Group #22571341 — subgroups

Subgroups — #22571341

Order 8576.

370 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16863953	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863799	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863792	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863742	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863742	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863717	C₂	1	4	yes	maximal, complemented
7	#16863696	C₂	1	4	yes	maximal, complemented
8	#16863449	C₂	1	1	yes	maximal, characteristic, complemented
9	#532×C₆₇	C₂	1	1	yes	maximal, characteristic
10	#14524043	C₂²	1	1	yes	characteristic
11	#14524043	C₂²	1	1	yes	characteristic
12	#14524035	C₂²	1	2	yes	—
13	#14524022	—	2	1	no	—
14	#14524022	—	2	1	no	—
15	C₄×Dic₁₃₄	—	2	1	no	—
16	C₄×Dic₁₃₄	—	2	1	no	—
17	#14524002	C₂²	1	2	yes	—
18	#14523976	C₂²	1	4	yes	—
19	#14523974	C₂²	1	4	yes	—
20	C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
21	#14523964	C₂²	1	4	yes	—
22	#14523964	C₂²	1	4	yes	—
23	C₂²⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
24	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
25	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×#41×C₆₇	—	2	1	no	—
27	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
28	#127×C₆₇	C₂²	1	1	yes	characteristic
29	#124×C₆₇	—	2	1	no	—
30	#121×C₆₇	—	2	1	no	—
31	#121×C₆₇	C₂²	1	2	yes	—
32	C₄×D₄×C₆₇	—	2	1	no	—
33	C₄×D₄×C₆₇	C₂²	1	2	yes	—
34	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×Dic₁₃₄	—	2	1	no	—
36	C₂×Dic₁₃₄	—	2	1	no	—
37	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
38	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
39	C₄⋊Dic₆₇	—	2	1	no	—
40	C₄⋊Dic₆₇	—	2	1	no	—
41	C₄⋊Dic₆₇	C₂³	1	2	yes	—
42	C₄⋊Dic₆₇	C₂³	1	2	yes	—
43	Dic₆₇⋊C₄	—	2	2	no	—
44	Dic₆₇⋊C₄	—	2	2	no	—
45	C₂²⋊Dic₆₇	—	2	2	no	—
46	C₄×Dic₆₇	—	2	2	no	—
47	#41×C₆₇	—	4	4	no	—
48	C₂×Q₈×C₆₇	—	2	1	no	—
49	C₂×D₄×C₆₇	—	2	2	no	—
50	C₂×D₄×C₆₇	—	2	1	no	—
51	C₂×D₄×C₆₇	—	2	1	no	—
52	C₂×D₄×C₆₇	C₂³	1	2	yes	—
53	C₂²×C₄×C₆₇	—	2	2	no	—
54	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
57	C₄⋊C₄×C₆₇	—	2	1	no	—
58	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
59	C₂²⋊C₄×C₆₇	—	2	2	no	—
60	C₂²⋊C₄×C₆₇	—	2	2	no	—
61	C₄²×C₆₇	D₄	1	1	yes	characteristic
62	C₄²×C₆₇	D₄	1	1	yes	characteristic
63	C₄²×C₆₇	C₂³	1	1	yes	characteristic
64	C₄²×C₆₇	C₂³	1	1	yes	characteristic
65	C₂³×C₆₇	—	2	2	no	—
66	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
67	Q₈×C₆₇	—	4	2	no	—
68	D₄×C₆₇	—	2	4	no	—
69	D₄×C₆₇	—	4	4	no	—
70	D₄×C₆₇	—	4	2	no	—
71	D₄×C₆₇	—	4	2	no	—
72	C₂×C₄×C₆₇	—	2	2	no	—
73	C₂×C₄×C₆₇	—	2	2	no	—
74	C₂×C₄×C₆₇	—	2	2	no	—
75	C₂×C₄×C₆₇	—	2	2	no	—
76	C₂×C₄×C₆₇	—	4	4	no	—
77	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
83	C₂×Dic₆₇	—	2	2	no	—
84	C₂×Dic₆₇	—	2	2	no	—
85	Dic₁₃₄	—	4	2	no	—
86	Dic₁₃₄	—	4	2	no	—
87	C₆₇⋊C₈	—	2	2	no	—
88	C₆₇⋊C₈	—	2	2	no	—
89	C₂²×C₆₇	—	2	1	no	—
90	C₂²×C₆₇	—	2	1	no	—
91	C₂²×C₆₇	—	2	1	no	—
92	C₂²×C₆₇	—	4	2	no	—
93	C₂²×C₆₇	—	4	2	no	—
94	C₂²×C₆₇	—	4	2	no	—
95	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
96	C₄×C₆₇	—	2	1	no	—
97	C₄×C₆₇	—	2	1	no	—
98	C₄×C₆₇	—	2	1	no	—
99	C₄×C₆₇	—	4	2	no	—
100	C₄×C₆₇	#135	1	2	yes	—
101	C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
102	C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
