Group #30748823 — subgroups

Subgroups — #30748823

Order 17152.

388 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 119 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	#22558972	C₂	1	2	yes	maximal, complemented
2	#22558844	C₂	1	2	yes	maximal
3	#22558120	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2931	C₂	1	1	yes	maximal, characteristic
5	#16863453	C₂²	1	4	yes	—
6	#16863149	—	2	2	no	—
7	#16863147	—	2	2	no	—
8	#16863136	—	2	4	no	—
9	C₂×#131×C₆₇	C₄	1	2	yes	—
10	#532×C₆₇	C₂²	1	1	yes	characteristic
11	#436×C₆₇	—	2	2	no	—
12	C₈×D₄×C₆₇	—	2	2	no	—
13	#423×C₆₇	C₂²	1	1	yes	characteristic
14	#408×C₆₇	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
17	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
18	#14523954	—	2	4	no	—
19	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
20	#131×C₆₇	—	4	4	no	—
21	C₂×M₁₆×C₆₇	—	2	2	no	—
22	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₆₇	—	2	2	no	—
24	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₄⋊Q₈×C₆₇	Q₈	1	1	yes	characteristic
27	#127×C₆₇	Q₈	1	1	yes	characteristic
28	#124×C₆₇	D₄	1	2	yes	—
29	#121×C₆₇	—	2	2	no	—
30	C₄×D₄×C₆₇	—	2	2	no	—
31	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₆₇	—	2	2	no	—
33	C₂²⋊C₈×C₆₇	—	2	4	no	—
34	C₄×C₈×C₆₇	D₄	1	2	yes	—
35	C₄×C₈×C₆₇	D₄	1	2	yes	—
36	C₂×C₆₇⋊C₈	—	4	4	no	—
37	#41×C₆₇	—	4	2	no	—
38	#41×C₆₇	—	4	2	no	—
39	C₂×Q₈×C₆₇	C₄⋊C₄	1	2	yes	—
40	C₂×D₄×C₆₇	—	2	2	no	—
41	C₂×D₄×C₆₇	—	2	2	no	—
42	C₂×D₄×C₆₇	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₆₇	—	2	2	no	—
44	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	#41	1	2	yes	—
46	M₁₆×C₆₇	—	4	4	no	—
47	M₁₆×C₆₇	—	4	2	no	—
48	C₂×C₈×C₆₇	—	2	2	no	—
49	C₂×C₈×C₆₇	—	2	2	no	—
50	C₂×C₈×C₆₇	—	4	1	no	—
51	C₂×C₈×C₆₇	—	4	4	no	—
52	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₄⋊C₄×C₆₇	—	2	2	no	—
56	C₂²⋊C₄×C₆₇	—	2	4	no	—
57	C₄²×C₆₇	C₂×D₄	1	2	yes	—
58	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
60	C₂³×C₆₇	—	2	2	no	—
61	C₂³×C₆₇	#117	1	1	yes	characteristic
62	Q₈×C₆₇	—	4	2	no	—
63	D₄×C₆₇	—	4	2	no	—
64	D₄×C₆₇	—	4	4	no	—
65	D₄×C₆₇	—	4	4	no	—
66	C₂×C₄×C₆₇	—	2	1	no	—
67	C₂×C₄×C₆₇	—	2	1	no	—
68	C₂×C₄×C₆₇	—	2	2	no	—
69	C₂×C₄×C₆₇	—	4	2	no	—
70	C₂×C₄×C₆₇	—	4	2	no	—
71	C₂×C₄×C₆₇	—	4	2	no	—
72	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₆₇	#121	1	2	yes	—
75	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
76	C₈×C₆₇	—	2	1	no	—
77	C₈×C₆₇	—	2	2	no	—
78	C₈×C₆₇	—	4	2	no	—
79	C₈×C₆₇	#100	1	2	yes	—
80	C₆₇⋊C₈	—	8	4	no	—
81	C₂²×C₆₇	—	2	1	no	—
82	C₂²×C₆₇	—	2	2	no	—
83	C₂²×C₆₇	—	4	2	no	—
84	C₂²×C₆₇	—	4	4	no	—
85	C₂²×C₆₇	#380	1	1	yes	characteristic
86	C₄×C₆₇	—	2	1	no	—
87	C₄×C₆₇	—	2	2	no	—
88	C₄×C₆₇	—	2	2	no	—
89	C₄×C₆₇	—	4	2	no	—
90	C₄×C₆₇	C₂×#100	1	1	yes	characteristic
91	C₄×C₆₇	#413	1	1	yes	characteristic
92	#8630	—	67	1	no	maximal, Sylow
93	C₂×C₆₇	—	4	1	no	—
94	C₂×C₆₇	—	8	2	no	—
