Group #22561693 — subgroups

Subgroups — #22561693

Order 8576.

214 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 85 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×#14523974	C₂	1	1	yes	maximal, characteristic
2	#16863295	C₂	1	1	yes	maximal, characteristic
3	#16863261	C₂	1	1	yes	maximal, characteristic
4	#16863159	C₂	1	1	yes	maximal, characteristic
5	#16863158	C₂	1	1	yes	maximal, characteristic
6	#16863150	C₂	1	1	yes	maximal, characteristic
7	C₂×#107×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₆₇	—	2	1	no	—
10	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
13	#14523974	—	2	2	no	—
14	C₆₇⋊#95	—	2	1	no	—
15	C₆₇⋊#95	—	2	1	no	—
16	C₆₇⋊#95	—	2	1	no	—
17	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
20	#107×C₆₇	—	2	2	no	—
21	C₂²×Dic₆₇	—	2	1	no	—
22	C₂²×Dic₆₇	—	2	1	no	—
23	C₂²×Dic₆₇	D₄	1	1	yes	characteristic
24	C₂²×Dic₆₇	D₄	1	1	yes	characteristic
25	C₂×C₆₇⋊C₈	—	2	1	no	—
26	C₂×C₆₇⋊C₈	—	2	1	no	—
27	C₂×C₆₇⋊C₈	Q₈	1	2	yes	—
28	C₄⋊Dic₆₇	—	2	1	no	—
29	C₄⋊Dic₆₇	—	2	1	no	—
30	C₄⋊Dic₆₇	—	2	1	no	—
31	C₄⋊Dic₆₇	—	2	1	no	—
32	C₄⋊Dic₆₇	—	4	2	no	—
33	C₄×Dic₆₇	—	2	1	no	—
34	C₄×Dic₆₇	—	2	1	no	—
35	C₂²×C₄×C₆₇	—	2	1	no	—
36	C₂²×C₄×C₆₇	—	2	1	no	—
37	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
38	C₂×C₈×C₆₇	—	2	1	no	—
39	C₂×C₈×C₆₇	—	2	1	no	—
40	C₂×C₈×C₆₇	Q₈	1	2	yes	—
41	C₄⋊C₄×C₆₇	—	2	1	no	—
42	C₄⋊C₄×C₆₇	—	2	1	no	—
43	C₄⋊C₄×C₆₇	—	2	1	no	—
44	C₄⋊C₄×C₆₇	D₄	1	2	yes	—
45	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₆₇	—	2	2	no	—
47	C₂×C₄×C₆₇	—	4	1	no	—
48	C₂×C₄×C₆₇	—	4	1	no	—
49	C₂×C₄×C₆₇	—	4	1	no	—
50	C₂×C₄×C₆₇	—	4	1	no	—
51	C₂×C₄×C₆₇	—	4	1	no	—
52	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
56	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
57	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
58	C₂×Dic₆₇	—	2	1	no	—
59	C₂×Dic₆₇	—	2	1	no	—
60	C₂×Dic₆₇	—	2	1	no	—
61	C₂×Dic₆₇	—	2	1	no	—
62	C₂×Dic₆₇	—	2	1	no	—
63	C₂×Dic₆₇	—	2	1	no	—
64	C₂×Dic₆₇	—	4	1	no	—
65	C₂×Dic₆₇	—	4	1	no	—
66	C₂×Dic₆₇	—	4	1	no	—
67	C₂×Dic₆₇	—	4	1	no	—
68	C₂×Dic₆₇	—	4	1	no	—
69	C₂×Dic₆₇	—	4	1	no	—
70	C₈×C₆₇	—	2	2	no	—
71	C₆₇⋊C₈	—	2	2	no	—
72	C₂²×C₆₇	#121	1	1	yes	characteristic
73	C₂²×C₆₇	#123	1	1	yes	characteristic
74	C₂²×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
75	C₂²×C₆₇	#135	1	1	yes	characteristic
76	C₂²×C₆₇	#135	1	1	yes	characteristic
77	C₂²×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂²×C₆₇	#137	1	1	yes	characteristic
79	C₄×C₆₇	—	4	2	no	—
80	C₄×C₆₇	—	8	1	no	—
81	C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
82	C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
83	C₄×C₆₇	#125	1	1	yes	characteristic
84	C₄×C₆₇	#125	1	1	yes	characteristic
85	Dic₆₇	—	4	1	no	—
86	Dic₆₇	—	4	1	no	—
87	Dic₆₇	—	8	1	no	—
88	Dic₆₇	—	8	1	no	—
89	C₂×C₆₇	#398	1	1	yes	characteristic
90	C₂×C₆₇	#463	1	1	yes	characteristic
91	C₂×C₆₇	#464	1	1	yes	characteristic
