Group #30772019 — subgroups

Subgroups — #30772019

Order 17152.

362 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 111 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×#16863814	C₂	1	1	yes	maximal, characteristic
2	#22561991	C₂	1	2	yes	maximal, complemented
3	#22560252	C₂	1	2	yes	maximal
4	#22558311	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1384	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14524049	C₂²	1	1	yes	characteristic
7	#16863814	—	2	2	no	—
8	#16863814	—	2	2	no	—
9	C₂×#14523982	C₄	1	2	yes	complemented
10	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×D₆₇	C₄	1	2	yes	complemented
12	#16863434	C₂²	1	2	yes	—
13	#16863369	—	2	2	no	—
14	#16863223	—	2	2	no	—
15	#16863221	—	2	2	no	—
16	#16863152	—	2	2	no	—
17	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
18	#427×C₆₇	C₂²	1	2	yes	—
19	C₂×#11761628	D₄	1	1	yes	characteristic
20	#14524049	—	2	2	no	—
21	#14524049	—	2	2	no	—
22	C₂×#11761612	D₄	1	1	yes	characteristic
23	C₂×#11761611	Q₈	1	2	yes	—
24	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
25	#14524002	—	2	2	no	—
26	#14523982	—	2	4	no	—
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	C₁₆×D₆₇	—	2	2	no	—
32	C₁₆×D₆₇	—	2	2	no	—
33	#14523977	—	2	2	no	—
34	#14523975	—	2	2	no	—
35	#14523957	—	4	4	no	—
36	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
37	#117×C₆₇	—	2	2	no	—
38	M₃₂×C₆₇	—	2	2	no	—
39	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
40	#107×C₆₇	—	2	2	no	—
41	#106×C₆₇	—	2	2	no	—
42	#11761628	—	2	2	no	—
43	#11761628	—	2	2	no	—
44	C₂×Dic₁₃₄	Q₁₆	1	1	yes	characteristic
45	C₂×D₂₆₈	Q₁₆	1	1	yes	characteristic
46	C₂×C₂²⋊D₆₇	D₈	1	2	yes	—
47	C₂×C₄×D₆₇	C₄⋊C₄	1	2	yes	—
48	#11761612	—	2	2	no	—
49	#11761611	—	2	4	no	—
50	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
51	C₈×D₆₇	—	2	2	no	—
52	C₈×D₆₇	—	2	2	no	—
53	C₄⋊Dic₆₇	—	4	2	no	—
54	C₂²⋊Dic₆₇	—	4	2	no	—
55	C₄×Dic₆₇	—	4	2	no	—
56	C₆₇⋊C₁₆	C₂²⋊C₄	1	1	yes	characteristic
57	C₆₇⋊C₁₆	C₂²⋊C₄	1	1	yes	characteristic
58	C₆₇⋊C₁₆	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
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	C₄⋊C₄×C₆₇	—	4	2	no	—
65	C₂²⋊C₄×C₆₇	—	4	2	no	—
66	C₄²×C₆₇	—	4	2	no	—
67	C₁₆×C₆₇	—	2	2	no	—
68	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
69	C₂²×D₆₇	#107	1	2	yes	—
70	C₂×C₄×C₆₇	—	8	2	no	—
71	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₇	#121	1	2	yes	—
73	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
74	C₂×C₄×C₆₇	#135	1	2	yes	—
75	C₂²⋊D₆₇	—	2	4	no	—
76	C₂×Dic₆₇	—	8	2	no	—
77	C₂×Dic₆₇	#107	1	2	yes	—
78	D₂₆₈	—	2	2	no	—
79	C₄×D₆₇	—	2	2	no	—
80	C₄×D₆₇	—	2	2	no	—
81	Dic₁₃₄	—	2	2	no	—
82	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₈×C₆₇	#117	1	1	yes	characteristic
84	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
85	C₆₇⋊C₈	—	2	2	no	—
86	C₂²×C₆₇	—	4	1	no	—
87	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
88	C₂²×C₆₇	#466	1	2	yes	—
89	C₂×D₆₇	—	2	2	no	—
90	C₂×D₆₇	—	4	2	no	—
91	C₄×C₆₇	—	16	2	no	—
92	C₄×C₆₇	#380	1	1	yes	characteristic
