Group #30755668 — subgroups

Subgroups — #30755668

Order 17152.

710 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 191 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×#16863773	C₂	1	1	yes	maximal, characteristic
2	#22561014	C₂	1	4	yes	maximal, complemented
3	C₂×#16863117	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1844	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₅₃₆	—	2	2	no	—
6	C₂×#14524040	C₂²	1	2	yes	—
7	#16863773	—	2	2	no	—
8	#16863773	C₄	1	4	yes	complemented
9	C₂×#14523963	—	2	8	no	—
10	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
11	#16863117	—	2	2	no	—
12	C₂×#106×C₆₇	—	2	2	no	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	#384×C₆₇	C₂²	1	2	yes	—
15	C₂²×D₂₆₈	—	2	4	no	—
16	C₂²×D₂₆₈	—	2	2	no	—
17	C₂×D₅₃₆	—	2	8	no	—
18	C₂×D₅₃₆	—	4	4	no	—
19	C₂×D₅₃₆	—	4	4	no	—
20	C₂²×C₆₇⋊C₈	D₄	1	4	yes	—
21	#14524040	—	2	2	no	—
22	#14524040	—	2	4	no	—
23	#14524040	C₂×C₄	1	4	yes	—
24	#14523963	—	4	16	no	—
25	C₄×C₆₇⋊C₈	—	2	2	no	—
26	C₄×C₆₇⋊C₈	—	2	2	no	—
27	C₂²×C₈×C₆₇	D₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
29	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
30	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
31	#106×C₆₇	—	4	4	no	—
32	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₆₇	D₄	1	2	yes	—
34	C₂³×D₆₇	—	4	2	no	—
35	C₂×D₂₆₈	—	2	4	no	—
36	C₂×D₂₆₈	—	2	8	no	—
37	C₂×D₂₆₈	—	4	4	no	—
38	C₂×D₂₆₈	—	4	4	no	—
39	C₂×D₂₆₈	—	4	8	no	—
40	C₂×D₂₆₈	—	4	8	no	—
41	C₂×D₂₆₈	—	4	4	no	—
42	C₂×D₂₆₈	—	4	2	no	—
43	D₅₃₆	—	4	16	no	—
44	C₂×C₆₇⋊C₈	—	2	4	no	—
45	C₂×C₆₇⋊C₈	—	2	8	no	—
46	C₂²×C₄×C₆₇	—	4	2	no	—
47	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
50	C₂×C₈×C₆₇	D₈	1	4	yes	—
51	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
52	C₄⋊C₄×C₆₇	—	4	4	no	—
53	C₄⋊C₄×C₆₇	—	4	8	no	—
54	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₆₇	#41	1	2	yes	—
57	C₂³×C₆₇	#127	1	1	yes	characteristic
58	C₂²×D₆₇	—	4	4	no	—
59	C₂²×D₆₇	—	8	2	no	—
60	C₂²×D₆₇	—	8	2	no	—
61	C₂²×D₆₇	—	8	4	no	—
62	C₂²×D₆₇	—	8	4	no	—
63	C₂×C₄×C₆₇	—	8	2	no	—
64	C₂×C₄×C₆₇	—	8	4	no	—
65	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
67	C₂×C₄×C₆₇	#121	1	2	yes	—
68	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
69	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
70	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
71	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	8	yes	—
72	D₂₆₈	—	4	8	no	—
73	D₂₆₈	—	4	16	no	—
74	D₂₆₈	—	8	8	no	—
75	D₂₆₈	—	8	4	no	—
76	C₈×C₆₇	#102	1	8	yes	complemented
77	C₆₇⋊C₈	—	4	4	no	—
78	C₂²×C₆₇	#390	1	1	yes	characteristic
79	C₂²×C₆₇	#492	1	1	yes	characteristic
80	C₂²×C₆₇	#493	1	1	yes	characteristic
81	C₂²×C₆₇	#496	1	4	yes	—
82	C₂×D₆₇	—	8	4	no	—
83	C₂×D₆₇	—	8	8	no	—
84	C₂×D₆₇	—	16	2	no	—
85	C₂×D₆₇	—	16	2	no	—
86	C₂×D₆₇	—	16	4	no	—
87	C₄×C₆₇	—	16	2	no	—
88	C₄×C₆₇	C₂×#102	1	2	yes	—
89	C₄×C₆₇	#442	1	4	yes	—
90	C₄×C₆₇	#469	1	4	yes	—
91	C₄×C₆₇	#486	1	2	yes	—
92	#9837	—	67	1	no	maximal, Sylow
