Group #30771987 — subgroups

Subgroups — #30771987

Order 17152.

358 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 143 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	#22561982	C₂	1	2	yes	maximal, complemented
2	#22560259	C₂	1	1	yes	maximal, characteristic
3	#22560255	C₂	1	1	yes	maximal, characteristic
4	#22558305	C₂	1	2	yes	maximal, complemented
5	C₂×#504×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523979	C₂²	1	2	yes	—
7	#16863434	C₂²	1	2	yes	—
8	#16863380	—	2	1	no	—
9	#16863370	—	2	1	no	—
10	#16863219	—	2	4	no	—
11	C₆₇⋊#338	—	2	1	no	—
12	#16863153	—	2	1	no	—
13	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
14	#504×C₆₇	—	2	4	no	—
15	C₂×M₃₂×C₆₇	C₄	1	2	yes	complemented
16	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₆₇	C₄	1	2	yes	complemented
18	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#11761612	—	2	2	no	—
20	#14524002	—	2	2	no	—
21	#14523979	—	2	2	no	—
22	#14523979	—	2	2	no	—
23	#14523977	—	2	2	no	—
24	#14523975	—	2	2	no	—
25	#14523962	—	4	1	no	—
26	#14523960	—	4	1	no	—
27	#14523958	—	4	2	no	—
28	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
29	#131×C₆₇	—	2	2	no	—
30	#131×C₆₇	—	2	2	no	—
31	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
32	C₂×M₁₆×C₆₇	Q₈	1	2	yes	—
33	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
35	M₃₂×C₆₇	—	2	4	no	—
36	M₃₂×C₆₇	D₄	1	4	yes	—
37	C₂×C₁₆×C₆₇	—	2	2	no	—
38	C₂×C₁₆×C₆₇	—	2	4	no	—
39	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
42	#11761612	—	4	2	no	—
43	#11761612	—	4	2	no	—
44	C₂×C₆₇⋊C₈	—	4	2	no	—
45	C₄⋊Dic₆₇	—	4	2	no	—
46	C₂²⋊Dic₆₇	—	4	2	no	—
47	C₄×Dic₆₇	—	4	2	no	—
48	#41×C₆₇	—	2	2	no	—
49	#41×C₆₇	—	2	2	no	—
50	C₂×Q₈×C₆₇	SD₁₆	1	1	yes	characteristic
51	C₂×D₄×C₆₇	SD₁₆	1	2	yes	—
52	C₂×D₄×C₆₇	SD₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
55	M₁₆×C₆₇	—	2	2	no	—
56	M₁₆×C₆₇	—	2	4	no	—
57	C₂×C₈×C₆₇	—	2	2	no	—
58	C₂×C₈×C₆₇	—	2	2	no	—
59	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
60	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₆₇	#41	1	2	yes	—
64	C₁₆×C₆₇	—	2	1	no	—
65	C₁₆×C₆₇	—	2	1	no	—
66	C₁₆×C₆₇	C₂²⋊C₄	1	4	yes	—
67	C₂³×C₆₇	#106	1	2	yes	—
68	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
69	Q₈×C₆₇	—	2	2	no	—
70	D₄×C₆₇	—	2	4	no	—
71	D₄×C₆₇	—	2	2	no	—
72	C₂×C₄×C₆₇	—	2	2	no	—
73	C₂×C₄×C₆₇	—	2	2	no	—
74	C₂×C₄×C₆₇	#106	1	2	yes	—
75	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₆₇	#121	1	2	yes	—
77	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₆₇	#135	1	2	yes	—
79	C₂×Dic₆₇	—	8	2	no	—
80	C₈×C₆₇	—	2	2	no	—
81	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₈×C₆₇	#117	1	1	yes	characteristic
83	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
84	C₆₇⋊C₈	—	8	2	no	—
85	C₂²×C₆₇	—	2	2	no	—
86	C₂²×C₆₇	—	4	1	no	—
87	C₂²×C₆₇	—	4	2	no	—
88	C₂²×C₆₇	C₂×#106	1	1	yes	characteristic
89	C₂²×C₆₇	#465	1	2	yes	—
90	C₄×C₆₇	—	2	2	no	—
91	C₄×C₆₇	#380	1	1	yes	characteristic
92	C₄×C₆₇	#427	1	1	yes	characteristic
93	C₄×C₆₇	#465	1	2	yes	—
94	Dic₆₇	—	16	2	no	—
95	#12832	—	67	1	no	maximal, Sylow
96	C₂×C₆₇	—	2	2	no	—
97	C₂×C₆₇	—	2	1	no	—
98	C₂×C₆₇	—	4	2	no	—
99	C₂×C₆₇	#1807	1	1	yes	characteristic
