Group #30772018 — subgroups

Subgroups — #30772018

Order 17152.

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

normal subgroups only

This group has 362 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	C₂×#16863813	C₂	1	1	yes	maximal, characteristic
2	#22561991	C₂	1	2	yes	maximal, complemented
3	#22560255	C₂	1	1	yes	maximal, characteristic
4	#22558310	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1812	C₂	1	1	yes	maximal, characteristic
6	#16863813	—	2	2	no	—
7	#16863813	—	2	2	no	—
8	C₂×#14523983	C₄	1	2	yes	complemented
9	C₂×#14523983	C₂²	1	1	yes	characteristic
10	C₂²×C₆₇⋊C₁₆	C₄	1	2	yes	complemented
11	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	#16863434	C₂²	1	2	yes	—
13	#16863370	—	2	1	no	—
14	#16863223	—	2	2	no	—
15	#16863221	—	2	2	no	—
16	#16863153	—	2	1	no	—
17	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
18	#427×C₆₇	C₂²	1	2	yes	—
19	#410×C₆₇	—	2	1	no	—
20	H(3,ℤ/4)×C₆₇	—	2	1	no	—
21	#14524002	—	2	2	no	—
22	#14523983	—	2	4	no	—
23	#14523983	—	2	2	no	—
24	C₂×C₆₇⋊C₁₆	—	2	2	no	—
25	C₂×C₆₇⋊C₁₆	—	2	2	no	—
26	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₆₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂×C₆₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₆₇⋊C₁₆	D₄	1	2	yes	—
30	C₂×C₆₇⋊C₁₆	D₄	1	2	yes	—
31	#14523977	—	2	2	no	—
32	#14523975	—	2	2	no	—
33	#14523958	—	4	2	no	—
34	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
35	#131×C₆₇	—	2	2	no	—
36	#131×C₆₇	—	2	2	no	—
37	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
38	C₂×M₁₆×C₆₇	Q₈	1	2	yes	—
39	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
41	#117×C₆₇	—	2	2	no	—
42	#107×C₆₇	—	2	2	no	—
43	#106×C₆₇	—	2	2	no	—
44	#99×C₆₇	—	4	2	no	—
45	C₄⋊Dic₆₇	—	4	2	no	—
46	C₂²⋊Dic₆₇	—	4	2	no	—
47	C₄×Dic₆₇	—	4	2	no	—
48	C₆₇⋊C₁₆	—	2	2	no	—
49	C₆₇⋊C₁₆	C₂²⋊C₄	1	1	yes	characteristic
50	C₆₇⋊C₁₆	C₂²⋊C₄	1	1	yes	characteristic
51	C₆₇⋊C₁₆	C₂²⋊C₄	1	2	yes	—
52	#41×C₆₇	—	2	2	no	—
53	#41×C₆₇	—	2	2	no	—
54	C₂×Q₈×C₆₇	Q₁₆	1	1	yes	characteristic
55	C₂×D₄×C₆₇	D₈	1	2	yes	—
56	C₂×D₄×C₆₇	Q₁₆	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
59	M₁₆×C₆₇	—	2	2	no	—
60	M₁₆×C₆₇	—	2	4	no	—
61	C₂×C₈×C₆₇	—	2	2	no	—
62	C₂×C₈×C₆₇	—	2	2	no	—
63	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
64	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₆₇	#41	1	2	yes	—
68	C₄⋊C₄×C₆₇	—	4	2	no	—
69	C₂²⋊C₄×C₆₇	—	4	2	no	—
70	C₄²×C₆₇	—	4	2	no	—
71	C₂³×C₆₇	#107	1	2	yes	—
72	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
73	Q₈×C₆₇	—	2	2	no	—
74	D₄×C₆₇	—	2	4	no	—
75	D₄×C₆₇	—	2	2	no	—
76	C₂×C₄×C₆₇	—	2	2	no	—
77	C₂×C₄×C₆₇	—	2	2	no	—
78	C₂×C₄×C₆₇	—	8	2	no	—
79	C₂×C₄×C₆₇	#107	1	2	yes	—
80	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₆₇	#121	1	2	yes	—
82	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
83	C₂×C₄×C₆₇	#135	1	2	yes	—
84	C₂×Dic₆₇	—	8	2	no	—
85	C₈×C₆₇	—	2	2	no	—
86	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₈×C₆₇	#117	1	1	yes	characteristic
88	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
89	C₂²×C₆₇	—	2	2	no	—
90	C₂²×C₆₇	—	4	1	no	—
91	C₂²×C₆₇	—	4	2	no	—
92	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
93	C₂²×C₆₇	#466	1	2	yes	—
94	C₄×C₆₇	—	2	2	no	—
95	C₄×C₆₇	—	16	2	no	—
96	C₄×C₆₇	#380	1	1	yes	characteristic
