Group #30775368 — subgroups

Subgroups — #30775368

Order 17152.

418 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 95 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	#22561967	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16863209	C₂	1	2	yes	maximal, complemented
3	#22561120	C₂	1	2	yes	maximal, complemented
4	#22558303	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₂²	1	1	yes	characteristic
7	C₂×#14523979	C₂²	1	2	yes	—
8	#16863401	—	2	2	no	—
9	C₂×#14523960	—	2	2	no	—
10	#16863351	C₂²	1	1	yes	characteristic
11	Dic₆₇⋊C₁₆	—	2	2	no	—
12	#16863209	—	2	4	no	—
13	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
14	C₂×D₁₆×C₆₇	—	2	4	no	—
15	C₂×D₁₆×C₆₇	—	2	2	no	—
16	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#11761612	—	2	2	no	—
18	C₂×#11761612	D₄	1	1	yes	characteristic
19	#14524002	D₄	1	1	yes	characteristic
20	#14523983	—	2	2	no	—
21	C₂×C₆₇⋊C₁₆	—	2	1	no	—
22	#14523979	—	2	2	no	—
23	#14523979	—	2	2	no	—
24	#14523964	—	4	4	no	—
25	#14523960	—	4	4	no	—
26	#14523953	—	2	1	no	—
27	C₈×Dic₆₇	—	2	1	no	—
28	C₂²×D₄×C₆₇	—	2	2	no	—
29	C₂×D₈×C₆₇	—	2	2	no	—
30	C₂×D₈×C₆₇	—	2	2	no	—
31	C₂×D₈×C₆₇	—	2	2	no	—
32	C₂×D₈×C₆₇	—	2	2	no	—
33	C₂×D₈×C₆₇	—	2	2	no	—
34	C₂×D₈×C₆₇	D₄	1	4	yes	—
35	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
36	D₁₆×C₆₇	—	2	8	no	—
37	C₂×C₁₆×C₆₇	—	2	2	no	—
38	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
39	#11761612	—	2	2	no	—
40	#11761612	—	4	2	no	—
41	#11761612	—	4	2	no	—
42	C₂×C₆₇⋊C₈	—	2	1	no	—
43	C₂×C₆₇⋊C₈	—	4	2	no	—
44	C₄⋊Dic₆₇	D₈	1	2	yes	—
45	C₂²⋊Dic₆₇	—	2	1	no	—
46	C₄×Dic₆₇	—	2	1	no	—
47	C₆₇⋊C₁₆	—	4	1	no	—
48	C₂⁴×C₆₇	—	4	2	no	—
49	C₂×D₄×C₆₇	—	2	4	no	—
50	C₂×D₄×C₆₇	—	4	2	no	—
51	C₂×D₄×C₆₇	—	4	2	no	—
52	C₂×D₄×C₆₇	—	4	2	no	—
53	C₂×D₄×C₆₇	—	4	2	no	—
54	C₂×D₄×C₆₇	—	4	2	no	—
55	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
56	D₈×C₆₇	—	2	4	no	—
57	D₈×C₆₇	—	2	4	no	—
58	D₈×C₆₇	—	4	2	no	—
59	D₈×C₆₇	—	4	2	no	—
60	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
63	C₂×C₈×C₆₇	#41	1	2	yes	—
64	C₁₆×C₆₇	—	2	2	no	—
65	C₂³×C₆₇	—	4	4	no	—
66	C₂³×C₆₇	—	8	2	no	—
67	C₂³×C₆₇	—	8	2	no	—
68	C₂³×C₆₇	—	16	4	no	—
69	C₂³×C₆₇	#135	1	1	yes	characteristic
70	D₄×C₆₇	—	4	4	no	—
71	D₄×C₆₇	—	4	4	no	—
72	D₄×C₆₇	—	8	2	no	—
73	D₄×C₆₇	—	8	2	no	—
74	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
75	C₂×C₄×C₆₇	#121	1	2	yes	—
76	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
78	C₂×Dic₆₇	—	2	2	no	—
79	C₈×C₆₇	#120	1	1	yes	characteristic
80	C₈×C₆₇	#123	1	2	yes	—
81	C₈×C₆₇	#124	1	1	yes	characteristic
82	C₆₇⋊C₈	—	4	1	no	—
83	C₆₇⋊C₈	—	8	2	no	—
84	C₂²×C₆₇	—	2	2	no	—
85	C₂²×C₆₇	—	8	4	no	—
86	C₂²×C₆₇	—	16	4	no	—
87	C₂²×C₆₇	—	16	4	no	—
88	C₂²×C₆₇	—	16	4	no	—
89	C₂²×C₆₇	#463	1	2	yes	—
90	C₂²×C₆₇	#466	1	1	yes	characteristic
91	C₄×C₆₇	#394	1	1	yes	characteristic
92	C₄×C₆₇	#459	1	2	yes	—
93	C₄×C₆₇	#469	1	1	yes	characteristic
94	Dic₆₇	—	4	2	no	—
