Group #30754457 — subgroups

Subgroups — #30754457

Order 17152.

294 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 123 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	C₄×#14523979	C₂	1	1	yes	maximal, characteristic
2	#22558962	C₂	1	2	yes	maximal
3	C₂×#16863149	C₂	1	1	yes	maximal, characteristic
4	#22558058	C₂	1	2	yes	maximal
5	C₆₇×#1452	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523979	C₄	1	2	yes	—
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
8	C₄×#11761612	C₂²	1	2	yes	—
9	#16863161	C₄	1	1	yes	characteristic
10	#16863161	C₄	1	1	yes	characteristic
11	#16863149	—	2	2	no	—
12	#16863148	—	2	2	no	—
13	#16863146	—	2	2	no	—
14	#431×C₆₇	C₂²	1	1	yes	characteristic
15	#431×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	#322×C₆₇	—	2	2	no	—
18	C₂×#11761612	C₂×C₄	1	2	yes	—
19	C₂×#11761612	C₂×C₄	1	2	yes	—
20	C₂²×C₆₇⋊C₈	—	2	2	no	—
21	#14523979	—	2	4	no	—
22	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	#14523954	—	2	2	no	—
25	C₄×C₆₇⋊C₈	—	2	2	no	—
26	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₆₇	—	2	1	no	—
30	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
31	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
32	C₂²⋊C₈×C₆₇	—	2	1	no	—
33	C₂²⋊C₈×C₆₇	Q₈	1	2	yes	—
34	C₄×C₈×C₆₇	—	2	1	no	—
35	C₄×C₈×C₆₇	—	2	1	no	—
36	C₄×C₈×C₆₇	D₄	1	2	yes	—
37	C₄×C₈×C₆₇	Q₈	1	2	yes	—
38	#11761612	—	2	2	no	—
39	#11761612	—	2	2	no	—
40	#11761612	—	2	2	no	—
41	#11761612	M₁₆	1	4	yes	—
42	C₂×C₆₇⋊C₈	—	2	2	no	—
43	C₂×C₆₇⋊C₈	—	2	2	no	—
44	C₂×C₆₇⋊C₈	—	4	2	no	—
45	C₂×C₆₇⋊C₈	—	8	2	no	—
46	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
49	C₂×C₈×C₆₇	—	2	2	no	—
50	C₂×C₈×C₆₇	—	2	2	no	—
51	C₂×C₈×C₆₇	—	2	1	no	—
52	C₂×C₈×C₆₇	—	2	1	no	—
53	C₂×C₈×C₆₇	—	2	2	no	—
54	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
55	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
56	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₆₇	#41	1	1	yes	characteristic
60	C₂³×C₆₇	C₄×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₆₇	—	2	1	no	—
62	C₂×C₄×C₆₇	—	2	1	no	—
63	C₂×C₄×C₆₇	—	4	1	no	—
64	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
67	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
69	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
71	C₂×C₄×C₆₇	#131	1	2	yes	—
72	C₈×C₆₇	—	2	2	no	—
73	C₈×C₆₇	—	2	2	no	—
74	C₈×C₆₇	—	4	2	no	—
75	C₈×C₆₇	—	4	1	no	—
76	C₈×C₆₇	—	4	1	no	—
77	C₆₇⋊C₈	—	2	4	no	—
78	C₆₇⋊C₈	—	4	2	no	—
79	C₆₇⋊C₈	—	8	2	no	—
80	C₂²×C₆₇	—	2	1	no	—
81	C₂²×C₆₇	—	2	1	no	—
82	C₂²×C₆₇	#389	1	1	yes	characteristic
83	C₂²×C₆₇	#446	1	2	yes	—
84	C₄×C₆₇	—	2	1	no	—
85	C₄×C₆₇	—	2	1	no	—
86	C₄×C₆₇	—	2	1	no	—
87	C₄×C₆₇	#423	1	1	yes	characteristic
88	C₄×C₆₇	#424	1	1	yes	characteristic
89	C₄×C₆₇	#435	1	2	yes	—
90	C₄×C₆₇	#472	1	1	yes	characteristic
91	C₄×C₆₇	#473	1	1	yes	characteristic
92	#9610	—	67	1	no	maximal, Sylow
93	C₂×C₆₇	—	2	2	no	—
94	C₂×C₆₇	#1940	1	1	yes	characteristic
95	C₂×C₆₇	#1952	1	1	yes	characteristic
96	C₂×C₆₇	#1953	1	1	yes	characteristic
97	C₄×#108	—	67	1	no	—