100	C₂×#504	D₆₇	1	1	yes	characteristic
101	#2155	—	67	2	no	—
102	#1813	—	67	1	no	—
103	#1812	—	67	1	no	—
104	#1382	—	67	2	no	—
105	C₆₇	#12832	1	1	yes	characteristic, complemented, Sylow
106	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
107	#504	—	2	4	no	—
108	C₂×M₃₂	Dic₆₇	1	2	yes	complemented
109	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
110	C₂²×C₁₆	Dic₆₇	1	2	yes	complemented
111	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
112	C₂×#108	—	67	2	no	—
113	#427	—	67	2	no	—
114	#413	—	134	1	no	—
115	#410	—	134	1	no	—
116	#365	—	134	4	no	—
117	#338	—	134	1	no	—
118	H(3,ℤ/4)	—	134	1	no	—
119	C₂×#41	D₂₆₈	1	1	yes	characteristic
120	#131	—	2	2	no	—
121	#131	—	2	2	no	—
122	C₂×M₁₆	—	134	2	no	—
123	C₂×M₁₆	Dic₁₃₄	1	2	yes	—
124	C₂×M₁₆	D₂₆₈	1	1	yes	characteristic
125	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
126	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
127	#117	—	134	2	no	—
128	M₃₂	—	2	4	no	—
129	M₃₂	C₂²⋊D₆₇	1	4	yes	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	2	4	no	—
132	C₂×C₁₆	C₂×Dic₆₇	1	1	yes	characteristic
133	C₂×C₁₆	C₂×Dic₆₇	1	1	yes	characteristic
134	C₂×C₁₆	C₂×Dic₆₇	1	2	yes	—
135	#108	—	134	2	no	—
136	#108	—	134	2	no	—
137	#107	—	134	2	no	—
138	#106	—	134	2	no	—
139	#101	—	268	1	no	—
140	#100	—	268	1	no	—
141	#99	—	268	2	no	—
142	#41	—	2	2	no	—
143	#41	—	2	2	no	—
144	C₂×Q₈	#11761617	1	1	yes	characteristic
145	C₂×D₄	#11761617	1	2	yes	—
146	C₂×D₄	#11761617	1	1	yes	characteristic
147	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
148	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
149	M₁₆	—	2	2	no	—
150	M₁₆	—	2	4	no	—
151	M₁₆	—	268	2	no	—
152	M₁₆	—	268	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	268	2	no	—
156	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
157	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
158	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
159	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
160	C₂×C₈	#11761628	1	2	yes	—
161	C₄⋊C₄	—	268	2	no	—
162	C₂²⋊C₄	—	268	2	no	—
163	C₄²	—	268	2	no	—
164	C₁₆	—	2	1	no	—
165	C₁₆	—	2	1	no	—
166	C₁₆	C₂²⋊Dic₆₇	1	4	yes	—
167	C₂³	#14523975	1	2	yes	—
168	C₂³	C₂×#11761617	1	1	yes	characteristic
169	Q₈	—	2	2	no	—
170	D₄	—	2	4	no	—
171	D₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	536	2	no	—
175	C₂×C₄	#14523975	1	2	yes	—
176	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
177	C₂×C₄	#14524017	1	2	yes	—
178	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
179	C₂×C₄	#14524065	1	2	yes	—
180	C₈	—	2	2	no	—
181	C₈	—	536	2	no	—
182	C₈	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
183	C₈	#14524002	1	1	yes	characteristic
184	C₈	#14524022	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	—	4	1	no	—
187	C₂²	—	4	2	no	—
188	C₂²	C₂×#14523975	1	1	yes	characteristic
189	C₂²	#16863625	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	1072	2	no	—
192	C₄	#16863256	1	1	yes	characteristic
193	C₄	#16863434	1	1	yes	characteristic
194	C₄	#16863625	1	2	yes	—
195	C₂	—	2	2	no	—
196	C₂	—	2	1	no	—
197	C₂	—	4	2	no	—
198	C₂	#22560234	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22561982