95	C₂×C₆₇	#1800	1	1	yes	characteristic
96	C₂×C₆₇	#1814	1	2	yes	—
97	#2931	D₆₇	1	1	yes	characteristic
98	#1568	—	67	2	no	—
99	#1536	—	67	2	no	—
100	#1307	—	67	2	no	—
101	C₆₇	#8630	1	1	yes	characteristic, complemented, Sylow
102	C₂×#131	Dic₆₇	1	2	yes	—
103	#532	C₂×D₆₇	1	1	yes	characteristic
104	#436	—	2	2	no	—
105	C₈×D₄	—	2	2	no	—
106	#432	—	67	4	no	—
107	#423	C₂×D₆₇	1	1	yes	characteristic
108	#408	Dic₆₇	1	2	yes	—
109	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
110	#335	—	134	2	no	—
111	#334	—	134	2	no	—
112	#329	—	134	4	no	—
113	C₂×#41	C₂×Dic₆₇	1	2	yes	—
114	#131	—	4	4	no	—
115	C₂×M₁₆	—	2	2	no	—
116	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
117	C₂²×C₈	—	2	2	no	—
118	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
119	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
120	C₄⋊Q₈	Dic₁₃₄	1	1	yes	characteristic
121	#127	Dic₁₃₄	1	1	yes	characteristic
122	#124	D₂₆₈	1	2	yes	—
123	#121	—	2	2	no	—
124	C₄×D₄	—	2	2	no	—
125	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	134	4	no	—
128	C₂²⋊C₈	—	2	4	no	—
129	C₂²⋊C₈	—	134	4	no	—
130	#97	—	134	4	no	—
131	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
132	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
133	#41	—	4	2	no	—
134	#41	—	4	2	no	—
135	C₂×Q₈	C₄⋊Dic₆₇	1	2	yes	—
136	C₂×D₄	—	2	2	no	—
137	C₂×D₄	—	2	2	no	—
138	C₂×D₄	C₄⋊Dic₆₇	1	2	yes	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
141	C₂²×C₄	#11761628	1	2	yes	—
142	M₁₆	—	4	4	no	—
143	M₁₆	—	4	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	4	1	no	—
147	C₂×C₈	—	4	4	no	—
148	C₂×C₈	—	268	4	no	—
149	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
150	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
151	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
152	C₄⋊C₄	—	2	2	no	—
153	C₂²⋊C₄	—	2	4	no	—
154	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
155	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
156	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
157	C₂³	—	2	2	no	—
158	C₂³	#14524002	1	1	yes	characteristic
159	Q₈	—	4	2	no	—
160	D₄	—	4	2	no	—
161	D₄	—	4	4	no	—
162	D₄	—	4	4	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
170	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
171	C₂×C₄	#14524017	1	2	yes	—
172	C₂×C₄	#14524022	1	2	yes	—
173	C₈	—	2	1	no	—
174	C₈	—	2	2	no	—
175	C₈	—	4	2	no	—
176	C₈	—	536	4	no	—
177	C₈	#14523960	1	2	yes	—
178	C₂²	—	2	2	no	—
179	C₂²	—	2	1	no	—
180	C₂²	—	4	4	no	—
181	C₂²	—	4	2	no	—
182	C₂²	#16863256	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	2	2	no	—
185	C₄	—	2	1	no	—
186	C₄	—	4	2	no	—
187	C₄	C₂×#14523960	1	1	yes	characteristic
188	C₄	#16863380	1	1	yes	characteristic
189	C₂	—	4	1	no	—
190	C₂	—	8	2	no	—
191	C₂	#22560201	1	1	yes	characteristic
192	C₂	#22560263	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22558972