98	#1565	—	67	2	no	—
99	C₂×#335	—	67	1	no	—
100	#1452	D₆₇	1	1	yes	characteristic
101	#1283	—	67	2	no	—
102	C₆₇	#9610	1	1	yes	characteristic, complemented, Sylow
103	#431	C₂×D₆₇	1	1	yes	characteristic
104	#431	C₂×D₆₇	1	1	yes	characteristic
105	C₂×#108	—	67	2	no	—
106	C₂×C₄⋊C₈	—	67	2	no	—
107	C₄×M₁₆	—	67	2	no	—
108	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
109	#341	—	67	1	no	—
110	#341	—	67	1	no	—
111	#335	—	134	2	no	—
112	#335	—	134	2	no	—
113	#334	—	134	2	no	—
114	#322	—	2	2	no	—
115	C₂×M₁₆	—	67	2	no	—
116	C₂×M₁₆	—	67	2	no	—
117	C₂²×C₈	—	134	2	no	—
118	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
119	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
120	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
121	#108	—	134	4	no	—
122	C₄⋊C₈	—	2	1	no	—
123	C₄⋊C₈	—	134	2	no	—
124	C₄⋊C₈	—	134	2	no	—
125	C₄⋊C₈	D₂₆₈	1	1	yes	characteristic
126	C₄⋊C₈	D₂₆₈	1	1	yes	characteristic
127	C₂²⋊C₈	—	2	1	no	—
128	C₂²⋊C₈	Dic₁₃₄	1	2	yes	—
129	#97	—	134	2	no	—
130	C₄×C₈	—	2	1	no	—
131	C₄×C₈	—	2	1	no	—
132	C₄×C₈	—	134	2	no	—
133	C₄×C₈	Dic₁₃₄	1	2	yes	—
134	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
135	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
136	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
137	C₂²×C₄	#11761628	1	1	yes	characteristic
138	M₁₆	—	67	4	no	—
139	M₁₆	—	134	2	no	—
140	M₁₆	—	134	2	no	—
141	M₁₆	—	134	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	268	2	no	—
148	C₂×C₈	—	536	2	no	—
149	C₂×C₈	—	134	2	no	—
150	C₂×C₈	—	134	2	no	—
151	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
152	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
153	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
154	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
155	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
156	C₄²	#11761628	1	1	yes	characteristic
157	C₂³	C₄×Dic₁₃₄	1	1	yes	characteristic
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
162	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
163	C₂×C₄	#14524017	1	1	yes	characteristic
164	C₂×C₄	#14524022	1	1	yes	characteristic
165	C₂×C₄	#14524035	1	1	yes	characteristic
166	C₂×C₄	#14524043	1	1	yes	characteristic
167	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
168	C₂×C₄	#14524051	1	2	yes	—
169	C₈	—	2	2	no	—
170	C₈	—	2	2	no	—
171	C₈	—	4	2	no	—
172	C₈	—	4	1	no	—
173	C₈	—	4	1	no	—
174	C₈	—	268	2	no	—
175	C₈	—	536	2	no	—
176	C₈	—	134	4	no	—
177	C₂²	—	2	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	#16863294	1	1	yes	characteristic
180	C₂²	#16863524	1	2	yes	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	#16863419	1	1	yes	characteristic
185	C₄	#16863424	1	1	yes	characteristic
186	C₄	#16863469	1	2	yes	—
187	C₄	#16863666	1	1	yes	characteristic
188	C₄	#16863671	1	1	yes	characteristic
189	C₂	—	2	2	no	—
190	C₂	#22560932	1	1	yes	characteristic
191	C₂	#22560994	1	1	yes	characteristic
192	C₂	#22560999	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14523979