97	C₄×C₆₇	#427	1	1	yes	characteristic
98	C₄×C₆₇	#467	1	2	yes	—
99	Dic₆₇	—	16	2	no	—
100	#12839	—	67	1	no	maximal, Sylow
101	C₂×C₆₇	—	2	2	no	—
102	C₂×C₆₇	—	2	1	no	—
103	C₂×C₆₇	—	4	2	no	—
104	C₂×C₆₇	#1808	1	1	yes	characteristic
105	C₂×#504	—	67	1	no	—
106	#2158	—	67	2	no	—
107	#1812	—	67	1	no	—
108	#1812	D₆₇	1	1	yes	characteristic
109	#1384	—	67	2	no	—
110	C₆₇	#12839	1	1	yes	characteristic, complemented, Sylow
111	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
112	#504	—	134	2	no	—
113	#504	—	134	2	no	—
114	C₂×M₃₂	—	67	2	no	—
115	C₂×M₃₂	—	67	1	no	—
116	C₂²×C₁₆	—	67	1	no	—
117	C₂²×C₁₆	—	67	2	no	—
118	#427	—	67	2	no	—
119	#427	C₂×D₆₇	1	2	yes	—
120	#410	—	2	1	no	—
121	#410	—	134	1	no	—
122	#367	—	134	2	no	—
123	#366	—	134	2	no	—
124	H(3,ℤ/4)	—	2	1	no	—
125	H(3,ℤ/4)	—	134	1	no	—
126	C₂×#41	C₂²⋊D₆₇	1	1	yes	characteristic
127	#131	—	2	2	no	—
128	#131	—	2	2	no	—
129	C₂×M₁₆	Dic₁₃₄	1	2	yes	—
130	C₂×M₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
131	C₂²×C₈	C₄×D₆₇	1	2	yes	—
132	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
133	#117	—	2	2	no	—
134	#117	—	134	2	no	—
135	M₃₂	—	134	4	no	—
136	M₃₂	—	134	2	no	—
137	C₂×C₁₆	—	67	1	no	—
138	C₂×C₁₆	—	67	1	no	—
139	C₂×C₁₆	—	67	2	no	—
140	C₂×C₁₆	—	67	2	no	—
141	C₂×C₁₆	—	67	2	no	—
142	C₂×C₁₆	—	134	2	no	—
143	C₂×C₁₆	—	134	2	no	—
144	#107	—	2	2	no	—
145	#107	—	134	2	no	—
146	#106	—	2	2	no	—
147	#106	—	134	2	no	—
148	#99	—	4	2	no	—
149	#99	—	268	2	no	—
150	#41	—	2	2	no	—
151	#41	—	2	2	no	—
152	C₂×Q₈	#11761618	1	1	yes	characteristic
153	C₂×D₄	#11761613	1	2	yes	—
154	C₂×D₄	#11761618	1	1	yes	characteristic
155	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
156	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
157	M₁₆	—	2	2	no	—
158	M₁₆	—	2	4	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
162	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
163	C₂×C₈	D₄×D₆₇	1	2	yes	—
164	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
165	C₂×C₈	#11761627	1	2	yes	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	268	2	no	—
168	C₂²⋊C₄	—	4	2	no	—
169	C₂²⋊C₄	—	268	2	no	—
170	C₄²	—	4	2	no	—
171	C₄²	—	268	2	no	—
172	C₁₆	—	67	1	no	—
173	C₁₆	—	67	1	no	—
174	C₁₆	—	67	2	no	—
175	C₁₆	—	134	2	no	—
176	C₂³	#14523976	1	2	yes	—
177	C₂³	C₂×#11761613	1	1	yes	characteristic
178	Q₈	—	2	2	no	—
179	D₄	—	2	4	no	—
180	D₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	8	2	no	—
184	C₂×C₄	—	536	2	no	—
185	C₂×C₄	#14523976	1	2	yes	—
186	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
187	C₂×C₄	#14524015	1	2	yes	—
188	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
189	C₂×C₄	#14524063	1	2	yes	—
190	C₈	—	2	2	no	—
191	C₈	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
192	C₈	#14523999	1	1	yes	characteristic
193	C₈	#14524020	1	2	yes	—
194	C₂²	—	2	2	no	—
195	C₂²	—	4	1	no	—
196	C₂²	—	4	2	no	—
197	C₂²	C₂×#14523976	1	1	yes	characteristic
198	C₂²	#16863630	1	2	yes	—
199	C₄	—	2	2	no	—
200	C₄	—	16	2	no	—
201	C₄	—	1072	2	no	—
202	C₄	#16863254	1	1	yes	characteristic
203	C₄	#16863432	1	1	yes	characteristic
204	C₄	#16863635	1	2	yes	—
205	C₂	—	2	2	no	—
206	C₂	—	2	1	no	—
207	C₂	—	4	2	no	—
208	C₂	#22560236	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863813