93	C₂×C₆₇	#1715	1	2	yes	—
94	C₂×C₆₇	#1721	1	2	yes	—
95	C₂×C₆₇	#1957	1	1	yes	characteristic
96	C₂×C₆₇	#1972	1	2	yes	—
97	D₆₇	—	16	4	no	—
98	C₂×#493	—	67	1	no	—
99	#1957	—	67	4	no	—
100	#1844	D₆₇	1	1	yes	characteristic, complemented
101	C₂×#322	—	67	1	no	—
102	C₆₇	#9837	1	1	yes	characteristic, complemented, Sylow
103	C₂²×D₈	—	134	2	no	—
104	C₂×#127	—	67	2	no	—
105	#493	—	67	4	no	—
106	#493	—	134	2	no	—
107	C₂×#106	—	2	2	no	—
108	C₂×#102	—	134	8	no	—
109	C₂×C₄×C₈	—	67	2	no	—
110	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
111	#384	C₂×D₆₇	1	2	yes	—
112	#322	—	134	2	no	—
113	C₂²×D₄	—	134	4	no	—
114	C₂²×D₄	—	134	2	no	—
115	C₂×D₈	—	268	4	no	—
116	C₂×D₈	—	268	4	no	—
117	C₂×D₈	—	134	8	no	—
118	C₂²×C₈	—	67	4	no	—
119	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
120	#127	—	67	4	no	—
121	#127	—	134	2	no	—
122	#127	—	134	4	no	—
123	C₂×C₄⋊C₄	—	2	4	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
126	#106	—	4	4	no	—
127	#102	—	268	16	no	—
128	C₄×C₈	—	134	2	no	—
129	C₄×C₈	—	134	2	no	—
130	C₄×C₈	C₄×D₆₇	1	2	yes	—
131	C₄×C₈	D₂₆₈	1	2	yes	—
132	C₂⁴	—	268	2	no	—
133	C₂×D₄	—	268	4	no	—
134	C₂×D₄	—	268	2	no	—
135	C₂×D₄	—	268	4	no	—
136	C₂×D₄	—	268	4	no	—
137	C₂×D₄	—	268	8	no	—
138	C₂×D₄	—	268	8	no	—
139	C₂×D₄	—	134	4	no	—
140	C₂×D₄	—	134	8	no	—
141	C₂²×C₄	—	4	2	no	—
142	C₂²×C₄	D₄×D₆₇	1	2	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
144	D₈	—	268	16	no	—
145	C₂×C₈	—	134	4	no	—
146	C₂×C₈	—	134	8	no	—
147	C₂×C₈	(C₂×D₆₇)⋊C₄	1	4	yes	—
148	C₂×C₈	#11761613	1	4	yes	—
149	C₂×C₈	#11761615	1	4	yes	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	4	8	no	—
152	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
153	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
154	C₄²	#11761629	1	2	yes	—
155	C₂³	—	268	4	no	—
156	C₂³	—	536	2	no	—
157	C₂³	—	536	2	no	—
158	C₂³	—	536	4	no	—
159	C₂³	—	536	4	no	—
160	C₂³	#14524039	1	1	yes	characteristic
161	D₄	—	268	16	no	—
162	D₄	—	268	8	no	—
163	D₄	—	536	4	no	—
164	D₄	—	536	8	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	8	4	no	—
167	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
168	C₂×C₄	#14524005	1	2	yes	—
169	C₂×C₄	#14524018	1	2	yes	—
170	C₂×C₄	#14524032	1	1	yes	characteristic
171	C₂×C₄	C₂×#11761613	1	2	yes	—
172	C₂×C₄	C₂×#11761615	1	2	yes	—
173	C₂×C₄	#14524070	1	8	yes	—
174	C₈	—	268	4	no	—
175	C₈	#14523963	1	8	yes	complemented
176	C₂²	—	536	8	no	—
177	C₂²	—	536	4	no	—
178	C₂²	—	1072	2	no	—
179	C₂²	—	1072	2	no	—
180	C₂²	—	1072	4	no	—
181	C₂²	#16863297	1	1	yes	characteristic
182	C₂²	#16863770	1	1	yes	characteristic
183	C₂²	#16863772	1	1	yes	characteristic
184	C₂²	#16863782	1	4	yes	—
185	C₄	—	16	2	no	—
186	C₄	C₂×#14523963	1	2	yes	—
187	C₄	#16863507	1	4	yes	—
188	C₄	#16863650	1	4	yes	—
189	C₄	#16863744	1	2	yes	—
190	C₂	—	1072	4	no	—
191	C₂	#22559838	1	2	yes	—
192	C₂	#22559873	1	2	yes	—
193	C₂	#22561014	1	1	yes	characteristic
194	C₂	#22561083	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863773

