Group #30760883 — subgroups

Subgroups — #30760883

Order 17152.

442 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 155 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	#22561171	C₂	1	2	yes	maximal
2	#22560369	C₂	1	1	yes	maximal, characteristic
3	#22560368	C₂	1	2	yes	maximal
4	C₆₇⋊#1273	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1956	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523977	—	2	2	no	—
7	C₂×#14523976	—	2	4	no	—
8	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
9	#16863266	C₂²	1	2	yes	—
10	#16863158	—	2	4	no	—
11	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#103×C₆₇	—	2	2	no	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	#384×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₆₇⋊C₈	D₄	1	4	yes	—
16	C₂×C₄⋊Dic₆₇	—	2	2	no	—
17	C₂×C₄⋊Dic₆₇	—	2	2	no	—
18	C₂×C₄⋊Dic₆₇	—	2	2	no	—
19	#14523977	—	4	4	no	—
20	#14523976	—	4	8	no	—
21	C₄×C₆₇⋊C₈	—	2	2	no	—
22	C₄×C₆₇⋊C₈	—	2	2	no	—
23	C₂²×Q₈×C₆₇	—	2	1	no	—
24	C₂²×C₈×C₆₇	D₄	1	2	yes	—
25	C₄⋊Q₈×C₆₇	—	2	2	no	—
26	C₄⋊Q₈×C₆₇	—	2	1	no	—
27	C₄⋊Q₈×C₆₇	Q₈	1	2	yes	—
28	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
29	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
31	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
32	#103×C₆₇	—	4	4	no	—
33	C₄×C₈×C₆₇	—	2	2	no	—
34	C₂²×Dic₆₇	—	4	2	no	—
35	C₂×C₆₇⋊C₈	—	2	4	no	—
36	C₂×C₆₇⋊C₈	—	2	4	no	—
37	C₂×C₆₇⋊C₈	Q₁₆	1	8	yes	—
38	C₄⋊Dic₆₇	—	4	4	no	—
39	C₄⋊Dic₆₇	—	4	4	no	—
40	C₄⋊Dic₆₇	—	4	4	no	—
41	C₂×Q₈×C₆₇	—	2	2	no	—
42	C₂×Q₈×C₆₇	—	4	1	no	—
43	C₂×Q₈×C₆₇	—	4	2	no	—
44	C₂×Q₈×C₆₇	—	4	2	no	—
45	C₂²×C₄×C₆₇	—	4	1	no	—
46	C₂²×C₄×C₆₇	—	4	1	no	—
47	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₆₇	—	2	2	no	—
50	C₂×C₈×C₆₇	—	2	2	no	—
51	C₂×C₈×C₆₇	D₈	1	4	yes	—
52	C₄⋊C₄×C₆₇	—	2	4	no	—
53	C₄⋊C₄×C₆₇	—	4	2	no	—
54	C₄⋊C₄×C₆₇	—	4	2	no	—
55	C₄⋊C₄×C₆₇	—	4	4	no	—
56	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₆₇	#41	1	2	yes	—
58	C₄²×C₆₇	#41	1	1	yes	characteristic
59	C₂³×C₆₇	#127	1	1	yes	characteristic
60	Q₈×C₆₇	—	4	4	no	—
61	Q₈×C₆₇	—	8	2	no	—
62	C₂×C₄×C₆₇	—	4	2	no	—
63	C₂×C₄×C₆₇	—	8	1	no	—
64	C₂×C₄×C₆₇	—	8	1	no	—
65	C₂×C₄×C₆₇	—	8	1	no	—
66	C₂×C₄×C₆₇	—	8	2	no	—
67	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
68	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
70	C₂×C₄×C₆₇	#123	1	2	yes	—
71	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
72	C₂×C₄×C₆₇	C₂×Q₁₆	1	4	yes	—
73	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
74	C₂×C₄×C₆₇	#137	1	2	yes	—
75	C₂×Dic₆₇	—	8	2	no	—
76	C₂×Dic₆₇	—	8	2	no	—
77	C₂×Dic₆₇	—	8	2	no	—
78	C₈×C₆₇	—	2	4	no	—
79	C₆₇⋊C₈	—	2	8	no	—
80	C₂²×C₆₇	#399	1	1	yes	characteristic
81	C₂²×C₆₇	#493	1	1	yes	characteristic
82	C₂²×C₆₇	#494	1	2	yes	—
83	C₂²×C₆₇	#496	1	2	yes	—
84	C₂²×C₆₇	#497	1	1	yes	characteristic
85	C₄×C₆₇	—	8	2	no	—
86	C₄×C₆₇	—	16	1	no	—
87	C₄×C₆₇	#474	1	2	yes	—
88	C₄×C₆₇	#477	1	4	yes	—
89	C₄×C₆₇	#480	1	2	yes	—
90	C₄×C₆₇	#484	1	4	yes	—
91	Dic₆₇	—	16	2	no	—