103	Dic₆₇	—	4	2	no	—
104	Dic₆₇	—	4	2	no	—
105	C₂×C₆₇	—	4	1	no	—
106	C₂×C₆₇	—	8	2	no	—
107	C₂×C₆₇	#536	1	1	yes	characteristic
108	C₂×C₆₇	C₂×#135	1	1	yes	characteristic
109	C₂×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
110	#3228	—	67	1	no	maximal, Sylow
111	C₆₇	#3228	1	1	yes	characteristic, complemented, Sylow
112	#533	—	67	1	no	—
113	#532	D₆₇	1	1	yes	characteristic
114	#500	—	67	1	no	—
115	#498	—	67	1	no	—
116	#486	—	67	1	no	—
117	#486	—	67	1	no	—
118	#482	—	67	4	no	—
119	#478	—	67	4	no	—
120	#431	—	67	1	no	—
121	C₂×#41	—	2	1	no	—
122	C₄⋊Q₈	—	67	1	no	—
123	C₄⋊Q₈	—	67	1	no	—
124	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
125	#127	C₂×D₆₇	1	1	yes	characteristic
126	#125	—	67	2	no	—
127	#124	—	2	1	no	—
128	C₂²⋊Q₈	—	134	1	no	—
129	C₂²⋊Q₈	—	134	1	no	—
130	#121	—	2	1	no	—
131	#121	C₂×D₆₇	1	2	yes	—
132	C₄×Q₈	—	134	1	no	—
133	C₄×Q₈	—	134	1	no	—
134	C₄×D₄	—	2	1	no	—
135	C₄×D₄	C₂×D₆₇	1	2	yes	—
136	#117	—	67	2	no	—
137	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
138	#107	—	67	4	no	—
139	#106	—	67	4	no	—
140	C₄⋊C₈	—	67	2	no	—
141	#102	—	67	4	no	—
142	#102	—	67	4	no	—
143	C₂²⋊C₈	—	67	2	no	—
144	C₄×C₈	—	67	1	no	—
145	C₄×C₈	—	67	1	no	—
146	#41	—	4	4	no	—
147	C₂×Q₈	—	2	1	no	—
148	C₂×Q₈	—	134	1	no	—
149	C₂×Q₈	—	134	1	no	—
150	C₂×D₄	—	2	2	no	—
151	C₂×D₄	—	2	1	no	—
152	C₂×D₄	—	2	1	no	—
153	C₂×D₄	C₂²×D₆₇	1	2	yes	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
156	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
157	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
158	C₂×C₈	—	67	2	no	—
159	C₂×C₈	—	67	2	no	—
160	C₄⋊C₄	—	2	1	no	—
161	C₄⋊C₄	—	67	2	no	—
162	C₄⋊C₄	—	67	2	no	—
163	C₄⋊C₄	—	134	2	no	—
164	C₄⋊C₄	—	134	2	no	—
165	C₄⋊C₄	—	134	1	no	—
166	C₄⋊C₄	—	134	1	no	—
167	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
168	C₂²⋊C₄	—	2	2	no	—
169	C₂²⋊C₄	—	2	2	no	—
170	C₂²⋊C₄	—	134	2	no	—
171	C₄²	—	134	2	no	—
172	C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
173	C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
174	C₄²	C₂²×D₆₇	1	1	yes	characteristic
175	C₄²	C₂²×D₆₇	1	1	yes	characteristic
176	C₂³	—	2	2	no	—
177	C₂³	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
178	Q₈	—	4	2	no	—
179	Q₈	—	268	2	no	—
180	Q₈	—	268	2	no	—
181	D₄	—	2	4	no	—
182	D₄	—	4	4	no	—
183	D₄	—	4	2	no	—
184	D₄	—	4	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	4	4	no	—
190	C₂×C₄	—	134	2	no	—
191	C₂×C₄	—	134	2	no	—
192	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
193	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
194	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
195	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
196	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
197	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
198	C₈	—	134	2	no	—
199	C₈	—	134	2	no	—
200	C₂²	—	2	1	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	4	2	no	—
204	C₂²	—	4	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	2	1	no	—
210	C₄	—	4	2	no	—
211	C₄	—	268	2	no	—
212	C₄	—	268	2	no	—
213	C₄	#14524063	1	2	yes	—
214	C₄	#14524069	1	2	yes	—
215	C₄	#14524090	1	2	yes	—
216	C₂	—	4	1	no	—
217	C₂	—	8	2	no	—
218	C₂	#16863967	1	1	yes	characteristic
219	C₂	C₂×#14524063	1	1	yes	characteristic
220	C₂	C₂×#14524069	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863953