95	#13439	—	67	1	no	maximal, Sylow
96	C₂×C₆₇	—	2	2	no	—
97	C₂×C₆₇	—	16	4	no	—
98	C₂×C₆₇	#2054	1	1	yes	characteristic
99	C₂×C₆₇	#2221	1	2	yes	—
100	C₂²×D₁₆	D₆₇	1	1	yes	characteristic
101	#2151	—	67	1	no	—
102	C₂×#361	—	67	2	no	—
103	#1978	—	67	2	no	—
104	#1381	—	67	1	no	—
105	C₆₇	#13439	1	1	yes	characteristic, complemented, Sylow
106	C₂²×D₈	C₂×D₆₇	1	2	yes	—
107	C₂×D₁₆	—	2	4	no	—
108	C₂×D₁₆	—	2	2	no	—
109	C₂×M₃₂	—	67	1	no	—
110	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
111	C₂×#108	—	67	2	no	—
112	#418	—	134	2	no	—
113	C₂×#100	—	134	2	no	—
114	#405	—	67	1	no	—
115	C₄⋊C₁₆	—	134	2	no	—
116	#361	—	134	4	no	—
117	C₂²×D₄	—	2	2	no	—
118	C₂×D₈	—	2	2	no	—
119	C₂×D₈	—	2	2	no	—
120	C₂×D₈	—	2	2	no	—
121	C₂×D₈	—	2	2	no	—
122	C₂×D₈	—	2	2	no	—
123	C₂×D₈	C₂²⋊D₆₇	1	4	yes	—
124	C₂×M₁₆	—	67	1	no	—
125	C₂×M₁₆	—	134	2	no	—
126	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
127	#117	—	67	1	no	—
128	D₁₆	—	2	8	no	—
129	M₃₂	—	134	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	134	1	no	—
132	C₂×C₁₆	C₂²⋊D₆₇	1	2	yes	—
133	#108	—	134	2	no	—
134	#108	—	134	2	no	—
135	#102	—	268	4	no	—
136	#100	—	268	4	no	—
137	#97	—	134	1	no	—
138	C₄×C₈	—	134	1	no	—
139	C₂⁴	—	4	2	no	—
140	C₂×D₄	—	2	4	no	—
141	C₂×D₄	—	4	2	no	—
142	C₂×D₄	—	4	2	no	—
143	C₂×D₄	—	4	2	no	—
144	C₂×D₄	—	4	2	no	—
145	C₂×D₄	—	4	2	no	—
146	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
147	D₈	—	2	4	no	—
148	D₈	—	2	4	no	—
149	D₈	—	4	2	no	—
150	D₈	—	4	2	no	—
151	M₁₆	—	268	2	no	—
152	M₁₆	—	268	2	no	—
153	M₁₆	—	134	2	no	—
154	C₂×C₈	—	268	2	no	—
155	C₂×C₈	—	134	1	no	—
156	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
157	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
158	C₂×C₈	#11761627	1	1	yes	characteristic
159	C₂×C₈	#11761627	1	2	yes	—
160	C₄⋊C₄	—	67	2	no	—
161	C₂²⋊C₄	—	134	1	no	—
162	C₄²	—	134	1	no	—
163	C₁₆	—	2	2	no	—
164	C₁₆	—	268	1	no	—
165	C₂³	—	4	4	no	—
166	C₂³	—	8	2	no	—
167	C₂³	—	8	2	no	—
168	C₂³	—	16	4	no	—
169	C₂³	#14524062	1	1	yes	characteristic
170	D₄	—	4	4	no	—
171	D₄	—	4	4	no	—
172	D₄	—	8	2	no	—
173	D₄	—	8	2	no	—
174	C₂×C₄	—	134	2	no	—
175	C₂×C₄	#14524014	1	1	yes	characteristic
176	C₂×C₄	#14524015	1	2	yes	—
177	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
178	C₂×C₄	#14524067	1	2	yes	—
179	C₈	—	268	1	no	—
180	C₈	—	536	2	no	—
181	C₈	#14524012	1	1	yes	characteristic
182	C₈	#14524024	1	2	yes	—
183	C₈	#14524030	1	1	yes	characteristic
184	C₂²	—	2	2	no	—
185	C₂²	—	8	4	no	—
186	C₂²	—	16	4	no	—
187	C₂²	—	16	4	no	—
188	C₂²	—	16	4	no	—
189	C₂²	#16863609	1	2	yes	—
190	C₂²	#16863629	1	1	yes	characteristic
191	C₄	—	268	2	no	—
192	C₄	#16863310	1	1	yes	characteristic
193	C₄	#16863589	1	2	yes	—
194	C₄	#16863646	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	16	4	no	—
197	C₂	#22561517	1	1	yes	characteristic
198	C₂	#22562326	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22561967