93	C₄×C₆₇	#427	1	1	yes	characteristic
94	C₄×C₆₇	#467	1	2	yes	—
95	Dic₆₇	—	2	2	no	—
96	Dic₆₇	—	16	2	no	—
97	#12839	—	67	1	no	maximal, Sylow
98	C₂×C₆₇	—	2	2	no	—
99	C₂×C₆₇	—	2	1	no	—
100	C₂×C₆₇	#1808	1	1	yes	characteristic
101	D₆₇	—	4	2	no	—
102	C₂×#504	—	67	1	no	—
103	#2158	—	67	2	no	—
104	#1812	—	67	2	no	—
105	#1384	—	67	1	no	—
106	#1384	D₆₇	1	1	yes	characteristic, complemented
107	C₆₇	#12839	1	1	yes	characteristic, complemented, Sylow
108	C₂×#131	—	67	1	no	—
109	#504	—	134	2	no	—
110	#504	—	134	2	no	—
111	C₂×M₃₂	—	67	2	no	—
112	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
113	C₂²×C₁₆	—	67	1	no	—
114	C₂²×C₁₆	—	67	2	no	—
115	#427	—	67	2	no	—
116	#427	C₂×D₆₇	1	2	yes	—
117	#410	—	134	2	no	—
118	#367	—	134	2	no	—
119	#366	—	134	2	no	—
120	H(3,ℤ/4)	—	134	2	no	—
121	C₂×#41	—	67	1	no	—
122	#131	—	134	2	no	—
123	#131	—	134	2	no	—
124	C₂×M₁₆	—	67	1	no	—
125	C₂×M₁₆	—	67	2	no	—
126	C₂²×C₈	—	67	2	no	—
127	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
128	#117	—	2	2	no	—
129	#117	—	134	2	no	—
130	M₃₂	—	2	2	no	—
131	M₃₂	—	134	4	no	—
132	C₂×C₁₆	—	67	1	no	—
133	C₂×C₁₆	—	67	1	no	—
134	C₂×C₁₆	—	67	2	no	—
135	C₂×C₁₆	—	67	2	no	—
136	C₂×C₁₆	—	134	2	no	—
137	C₂×C₁₆	—	134	2	no	—
138	C₂×C₁₆	C₄×D₆₇	1	2	yes	—
139	#107	—	2	2	no	—
140	#107	—	134	2	no	—
141	#106	—	2	2	no	—
142	#106	—	134	2	no	—
143	#99	—	268	4	no	—
144	#41	—	134	2	no	—
145	#41	—	134	2	no	—
146	C₂×Q₈	—	67	1	no	—
147	C₂×D₄	—	67	2	no	—
148	C₂×D₄	—	67	1	no	—
149	C₂²×C₄	—	67	2	no	—
150	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
151	M₁₆	—	134	2	no	—
152	M₁₆	—	134	4	no	—
153	C₂×C₈	—	67	2	no	—
154	C₂×C₈	—	134	2	no	—
155	C₂×C₈	—	134	2	no	—
156	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
157	C₂×C₈	D₄×D₆₇	1	2	yes	—
158	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
159	C₂×C₈	#11761628	1	2	yes	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	268	2	no	—
162	C₂²⋊C₄	—	4	2	no	—
163	C₂²⋊C₄	—	268	2	no	—
164	C₄²	—	4	2	no	—
165	C₄²	—	268	2	no	—
166	C₁₆	—	2	2	no	—
167	C₁₆	—	67	1	no	—
168	C₁₆	—	67	1	no	—
169	C₁₆	—	67	2	no	—
170	C₂³	—	67	2	no	—
171	C₂³	D₈×D₆₇	1	1	yes	characteristic
172	Q₈	—	134	2	no	—
173	D₄	—	134	4	no	—
174	D₄	—	134	2	no	—
175	C₂×C₄	—	8	2	no	—
176	C₂×C₄	—	536	2	no	—
177	C₂×C₄	—	67	2	no	—
178	C₂×C₄	—	134	2	no	—
179	C₂×C₄	—	134	2	no	—
180	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
181	C₂×C₄	#14524016	1	2	yes	—
182	C₂×C₄	Q₁₆×D₆₇	1	1	yes	characteristic
183	C₂×C₄	#14524064	1	2	yes	—
184	C₈	—	134	2	no	—
185	C₈	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
186	C₈	#14524000	1	1	yes	characteristic
187	C₈	#14524021	1	2	yes	—
188	C₂²	—	4	1	no	—
189	C₂²	—	268	2	no	—
190	C₂²	—	134	2	no	—
191	C₂²	#107×D₆₇	1	1	yes	characteristic
192	C₂²	#16863631	1	2	yes	—
193	C₄	—	16	2	no	—
194	C₄	—	1072	2	no	—
195	C₄	—	134	2	no	—
196	C₄	#16863255	1	1	yes	characteristic
197	C₄	#16863431	1	1	yes	characteristic
198	C₄	#16863636	1	2	yes	—
199	C₂	—	2	2	no	—
200	C₂	—	2	1	no	—
201	C₂	—	268	2	no	—
202	C₂	#22560237	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863814