C₂

yes

maximal, complemented

#22560259

C₂

yes

maximal, characteristic

#22560255

C₂

yes

maximal, characteristic

#22558305

C₂

yes

maximal, complemented

C₂×#504×C₆₇

C₂

yes

maximal, characteristic

C₂×#14523979

C₂²

yes

—

#16863434

C₂²

yes

—

#16863380

—

#16863370

—

#16863219

—

C₆₇⋊#338

—

#16863153

—

C₂×#131×C₆₇

C₂²

yes

characteristic

#504×C₆₇

—

C₂×M₃₂×C₆₇

C₄

yes

complemented

C₂×M₃₂×C₆₇

C₂²

yes

characteristic

C₂²×C₁₆×C₆₇

C₄

yes

complemented

C₂²×C₁₆×C₆₇

C₂²

yes

—

—

—

—

—

—

—

—

—

C₂×#41×C₆₇

D₄

yes

characteristic

#131×C₆₇

—

#131×C₆₇

—

C₂×M₁₆×C₆₇

D₄

yes

characteristic

C₂×M₁₆×C₆₇

Q₈

yes

—

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

M₃₂×C₆₇

—

M₃₂×C₆₇

D₄

yes

—

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₇

C₂×C₄

yes

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

SD₁₆

yes

characteristic

C₂×D₄×C₆₇

SD₁₆

yes

—

C₂×D₄×C₆₇

SD₁₆

yes

characteristic

C₂²×C₄×C₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×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

—

C₁₆×C₆₇

—

C₁₆×C₆₇

—

C₁₆×C₆₇

C₂²⋊C₄

yes

—

C₂³×C₆₇

#106

yes

—

C₂³×C₆₇

C₂×SD₁₆

yes

characteristic

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#106

yes

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

#135

yes

—

C₂×Dic₆₇

—

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

C₂×#106

yes

characteristic

C₂²×C₆₇

#465

yes

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

#12832

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#1807

yes

characteristic

100

C₂×#504

D₆₇

yes

characteristic

101

#2155

—

102

#1813

—

103

#1812

—

104

#1382

—

105

C₆₇

#12832

yes

characteristic, complemented, Sylow

106

C₂×#131

C₂×D₆₇

yes

characteristic

107

#504

—

108

C₂×M₃₂

Dic₆₇

yes

complemented

109

C₂×M₃₂

C₂×D₆₇

yes

characteristic

110

C₂²×C₁₆

Dic₆₇

yes

complemented

111

C₂²×C₁₆

C₂×D₆₇

yes

characteristic

112

C₂×#108

—

113

#427

—

114

#413

—

134

—

115

#410

—

134

—

116

#365

—

134

—

117

#338

—

134

—

118

H(3,ℤ/4)

