Group #30689883 — subgroups

Subgroups — C₆₇×#10632 #30689883

Order 17152.

310 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 106 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	C₆₇×#1960	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1849	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1546	C₂	1	2	yes	maximal
4	C₆₇×#1286	C₂	1	2	yes	maximal
5	C₆₇×#1281	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
7	#431×C₆₇	C₂²	1	2	yes	—
8	C₂×#108×C₆₇	—	2	1	no	—
9	C₂×#108×C₆₇	—	2	1	no	—
10	#423×C₆₇	C₂²	1	2	yes	—
11	C₂×#103×C₆₇	—	2	1	no	—
12	C₂×#103×C₆₇	—	2	1	no	—
13	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
14	#382×C₆₇	C₂²	1	1	yes	characteristic
15	#332×C₆₇	—	2	4	no	—
16	C₂²×Q₈×C₆₇	—	2	1	no	—
17	C₂×M₁₆×C₆₇	—	2	2	no	—
18	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
20	C₄⋊Q₈×C₆₇	—	2	2	no	—
21	C₄⋊Q₈×C₆₇	—	2	1	no	—
22	C₄⋊Q₈×C₆₇	D₄	1	2	yes	—
23	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
24	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
25	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
26	#108×C₆₇	—	4	2	no	—
27	#108×C₆₇	—	4	2	no	—
28	C₄⋊C₈×C₆₇	—	2	2	no	—
29	C₄⋊C₈×C₆₇	—	2	2	no	—
30	C₄⋊C₈×C₆₇	—	2	2	no	—
31	#103×C₆₇	—	4	2	no	—
32	#103×C₆₇	—	4	2	no	—
33	C₂²⋊C₈×C₆₇	—	2	2	no	—
34	#97×C₆₇	—	2	2	no	—
35	C₄×C₈×C₆₇	—	2	2	no	—
36	#95×C₆₇	—	2	2	no	—
37	C₂×Q₈×C₆₇	—	2	2	no	—
38	C₂×Q₈×C₆₇	—	4	1	no	—
39	C₂×Q₈×C₆₇	—	4	1	no	—
40	C₂×Q₈×C₆₇	—	4	1	no	—
41	C₂×Q₈×C₆₇	—	4	1	no	—
42	C₂×Q₈×C₆₇	—	4	1	no	—
43	C₂²×C₄×C₆₇	—	4	1	no	—
44	C₂²×C₄×C₆₇	—	4	1	no	—
45	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	#41	1	2	yes	—
47	M₁₆×C₆₇	—	4	2	no	—
48	M₁₆×C₆₇	—	4	2	no	—
49	C₂×C₈×C₆₇	—	2	1	no	—
50	C₂×C₈×C₆₇	—	2	1	no	—
51	C₂×C₈×C₆₇	—	2	1	no	—
52	C₂×C₈×C₆₇	—	2	1	no	—
53	C₂×C₈×C₆₇	—	4	2	no	—
54	C₂×C₈×C₆₇	—	4	2	no	—
55	C₂×C₈×C₆₇	D₈	1	2	yes	—
56	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
57	C₄⋊C₄×C₆₇	—	2	4	no	—
58	C₄⋊C₄×C₆₇	—	4	2	no	—
59	C₄⋊C₄×C₆₇	—	4	1	no	—
60	C₄⋊C₄×C₆₇	—	4	1	no	—
61	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₆₇	#41	1	1	yes	characteristic
63	C₄²×C₆₇	#41	1	2	yes	—
64	C₂³×C₆₇	#124	1	1	yes	characteristic
65	Q₈×C₆₇	—	4	2	no	—
66	Q₈×C₆₇	—	4	2	no	—
67	Q₈×C₆₇	—	8	1	no	—
68	Q₈×C₆₇	—	8	1	no	—
69	C₂×C₄×C₆₇	—	4	2	no	—
70	C₂×C₄×C₆₇	—	4	2	no	—
71	C₂×C₄×C₆₇	—	8	1	no	—
72	C₂×C₄×C₆₇	—	8	1	no	—
73	C₂×C₄×C₆₇	—	8	2	no	—
74	C₂×C₄×C₆₇	—	8	1	no	—
75	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
76	C₂×C₄×C₆₇	#123	1	2	yes	—
77	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
78	C₂×C₄×C₆₇	#126	1	2	yes	—
79	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
83	C₈×C₆₇	—	2	2	no	—
84	C₈×C₆₇	—	2	2	no	—
85	C₈×C₆₇	—	8	2	no	—
86	C₈×C₆₇	—	8	2	no	—
87	C₂²×C₆₇	—	2	2	no	—
88	C₂²×C₆₇	#397	1	1	yes	characteristic
89	C₂²×C₆₇	#486	1	1	yes	characteristic
90	C₂²×C₆₇	#489	1	1	yes	characteristic
91	C₄×C₆₇	—	2	2	no	—
92	C₄×C₆₇	—	2	2	no	—
93	C₄×C₆₇	—	8	2	no	—
94	C₄×C₆₇	—	16	1	no	—
95	C₄×C₆₇	#459	1	1	yes	characteristic
96	C₄×C₆₇	#461	1	1	yes	characteristic
97	C₄×C₆₇	#480	1	1	yes	characteristic
98	C₄×C₆₇	#481	1	1	yes	characteristic
99	#10632	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂×C₆₇	—	2	1	no	—