92	#10674	—	67	1	no	maximal, Sylow
93	C₂×C₆₇	#1713	1	1	yes	characteristic
94	C₂×C₆₇	#1721	1	1	yes	characteristic
95	C₂×C₆₇	#1723	1	2	yes	—
96	C₂×C₆₇	#2093	1	1	yes	characteristic
97	C₂×C₆₇	#2095	1	2	yes	—
98	#1990	—	67	2	no	—
99	#1956	D₆₇	1	1	yes	characteristic
100	#1845	—	67	1	no	—
101	#1845	—	67	2	no	—
102	#1273	—	67	1	no	—
103	C₆₇	#10674	1	1	yes	characteristic, complemented, Sylow
104	C₂×C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
105	C₂×#107	—	134	4	no	—
106	C₂×#107	—	134	2	no	—
107	C₂×#103	—	2	2	no	—
108	C₂×C₄×C₈	—	67	2	no	—
109	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
110	#384	—	67	2	no	—
111	#384	C₂×D₆₇	1	1	yes	characteristic
112	#340	—	134	4	no	—
113	C₂²×Q₈	—	2	1	no	—
114	C₂²×C₈	—	67	4	no	—
115	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
116	C₄⋊Q₈	—	2	2	no	—
117	C₄⋊Q₈	—	2	1	no	—
118	C₄⋊Q₈	Dic₁₃₄	1	2	yes	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	1	no	—
122	C₂×C₄⋊C₄	—	134	2	no	—
123	C₂×C₄⋊C₄	—	134	2	no	—
124	C₂×C₄⋊C₄	—	134	2	no	—
125	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
126	#107	—	268	8	no	—
127	#107	—	268	4	no	—
128	#103	—	4	4	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	134	2	no	—
131	C₄×C₈	—	134	2	no	—
132	C₂×Q₈	—	2	2	no	—
133	C₂×Q₈	—	4	1	no	—
134	C₂×Q₈	—	4	2	no	—
135	C₂×Q₈	—	4	2	no	—
136	C₂²×C₄	—	4	1	no	—
137	C₂²×C₄	—	4	1	no	—
138	C₂²×C₄	—	268	2	no	—
139	C₂²×C₄	D₄×D₆₇	1	2	yes	—
140	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	67	8	no	—
144	C₂×C₈	—	134	4	no	—
145	C₂×C₈	—	134	4	no	—
146	C₂×C₈	#11761613	1	4	yes	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	268	4	no	—
152	C₄⋊C₄	—	268	4	no	—
153	C₄⋊C₄	—	268	4	no	—
154	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
155	C₄²	#11761627	1	2	yes	—
156	C₄²	#11761628	1	1	yes	characteristic
157	C₂³	#14524039	1	1	yes	characteristic
158	Q₈	—	4	4	no	—
159	Q₈	—	8	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	8	1	no	—
163	C₂×C₄	—	8	1	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	536	2	no	—
166	C₂×C₄	—	536	2	no	—
167	C₂×C₄	—	536	2	no	—
168	C₂×C₄	#14524020	1	2	yes	—
169	C₂×C₄	#14524022	1	1	yes	characteristic
170	C₂×C₄	#14524024	1	1	yes	characteristic
171	C₂×C₄	#14524025	1	2	yes	—
172	C₂×C₄	C₂×#11761613	1	2	yes	—
173	C₂×C₄	Q₁₆×D₆₇	1	4	yes	—
174	C₂×C₄	#14524070	1	4	yes	—
175	C₂×C₄	#14524073	1	2	yes	—
176	C₈	—	2	4	no	—
177	C₈	—	134	8	no	—
178	C₂²	#16863332	1	1	yes	characteristic
179	C₂²	#16863772	1	1	yes	characteristic
180	C₂²	#16863774	1	2	yes	—
181	C₂²	#16863782	1	2	yes	—
182	C₂²	#16863791	1	1	yes	characteristic
183	C₄	—	8	2	no	—
184	C₄	—	16	1	no	—
185	C₄	—	1072	2	no	—
186	C₄	#16863676	1	2	yes	—
187	C₄	#16863689	1	4	yes	—
188	C₄	#16863707	1	2	yes	—
189	C₄	#16863730	1	4	yes	—
190	C₂	#22559825	1	1	yes	characteristic
191	C₂	#22559873	1	1	yes	characteristic
192	C₂	#22559889	1	2	yes	—
193	C₂	#22561737	1	1	yes	characteristic
194	C₂	#22561745	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22561171