92	C₂×C₆₇	#482	1	1	yes	characteristic
93	C₂×C₆₇	#485	1	1	yes	characteristic
94	C₂×C₆₇	#499	1	1	yes	characteristic
95	C₂×C₆₇	#501	1	1	yes	characteristic
96	#2085	—	67	1	no	maximal, Sylow
97	C₆₇	#2085	1	1	yes	characteristic, complemented, Sylow
98	C₂×#107	D₆₇	1	1	yes	characteristic
99	C₂×#106	—	67	1	no	—
100	#389	—	67	1	no	—
101	#382	—	67	1	no	—
102	#340	—	67	1	no	—
103	#340	—	67	1	no	—
104	#336	—	67	1	no	—
105	C₂²×C₈	—	67	1	no	—
106	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	67	1	no	—
108	C₂×C₄⋊C₄	—	67	1	no	—
109	C₂×C₄⋊C₄	—	134	1	no	—
110	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
111	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
112	C₂×C₄²	—	67	1	no	—
113	#107	—	2	2	no	—
114	#106	—	134	2	no	—
115	#95	—	134	1	no	—
116	#95	—	134	1	no	—
117	#95	—	134	1	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	—	67	1	no	—
121	C₂²×C₄	—	67	1	no	—
122	C₂²×C₄	—	134	1	no	—
123	C₂²×C₄	—	134	1	no	—
124	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	67	2	no	—
128	C₂×C₈	—	134	1	no	—
129	C₂×C₈	—	134	1	no	—
130	C₂×C₈	Dic₁₃₄	1	2	yes	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	2	1	no	—
134	C₄⋊C₄	—	268	2	no	—
135	C₄⋊C₄	—	134	1	no	—
136	C₄⋊C₄	—	134	1	no	—
137	C₄⋊C₄	—	134	1	no	—
138	C₄⋊C₄	—	134	1	no	—
139	C₄⋊C₄	C₂²⋊D₆₇	1	2	yes	—
140	C₄²	—	134	1	no	—
141	C₄²	—	134	1	no	—
142	C₂³	D₄×D₆₇	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	268	1	no	—
150	C₂×C₄	—	268	1	no	—
151	C₂×C₄	—	268	1	no	—
152	C₂×C₄	—	268	1	no	—
153	C₂×C₄	—	268	1	no	—
154	C₂×C₄	—	268	1	no	—
155	C₂×C₄	—	134	1	no	—
156	C₂×C₄	—	134	1	no	—
157	C₂×C₄	—	134	1	no	—
158	C₂×C₄	—	134	1	no	—
159	C₂×C₄	—	134	1	no	—
160	C₂×C₄	—	134	1	no	—
161	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
162	C₂×C₄	Q₈×D₆₇	1	1	yes	characteristic
163	C₂×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
164	C₂×C₄	#11761627	1	1	yes	characteristic
165	C₂×C₄	#11761628	1	1	yes	characteristic
166	C₂×C₄	#11761629	1	1	yes	characteristic
167	C₈	—	2	2	no	—
168	C₈	—	134	2	no	—
169	C₂²	#14524015	1	1	yes	characteristic
170	C₂²	#14524028	1	1	yes	characteristic
171	C₂²	#14524042	1	1	yes	characteristic
172	C₂²	#14524062	1	1	yes	characteristic
173	C₂²	#14524066	1	1	yes	characteristic
174	C₂²	#14524067	1	1	yes	characteristic
175	C₂²	#14524075	1	1	yes	characteristic
176	C₄	—	4	2	no	—
177	C₄	—	8	1	no	—
178	C₄	—	268	1	no	—
179	C₄	—	268	1	no	—
180	C₄	—	536	1	no	—
181	C₄	—	536	1	no	—
182	C₄	#14524019	1	1	yes	characteristic
183	C₄	#14524022	1	1	yes	characteristic
184	C₄	#14524033	1	1	yes	characteristic
185	C₄	#14524034	1	1	yes	characteristic
186	C₂	#16863329	1	1	yes	characteristic
187	C₂	#16863609	1	1	yes	characteristic
188	C₂	#16863620	1	1	yes	characteristic
189	C₂	#16863723	1	1	yes	characteristic
190	C₂	#16863739	1	1	yes	characteristic
191	C₂	#16863797	1	1	yes	characteristic
192	C₂	#16863804	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523974

