Group #30772010 — subgroups

Subgroups — #30772010

Order 17152.

394 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 139 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	C₂³×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863225	C₂	1	2	yes	maximal, complemented
3	#22560247	C₂	1	1	yes	maximal, characteristic
4	#22558302	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1810	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₁₆	—	2	4	no	—
7	C₂²×C₆₇⋊C₁₆	C₄	1	4	yes	—
8	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×#14523979	C₂²	1	2	yes	—
11	#16863360	—	2	1	no	—
12	#16863225	—	2	4	no	—
13	#16863161	—	2	1	no	—
14	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#108×C₆₇	C₂²	1	2	yes	—
16	#407×C₆₇	—	2	1	no	—
17	#341×C₆₇	—	2	1	no	—
18	C₂×#11761612	—	2	2	no	—
19	C₂×C₆₇⋊C₁₆	—	2	8	no	—
20	C₂×C₆₇⋊C₁₆	—	2	2	no	—
21	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
22	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₆₇⋊C₁₆	D₄	1	4	yes	—
24	#14523979	—	2	2	no	—
25	#14523979	—	2	2	no	—
26	C₂²⋊(C₆₇⋊C₈)	—	4	2	no	—
27	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
28	C₂×M₁₆×C₆₇	—	2	2	no	—
29	C₂²×C₈×C₆₇	—	2	2	no	—
30	C₂²×C₈×C₆₇	—	2	2	no	—
31	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
34	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
35	#108×C₆₇	—	2	2	no	—
36	#108×C₆₇	—	2	2	no	—
37	C₂²⋊C₈×C₆₇	—	4	2	no	—
38	#11761612	—	4	2	no	—
39	#11761612	—	4	2	no	—
40	C₂×C₆₇⋊C₈	—	4	2	no	—
41	C₆₇⋊C₁₆	—	2	2	no	—
42	C₆₇⋊C₁₆	C₂²⋊C₄	1	4	yes	—
43	C₂⁴×C₆₇	Q₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	—	2	2	no	—
45	C₂²×C₄×C₆₇	—	2	2	no	—
46	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₆₇	D₈	1	2	yes	—
48	C₂²×C₄×C₆₇	Q₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
50	M₁₆×C₆₇	—	4	2	no	—
51	M₁₆×C₆₇	—	4	2	no	—
52	C₂×C₈×C₆₇	—	2	2	no	—
53	C₂×C₈×C₆₇	—	2	2	no	—
54	C₂×C₈×C₆₇	—	2	2	no	—
55	C₂×C₈×C₆₇	—	2	4	no	—
56	C₂×C₈×C₆₇	—	4	2	no	—
57	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
58	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₆₇	#41	1	2	yes	—
62	C₂³×C₆₇	—	2	4	no	—
63	C₂³×C₆₇	—	2	2	no	—
64	C₂³×C₆₇	#107	1	2	yes	—
65	C₂³×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₆₇	—	2	2	no	—
67	C₂×C₄×C₆₇	—	2	2	no	—
68	C₂×C₄×C₆₇	—	2	2	no	—
69	C₂×C₄×C₆₇	—	2	4	no	—
70	C₂×C₄×C₆₇	#107	1	2	yes	—
71	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₇	#121	1	2	yes	—
73	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₆₇	#135	1	2	yes	—
75	C₈×C₆₇	—	2	2	no	—
76	C₈×C₆₇	—	8	2	no	—
77	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₈×C₆₇	#117	1	1	yes	characteristic
79	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
80	C₆₇⋊C₈	—	8	2	no	—
81	C₂²×C₆₇	—	2	8	no	—
82	C₂²×C₆₇	—	2	2	no	—
83	C₂²×C₆₇	—	2	2	no	—
84	C₂²×C₆₇	—	2	2	no	—
85	C₂²×C₆₇	#368	1	4	yes	complemented
86	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
87	C₂²×C₆₇	#467	1	2	yes	—
88	C₄×C₆₇	—	2	2	no	—
89	C₄×C₆₇	#380	1	1	yes	characteristic