C₂

yes

maximal, characteristic, complemented

#16863799

C₂

yes

maximal, characteristic, complemented

#16863792

C₂

yes

maximal, characteristic, complemented

#16863742

C₂

yes

maximal, characteristic, complemented

#16863742

C₂

yes

maximal, characteristic, complemented

#16863717

C₂

yes

maximal, complemented

#16863696

C₂

yes

maximal, complemented

#16863449

C₂

yes

maximal, characteristic, complemented

#532×C₆₇

C₂

yes

maximal, characteristic

#14524043

C₂²

yes

characteristic

#14524043

C₂²

yes

characteristic

#14524035

C₂²

yes

—

#14524022

—

#14524022

—

C₄×Dic₁₃₄

—

C₄×Dic₁₃₄

—

#14524002

C₂²

yes

—

#14523976

C₂²

yes

—

#14523974

C₂²

yes

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

#14523964

C₂²

yes

—

#14523964

C₂²

yes

—

C₂²⋊(C₆₇⋊C₈)

C₂²

yes

—

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×#41×C₆₇

—

C₄⋊Q₈×C₆₇

C₂²

yes

characteristic

#127×C₆₇

C₂²

yes

characteristic

#124×C₆₇

—

#121×C₆₇

—

#121×C₆₇

C₂²

yes

—

C₄×D₄×C₆₇

—

C₄×D₄×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂³

yes

—

C₂²⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

C₂³

yes

characteristic

C₄²×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×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×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

100

C₄×C₆₇

#135

yes

—

101

C₄×C₆₇

AGL(1,ℤ/8)