—

134

—

119

C₂×#41

D₂₆₈

yes

characteristic

120

#131

—

121

#131

—

122

C₂×M₁₆

—

134

—

123

C₂×M₁₆

Dic₁₃₄

yes

—

124

C₂×M₁₆

D₂₆₈

yes

characteristic

125

C₂²×C₈

C₂×Dic₆₇

yes

—

126

C₂²×C₈

C₂²×D₆₇

yes

characteristic

127

#117

—

134

—

128

M₃₂

—

129

M₃₂

C₂²⋊D₆₇

yes

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

132

C₂×C₁₆

C₂×Dic₆₇

yes

characteristic

133

C₂×C₁₆

C₂×Dic₆₇

yes

characteristic

134

C₂×C₁₆

C₂×Dic₆₇

yes

—

135

#108

—

134

—

136

#108

—

134

—

137

#107

—

134

—

138

#106

—

134

—

139

#101

—

268

—

140

#100

—

268

—

141

#99

—

268

—

142

#41

—

143

#41

—

144

C₂×Q₈

#11761617

yes

characteristic

145

C₂×D₄

#11761617

yes

—

146

C₂×D₄

#11761617

yes

characteristic

147

C₂²×C₄

C₄⋊Dic₆₇

yes

—

148

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

149

M₁₆

—

150

M₁₆

—

151

M₁₆

—

268

—

152

M₁₆

—

268

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

268

—

156

C₂×C₈

C₄⋊Dic₆₇

yes

—

157

C₂×C₈

C₂²×Dic₆₇

yes

characteristic

158

C₂×C₈

C₂×C₂²⋊D₆₇

yes

—

159

C₂×C₈

C₂×Dic₁₃₄

yes

characteristic

160

C₂×C₈

#11761628

yes

—

161

C₄⋊C₄

—

268

—

162

C₂²⋊C₄

—

268

—

163

C₄²

—

268

—

164

C₁₆

—

165

C₁₆

—

166

C₁₆

C₂²⋊Dic₆₇

yes

—

167

C₂³

#14523975

yes

—

168

C₂³

C₂×#11761617

yes

characteristic

169

Q₈

—

170

D₄

—

171

D₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

536

—

175

C₂×C₄

#14523975

yes

—

176

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

characteristic

177

C₂×C₄

#14524017

yes

—

178

C₂×C₄

C₂×#11761617

yes

characteristic

179

C₂×C₄

#14524065

yes

—

180

C₈

—

181

C₈

—

536

—

182

C₈

C₂×C₂²⋊Dic₆₇

yes

characteristic

183

C₈

#14524002

yes

characteristic

184

C₈

#14524022

yes

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

C₂×#14523975

yes

characteristic

189

C₂²

#16863625

yes

—

190

C₄

—

191

C₄

—

1072

—

192

C₄

#16863256

yes

characteristic

193

C₄

#16863434

yes

characteristic

194

C₄

#16863625

yes

—

195

C₂

—

196

C₂

—

197

C₂

—

198

C₂

#22560234

yes

characteristic

	subgroup	G/H
center	C₄	#16863434
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#504×C₆₇	C₂
socle	C₂×C₆₇	#1807
prime radical	C₂×#41×C₆₇	D₄
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#504	D₆₇
hypocenter	C₆₇	#12832
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ C₂×#504

subgroup

G/H

center

C₄

#16863434

commutator

C₂×C₈×C₆₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₂×#504×C₆₇

C₂

socle

C₂×C₆₇

#1807

prime radical

C₂×#41×C₆₇

D₄

cosocle

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ C₂×#504

N	H ≅ G/N	type
1. #22561982	C₂	semidirect
4. #22558305	C₂	semidirect
15. C₂×M₃₂×C₆₇	C₄	semidirect
17. C₂²×C₁₆×C₆₇	C₄	semidirect
105. C₆₇	#12832	semidirect
108. C₂×M₃₂	Dic₆₇	semidirect
110. C₂²×C₁₆	Dic₆₇	semidirect

H ≅ G/N

type

1. #22561982

C₂

semidirect

4. #22558305

C₂

semidirect

15. C₂×M₃₂×C₆₇

C₄

semidirect

17. C₂²×C₁₆×C₆₇

C₄

semidirect

105. C₆₇

#12832

semidirect

108. C₂×M₃₂

Dic₆₇

semidirect

110. C₂²×C₁₆

Dic₆₇

semidirect

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

subgroup	index
#22561982	2
#22560259	2
#22560255	2
#22558305	2
C₂×#504×C₆₇	2
#12832	67

subgroup

index

C₂×#504×C₆₇

Subgroups — #30771987

Conjugacy classes of subgroups

Complements

Maximal subgroups