C₂

yes

maximal, characteristic

#22561991

C₂

yes

maximal, complemented

#22560255

C₂

yes

maximal, characteristic

#22558310

C₂

yes

maximal, complemented

C₆₇×#1812

C₂

yes

maximal, characteristic

#16863813

—

#16863813

—

C₂×#14523983

C₄

yes

complemented

C₂×#14523983

C₂²

yes

characteristic

C₂²×C₆₇⋊C₁₆

C₄

yes

complemented

C₂²×C₆₇⋊C₁₆

C₂²

yes

characteristic

#16863434

C₂²

yes

—

—

—

—

—

C₂×#131×C₆₇

C₂²

yes

characteristic

#427×C₆₇

C₂²

yes

—

#410×C₆₇

—

H(3,ℤ/4)×C₆₇

—

#14524002

—

#14523983

—

#14523983

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

—

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₁₆

D₄

yes

—

C₂×C₆₇⋊C₁₆

D₄

yes

—

#14523977

—

#14523975

—

#14523958

—

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

#117×C₆₇

—

#107×C₆₇

—

#106×C₆₇

—

#99×C₆₇

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

—

C₆₇⋊C₁₆

C₂²⋊C₄

yes

characteristic

C₆₇⋊C₁₆

C₂²⋊C₄

yes

characteristic

C₆₇⋊C₁₆

C₂²⋊C₄

yes

—

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

Q₁₆

yes

characteristic

C₂×D₄×C₆₇

D₈

yes

—

C₂×D₄×C₆₇

Q₁₆

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

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

#107

yes

—

C₂³×C₆₇

C₂×D₈

yes

characteristic

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×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×Q₁₆

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

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×#107

yes

characteristic

C₂²×C₆₇

#466

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

100

#12839

—

maximal, Sylow

101

C₂×C₆₇

—

102

C₂×C₆₇

—

103

C₂×C₆₇

—

104

C₂×C₆₇

#1808

yes

characteristic

105

C₂×#504

—

106

#2158

—

107

#1812

—

108

#1812

D₆₇

yes

characteristic

109

#1384

—

110

C₆₇

#12839

yes

characteristic, complemented, Sylow

111

C₂×#131

C₂×D₆₇

yes

characteristic

112

#504

—

134

—

113

#504

—

134

—

114

C₂×M₃₂

—

115

C₂×M₃₂

—

116

C₂²×C₁₆

—

117

C₂²×C₁₆

—

118

#427

—

119

#427

C₂×D₆₇

yes

—

120

#410

—

121

#410

—

134

—

122

#367

—

134

—

123

#366

—

134

—

124

H(3,ℤ/4)

—

125

H(3,ℤ/4)