yes

—

102

C₄×C₆₇

2₋¹⁺⁴

yes

—

103

Dic₆₇

—

104

Dic₆₇

—

105

C₂×C₆₇

—

106

C₂×C₆₇

—

107

C₂×C₆₇

#536

yes

characteristic

108

C₂×C₆₇

C₂×#135

yes

characteristic

109

C₂×C₆₇

yes

110

—

111

C₆₇

#3228

yes

characteristic, complemented, Sylow

112

#533

—

113

#532

D₆₇

yes

characteristic

114

#500

—

115

#498

—

116

#486

—

117

#486

—

118

#482

—

119

#478

—

120

#431

—

121

C₂×#41

—

122

C₄⋊Q₈

—

123

C₄⋊Q₈

—

124

C₄⋊Q₈

C₂×D₆₇

yes

characteristic

125

#127

C₂×D₆₇

yes

characteristic

126

#125

—

127

#124

—

128

C₂²⋊Q₈

—

134

—

129

C₂²⋊Q₈

—

134

—

130

#121

—

131

#121

C₂×D₆₇

yes

—

132

C₄×Q₈

—

134

—

133

C₄×Q₈

—

134

—

134

C₄×D₄

—

135

C₄×D₄

C₂×D₆₇

yes

—

136

#117

—

137

C₂×C₄²

C₂×D₆₇

yes

characteristic

138

#107

—

139

#106

—

140

C₄⋊C₈

—

141

#102

—

142

#102

—

143

C₂²⋊C₈

—

144

C₄×C₈

—

145

C₄×C₈

—

146

#41

—

147

C₂×Q₈

—

148

C₂×Q₈

—

134

—

149

C₂×Q₈

—

134

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂×D₄

—

153

C₂×D₄

C₂²×D₆₇

yes

—

154

C₂²×C₄

—

155

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

156

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

157

C₂²×C₄

C₂²×D₆₇

yes

characteristic

158

C₂×C₈

—

159

C₂×C₈

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

134

—

164

C₄⋊C₄

—

134

—

165

C₄⋊C₄

—

134

—

166

C₄⋊C₄

—

134

—

167

C₄⋊C₄

C₂²×D₆₇

yes

—

168

C₂²⋊C₄

—

169

C₂²⋊C₄

—

170

C₂²⋊C₄

—

134

—

171

C₄²

—

134

—

172

C₄²

C₂²⋊D₆₇

yes

characteristic

173

C₄²

C₂²⋊D₆₇

yes

characteristic

174

C₄²

C₂²×D₆₇

yes

characteristic

175

C₄²

C₂²×D₆₇

yes

characteristic

176

C₂³

—

177

C₂³

C₂×C₂²⋊D₆₇

yes

characteristic

178

Q₈

—

179

Q₈

—

268

—

180

Q₈

—

268

—

181

D₄

—

182

D₄

—

183

D₄

—

184

D₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

134

—

191

C₂×C₄

—

134

—

192

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

193

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

194

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

195

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

196

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

197

C₂×C₄

C₂³×D₆₇

yes

characteristic

198

C₈

—

134

—

199

C₈

—

134

—

200

C₂²

—

201

C₂²

—

202

C₂²

—

203

C₂²

—

204

C₂²

—

205

C₂²

—

206

C₂²

C₂²×C₂²⋊D₆₇

yes

characteristic

207

C₄

—

208

C₄

—

209

C₄

—

210

C₄

—

211

C₄

—

268

—

212

C₄

—

268

—

213

C₄

#14524063

yes

—

214

C₄

#14524069

yes

—

215

C₄

#14524090

yes

—

216

C₂

—

217

C₂

—

218

C₂

#16863967

yes

characteristic

219

C₂

C₂×#14524063

yes

characteristic

220

C₂

C₂×#14524069

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	#532×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	#532	D₆₇
hypocenter	C₆₇	#3228
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₆₇

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

#532×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #16863953	C₂	semidirect
2. #16863799	C₂	semidirect
3. #16863792	C₂	semidirect
4. #16863742	C₂	semidirect
5. #16863742	C₂	semidirect
6. #16863717	C₂	semidirect
7. #16863696	C₂	semidirect
8. #16863449	C₂	semidirect
111. C₆₇	#3228	semidirect

H ≅ G/N

type

1. #16863953

C₂

semidirect

2. #16863799

C₂

semidirect

3. #16863792

C₂

semidirect

4. #16863742

C₂

semidirect

5. #16863742

C₂

semidirect

6. #16863717

C₂

semidirect

7. #16863696

C₂

semidirect

8. #16863449

C₂

semidirect

111. C₆₇

#3228

semidirect

Subgroups — #22571341

Conjugacy classes of subgroups

Complements