101	C₂×C₆₇	—	2	1	no	—
102	C₂×C₆₇	#2071	1	1	yes	characteristic
103	C₂×C₆₇	#2082	1	2	yes	—
104	#1960	C₂×C₆₇	1	1	yes	characteristic
105	#1849	C₂×C₆₇	1	1	yes	characteristic
106	#1546	C₂×C₆₇	1	2	yes	—
107	#1286	C₂×C₆₇	1	2	yes	—
108	#1281	C₂×C₆₇	1	1	yes	characteristic
109	C₆₇	#10632	1	1	yes	characteristic, direct product factor, Sylow
110	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
111	#431	C₂²×C₆₇	1	2	yes	—
112	C₂×#108	—	2	1	no	—
113	C₂×#108	—	2	1	no	—
114	#423	C₂²×C₆₇	1	2	yes	—
115	C₂×#103	—	2	1	no	—
116	C₂×#103	—	2	1	no	—
117	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
118	#382	C₂²×C₆₇	1	1	yes	characteristic
119	#332	—	2	4	no	—
120	C₂²×Q₈	—	2	1	no	—
121	C₂×M₁₆	—	2	2	no	—
122	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
123	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
124	C₄⋊Q₈	—	2	2	no	—
125	C₄⋊Q₈	—	2	1	no	—
126	C₄⋊Q₈	D₄×C₆₇	1	2	yes	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	—	2	1	no	—
129	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
130	#108	—	4	2	no	—
131	#108	—	4	2	no	—
132	C₄⋊C₈	—	2	2	no	—
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	2	2	no	—
135	#103	—	4	2	no	—
136	#103	—	4	2	no	—
137	C₂²⋊C₈	—	2	2	no	—
138	#97	—	2	2	no	—
139	C₄×C₈	—	2	2	no	—
140	#95	—	2	2	no	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	—	4	1	no	—
143	C₂×Q₈	—	4	1	no	—
144	C₂×Q₈	—	4	1	no	—
145	C₂×Q₈	—	4	1	no	—
146	C₂×Q₈	—	4	1	no	—
147	C₂²×C₄	—	4	1	no	—
148	C₂²×C₄	—	4	1	no	—
149	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
150	C₂²×C₄	#41×C₆₇	1	2	yes	—
151	M₁₆	—	4	2	no	—
152	M₁₆	—	4	2	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	D₈×C₆₇	1	2	yes	—
160	C₂×C₈	SD₁₆×C₆₇	1	2	yes	—
161	C₄⋊C₄	—	2	4	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	1	no	—
164	C₄⋊C₄	—	4	1	no	—
165	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
166	C₄²	#41×C₆₇	1	1	yes	characteristic
167	C₄²	#41×C₆₇	1	2	yes	—
168	C₂³	#124×C₆₇	1	1	yes	characteristic
169	Q₈	—	4	2	no	—
170	Q₈	—	4	2	no	—
171	Q₈	—	8	1	no	—
172	Q₈	—	8	1	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	8	1	no	—
176	C₂×C₄	—	8	1	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	8	1	no	—
179	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
180	C₂×C₄	#123×C₆₇	1	2	yes	—
181	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
182	C₂×C₄	#126×C₆₇	1	2	yes	—
183	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
184	C₂×C₄	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
185	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
186	C₂×C₄	#137×C₆₇	1	1	yes	characteristic
187	C₈	—	2	2	no	—
188	C₈	—	2	2	no	—
189	C₈	—	8	2	no	—
190	C₈	—	8	2	no	—
191	C₂²	—	2	2	no	—
192	C₂²	#397×C₆₇	1	1	yes	characteristic
193	C₂²	#486×C₆₇	1	1	yes	characteristic
194	C₂²	#489×C₆₇	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	8	2	no	—
198	C₄	—	16	1	no	—
199	C₄	#459×C₆₇	1	1	yes	characteristic
200	C₄	#461×C₆₇	1	1	yes	characteristic
201	C₄	#480×C₆₇	1	1	yes	characteristic
202	C₄	#481×C₆₇	1	1	yes	characteristic
203	C₂	—	2	1	no	—
204	C₂	—	2	1	no	—
205	C₂	C₆₇×#2071	1	1	yes	characteristic
206	C₂	C₆₇×#2082	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₇×#1960