C₂

yes

maximal, characteristic

#22561014

C₂

yes

maximal, complemented

C₂×#16863117

C₂

yes

maximal, characteristic, complemented

C₆₇×#1844

C₂

yes

maximal, characteristic, complemented

C₂²×D₅₃₆

—

C₂×#14524040

C₂²

yes

—

#16863773

—

#16863773

C₄

yes

complemented

C₂×#14523963

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

#16863117

—

C₂×#106×C₆₇

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#384×C₆₇

C₂²

yes

—

C₂²×D₂₆₈

—

C₂²×D₂₆₈

—

C₂×D₅₃₆

—

C₂×D₅₃₆

—

C₂×D₅₃₆

—

C₂²×C₆₇⋊C₈

D₄

yes

—

#14524040

—

#14524040

—

#14524040

C₂×C₄

yes

—

#14523963

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#106×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₂³×D₆₇

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

D₅₃₆

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

SD₁₆

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

#127

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

#124

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

—

C₂×C₄×C₆₇

C₂×SD₁₆

yes

—

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₈×C₆₇

#102

yes

complemented

C₆₇⋊C₈

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×#102

yes

—

C₄×C₆₇

#442

yes

—

C₄×C₆₇

#469

yes

—

C₄×C₆₇

#486

yes

—

#9837

—

maximal, Sylow

C₂×C₆₇

#1715

yes

—

C₂×C₆₇

#1721

yes

—

C₂×C₆₇

#1957

yes

characteristic

C₂×C₆₇

#1972

yes

—

D₆₇

—

C₂×#493

—

#1957

—

100

#1844

D₆₇

yes

characteristic, complemented

101

C₂×#322

—

102

C₆₇

#9837

yes

characteristic, complemented, Sylow

103

C₂²×D₈

—

134

—

104

C₂×#127

—

105

#493

—

106

#493

—

134

—

107

C₂×#106

—

108

C₂×#102

—

134

—

109

C₂×C₄×C₈

—

110

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

111

#384

C₂×D₆₇

yes

—

112

#322

—

134

—

113

C₂²×D₄

—

134

—

114

C₂²×D₄

—

134

—

115

C₂×D₈

—

268

—

116

C₂×D₈

—

268

—

117

C₂×D₈

—

134

—

118

C₂²×C₈

—

119

C₂²×C₈

C₂²⋊D₆₇

yes

—

120

#127

—

121

#127

—

134

—

122

#127

—

134

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄²

C₂²×D₆₇

yes

characteristic

126

#106

—

127

#102

—

268

—

128

C₄×C₈

—

134

—

129

C₄×C₈

—

134

—

130

C₄×C₈

C₄×D₆₇

yes

—

131

C₄×C₈

D₂₆₈

yes

—

132

C₂⁴

—

268

—

133

C₂×D₄

—

268

—

134

C₂×D₄

—

268

—

135

C₂×D₄

—

268

—

136

C₂×D₄

—

268

—

137

C₂×D₄

—

268

—

138

C₂×D₄

—

268

—

139

C₂×D₄

—

134

—

140

C₂×D₄

—

134

—

141

C₂²×C₄

—

142

C₂²×C₄

D₄×D₆₇

yes

—

143

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

144

D₈

—

268

—

