Group #22561089 — subgroups

Subgroups — #22561089

Order 8576.

340 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 99 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	C₂²×#11761616	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523976	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14523966	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14523964	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523954	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863302	C₂	1	1	yes	maximal, characteristic, complemented
7	#390×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₃₄	C₂²	1	1	yes	characteristic
9	C₂×#11761616	—	2	2	no	—
10	C₂×#11761616	—	2	2	no	—
11	C₂×#11761616	C₄	1	4	yes	complemented
12	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
15	#14523976	—	2	2	no	—
16	#14523966	—	2	2	no	—
17	#14523964	—	2	2	no	—
18	#14523954	—	2	2	no	—
19	C₆₇⋊#95	—	2	2	no	—
20	C₂²×D₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
23	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×Dic₁₃₄	—	2	1	no	—
25	C₂×Dic₁₃₄	—	2	1	no	—
26	C₂×Dic₁₃₄	—	2	1	no	—
27	C₂×Dic₁₃₄	—	2	1	no	—
28	C₂×Dic₁₃₄	—	2	1	no	—
29	C₂×Dic₁₃₄	C₂×C₄	1	2	yes	—
30	C₂²×Dic₆₇	—	2	1	no	—
31	C₂²×Dic₆₇	—	2	1	no	—
32	#11761616	—	2	8	no	—
33	C₂×C₆₇⋊C₈	—	2	1	no	—
34	C₂×C₆₇⋊C₈	—	2	1	no	—
35	C₂×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
36	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
37	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
38	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
39	C₄⋊Dic₆₇	—	2	1	no	—
40	C₄⋊Dic₆₇	—	2	1	no	—
41	C₂⁴×C₆₇	—	2	1	no	—
42	C₂×D₄×C₆₇	—	2	1	no	—
43	C₂×D₄×C₆₇	—	2	1	no	—
44	C₂×D₄×C₆₇	—	2	1	no	—
45	C₂×D₄×C₆₇	—	2	1	no	—
46	C₂×D₄×C₆₇	—	2	1	no	—
47	C₂×D₄×C₆₇	C₂×C₄	1	2	yes	—
48	C₂²×C₄×C₆₇	—	2	1	no	—
49	C₂²×C₄×C₆₇	D₄	1	2	yes	—
50	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₆₇	—	2	1	no	—
52	C₄⋊C₄×C₆₇	—	2	1	no	—
53	C₂²⋊C₄×C₆₇	—	4	4	no	—
54	C₄²×C₆₇	—	2	1	no	—
55	C₄²×C₆₇	—	2	1	no	—
56	C₂³×C₆₇	—	2	2	no	—
57	C₂³×C₆₇	—	4	1	no	—
58	C₂³×C₆₇	—	4	1	no	—
59	C₂³×C₆₇	—	4	2	no	—
60	C₂³×C₆₇	—	4	2	no	—
61	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
62	D₄×C₆₇	—	2	2	no	—
63	D₄×C₆₇	—	2	2	no	—
64	D₄×C₆₇	—	4	1	no	—
65	D₄×C₆₇	—	4	1	no	—