—

134

—

126

C₂×#41

C₂²⋊D₆₇

yes

characteristic

127

#131

—

128

#131

—

129

C₂×M₁₆

Dic₁₃₄

yes

—

130

C₂×M₁₆

C₂²⋊D₆₇

yes

characteristic

131

C₂²×C₈

C₄×D₆₇

yes

—

132

C₂²×C₈

C₂²×D₆₇

yes

characteristic

133

#117

—

134

#117

—

134

—

135

M₃₂

—

134

—

136

M₃₂

—

134

—

137

C₂×C₁₆

—

138

C₂×C₁₆

—

139

C₂×C₁₆

—

140

C₂×C₁₆

—

141

C₂×C₁₆

—

142

C₂×C₁₆

—

134

—

143

C₂×C₁₆

—

134

—

144

#107

—

145

#107

—

134

—

146

#106

—

147

#106

—

134

—

148

#99

—

149

#99

—

268

—

150

#41

—

151

#41

—

152

C₂×Q₈

#11761618

yes

characteristic

153

C₂×D₄

#11761613

yes

—

154

C₂×D₄

#11761618

yes

characteristic

155

C₂²×C₄

Dic₆₇⋊C₄

yes

—

156

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

157

M₁₆

—

158

M₁₆

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

Dic₆₇⋊C₄

yes

—

162

C₂×C₈

C₂×C₄×D₆₇

yes

characteristic

163

C₂×C₈

D₄×D₆₇

yes

—

164

C₂×C₈

C₂×Dic₁₃₄

yes

characteristic

165

C₂×C₈

#11761627

yes

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

268

—

168

C₂²⋊C₄

—

169

C₂²⋊C₄

—

268

—

170

C₄²

—

171

C₄²

—

268

—

172

C₁₆

—

173

C₁₆

—

174

C₁₆

—

175

C₁₆

—

134

—

176

C₂³

#14523976

yes

—

177

C₂³

C₂×#11761613

yes

characteristic

178

Q₈

—

179

D₄

—

180

D₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

536

—

185

C₂×C₄

#14523976

yes

—

186

C₂×C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

187

C₂×C₄

#14524015

yes

—

188

C₂×C₄

C₂×#11761618

yes

characteristic

189

C₂×C₄

#14524063

yes

—

190

C₈

—

191

C₈

C₂²⋊C₄×D₆₇

yes

characteristic

192

C₈

#14523999

yes

characteristic

193

C₈

#14524020

yes

—

194

C₂²

—

195

C₂²

—

196

C₂²

—

197

C₂²

C₂×#14523976

yes

characteristic

198

C₂²

#16863630

yes

—

199

C₄

—

200

C₄

—

201

C₄

—

1072

—

202

C₄

#16863254

yes

characteristic

203

C₄

#16863432

yes

characteristic

204

C₄

#16863635

yes

—

205

C₂

—

206

C₂

—

207

C₂

—

208

C₂

#22560236

yes

characteristic

	subgroup	G/H
center	C₄	#16863432
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1812	C₂
socle	C₂×C₆₇	#1808
prime radical	C₂×#41×C₆₇	D₄
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1812	D₆₇
hypocenter	C₆₇	#12839
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ #1812

subgroup

G/H

center

C₄

#16863432

commutator

C₂×C₈×C₆₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#1812

C₂

socle

C₂×C₆₇

#1808

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 ◁ #1812

N	H ≅ G/N	type
2. #22561991	C₂	semidirect
4. #22558310	C₂	semidirect
8. C₂×#14523983	C₄	semidirect
10. C₂²×C₆₇⋊C₁₆	C₄	semidirect
110. C₆₇	#12839	semidirect

H ≅ G/N

type

2. #22561991

C₂

semidirect

4. #22558310

C₂

semidirect

8. C₂×#14523983

C₄

semidirect

10. C₂²×C₆₇⋊C₁₆

C₄

semidirect

110. C₆₇

#12839

semidirect

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

subgroup	index
C₂×#16863813	2
#22561991	2
#22560255	2
#22558310	2
C₆₇×#1812	2
#12839	67

subgroup

index

C₆₇×#1812

Subgroups — #30772018

Conjugacy classes of subgroups

Complements

Maximal subgroups