C₂

yes

maximal, characteristic, complemented

C₂×#16863209

C₂

yes

maximal, complemented

#22561120

C₂

yes

maximal, complemented

#22558303

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₆×C₆₇

C₂

yes

maximal, characteristic

C₂×#14523983

C₂²

yes

characteristic

C₂×#14523979

C₂²

yes

—

#16863401

—

C₂×#14523960

—

#16863351

C₂²

yes

characteristic

Dic₆₇⋊C₁₆

—

#16863209

—

C₂²×D₈×C₆₇

C₂²

yes

—

C₂×D₁₆×C₆₇

—

C₂×D₁₆×C₆₇

—

C₂²×C₁₆×C₆₇

C₂²

yes

characteristic

C₂×#11761612

—

C₂×#11761612

D₄

yes

characteristic

#14524002

D₄

yes

characteristic

#14523983

—

C₂×C₆₇⋊C₁₆

—

—

—

—

—

—

C₈×Dic₆₇

—

C₂²×D₄×C₆₇

—

C₂×D₈×C₆₇

—

C₂×D₈×C₆₇

—

C₂×D₈×C₆₇

—

C₂×D₈×C₆₇

—

C₂×D₈×C₆₇

—

C₂×D₈×C₆₇

D₄

yes

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

D₁₆×C₆₇

—

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

D₄

yes

—

#11761612

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

D₈

yes

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

—

C₂⁴×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

D₈×C₆₇

—

D₈×C₆₇

—

D₈×C₆₇

—

D₈×C₆₇

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

C₂×C₈×C₆₇

yes

C₂×C₈×C₆₇

yes

—

C₁₆×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

#135

yes

characteristic

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

#121

yes

characteristic

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₆₇

—

C₈×C₆₇

#120

yes

characteristic

C₈×C₆₇

#123

yes

—

C₈×C₆₇

#124

yes

characteristic

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#463

yes

—

C₂²×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₄×C₆₇

#469

yes

characteristic

Dic₆₇

—

#13439

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#2054

yes

characteristic

C₂×C₆₇

#2221

yes

—

100

C₂²×D₁₆

D₆₇

yes

characteristic

101

#2151

—

102

C₂×#361

—

103

#1978

—

104

#1381

—

105

C₆₇

#13439

yes

characteristic, complemented, Sylow

106

C₂²×D₈

C₂×D₆₇

yes

—

107

C₂×D₁₆

—

108

C₂×D₁₆

—

109

C₂×M₃₂

—

110

C₂²×C₁₆

C₂×D₆₇

yes

characteristic

111

C₂×#108

—

112

#418

—

134

—

113

C₂×#100

—

134

—

114

#405

—

115

C₄⋊C₁₆

—

134

—

116

#361

—

134

—

117

C₂²×D₄

—

118

C₂×D₈

—

119

C₂×D₈

—

120

C₂×D₈

—

121

C₂×D₈

—

122

C₂×D₈

—

123

C₂×D₈

C₂²⋊D₆₇

yes

—

124

C₂×M₁₆

—

125

C₂×M₁₆

—

134

—

126

C₂²×C₈

C₂²×D₆₇

yes

characteristic

127

#117

—

128

D₁₆

—

129

M₃₂

—

134

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

134

—

132

C₂×C₁₆

C₂²⋊D₆₇

yes

—

133

#108

—

134

—

134

#108

—

134

—

135