C₂

yes

maximal, characteristic

#22561991

C₂

yes

maximal, complemented

#22560252

C₂

yes

maximal

#22558311

C₂

yes

maximal, characteristic, complemented

C₆₇×#1384

C₂

yes

maximal, characteristic, complemented

C₂×#14524049

C₂²

yes

—

—

C₄

yes

complemented

C₂²×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₂×C₁₆×D₆₇

C₄

yes

complemented

#16863434

C₂²

yes

—

—

—

—

—

C₂×M₃₂×C₆₇

C₂²

yes

characteristic

#427×C₆₇

C₂²

yes

—

C₂×#11761628

D₄

yes

—

—

D₄

yes

characteristic

C₂×#11761611

Q₈

yes

—

C₂×C₈×D₆₇

C₂×C₄

yes

—

#14524002

—

#14523982

—

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₁₆

D₄

yes

—

C₂×C₆₇⋊C₁₆

D₄

yes

—

C₁₆×D₆₇

—

C₁₆×D₆₇

—

#14523977

—

#14523975

—

#14523957

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

#117×C₆₇

—

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

C₂×C₄

yes

—

#107×C₆₇

—

#106×C₆₇

—

#11761628

—

#11761628

—

C₂×Dic₁₃₄

Q₁₆

yes

characteristic

C₂×D₂₆₈

Q₁₆

yes

characteristic

C₂×C₂²⋊D₆₇

D₈

yes

—

C₂×C₄×D₆₇

C₄⋊C₄

yes

—

#11761612

—

#11761611

—

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

—

C₈×D₆₇

—

C₈×D₆₇

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

C₂²⋊C₄

yes

characteristic

C₆₇⋊C₁₆

C₂²⋊C₄

yes

characteristic

C₆₇⋊C₁₆

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

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

—

C₁₆×C₆₇

—

C₂³×C₆₇

C₂×D₈

yes

characteristic

C₂²×D₆₇

#107

yes

—

C₂×C₄×C₆₇

—

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

—

C₂²⋊D₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

#107

yes

—

D₂₆₈

—

C₄×D₆₇

—

C₄×D₆₇

—

Dic₁₃₄

—

C₈×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₆₇

#117

yes

characteristic

C₈×C₆₇

C₂²⋊Q₈

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×#107

yes

characteristic

C₂²×C₆₇

#466

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

Dic₆₇

—

#12839

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

100

C₂×C₆₇

#1808

yes

characteristic

101

D₆₇

—

102

C₂×#504

—

103

#2158

—

104

#1812

—

105

#1384

—

106

#1384

D₆₇

yes

characteristic, complemented

107

C₆₇

#12839

yes

characteristic, complemented, Sylow

108

C₂×#131

—

109

#504

—

134

—

110

#504

—

134

—

111

C₂×M₃₂

—

112

C₂×M₃₂

C₂×D₆₇

yes

characteristic

113

C₂²×C₁₆

—

114

C₂²×C₁₆

—

115

#427

—

116

#427

C₂×D₆₇

yes

—

117

#410

—

134

—

118

#367

—

134

—

119

#366

—

134

—

120

H(3,ℤ/4)