C₂

yes

maximal

#22560369

C₂

yes

maximal, characteristic

#22560368

C₂

yes

maximal

C₆₇⋊#1273

C₂

yes

maximal, characteristic

C₆₇×#1956

C₂

yes

maximal, characteristic

C₂×#14523977

—

C₂×#14523976

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

#16863266

C₂²

yes

—

#16863158

—

C₂×C₄⋊Q₈×C₆₇

C₂²

yes

characteristic

C₂×#103×C₆₇

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#384×C₆₇

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

D₄

yes

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

#14523977

—

#14523976

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₂²×Q₈×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

Q₈

yes

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#103×C₆₇

—

C₄×C₈×C₆₇

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

Q₁₆

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₈

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₄²×C₆₇

#41

yes

—

C₄²×C₆₇

yes

C₂³×C₆₇

yes

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

—

C₂×C₄×C₆₇

C₂×Q₁₆

yes

—

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₆₇

#137

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₂²×C₆₇

#496

yes

—

C₂²×C₆₇

#497

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#474

yes

—

C₄×C₆₇

#477

yes

—

C₄×C₆₇

#480

yes

—

C₄×C₆₇

#484

yes

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×C₆₇

#2093

yes

characteristic

C₂×C₆₇

#2095

yes

—

#1990

—

#1956

D₆₇

yes

characteristic

100

#1845

—

101

#1845

—

102

#1273

—

103

C₆₇

#10674

yes

characteristic, complemented, Sylow

104

C₂×C₄⋊Q₈

C₂×D₆₇

yes

characteristic

105

C₂×#107

—

134

—

106

C₂×#107

—

134

—

107

C₂×#103

—

108

C₂×C₄×C₈

—

109

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

110

#384

—

111

#384

C₂×D₆₇

yes

characteristic

112

#340

—

134

—

113

C₂²×Q₈

—

114

C₂²×C₈

—

115

C₂²×C₈

C₂²⋊D₆₇

yes

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

Dic₁₃₄

yes

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

134

—

123

C₂×C₄⋊C₄

—

134

—

124

C₂×C₄⋊C₄

—

134

—

125

C₂×C₄²

C₂²×D₆₇

yes

characteristic

126

#107

—

268

—

127

#107

—

268

—

128

#103

—

129

C₄×C₈

—

130

C₄×C₈

—

134

—

131

C₄×C₈

—

134

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

268

—

139

C₂²×C₄

D₄×D₆₇

yes

—

140

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

134

—

145

C₂×C₈

—

134

—

146

C₂×C₈

#11761613

yes

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

268

—

152

C₄⋊C₄

—

268

—

153

C₄⋊C₄

—

268

—

154

C₄²

C₂×Dic₁₃₄

yes

characteristic

155

C₄²

#11761627

yes

—

156

C₄²

#11761628

yes

characteristic

157

C₂³

#14524039

yes

characteristic

158

Q₈

—

159

Q₈

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

536

—

166

C₂×C₄

—

536

—

167

C₂×C₄

—

536

—

168

C₂×C₄

#14524020

yes

—

169

C₂×C₄

#14524022

yes

characteristic

170

C₂×C₄

#14524024

yes

characteristic

171

C₂×C₄

#14524025

yes

—

172

C₂×C₄

C₂×#11761613

yes

—

173

C₂×C₄

Q₁₆×D₆₇

yes

—

174

C₂×C₄

#14524070

yes

—

175

C₂×C₄

#14524073

yes

—

176

C₈

—

177

C₈

—

134

—

178

C₂²

#16863332

yes

characteristic

179

C₂²

#16863772

yes

characteristic

180

C₂²

#16863774

yes

—

181

C₂²

#16863782

yes

—

182

C₂²

#16863791

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

1072

—

186

C₄

#16863676

yes

—

187

C₄

#16863689

yes

—

188

C₄

#16863707

yes

—

189

C₄

#16863730

yes

—

190

C₂

#22559825

yes

characteristic

191

C₂

#22559873

yes

characteristic

192

C₂

#22559889

yes

—

193

C₂

#22561737

yes

characteristic

194

C₂

#22561745

yes

—

	subgroup	G/H
center	C₂³	#14524039
commutator, cosocle	C₂×C₄²×C₆₇	C₂³
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1956	C₂
socle, prime radical	C₂³×C₆₇	#127
hypercenter	#1956	D₆₇
hypocenter	C₆₇	#10674
LCS	G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

subgroup

G/H

center

C₂³

#14524039

commutator, cosocle

C₂×C₄²×C₆₇

C₂³

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1956

C₂

C₂³×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

N	H ≅ G/N	type
103. C₆₇	#10674	semidirect

H ≅ G/N

type

103. C₆₇

#10674

semidirect

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

subgroup	index
#22561171	2
#22560369	2
#22560368	2
C₆₇⋊#1273	2
C₆₇×#1956	2
#10674	67

subgroup

index

#22561171

#22560369

#22560368

C₆₇⋊#1273

C₆₇×#1956

#10674

Subgroups — #30760883

Conjugacy classes of subgroups

Complements

Maximal subgroups