C₂

yes

maximal, characteristic

#22558962

C₂

yes

maximal

C₂×#16863149

C₂

yes

maximal, characteristic

#22558058

C₂

yes

maximal

C₆₇×#1452

C₂

yes

maximal, characteristic

C₂×#14523979

C₄

yes

—

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₄×#11761612

C₂²

yes

—

#16863161

C₄

yes

characteristic

#16863161

C₄

yes

—

—

—

#431×C₆₇

C₂²

yes

characteristic

#431×C₆₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#322×C₆₇

—

C₂×#11761612

C₂×C₄

yes

—

C₂×#11761612

C₂×C₄

yes

—

C₂²×C₆₇⋊C₈

—

#14523979

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

#14523954

—

C₄×C₆₇⋊C₈

—

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

D₄

yes

characteristic

C₄⋊C₈×C₆₇

D₄

yes

characteristic

C₂²⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

Q₈

yes

—

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

Q₈

yes

—

—

—

—

M₁₆

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₄²×C₆₇

#41

yes

characteristic

C₂³×C₆₇

C₄×Q₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄×D₄

yes

characteristic

C₂×C₄×C₆₇

#121

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

#125

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₂×M₁₆

yes

—

C₂×C₄×C₆₇

#131

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#389

yes

characteristic

C₂²×C₆₇

#446

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

—

—

100

#1452

D₆₇

yes

characteristic

101

#1283

—

102

C₆₇

#9610

yes

characteristic, complemented, Sylow

103

#431

C₂×D₆₇

yes

characteristic

104

#431

C₂×D₆₇

yes

characteristic

105

C₂×#108

—

106

C₂×C₄⋊C₈

—

107

C₄×M₁₆

—

108

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

109

#341

—

110

#341

—

111

#335

—

134

—

112

#335

—

134

—

113

#334

—

134

—

114

#322

—

115

C₂×M₁₆

—

116

C₂×M₁₆

—

117

C₂²×C₈

—

134

—

118

C₂²×C₈

C₄×D₆₇

yes

characteristic

119

C₂²×C₈

C₄×D₆₇

yes

characteristic

120

C₂×C₄²

C₂²×D₆₇

yes

characteristic

121

#108

—

134

—

122

C₄⋊C₈

—

123

C₄⋊C₈

—

134

—

124

C₄⋊C₈

—

134

—

125

C₄⋊C₈

D₂₆₈

yes

characteristic

126

C₄⋊C₈

D₂₆₈

yes

characteristic

127

C₂²⋊C₈

—

128

C₂²⋊C₈

Dic₁₃₄

yes

—

129

#97

—

134

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₄×C₈

—

134

—

133

C₄×C₈

Dic₁₃₄

yes

—

134

C₄×C₈

C₂²⋊D₆₇

yes

—

135

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

136

C₂²×C₄

C₂×Dic₁₃₄

yes

characteristic

137

C₂²×C₄

#11761628

yes

characteristic

138

M₁₆

—

139

M₁₆

—

134

—

140

M₁₆

—

134

—

141

M₁₆

—

134

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

268

—

148

C₂×C₈

—

536

—

149

C₂×C₈

—

134

—

150

C₂×C₈

—

134

—

151

C₂×C₈

Dic₆₇⋊C₄

yes

—

152

C₂×C₈

Dic₆₇⋊C₄

yes

—

153

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

154

C₄²

C₂×D₂₆₈

yes

characteristic

155

C₄²

C₂×Dic₁₃₄

yes

characteristic

156

C₄²

#11761628

yes

characteristic

157

C₂³

C₄×Dic₁₃₄

yes

characteristic

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

162

C₂×C₄

C₄×D₂₆₈

yes

characteristic

163

C₂×C₄

#14524017

yes

characteristic

164

C₂×C₄

#14524022

yes

characteristic

165

C₂×C₄

#14524035

yes

characteristic

166

C₂×C₄

#14524043

yes

characteristic

167

C₂×C₄

M₁₆×D₆₇

yes

—

168

C₂×C₄

#14524051

yes

—

169

C₈

—

170

C₈

—

171

C₈

—

172

C₈

—

173

C₈

—

174

C₈

—

268

—

175

C₈

—

536

—

176

C₈

—

134

—

177

C₂²

—

178

C₂²

—

179

C₂²

#16863294

yes

characteristic

180

C₂²

#16863524

yes

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

#16863419

yes

characteristic

185

C₄

#16863424

yes

characteristic

186

C₄

#16863469

yes

—

187

C₄

#16863666

yes

characteristic

188

C₄

#16863671

yes

characteristic

189

C₂

—

190

C₂

#22560932

yes

characteristic

191

C₂

#22560994

yes

characteristic

192

C₂

#22560999

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14524017
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1452	C₂
socle	C₂²×C₆₇	#389
prime radical	C₂³×C₆₇	C₄×Q₈
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1452	D₆₇
hypocenter	C₆₇	#9610
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1452

subgroup

G/H

center

C₂×C₄

#14524017

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1452

C₂

socle

C₂²×C₆₇

#389

prime radical

C₂³×C₆₇

C₄×Q₈

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1452

N	H ≅ G/N	type
102. C₆₇	#9610	semidirect

H ≅ G/N

type

102. C₆₇

#9610

semidirect

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

subgroup	index
C₄×#14523979	2
#22558962	2
C₂×#16863149	2
#22558058	2
C₆₇×#1452	2
#9610	67

subgroup

index

C₆₇×#1452

Subgroups — #30754457

Conjugacy classes of subgroups

Complements

Maximal subgroups