Group #30748849 — subgroups

Subgroups — #30748849

Order 17152.

404 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 127 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	#22558965	C₂	1	2	yes	maximal, complemented
2	#22558944	C₂	1	2	yes	maximal
3	C₆₇⋊#1309	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2931	C₂	1	1	yes	maximal, characteristic
5	#16863420	C₂²	1	4	yes	—
6	#16863147	—	2	2	no	—
7	#16863147	—	2	2	no	—
8	#16863145	—	2	2	no	—
9	#16863140	—	2	2	no	—
10	C₂×#131×C₆₇	C₄	1	2	yes	—
11	#532×C₆₇	C₂²	1	1	yes	characteristic
12	#436×C₆₇	—	2	2	no	—
13	C₈×D₄×C₆₇	—	2	2	no	—
14	#423×C₆₇	C₂²	1	1	yes	characteristic
15	#408×C₆₇	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#11761612	—	2	4	no	—
18	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
19	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
20	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
21	#131×C₆₇	—	4	4	no	—
22	C₂×M₁₆×C₆₇	—	2	2	no	—
23	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₆₇	—	2	2	no	—
25	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
28	#127×C₆₇	D₄	1	1	yes	characteristic
29	#124×C₆₇	Q₈	1	2	yes	—
30	#121×C₆₇	—	2	2	no	—
31	C₄×D₄×C₆₇	—	2	2	no	—
32	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₆₇	—	2	2	no	—
34	C₂²⋊C₈×C₆₇	—	2	4	no	—
35	C₄×C₈×C₆₇	D₄	1	2	yes	—
36	C₄×C₈×C₆₇	D₄	1	2	yes	—
37	#11761612	—	8	4	no	—
38	C₂×C₆₇⋊C₈	—	4	4	no	—
39	#41×C₆₇	—	4	2	no	—
40	#41×C₆₇	—	4	2	no	—
41	C₂×Q₈×C₆₇	C₄⋊C₄	1	2	yes	—
42	C₂×D₄×C₆₇	—	2	2	no	—
43	C₂×D₄×C₆₇	—	2	2	no	—
44	C₂×D₄×C₆₇	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₆₇	—	2	2	no	—
46	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₇	#41	1	2	yes	—
48	M₁₆×C₆₇	—	4	4	no	—
49	M₁₆×C₆₇	—	4	2	no	—
50	C₂×C₈×C₆₇	—	2	2	no	—
51	C₂×C₈×C₆₇	—	2	2	no	—
52	C₂×C₈×C₆₇	—	4	1	no	—
53	C₂×C₈×C₆₇	—	4	4	no	—
54	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₄⋊C₄×C₆₇	—	2	2	no	—
58	C₂²⋊C₄×C₆₇	—	2	4	no	—
59	C₄²×C₆₇	C₂×D₄	1	2	yes	—
60	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
62	C₂³×C₆₇	—	2	2	no	—
63	C₂³×C₆₇	#117	1	1	yes	characteristic
64	Q₈×C₆₇	—	4	2	no	—
65	D₄×C₆₇	—	4	2	no	—
66	D₄×C₆₇	—	4	4	no	—
67	D₄×C₆₇	—	4	4	no	—
68	C₂×C₄×C₆₇	—	2	1	no	—
69	C₂×C₄×C₆₇	—	2	1	no	—
70	C₂×C₄×C₆₇	—	2	4	no	—
71	C₂×C₄×C₆₇	—	2	2	no	—
72	C₂×C₄×C₆₇	—	4	2	no	—
73	C₂×C₄×C₆₇	—	4	2	no	—
74	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₆₇	#121	1	2	yes	—
77	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
78	C₈×C₆₇	—	2	1	no	—
79	C₈×C₆₇	—	2	2	no	—
80	C₈×C₆₇	—	4	2	no	—
81	C₈×C₆₇	#100	1	2	yes	—
82	C₆₇⋊C₈	—	8	4	no	—
83	C₂²×C₆₇	—	2	1	no	—
84	C₂²×C₆₇	—	2	2	no	—
85	C₂²×C₆₇	—	4	2	no	—
86	C₂²×C₆₇	—	4	4	no	—
87	C₂²×C₆₇	#380	1	1	yes	characteristic
88	C₄×C₆₇	—	2	1	no	—
89	C₄×C₆₇	—	2	2	no	—
90	C₄×C₆₇	—	4	2	no	—
91	C₄×C₆₇	C₂×#100	1	1	yes	characteristic
92	C₄×C₆₇	#413	1	1	yes	characteristic
93	C₄×C₆₇	#471	1	4	yes	—
94	#8636	—	67	1	no	maximal, Sylow
95	C₂×C₆₇	—	4	1	no	—
96	C₂×C₆₇	—	8	2	no	—