66	C₂×C₄×C₆₇	—	2	2	no	—
67	C₂×C₄×C₆₇	—	2	2	no	—
68	C₂×C₄×C₆₇	—	2	2	no	—
69	C₂×C₄×C₆₇	—	4	1	no	—
70	C₂×C₄×C₆₇	—	4	2	no	—
71	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
76	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
77	C₂×Dic₆₇	—	2	2	no	—
78	C₂×Dic₆₇	—	4	1	no	—
79	C₂×Dic₆₇	—	4	1	no	—
80	C₂×Dic₆₇	—	4	1	no	—
81	C₂×Dic₆₇	—	4	2	no	—
82	Dic₁₃₄	—	2	2	no	—
83	Dic₁₃₄	—	2	2	no	—
84	Dic₁₃₄	—	4	1	no	—
85	Dic₁₃₄	—	4	1	no	—
86	C₆₇⋊C₈	—	2	2	no	—
87	C₆₇⋊C₈	C₂²⋊C₄	1	4	yes	complemented
88	C₂²×C₆₇	—	4	2	no	—
89	C₂²×C₆₇	—	4	4	no	—
90	C₂²×C₆₇	—	8	2	no	—
91	C₂²×C₆₇	—	8	1	no	—
92	C₂²×C₆₇	—	8	1	no	—
93	C₂²×C₆₇	C₄×D₄	1	1	yes	characteristic
94	C₂²×C₆₇	#121	1	1	yes	characteristic
95	C₂²×C₆₇	#127	1	1	yes	characteristic
96	C₂²×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
97	C₂²×C₆₇	#137	1	2	yes	—
98	C₄×C₆₇	—	4	2	no	—
99	C₄×C₆₇	—	8	1	no	—
100	C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
102	C₄×C₆₇	#121	1	1	yes	characteristic
103	C₄×C₆₇	#124	1	1	yes	characteristic
104	Dic₆₇	—	4	2	no	—
105	Dic₆₇	—	8	1	no	—
106	C₂×C₆₇	—	8	2	no	—
107	C₂×C₆₇	#390	1	1	yes	characteristic
108	C₂×C₆₇	#440	1	2	yes	—
109	C₂×C₆₇	#468	1	2	yes	—
110	C₂×C₆₇	#496	1	1	yes	characteristic
111	C₂×C₆₇	#497	1	1	yes	characteristic
112	#1973	—	67	1	no	maximal, Sylow
113	C₆₇	#1973	1	1	yes	characteristic, complemented, Sylow
114	C₂²×SD₁₆	—	67	1	no	—
115	C₂×#107	—	67	1	no	—
116	C₂×#103	—	67	1	no	—
117	C₂×#102	—	67	1	no	—
118	C₂×#97	—	67	1	no	—
119	#391	—	67	1	no	—
120	#390	D₆₇	1	1	yes	characteristic
121	C₂²×Q₈	—	67	1	no	—
122	C₂²×D₄	C₂×D₆₇	1	1	yes	characteristic
123	C₂×SD₁₆	—	67	4	no	—
124	C₂×SD₁₆	—	134	2	no	—
125	C₂×SD₁₆	—	134	2	no	—
126	C₂²×C₈	—	67	1	no	—
127	C₂²×C₈	—	67	1	no	—
128	C₂×C₄⋊C₄	—	67	1	no	—
129	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
130	C₂×C₂²⋊C₄	—	2	2	no	—
131	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
132	#107	—	134	2	no	—
133	#103	—	134	2	no	—
134	#102	—	134	2	no	—
135	#97	—	134	2	no	—
136	#95	—	134	2	no	—
137	C₂⁴	—	2	1	no	—
138	C₂×Q₈	—	67	2	no	—
139	C₂×Q₈	—	134	1	no	—
140	C₂×Q₈	—	134	1	no	—
141	C₂×Q₈	—	134	1	no	—
142	C₂×Q₈	—	134	1	no	—
143	C₂×Q₈	—	134	1	no	—
144	C₂×D₄	—	2	1	no	—
145	C₂×D₄	—	2	1	no	—
146	C₂×D₄	—	2	1	no	—
147	C₂×D₄	—	2	1	no	—