C₂

yes

maximal, characteristic

#16863295

C₂

yes

maximal, characteristic

#16863261

C₂

yes

maximal, characteristic

#16863159

C₂

yes

maximal, characteristic

#16863158

C₂

yes

maximal, characteristic

#16863150

C₂

yes

maximal, characteristic

C₂×#107×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×C₄×Dic₆₇

C₂²

yes

characteristic

#14523974

—

C₆₇⋊#95

—

C₆₇⋊#95

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₇

C₂²

yes

characteristic

#107×C₆₇

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

D₄

yes

characteristic

C₂²×Dic₆₇

D₄

yes

characteristic

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

Q₈

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

Q₈

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

D₄

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

C₄⋊Q₈

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₄×C₆₇

C₂²⋊Q₈

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×#107

D₆₇

yes

characteristic

C₂×#106

—

100

#389

—

101

#382

—

102

#340

—

103

#340

—

104

#336

—

105

C₂²×C₈

—

106

C₂²×C₈

C₂×D₆₇

yes

characteristic

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

134

—

110

C₂×C₄⋊C₄

C₂×D₆₇

yes

characteristic

111

C₂×C₄⋊C₄

C₂×D₆₇

yes

characteristic

112

C₂×C₄²

—

113

#107

—

114

#106

—

134

—

115

#95

—

134

—

116

#95

—

134

—

117

#95

—

134

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

134

—

123

C₂²×C₄

—

134

—

124

C₂²×C₄

C₂²×D₆₇

yes

characteristic

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

134

—

129

C₂×C₈

—

134

—

130

C₂×C₈

Dic₁₃₄

yes

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

268

—

135

C₄⋊C₄

—

134

—

136

C₄⋊C₄

—

134

—

137

C₄⋊C₄

—

134

—

138

C₄⋊C₄

—

134

—

139

C₄⋊C₄

C₂²⋊D₆₇

yes

—

140

C₄²

—

134

—

141

C₄²

—

134

—

142

C₂³

D₄×D₆₇

yes

characteristic

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

268

—

150

C₂×C₄

—

268

—

151

C₂×C₄

—

268

—

152

C₂×C₄

—

268

—

153

C₂×C₄

—

268

—

154

C₂×C₄

—

268

—

155

C₂×C₄

—

134

—

156

C₂×C₄

—

134

—

157

C₂×C₄

—

134

—

158

C₂×C₄

—

134

—

159

C₂×C₄

—

134

—

160

C₂×C₄

—

134

—

161

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

162

C₂×C₄

Q₈×D₆₇

yes

characteristic

163

C₂×C₄

C₂×Dic₁₃₄

yes

characteristic

164

C₂×C₄

#11761627

yes

characteristic

165

C₂×C₄

#11761628

yes

characteristic

166

C₂×C₄

#11761629

yes

characteristic

167

C₈

—

168

C₈

—

134

—

169

C₂²

#14524015

yes

characteristic

170

C₂²

#14524028

yes

characteristic

171

C₂²

#14524042

yes

characteristic

172

C₂²

#14524062

yes

characteristic

173

C₂²

#14524066

yes

characteristic

174

C₂²

#14524067

yes

characteristic

175

C₂²

#14524075

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

268

—

179

C₄

—

268

—

180

C₄

—

536

—

181

C₄

—

536

—

182

C₄

#14524019

yes

characteristic

183

C₄

#14524022

yes

characteristic

184

C₄

#14524033

yes

characteristic

185

C₄

#14524034

yes

characteristic

186

C₂

#16863329

yes

characteristic

187

C₂

#16863609

yes

characteristic

188

C₂

#16863620

yes

characteristic

189

C₂

#16863723

yes

characteristic

190

C₂

#16863739

yes

characteristic

191

C₂

#16863797

yes

characteristic

192

C₂

#16863804

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₆₇
commutator	C₂²×C₄×C₆₇	C₂³
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	C₂×#107×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	C₂×#107	D₆₇
hypocenter	C₆₇	#2085
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₆₇

commutator

C₂²×C₄×C₆₇

C₂³

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

C₂×#107×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
97. C₆₇	#2085	semidirect

H ≅ G/N

type

97. C₆₇

#2085

semidirect

Subgroups — #22561693

Conjugacy classes of subgroups

Complements