C₂

yes

maximal, characteristic

C₆₇×#1849

C₂

yes

maximal, characteristic

C₆₇×#1546

C₂

yes

maximal

C₆₇×#1286

C₂

yes

maximal

C₆₇×#1281

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₆₇

C₂²

yes

characteristic

#431×C₆₇

C₂²

yes

—

C₂×#108×C₆₇

—

C₂×#108×C₆₇

—

#423×C₆₇

C₂²

yes

—

C₂×#103×C₆₇

—

C₂×#103×C₆₇

—

C₂×#97×C₆₇

C₂²

yes

characteristic

#382×C₆₇

C₂²

yes

characteristic

#332×C₆₇

—

C₂²×Q₈×C₆₇

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

D₄

yes

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#108×C₆₇

—

#108×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

#103×C₆₇

—

#103×C₆₇

—

C₂²⋊C₈×C₆₇

—

#97×C₆₇

—

C₄×C₈×C₆₇

—

#95×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

SD₁₆

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

C₄²×C₆₇

yes

C₄²×C₆₇

yes

—

C₂³×C₆₇

#124

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#121

yes

characteristic

C₂×C₄×C₆₇

#123

yes

—

C₂×C₄×C₆₇

#123

yes

characteristic

C₂×C₄×C₆₇

#126

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

C₂×SD₁₆

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

100

C₂×C₆₇

—

101

C₂×C₆₇

—

102

C₂×C₆₇

#2071

yes

characteristic

103

C₂×C₆₇

#2082

yes

—

104

#1960

C₂×C₆₇

yes

characteristic

105

#1849

C₂×C₆₇

yes

characteristic

106

#1546

C₂×C₆₇

yes

—

107

#1286

C₂×C₆₇

yes

—

108

#1281

C₂×C₆₇

yes

characteristic

109

C₆₇

#10632

yes

characteristic, direct product factor, Sylow

110

C₂×C₄⋊Q₈

C₂²×C₆₇

yes

characteristic

111

#431

C₂²×C₆₇

yes

—

112

C₂×#108

—

113

C₂×#108

—

114

#423

C₂²×C₆₇

yes

—

115

C₂×#103

—

116

C₂×#103

—

117

C₂×#97

C₂²×C₆₇

yes

characteristic

118

#382

C₂²×C₆₇

yes

characteristic

119

#332

—

120

C₂²×Q₈

—

121

C₂×M₁₆

—

122

C₂²×C₈

D₄×C₆₇

yes

characteristic

123

C₂²×C₈

D₄×C₆₇

yes

characteristic

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

D₄×C₆₇

yes

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄²

C₂³×C₆₇

yes

characteristic

130

#108

—

131

#108

—

132

C₄⋊C₈

—

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

#103

—

136

#103

—

137

C₂²⋊C₈

—

138

#97

—

139

C₄×C₈

—

140

#95

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂×Q₈

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

C₂×D₄×C₆₇

yes

characteristic

150

C₂²×C₄

#41×C₆₇

yes

—

151

M₁₆

—

152

M₁₆

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

D₈×C₆₇

yes

—

160

C₂×C₈

SD₁₆×C₆₇

yes

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄²

C₂×D₄×C₆₇

yes

characteristic

166

C₄²

#41×C₆₇

yes

characteristic

167

C₄²

#41×C₆₇

yes

—

168

C₂³

#124×C₆₇

yes

characteristic

169

Q₈

—

170

Q₈

—

171

Q₈

—

172

Q₈

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

#121×C₆₇

yes

characteristic

180

C₂×C₄

#123×C₆₇

yes

—

181

C₂×C₄

#123×C₆₇

yes

characteristic

182

C₂×C₄

#126×C₆₇

yes

—

183

C₂×C₄

C₂×D₈×C₆₇

yes

characteristic

184

C₂×C₄

C₂×SD₁₆×C₆₇

yes

characteristic

185

C₂×C₄

AGL(1,ℤ/8)×C₆₇

yes

characteristic

186

C₂×C₄

#137×C₆₇

yes

characteristic

187

C₈

—

188

C₈

—

189

C₈

—

190

C₈

—

191

C₂²

—

192

C₂²

#397×C₆₇

yes

characteristic

193

C₂²

#486×C₆₇

yes

characteristic

194

C₂²

#489×C₆₇

yes

characteristic

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

#459×C₆₇

yes

characteristic

200

C₄

#461×C₆₇

yes

characteristic

201

C₄

#480×C₆₇

yes

characteristic

202

C₄

#481×C₆₇

yes

characteristic

203

C₂

—

204

C₂

—

205

C₂

C₆₇×#2071

yes

characteristic

206

C₂

C₆₇×#2082

yes

—

	subgroup	G/H
center, socle	C₂²×C₆₇	#397
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
prime radical	C₂³×C₆₇	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₆₇

#397

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₆₇

prime radical

C₂³×C₆₇

#124

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

N	H ≅ G/N	type
99. #10632	C₆₇	direct (N × H)
109. C₆₇	#10632	direct (N × H)

H ≅ G/N

type

99. #10632

C₆₇

direct (N × H)

109. C₆₇

#10632

direct (N × H)

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

subgroup	index
C₆₇×#1960	2
C₆₇×#1849	2
C₆₇×#1546	2
C₆₇×#1286	2
C₆₇×#1281	2
#10632	67

subgroup

index

C₆₇×#1960

C₆₇×#1849

C₆₇×#1546

C₆₇×#1286

C₆₇×#1281

#10632

Subgroups — C₆₇×#10632 #30689883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#10632 #30689883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#10632 #30689883