148	C₂×D₄	—	2	1	no	—
149	C₂×D₄	C₄×D₆₇	1	2	yes	—
150	C₂²×C₄	—	2	1	no	—
151	C₂²×C₄	—	134	1	no	—
152	C₂²×C₄	—	134	1	no	—
153	C₂²×C₄	C₂²⋊D₆₇	1	2	yes	—
154	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
155	SD₁₆	—	134	8	no	—
156	C₂×C₈	—	67	2	no	—
157	C₂×C₈	—	67	2	no	—
158	C₂×C₈	—	67	2	no	—
159	C₂×C₈	—	67	2	no	—
160	C₂×C₈	—	134	1	no	—
161	C₂×C₈	—	134	1	no	—
162	C₄⋊C₄	—	2	1	no	—
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	—	134	1	no	—
165	C₄⋊C₄	—	134	1	no	—
166	C₂²⋊C₄	—	4	4	no	—
167	C₄²	—	2	1	no	—
168	C₄²	—	2	1	no	—
169	C₂³	—	2	2	no	—
170	C₂³	—	4	1	no	—
171	C₂³	—	4	1	no	—
172	C₂³	—	4	2	no	—
173	C₂³	—	4	2	no	—
174	C₂³	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
175	Q₈	—	268	1	no	—
176	Q₈	—	268	1	no	—
177	Q₈	—	134	2	no	—
178	Q₈	—	134	2	no	—
179	D₄	—	2	2	no	—
180	D₄	—	2	2	no	—
181	D₄	—	4	1	no	—
182	D₄	—	4	1	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	4	1	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	268	1	no	—
189	C₂×C₄	—	268	1	no	—
190	C₂×C₄	—	268	1	no	—
191	C₂×C₄	—	268	2	no	—
192	C₂×C₄	—	134	2	no	—
193	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
194	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
195	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
196	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
197	C₂×C₄	#11761627	1	1	yes	characteristic
198	C₂×C₄	#11761628	1	1	yes	characteristic
199	C₈	—	67	4	no	—
200	C₈	—	134	2	no	—
201	C₂²	—	4	4	no	—
202	C₂²	—	4	2	no	—
203	C₂²	—	8	1	no	—
204	C₂²	—	8	1	no	—
205	C₂²	—	8	2	no	—
206	C₂²	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
207	C₂²	#14524015	1	1	yes	characteristic
208	C₂²	#14524039	1	1	yes	characteristic
209	C₂²	#14524069	1	2	yes	—
210	C₂²	#14524073	1	2	yes	—
211	C₄	—	4	2	no	—
212	C₄	—	8	1	no	—
213	C₄	—	268	2	no	—
214	C₄	—	536	1	no	—
215	C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
216	C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
217	C₄	#14524016	1	1	yes	characteristic
218	C₄	#14524029	1	1	yes	characteristic
219	C₂	—	8	2	no	—
220	C₂	#16863296	1	1	yes	characteristic
221	C₂	#16863496	1	2	yes	—
222	C₂	#16863640	1	2	yes	—
223	C₂	#16863783	1	1	yes	characteristic
224	C₂	#16863791	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11761616