—

134

—

121

C₂×#41

—

122

#131

—

134

—

123

#131

—

134

—

124

C₂×M₁₆

—

125

C₂×M₁₆

—

126

C₂²×C₈

—

127

C₂²×C₈

C₂²×D₆₇

yes

characteristic

128

#117

—

129

#117

—

134

—

130

M₃₂

—

131

M₃₂

—

134

—

132

C₂×C₁₆

—

133

C₂×C₁₆

—

134

C₂×C₁₆

—

135

C₂×C₁₆

—

136

C₂×C₁₆

—

134

—

137

C₂×C₁₆

—

134

—

138

C₂×C₁₆

C₄×D₆₇

yes

—

139

#107

—

140

#107

—

134

—

141

#106

—

142

#106

—

134

—

143

#99

—

268

—

144

#41

—

134

—

145

#41

—

134

—

146

C₂×Q₈

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂²×C₄

—

150

C₂²×C₄

D₄×D₆₇

yes

characteristic

151

M₁₆

—

134

—

152

M₁₆

—

134

—

153

C₂×C₈

—

154

C₂×C₈

—

134

—

155

C₂×C₈

—

134

—

156

C₂×C₈

C₂×C₄×D₆₇

yes

characteristic

157

C₂×C₈

D₄×D₆₇

yes

—

158

C₂×C₈

Q₈×D₆₇

yes

characteristic

159

C₂×C₈

#11761628

yes

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

268

—

162

C₂²⋊C₄

—

163

C₂²⋊C₄

—

268

—

164

C₄²

—

165

C₄²

—

268

—

166

C₁₆

—

167

C₁₆

—

168

C₁₆

—

169

C₁₆

—

170

C₂³

—

171

C₂³

D₈×D₆₇

yes

characteristic

172

Q₈

—

134

—

173

D₄

—

134

—

174

D₄

—

134

—

175

C₂×C₄

—

176

C₂×C₄

—

536

—

177

C₂×C₄

—

178

C₂×C₄

—

134

—

179

C₂×C₄

—

134

—

180

C₂×C₄

C₄⋊C₄×D₆₇

yes

characteristic

181

C₂×C₄

#14524016

yes

—

182

C₂×C₄

Q₁₆×D₆₇

yes

characteristic

183

C₂×C₄

#14524064

yes

—

184

C₈

—

134

—

185

C₈

C₂²⋊C₄×D₆₇

yes

characteristic

186

C₈

#14524000

yes

characteristic

187

C₈

#14524021

yes

—

188

C₂²

—

189

C₂²

—

268

—

190

C₂²

—

134

—

191

C₂²

#107×D₆₇

yes

characteristic

192

C₂²

#16863631

yes

—

193

C₄

—

194

C₄

—

1072

—

195

C₄

—

134

—

196

C₄

#16863255

yes

characteristic

197

C₄

#16863431

yes

characteristic

198

C₄

#16863636

yes

—

199

C₂

—

200

C₂

—

201

C₂

—

268

—

202

C₂

#22560237

yes

characteristic

	subgroup	G/H
center	C₄	#16863431
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1384	C₂
socle	C₂×C₆₇	#1808
prime radical	C₂×#11761628	D₄
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1384	D₆₇
hypocenter	C₆₇	#12839
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1384

subgroup

G/H

center

C₄

#16863431

commutator

C₂×C₈×C₆₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#1384

C₂

C₂×C₆₇

D₄

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1384

N	H ≅ G/N	type
2. #22561991	C₂	semidirect
4. #22558311	C₂	semidirect
5. C₆₇×#1384	C₂	semidirect
9. C₂×#14523982	C₄	semidirect
11. C₂×C₁₆×D₆₇	C₄	semidirect
106. #1384	D₆₇	semidirect
107. C₆₇	#12839	semidirect

H ≅ G/N

type

2. #22561991

C₂

semidirect

4. #22558311

C₂

semidirect

5. C₆₇×#1384

C₂

semidirect

9. C₂×#14523982

C₄

semidirect

11. C₂×C₁₆×D₆₇

C₄

semidirect

106. #1384

D₆₇

semidirect

107. C₆₇

#12839

semidirect

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

subgroup	index
C₂×#16863814	2
#22561991	2
#22560252	2
#22558311	2
C₆₇×#1384	2
#12839	67

subgroup

index

C₆₇×#1384

Subgroups — #30772019

Conjugacy classes of subgroups

Complements

Maximal subgroups