97	C₂×C₆₇	#1800	1	1	yes	characteristic
98	C₂×C₆₇	#1815	1	2	yes	—
99	#2931	D₆₇	1	1	yes	characteristic
100	#1566	—	67	2	no	—
101	#1560	—	67	2	no	—
102	#1309	—	67	2	no	—
103	C₆₇	#8636	1	1	yes	characteristic, complemented, Sylow
104	C₂×#131	Dic₆₇	1	2	yes	—
105	#532	C₂×D₆₇	1	1	yes	characteristic
106	#436	—	2	2	no	—
107	C₈×D₄	—	2	2	no	—
108	#423	—	67	4	no	—
109	#423	C₂×D₆₇	1	1	yes	characteristic
110	#408	Dic₆₇	1	2	yes	—
111	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
112	#334	—	134	2	no	—
113	#334	—	134	2	no	—
114	#333	—	134	2	no	—
115	#331	—	134	2	no	—
116	C₂×#41	C₂×Dic₆₇	1	2	yes	—
117	#131	—	4	4	no	—
118	C₂×M₁₆	—	2	2	no	—
119	C₂×M₁₆	—	134	4	no	—
120	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
121	C₂²×C₈	—	2	2	no	—
122	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
123	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
124	C₄⋊Q₈	D₂₆₈	1	1	yes	characteristic
125	#127	D₂₆₈	1	1	yes	characteristic
126	#124	Dic₁₃₄	1	2	yes	—
127	#121	—	2	2	no	—
128	C₄×D₄	—	2	2	no	—
129	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
130	C₄⋊C₈	—	2	2	no	—
131	C₄⋊C₈	—	134	4	no	—
132	C₄⋊C₈	—	134	4	no	—
133	C₂²⋊C₈	—	2	4	no	—
134	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
135	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
136	#41	—	4	2	no	—
137	#41	—	4	2	no	—
138	C₂×Q₈	C₄⋊Dic₆₇	1	2	yes	—
139	C₂×D₄	—	2	2	no	—
140	C₂×D₄	—	2	2	no	—
141	C₂×D₄	C₄⋊Dic₆₇	1	2	yes	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
144	C₂²×C₄	#11761628	1	2	yes	—
145	M₁₆	—	4	4	no	—
146	M₁₆	—	4	2	no	—
147	M₁₆	—	536	4	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	4	1	no	—
151	C₂×C₈	—	4	4	no	—
152	C₂×C₈	—	268	4	no	—
153	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
154	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
155	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
156	C₄⋊C₄	—	2	2	no	—
157	C₂²⋊C₄	—	2	4	no	—
158	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
159	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
160	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
161	C₂³	—	2	2	no	—
162	C₂³	#14524002	1	1	yes	characteristic
163	Q₈	—	4	2	no	—
164	D₄	—	4	2	no	—
165	D₄	—	4	4	no	—
166	D₄	—	4	4	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
174	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
175	C₂×C₄	#14524017	1	2	yes	—
176	C₂×C₄	#14524022	1	2	yes	—
177	C₈	—	2	1	no	—
178	C₈	—	2	2	no	—
179	C₈	—	4	2	no	—
180	C₈	—	536	4	no	—
181	C₈	#14523960	1	2	yes	—
182	C₂²	—	2	2	no	—
183	C₂²	—	2	1	no	—
184	C₂²	—	4	4	no	—
185	C₂²	—	4	2	no	—
186	C₂²	#16863256	1	1	yes	characteristic
187	C₄	—	2	2	no	—
188	C₄	—	2	1	no	—
189	C₄	—	4	2	no	—
190	C₄	C₂×#14523960	1	1	yes	characteristic
191	C₄	#16863380	1	1	yes	characteristic
192	C₄	#16863659	1	4	yes	—
193	C₂	—	4	1	no	—
194	C₂	—	8	2	no	—
195	C₂	#22560201	1	1	yes	characteristic
196	C₂	#22560268	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22558965