C₂

yes

maximal, characteristic

C₂×#14523976

C₂

yes

maximal, characteristic, complemented

C₂×#14523966

C₂

yes

maximal, characteristic, complemented

C₂×#14523964

C₂

yes

maximal, characteristic

C₂×#14523954

C₂

yes

maximal, characteristic, complemented

#16863302

C₂

yes

maximal, characteristic, complemented

#390×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

C₂²

yes

—

—

C₄

yes

complemented

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

—

—

—

—

C₆₇⋊#95

—

C₂²×D₄×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

C₂×C₄

yes

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

#11761616

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

—

C₂×C₆₇⋊C₈

D₄

yes

—

C₂×C₆₇⋊C₈

D₄

yes

—

C₂×C₆₇⋊C₈

D₄

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂⁴×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

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

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₆₇⋊C₈

—

C₆₇⋊C₈

C₂²⋊C₄

yes

complemented

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₄×D₄

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₂²×C₆₇

#137

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

100

C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

101

C₄×C₆₇

C₄×D₄

yes

characteristic

102

C₄×C₆₇

#121

yes

characteristic

103

C₄×C₆₇

#124

yes

characteristic

104

Dic₆₇

—

105

Dic₆₇

—

106

C₂×C₆₇

—

107

C₂×C₆₇

#390

yes

characteristic

108

C₂×C₆₇

#440

yes

—

109

C₂×C₆₇

#468

yes

—

110

C₂×C₆₇

#496

yes

characteristic

111

C₂×C₆₇

yes

112

—

113

C₆₇

#1973

yes

characteristic, complemented, Sylow

114

C₂²×SD₁₆

—

115

C₂×#107

—

116

C₂×#103

—

117

C₂×#102

—

118

C₂×#97

—

119

#391

—

120

#390

D₆₇

yes

characteristic

121

C₂²×Q₈

—

122

C₂²×D₄

C₂×D₆₇

yes

characteristic

123

C₂×SD₁₆

—

124

C₂×SD₁₆

—

134

—

125

C₂×SD₁₆

—

134

—

126

C₂²×C₈

—

127

C₂²×C₈

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

C₂×D₆₇

yes

characteristic

130

C₂×C₂²⋊C₄

—

131

C₂×C₄²

C₂×D₆₇

yes

characteristic

132

#107

—

134

—

133

#103

—

134

—

134

#102

—

134

—

135

#97

—

134

—

136

#95

—

134

—

137

C₂⁴

—

138

C₂×Q₈

—

139

C₂×Q₈

—

134

—

140

C₂×Q₈

—

134

—

141

C₂×Q₈

—

134

—

142

C₂×Q₈

—

134

—

143

C₂×Q₈

—

134

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

C₄×D₆₇

yes

—

150

C₂²×C₄

—

151

C₂²×C₄

—

134

—

152

C₂²×C₄

—

134

—

153

C₂²×C₄

C₂²⋊D₆₇

yes

—

154

C₂²×C₄

C₂²×D₆₇

yes

characteristic

155

SD₁₆

—

134

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

134

—

161

C₂×C₈

—

134

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

134

—

165

C₄⋊C₄

—

134

—

166

C₂²⋊C₄

—

167

C₄²

—

168

C₄²

—

169

C₂³

—

170

C₂³

—

171

C₂³

—

172

C₂³

—

173

C₂³

—

174

C₂³

C₂×C₂²⋊D₆₇

yes

characteristic

175

Q₈

—

268

—

176

Q₈

—

268

—

177

Q₈

—

134

—

178

Q₈

—

134

—

179

D₄

—

180

D₄

—

181

D₄

—

182

D₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

268

—

189

C₂×C₄

—

268

—

190

C₂×C₄

—

268

—

191

C₂×C₄

—

268

—

192

C₂×C₄

—

134

—

193

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

194

C₂×C₄

D₄×D₆₇

yes

characteristic

195

C₂×C₄

D₄×D₆₇

yes

characteristic

196

C₂×C₄

D₄×D₆₇

yes

characteristic

197

C₂×C₄

#11761627

yes

characteristic

198

C₂×C₄

#11761628

yes

characteristic

199

C₈

—

200

C₈

—

134

—

201

C₂²

—

202

C₂²

—

203

C₂²

—

204

C₂²

—

205

C₂²

—

206

C₂²

C₄×C₂²⋊D₆₇

yes

characteristic

207

C₂²

#14524015

yes

characteristic

208

C₂²

#14524039

yes

characteristic

209

C₂²

#14524069

yes

—

210

C₂²

#14524073

yes

—

211

C₄

—

212

C₄

—

213

C₄

—

268

—

214

C₄

—

536

—

215

C₄

C₂²⋊C₄×D₆₇

yes

characteristic

216

C₄

(C₂²⋊D₆₇)⋊C₄

yes

characteristic

217

C₄

#14524016

yes

characteristic

218

C₄

#14524029

yes

characteristic

219

C₂

—

220

C₂

#16863296

yes

characteristic

221

C₂

#16863496

yes

—

222

C₂

#16863640

yes

—

223

C₂

#16863783

yes

characteristic

224

C₂

#16863791

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#390×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	#390	D₆₇
hypocenter	C₆₇	#1973
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₆₇

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#390×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

N	H ≅ G/N	type
2. C₂×#14523976	C₂	semidirect
3. C₂×#14523966	C₂	semidirect
5. C₂×#14523954	C₂	semidirect
6. #16863302	C₂	semidirect
11. C₂×#11761616	C₄	semidirect
87. C₆₇⋊C₈	C₂²⋊C₄	semidirect
113. C₆₇	#1973	semidirect

H ≅ G/N

type

2. C₂×#14523976

C₂

semidirect

3. C₂×#14523966

C₂

semidirect

5. C₂×#14523954

C₂

semidirect

6. #16863302

C₂

semidirect

11. C₂×#11761616

C₄

semidirect

87. C₆₇⋊C₈

C₂²⋊C₄

semidirect

113. C₆₇

#1973

semidirect

Subgroups — #22561089

Conjugacy classes of subgroups

Complements