90	C₄×C₆₇	#427	1	1	yes	characteristic
91	C₄×C₆₇	#466	1	2	yes	—
92	#12837	—	67	1	no	maximal, Sylow
93	C₂×C₆₇	—	2	4	no	—
94	C₂×C₆₇	—	2	2	no	—
95	C₂×C₆₇	#1808	1	1	yes	characteristic
96	C₂×C₆₇	C₂×#368	1	2	yes	—
97	C₂³×C₁₆	—	67	1	no	—
98	C₂×#368	—	67	2	no	—
99	#1810	—	67	1	no	—
100	#1810	D₆₇	1	1	yes	characteristic
101	#1381	—	67	2	no	—
102	C₆₇	#12837	1	1	yes	characteristic, complemented, Sylow
103	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
104	C₂²×C₁₆	—	67	1	no	—
105	C₂²×C₁₆	—	67	4	no	—
106	C₂²×C₁₆	—	67	1	no	—
107	C₂²×C₁₆	—	134	4	no	—
108	C₂×#108	—	67	2	no	—
109	C₂×#108	C₂×D₆₇	1	2	yes	—
110	#407	—	2	1	no	—
111	#407	—	134	1	no	—
112	#368	—	134	4	no	—
113	#341	—	2	1	no	—
114	#341	—	134	1	no	—
115	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
116	C₂×M₁₆	—	2	2	no	—
117	C₂×M₁₆	—	134	2	no	—
118	C₂²×C₈	—	2	2	no	—
119	C₂²×C₈	—	2	2	no	—
120	C₂²×C₈	Dic₁₃₄	1	2	yes	—
121	C₂²×C₈	C₄×D₆₇	1	2	yes	—
122	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
124	C₂×C₁₆	—	67	2	no	—
125	C₂×C₁₆	—	67	4	no	—
126	C₂×C₁₆	—	67	2	no	—
127	C₂×C₁₆	—	134	8	no	—
128	C₂×C₁₆	—	134	2	no	—
129	#108	—	2	2	no	—
130	#108	—	2	2	no	—
131	#108	—	134	2	no	—
132	#108	—	134	2	no	—
133	C₂²⋊C₈	—	4	2	no	—
134	C₂²⋊C₈	—	268	2	no	—
135	C₂⁴	#11761618	1	1	yes	characteristic
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	—	2	2	no	—
138	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
139	C₂²×C₄	#11761613	1	2	yes	—
140	C₂²×C₄	#11761618	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
142	M₁₆	—	4	2	no	—
143	M₁₆	—	4	2	no	—
144	M₁₆	—	268	2	no	—
145	M₁₆	—	268	2	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	268	2	no	—
152	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
153	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
154	C₂×C₈	D₄×D₆₇	1	2	yes	—
155	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
156	C₂×C₈	#11761627	1	2	yes	—
157	C₁₆	—	67	4	no	—
158	C₁₆	—	134	2	no	—
159	C₂³	—	2	2	no	—
160	C₂³	—	2	4	no	—
161	C₂³	#14523976	1	2	yes	—
162	C₂³	C₂×#11761618	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	#14523976	1	2	yes	—
168	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
169	C₂×C₄	#14524015	1	2	yes	—
170	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
171	C₂×C₄	#14524063	1	2	yes	—
172	C₈	—	2	2	no	—
173	C₈	—	8	2	no	—
174	C₈	—	536	2	no	—
175	C₈	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
176	C₈	#14523999	1	1	yes	characteristic
177	C₈	#14524020	1	2	yes	—
178	C₂²	—	2	8	no	—
179	C₂²	—	2	2	no	—
180	C₂²	—	2	2	no	—
181	C₂²	—	2	2	no	—
182	C₂²	#16863225	1	4	yes	complemented
183	C₂²	C₂×#14523976	1	1	yes	characteristic
184	C₂²	#16863635	1	2	yes	—
185	C₄	—	2	2	no	—
186	C₄	#16863254	1	1	yes	characteristic
187	C₄	#16863432	1	1	yes	characteristic
188	C₄	#16863630	1	2	yes	—
189	C₂	—	2	2	no	—
190	C₂	—	2	4	no	—
191	C₂	#22560236	1	1	yes	characteristic
192	C₂	C₂×#16863225	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₆₇⋊C₁₆