C₂

yes

maximal, complemented

#22558944

C₂

yes

maximal

C₆₇⋊#1309

C₂

yes

maximal, complemented

C₆₇×#2931

C₂

yes

maximal, characteristic

#16863420

C₂²

yes

—

—

—

—

—

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

C₂×#11761612

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

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

D₄

yes

characteristic

#127×C₆₇

D₄

yes

characteristic

#124×C₆₇

Q₈

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

—

#11761612

—

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

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

#8636

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#1800

yes

characteristic

C₂×C₆₇

#1815

yes

—

#2931

D₆₇

yes

characteristic

100

#1566

—

101

#1560

—

102

#1309

—

103

C₆₇

#8636

yes

characteristic, complemented, Sylow

104

C₂×#131

Dic₆₇

yes

—

105

#532

C₂×D₆₇

yes

characteristic

106

#436

—

107

C₈×D₄

—

108

#423

—

109

#423

C₂×D₆₇

yes

characteristic

110

#408

Dic₆₇

yes

—

111

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

112

#334

—

134

—

113

#334

—

134

—

114

#333

—

134

—

115

#331

—

134

—

116

C₂×#41

C₂×Dic₆₇

yes

—

117

#131

—

118

C₂×M₁₆

—

119

C₂×M₁₆

—

134

—

120

C₂×M₁₆

C₂×Dic₆₇

yes

—

121

C₂²×C₈

—

122

C₂²×C₈

C₂×Dic₆₇

yes

characteristic

123

C₂²×C₈

C₂×Dic₆₇

yes

characteristic

124

C₄⋊Q₈

D₂₆₈

yes

characteristic

125

#127

D₂₆₈

yes

characteristic

126

#124

Dic₁₃₄

yes

—

127

#121

—

128

C₄×D₄

—

129

C₂×C₄²

C₂²×D₆₇

yes

characteristic

130

C₄⋊C₈

—

131

C₄⋊C₈

—

134

—

132

C₄⋊C₈

—

134

—

133

C₂²⋊C₈

—

134

C₄×C₈

C₂²⋊D₆₇

yes

—

135

C₄×C₈

C₂²⋊D₆₇

yes

—

136

#41

—

137

#41

—

138

C₂×Q₈

C₄⋊Dic₆₇

yes

—

139

C₂×D₄

—

140

C₂×D₄

—

141

C₂×D₄

C₄⋊Dic₆₇

yes

—

142

C₂²×C₄

—

143

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

144

C₂²×C₄

#11761628

yes

—

145

M₁₆

—

146

M₁₆

—

147

M₁₆

—

536

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

268

—

153

C₂×C₈

C₂²⋊Dic₆₇

yes

—

154

C₂×C₈

C₂²⋊Dic₆₇

yes

characteristic

155

C₂×C₈

C₂²⋊Dic₆₇

yes

characteristic

156

C₄⋊C₄

—

157

C₂²⋊C₄

—

158

C₄²

C₂×C₂²⋊D₆₇

yes

—

159

C₄²

C₂×D₂₆₈

yes

characteristic

160

C₄²

C₂×Dic₁₃₄

yes

characteristic

161

C₂³

—

162

C₂³

#14524002

yes

characteristic

163

Q₈

—

164

D₄

—

165

D₄

—

166

D₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

C₂×C₂²⋊Dic₆₇

yes

characteristic

174

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

characteristic

175

C₂×C₄

#14524017

yes

—

176

C₂×C₄

#14524022

yes

—

177

C₈

—

178

C₈

—

179

C₈

—

180

C₈

—

536

—

181

C₈

#14523960

yes

—

182

C₂²

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

#16863256

yes

characteristic

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

C₂×#14523960

yes

characteristic

191

C₄

#16863380

yes

characteristic

192

C₄

#16863659

yes

—

193

C₂

—

194

C₂

—

195

C₂

#22560201

yes

characteristic

196

C₂

#22560268

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	#8636
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. #22558965	C₂	semidirect
3. C₆₇⋊#1309	C₂	semidirect
103. C₆₇	#8636	semidirect

H ≅ G/N

type

1. #22558965

C₂

semidirect

3. C₆₇⋊#1309

C₂

semidirect

103. C₆₇

#8636

semidirect

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

subgroup	index
#22558965	2
#22558944	2
C₆₇⋊#1309	2
C₆₇×#2931	2
#8636	67

subgroup

index

#22558965

#22558944

C₆₇⋊#1309

C₆₇×#2931

#8636

Subgroups — #30748849

Conjugacy classes of subgroups

Complements

Maximal subgroups