊(C₆₇⋊C₈)

—

C₂²×C₈×C₆₇

—

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

D₄

yes

—

#97×C₆₇

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

SD₁₆

yes

—

C₄²×C₆₇

Q₁₆

yes

—

C₄²×C₆₇

C₂×D₄

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

#101

yes

—

C₂×C₄×C₆₇

#103

yes

—

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₆₇

#137

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

#390

yes

characteristic

C₂²×C₆₇

C₂×#101

yes

—

C₂²×C₆₇

C₂×#103

yes

characteristic

C₂²×C₆₇

#418

yes

—

C₂²×C₆₇

#419

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#461

yes

—

C₄×C₆₇

#462

yes

—

C₄×C₆₇

#490

yes

—

C₄×C₆₇

#496

yes

—

Dic₆₇

—

#9703

—

maximal, Sylow

C₂×C₆₇

#1549

yes

—

C₂×C₆₇

#1556

yes

—

C₂×C₆₇

#1961

yes

characteristic

100

C₂×C₆₇

#1966

yes

characteristic

101

C₂×C₆₇

#1981

yes

characteristic

102

#2868

—

103

#1831

—

104

#1767

D₆₇

yes

characteristic

105

C₂×#332

—

106

C₆₇

#9703

yes

characteristic, complemented, Sylow

107

C₂×C₄⋊Q₈

—

108

C₂×C₄⋊Q₈

—

134

—

109

C₂×C₄⋊C₈

—

110

C₂×C₄⋊C₈

C₂×D₆₇

yes

—

111

C₂×#97

—

112

#384

—

113

#384

—

134

—

114

#384

—

134

—

115

C₄³

C₂×D₆₇

yes

characteristic

116

#336

—

117

#336

—

134

—

118

#332

—

134

—

119

C₂²×Q₈

—

134

—

120

C₂²×C₈

—

121

C₂²×C₈

—

134

—

122

C₄⋊Q₈

—

268

—

123

C₄⋊Q₈

—

134

—

124

C₄⋊Q₈

—

134

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

134

—

127

C₂×C₄⋊C₄

—

134

—

128

C₂×C₄⋊C₄

—

134

—

129

C₂×C₄²

C₄×D₆₇

yes

—

130

C₂×C₄²

D₂₆₈

yes

—

131

C₂×C₄²

C₂²⋊D₆₇

yes

characteristic

132

C₂×C₄²

C₂²⋊D₆₇

yes

characteristic

133

C₂×C₄²

C₂²×D₆₇

yes

characteristic

134

C₄⋊C₈

—

135

C₄⋊C₈

—

134

—

136

C₄⋊C₈

D₂₆₈

yes

—

137

#97

—

138

C₂×Q₈

—

268

—

139

C₂×Q₈

—

268

—

140

C₂×Q₈

—

268

—

141

C₂²×C₄

—

134

—

142

C₂²×C₄

(C₂×D₆₇)⋊C₄

yes

—

143

C₂²×C₄

(C₂×D₆₇)⋊C₄

yes

—

144

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

145

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

146

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

268

—

151

C₂×C₈

—

268

—

152

C₄⋊C₄

—

268

—

153

C₄⋊C₄

—

268

—

154

C₄⋊C₄

—

268

—

155

C₄⋊C₄

—

134

—

156

C₄²

—

157

C₄²

—

158

C₄²

—

159

C₄²

#11761616

yes

—

160

C₄²

#11761618

yes

—

161

C₄²

C₂×D₂₆₈

yes

—

162

C₄²

#11761628

yes

—

163

C₂³

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

164

Q₈

—

536

—

165

Q₈

—

536

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

268

—

172

C₂×C₄

—

268

—

173

C₂×C₄

#14523961

yes

—

174

C₂×C₄

#14523966

yes

—

175

C₂×C₄

C₄×D₂₆₈

yes

—

176

C₂×C₄

#14524017

yes

—

177

C₂×C₄

#14524031

yes

characteristic

178

C₂×C₄

#14524040

yes

characteristic

179

C₂×C₄

C₂×#11761616

yes

characteristic

180

C₂×C₄

C₂×#11761618

yes

characteristic

181

C₂×C₄

#14524071

yes

—

182

C₂×C₄

#14524073

yes

—

183

C₈

—

184

C₈

—

536

—

185

C₂²

#16863298

yes

characteristic

186

C₂²

C₂×#14523961

yes

—

187

C₂²

C₂×#14523966

yes

characteristic

188

C₂²

#16863402

yes

—

189

C₂²

#16863404

yes

characteristic

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

536

—

194

C₄

#16863601

yes

—

195

C₄

#16863606

yes

—

196

C₄

#16863761

yes

—

197

C₄

#16863785

yes

—

198

C₂

#22558895

yes

—

199

C₂

#22558925

yes

—

200

C₂

#22561033

yes

characteristic

201

C₂

#22561053

yes

characteristic

202

C₂

#22561132

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₆₇)⋊C₄
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1767	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1767	D₆₇
hypocenter	C₆₇	#9703
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

subgroup

G/H

center

C₂³

C₂×(C₂×D₆₇)⋊C₄

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1767

C₂

socle, prime radical

C₂³×C₆₇

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

N	H ≅ G/N	type
106. C₆₇	#9703	semidirect

H ≅ G/N

type

106. C₆₇

#9703

semidirect

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

subgroup	index
#22568607	2
#22560326	2
C₂×#16863141	2
C₆₇×#1767	2
#9703	67

subgroup

index

C₆₇×#1767

Subgroups — #30754916

Conjugacy classes of subgroups

Complements

Maximal subgroups