#102

—

268

—

136

#100

—

268

—

137

#97

—

134

—

138

C₄×C₈

—

134

—

139

C₂⁴

—

140

C₂×D₄

—

141

C₂×D₄

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂²×C₄

D₄×D₆₇

yes

characteristic

147

D₈

—

148

D₈

—

149

D₈

—

150

D₈

—

151

M₁₆

—

268

—

152

M₁₆

—

268

—

153

M₁₆

—

134

—

154

C₂×C₈

—

268

—

155

C₂×C₈

—

134

—

156

C₂×C₈

C₂×C₂²⋊D₆₇

yes

—

157

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

158

C₂×C₈

#11761627

yes

characteristic

159

C₂×C₈

#11761627

yes

—

160

C₄⋊C₄

—

161

C₂²⋊C₄

—

134

—

162

C₄²

—

134

—

163

C₁₆

—

164

C₁₆

—

268

—

165

C₂³

—

166

C₂³

—

167

C₂³

—

168

C₂³

—

169

C₂³

#14524062

yes

characteristic

170

D₄

—

171

D₄

—

172

D₄

—

173

D₄

—

174

C₂×C₄

—

134

—

175

C₂×C₄

#14524014

yes

characteristic

176

C₂×C₄

#14524015

yes

—

177

C₂×C₄

D₈×D₆₇

yes

characteristic

178

C₂×C₄

#14524067

yes

—

179

C₈

—

268

—

180

C₈

—

536

—

181

C₈

#14524012

yes

characteristic

182

C₈

#14524024

yes

—

183

C₈

#14524030

yes

characteristic

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

#16863609

yes

—

190

C₂²

#16863629

yes

characteristic

191

C₄

—

268

—

192

C₄

#16863310

yes

characteristic

193

C₄

#16863589

yes

—

194

C₄

#16863646

yes

characteristic

195

C₂

—

196

C₂

—

197

C₂

#22561517

yes

characteristic

198

C₂

#22562326

yes

—

	subgroup	G/H
center	C₂²	#16863629
commutator, cosocle	C₂²×C₈×C₆₇	C₂³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting, prime radical	C₂²×D₁₆×C₆₇	C₂
socle	C₂²×C₆₇	#466
hypercenter	C₂²×D₁₆	D₆₇
hypocenter	C₆₇	#13439
LCS	G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₁₆

subgroup

G/H

center

C₂²

#16863629

commutator, cosocle

C₂²×C₈×C₆₇

C₂³

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting, prime radical

C₂²×D₁₆×C₆₇

C₂

socle

C₂²×C₆₇

#466

hypercenter

C₂²×D₁₆

D₆₇

hypocenter

C₆₇

#13439

LCS

G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₁₆

N	H ≅ G/N	type
1. #22561967	C₂	semidirect
2. C₂×#16863209	C₂	semidirect
3. #22561120	C₂	semidirect
4. #22558303	C₂	semidirect
105. C₆₇	#13439	semidirect

H ≅ G/N

type

1. #22561967

C₂

semidirect

2. C₂×#16863209

C₂

semidirect

3. #22561120

C₂

semidirect

4. #22558303

C₂

semidirect

105. C₆₇

#13439

semidirect

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

subgroup	index
#22561967	2
C₂×#16863209	2
#22561120	2
#22558303	2
C₂²×D₁₆×C₆₇	2
#13439	67

subgroup

index

C₂²×D₁₆×C₆₇

Subgroups — #30775368

Conjugacy classes of subgroups

Complements

Maximal subgroups