145

C₂×C₈

—

134

—

146

C₂×C₈

—

134

—

147

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

148

C₂×C₈

#11761613

yes

—

149

C₂×C₈

#11761615

yes

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄²

C₂×C₄×D₆₇

yes

characteristic

153

C₄²

C₂×D₂₆₈

yes

characteristic

154

C₄²

#11761629

yes

—

155

C₂³

—

268

—

156

C₂³

—

536

—

157

C₂³

—

536

—

158

C₂³

—

536

—

159

C₂³

—

536

—

160

C₂³

#14524039

yes

characteristic

161

D₄

—

268

—

162

D₄

—

268

—

163

D₄

—

536

—

164

D₄

—

536

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

168

C₂×C₄

#14524005

yes

—

169

C₂×C₄

#14524018

yes

—

170

C₂×C₄

#14524032

yes

characteristic

171

C₂×C₄

C₂×#11761613

yes

—

172

C₂×C₄

C₂×#11761615

yes

—

173

C₂×C₄

#14524070

yes

—

174

C₈

—

268

—

175

C₈

#14523963

yes

complemented

176

C₂²

—

536

—

177

C₂²

—

536

—

178

C₂²

—

1072

—

179

C₂²

—

1072

—

180

C₂²

—

1072

—

181

C₂²

#16863297

yes

characteristic

182

C₂²

#16863770

yes

characteristic

183

C₂²

#16863772

yes

characteristic

184

C₂²

#16863782

yes

—

185

C₄

—

186

C₄

C₂×#14523963

yes

—

187

C₄

#16863507

yes

—

188

C₄

#16863650

yes

—

189

C₄

#16863744

yes

—

190

C₂

—

1072

—

191

C₂

#22559838

yes

—

192

C₂

#22559873

yes

—

193

C₂

#22561014

yes

characteristic

194

C₂

#22561083

yes

—

	subgroup	G/H
center	C₂³	#14524039
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1844	C₂
socle	C₂³×C₆₇	#127
prime radical	C₂×#16863773	C₂
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1844	D₆₇
hypocenter	C₆₇	#9837
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

subgroup

G/H

center

C₂³

#14524039

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1844

C₂

C₂³×C₆₇

C₂

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

N	H ≅ G/N	type
2. #22561014	C₂	semidirect
3. C₂×#16863117	C₂	semidirect
4. C₆₇×#1844	C₂	semidirect
8. #16863773	C₄	semidirect
76. C₈×C₆₇	#102	semidirect
100. #1844	D₆₇	semidirect
102. C₆₇	#9837	semidirect
175. C₈	#14523963	semidirect

H ≅ G/N

type

2. #22561014

C₂

semidirect

3. C₂×#16863117

C₂

semidirect

4. C₆₇×#1844

C₂

semidirect

8. #16863773

C₄

semidirect

76. C₈×C₆₇

#102

semidirect

100. #1844

D₆₇

semidirect

102. C₆₇

#9837

semidirect

175. C₈

#14523963

semidirect

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

subgroup	index
C₂×#16863773	2
#22561014	2
C₂×#16863117	2
C₆₇×#1844	2
#9837	67

subgroup

index

C₆₇×#1844

Subgroups — #30755668

Conjugacy classes of subgroups

Complements

Maximal subgroups