C₂

yes

maximal, complemented

#22558844

C₂

yes

maximal

#22558120

C₂

yes

maximal, complemented

C₆₇×#2931

C₂

yes

maximal, characteristic

#16863453

C₂²

yes

—

#16863149

—

#16863147

—

#16863136

—

C₂×#131×C₆₇

C₄

yes

—

#532×C₆₇

C₂²

yes

characteristic

#436×C₆₇

—

C₈×D₄×C₆₇

—

#423×C₆₇

C₂²

yes

characteristic

#408×C₆₇

C₄

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

—

#14523954

—

C₂×#41×C₆₇

C₂×C₄

yes

—

#131×C₆₇

—

C₂×M₁₆×C₆₇

—

C₂×M₁₆×C₆₇

C₂×C₄

yes

—

C₂²×C₈×C₆₇

—

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄⋊Q₈×C₆₇

Q₈

yes

characteristic

#127×C₆₇

Q₈

yes

characteristic

#124×C₆₇

D₄

yes

—

#121×C₆₇

—

C₄×D₄×C₆₇

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₂×C₆₇⋊C₈

—

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

C₄⋊C₄

yes

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂²×C₄

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

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₂³×C₆₇

—

C₂³×C₆₇

#117

yes

characteristic

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#100

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#380

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#1800

yes

characteristic

C₂×C₆₇

#1814

yes

—

#2931

D₆₇

yes

characteristic

#1568

—

#1536

—

100

#1307

—

101

C₆₇

#8630

yes

characteristic, complemented, Sylow

102

C₂×#131

Dic₆₇

yes

—

103

#532

C₂×D₆₇

yes

characteristic

104

#436

—

105

C₈×D₄

—

106

#432

—

107

#423

C₂×D₆₇

yes

characteristic

108

#408

Dic₆₇

yes

—

109

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

110

#335

—

134

—

111

#334

—

134

—

112

#329

—

134

—

113

C₂×#41

C₂×Dic₆₇

yes

—

114

#131

—

115

C₂×M₁₆

—

116

C₂×M₁₆

C₂×Dic₆₇

yes

—

117

C₂²×C₈

—

118

C₂²×C₈

C₂×Dic₆₇

yes

characteristic

119

C₂²×C₈

C₂×Dic₆₇

yes

characteristic

120

C₄⋊Q₈

Dic₁₃₄

yes

characteristic

121

#127

Dic₁₃₄

yes

characteristic

122

#124

D₂₆₈

yes

—

123

#121

—

124

C₄×D₄

—

125

C₂×C₄²

C₂²×D₆₇

yes

characteristic

126

C₄⋊C₈

—

127

C₄⋊C₈

—

134

—

128

C₂²⋊C₈

—

129

C₂²⋊C₈

—

134

—

130

#97

—

134

—

131

C₄×C₈

C₂²⋊D₆₇

yes

—

132

C₄×C₈

C₂²⋊D₆₇

yes

—

133

#41

—

134

#41

—

135

C₂×Q₈

C₄⋊Dic₆₇

yes

—

136

C₂×D₄

—

137

C₂×D₄

—

138

C₂×D₄

C₄⋊Dic₆₇

yes

—

139

C₂²×C₄

—

140

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

141

C₂²×C₄

#11761628

yes

—

142

M₁₆

—

143

M₁₆

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

268

—

149

C₂×C₈

C₂²⋊Dic₆₇

yes

—

150

C₂×C₈

C₂²⋊Dic₆₇

yes

characteristic

151

C₂×C₈

C₂²⋊Dic₆₇

yes

characteristic

152

C₄⋊C₄

—

153

C₂²⋊C₄

—

154

C₄²

C₂×C₂²⋊D₆₇

yes

—

155

C₄²

C₂×D₂₆₈

yes

characteristic

156

C₄²

C₂×Dic₁₃₄

yes

characteristic

157

C₂³

—

158

C₂³

#14524002

yes

characteristic

159

Q₈

—

160

D₄

—

161

D₄

—

162

D₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

C₂×C₂²⋊Dic₆₇

yes

characteristic

170

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

characteristic

171

C₂×C₄

#14524017

yes

—

172

C₂×C₄

#14524022

yes

—

173

C₈

—

174

C₈

—

175

C₈

—

176

C₈

—

536

—

177

C₈

#14523960

yes

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

—

182

C₂²

#16863256

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

C₂×#14523960

yes

characteristic

188

C₄

#16863380

yes

characteristic

189

C₂

—

190

C₂

—

191

C₂

#22560201

yes

characteristic

192

C₂

#22560263

yes

—

	subgroup	G/H
center	C₂²	#16863256
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2931	C₂
socle	C₂²×C₆₇	#380
prime radical	#532×C₆₇	C₂²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#2931	D₆₇
hypocenter	C₆₇	#8630
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931

subgroup

G/H

center

C₂²

#16863256

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#2931

C₂

socle

C₂²×C₆₇

#380

prime radical

#532×C₆₇

C₂²

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931

N	H ≅ G/N	type
1. #22558972	C₂	semidirect
3. #22558120	C₂	semidirect
101. C₆₇	#8630	semidirect

H ≅ G/N

type

1. #22558972

C₂

semidirect

3. #22558120

C₂

semidirect

101. C₆₇

#8630

semidirect

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

subgroup	index
#22558972	2
#22558844	2
#22558120	2
C₆₇×#2931	2
#8630	67

subgroup

index

C₆₇×#2931

Subgroups — #30748823

Conjugacy classes of subgroups

Complements

Maximal subgroups