C₂

yes

maximal, characteristic

C₂×#16863225

C₂

yes

maximal, complemented

#22560247

C₂

yes

maximal, characteristic

#22558302

C₂

yes

maximal, complemented

C₆₇×#1810

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₁₆

—

C₂²×C₆₇⋊C₁₆

C₄

yes

—

C₂²×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₂²×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₂×#14523979

C₂²

yes

—

#16863360

—

#16863225

—

#16863161

—

C₂³×C₈×C₆₇

C₂²

yes

characteristic

C₂×#108×C₆₇

C₂²

yes

—

#407×C₆₇

—

#341×C₆₇

—

C₂×#11761612

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

—

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

—

C₂×C₆₇⋊C₁₆

D₄

yes

—

#14523979

—

#14523979

—

C₂²⋊(C₆₇⋊C₈)

—

C₂³×C₄×C₆₇

D₄

yes

characteristic

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

—

C₂²×C₈×C₆₇

—

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

Q₈

yes

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

#108×C₆₇

—

#108×C₆₇

—

C₂²⋊C₈×C₆₇

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₆₇⋊C₁₆

—

C₆₇⋊C₁₆

C₂²⋊C₄

yes

—

C₂⁴×C₆₇

Q₁₆

yes

characteristic

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

D₈

yes

—

C₂²×C₄×C₆₇

Q₁₆

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

#107

yes

—

C₂³×C₆₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#107

yes

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

#135

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₆₇

#117

yes

characteristic

C₈×C₆₇

C₂²⋊Q₈

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

#12837

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#1808

yes

characteristic

C₂×C₆₇

C₂×#368

yes

—

C₂³×C₁₆

—

C₂×#368

—

#1810

—

100

#1810

D₆₇

yes

characteristic

101

#1381

—

102

C₆₇

#12837

yes

characteristic, complemented, Sylow

103

C₂³×C₈

C₂×D₆₇

yes

characteristic

104

C₂²×C₁₆

—

105

C₂²×C₁₆

—

106

C₂²×C₁₆

—

107

C₂²×C₁₆

—

134

—

108

C₂×#108

—

109

C₂×#108

C₂×D₆₇

yes

—

110

#407

—

111

#407

—

134

—

112

#368

—

134

—

113

#341

—

114

#341

—

134

—

115

C₂³×C₄

C₂²⋊D₆₇

yes

characteristic

116

C₂×M₁₆

—

117

C₂×M₁₆

—

134

—

118

C₂²×C₈

—

119

C₂²×C₈

—

120

C₂²×C₈

Dic₁₃₄

yes

—

121

C₂²×C₈

C₄×D₆₇

yes

—

122

C₂²×C₈

C₂²⋊D₆₇

yes

characteristic

123

C₂²×C₈

C₂²×D₆₇

yes

characteristic

124

C₂×C₁₆

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

127

C₂×C₁₆

—

134

—

128

C₂×C₁₆

—

134

—

129

#108

—

130

#108

—

131

#108

—

134

—

132

#108

—

134

—

133

C₂²⋊C₈

—

134

C₂²⋊C₈

—

268

—

135

C₂⁴

#11761618

yes

characteristic

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

Dic₆₇⋊C₄

yes

—

139

C₂²×C₄

#11761613

yes

—

140

C₂²×C₄

#11761618

yes

characteristic

141

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

142

M₁₆

—

143

M₁₆

—

144

M₁₆

—

268

—

145

M₁₆

—

268

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

268

—

152

C₂×C₈

Dic₆₇⋊C₄

yes

—

153

C₂×C₈

C₂×C₄×D₆₇

yes

characteristic

154

C₂×C₈

D₄×D₆₇

yes

—

155

C₂×C₈

C₂×Dic₁₃₄

yes

characteristic

156

C₂×C₈

#11761627

yes

—

157

C₁₆

—

158

C₁₆

—

134

—

159

C₂³

—

160

C₂³

—

161

C₂³

#14523976

yes

—

162

C₂³

C₂×#11761618

yes

characteristic

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

#14523976

yes

—

168

C₂×C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

169

C₂×C₄

#14524015

yes

—

170

C₂×C₄

C₂×#11761613

yes

characteristic

171

C₂×C₄

#14524063

yes

—

172

C₈

—

173

C₈

—

174

C₈

—

536

—

175

C₈

C₂²⋊C₄×D₆₇

yes

characteristic

176

C₈

#14523999

yes

characteristic

177

C₈

#14524020

yes

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

—

182

C₂²

#16863225

yes

complemented

183

C₂²

C₂×#14523976

yes

characteristic

184

C₂²

#16863635

yes

—

185

C₄

—

186

C₄

#16863254

yes

characteristic

187

C₄

#16863432

yes

characteristic

188

C₄

#16863630

yes

—

189

C₂

—

190

C₂

—

191

C₂

#22560236

yes

characteristic

192

C₂

C₂×#16863225

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×#11761613
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1810	C₂
socle	C₂²×C₆₇	C₂×#107
prime radical	C₂⁴×C₆₇	Q₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1810	D₆₇
hypocenter	C₆₇	#12837
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1810

subgroup

G/H

center

C₂×C₄

C₂×#11761613

commutator

C₂×C₈×C₆₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#1810

C₂

socle

C₂²×C₆₇

C₂×#107

prime radical

C₂⁴×C₆₇

Q₁₆

cosocle

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1810

N	H ≅ G/N	type
2. C₂×#16863225	C₂	semidirect
4. #22558302	C₂	semidirect
85. C₂²×C₆₇	#368	semidirect
102. C₆₇	#12837	semidirect
182. C₂²	#16863225	semidirect

H ≅ G/N

type

2. C₂×#16863225

C₂

semidirect

4. #22558302

C₂

semidirect

85. C₂²×C₆₇

#368

semidirect

102. C₆₇

#12837

semidirect

182. C₂²

#16863225

semidirect

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

subgroup	index
C₂³×C₆₇⋊C₁₆	2
C₂×#16863225	2
#22560247	2
#22558302	2
C₆₇×#1810	2
#12837	67

subgroup

index

C₂³×C₆₇⋊C₁₆

C₆₇×#1810

Subgroups — #30772010

Conjugacy classes of subgroups

Complements

Maximal subgroups