Group #22565162 — subgroups

Subgroups — #22565162

Order 8576.

676 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 197 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×#14524043	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14524022	C₂	1	4	yes	maximal, complemented
3	#16863332	C₂	1	4	yes	maximal, complemented
4	#16863299	C₂	1	2	yes	maximal
5	#16863265	C₂	1	2	yes	maximal, complemented
6	C₄×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₃₄	C₂²	1	2	yes	—
9	#14524043	—	2	4	no	—
10	#14524022	—	2	16	no	—
11	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
14	C₂×Dic₆₇⋊C₄	C₂²	1	8	yes	—
15	C₂×C₂²⋊Dic₆₇	C₂²	1	8	yes	—
16	C₂×C₄×Dic₆₇	C₂²	1	2	yes	—
17	C₆₇⋊#95	C₂²	1	2	yes	—
18	C₂²×D₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂³×C₄×C₆₇	C₂²	1	2	yes	—
20	C₄×D₄×C₆₇	—	2	4	no	—
21	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×Dic₁₃₄	—	2	8	no	—
25	C₂×Dic₁₃₄	—	2	4	no	—
26	C₂²×Dic₆₇	C₂³	1	4	yes	—
27	C₂²×Dic₆₇	C₂³	1	4	yes	—
28	C₄⋊Dic₆₇	—	2	4	no	—
29	C₄⋊Dic₆₇	—	2	4	no	—
30	C₄⋊Dic₆₇	D₄	1	8	yes	complemented
31	Dic₆₇⋊C₄	—	2	16	no	—
32	C₂²⋊Dic₆₇	—	2	16	no	—
33	C₄×Dic₆₇	—	2	4	no	—
34	C₂⁴×C₆₇	C₂³	1	2	yes	—
35	C₂×D₄×C₆₇	—	2	2	no	—
36	C₂×D₄×C₆₇	—	2	2	no	—
37	C₂×D₄×C₆₇	Q₈	1	4	yes	complemented
38	C₂²×C₄×C₆₇	—	2	4	no	—
39	C₂²×C₄×C₆₇	—	2	8	no	—
40	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
41	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₆₇	—	2	2	no	—
45	C₂²⋊C₄×C₆₇	—	2	4	no	—
46	C₄²×C₆₇	D₄	1	4	yes	—
47	C₂³×C₆₇	—	2	2	no	—
48	C₂³×C₆₇	—	2	4	no	—
49	C₂³×C₆₇	—	2	2	no	—
50	C₂³×C₆₇	—	2	4	no	—
51	C₂³×C₆₇	C₂×Q₈	1	4	yes	—
52	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
53	D₄×C₆₇	—	2	8	no	—
54	C₂×C₄×C₆₇	—	2	2	no	—
55	C₂×C₄×C₆₇	—	2	2	no	—
56	C₂×C₄×C₆₇	—	2	4	no	—
57	C₂×C₄×C₆₇	—	4	8	no	—
58	C₂×C₄×C₆₇	C₂×D₄	1	4	yes	—
59	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₆₇	#41	1	2	yes	—
64	C₂×Dic₆₇	—	2	4	no	—
65	C₂×Dic₆₇	—	2	4	no	—
66	C₂×Dic₆₇	—	2	8	no	—
67	C₂×Dic₆₇	—	2	4	no	—
68	C₂×Dic₆₇	C₂×D₄	1	8	yes	—
69	Dic₁₃₄	—	4	8	no	—
70	C₂²×C₆₇	—	2	4	no	—
71	C₂²×C₆₇	—	2	8	no	—
72	C₂²×C₆₇	—	4	2	no	—
73	C₂²×C₆₇	—	4	2	no	—
74	C₂²×C₆₇	—	4	4	no	—
75	C₂²×C₆₇	C₂²×D₄	1	2	yes	—
76	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
77	C₂²×C₆₇	C₂²×Q₈	1	1	yes	characteristic
78	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
79	C₂²×C₆₇	2₋¹⁺⁴	1	2	yes	—
80	C₄×C₆₇	—	2	2	no	—
81	C₄×C₆₇	—	4	2	no	—
82	C₄×C₆₇	C₂²⋊Q₈	1	4	yes	complemented
83	C₄×C₆₇	#127	1	4	yes	—
84	Dic₆₇	—	2	8	no	—
85	Dic₆₇	—	4	4	no	—
86	C₂×C₆₇	—	4	4	no	—
87	C₂×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×C₆₇	C₂×#127	1	1	yes	characteristic
89	C₂×C₆₇	#536	1	1	yes	characteristic
90	C₂×C₆₇	#547	1	2	yes	—
91	C₂×C₆₇	D₄×Q₈	1	2	yes	—
92	#2542	—	67	1	no	maximal, Sylow
93	C₆₇	#2542	1	1	yes	characteristic, complemented, Sylow
94	C₂×C₄⋊Q₈	—	67	1	no	—
95	C₂×C₂²⋊Q₈	—	67	4	no	—
96	C₂×C₄×D₄	D₆₇	1	1	yes	characteristic
97	#399	—	67	4	no	—
98	#390	—	67	2	no	—
99	#384	—	67	2	no	—
100	C₄×C₄⋊C₄	—	67	1	no	—
101	C₂²×Q₈	—	67	2	no	—
102	C₂²×D₄	C₂×D₆₇	1	1	yes	characteristic
103	C₂³×C₄	C₂×D₆₇	1	2	yes	—
104	C₄⋊Q₈	—	134	4	no	—
105	C₂²⋊Q₈	—	134	16	no	—
106	C₄×D₄	—	2	4	no	—
107	C₂×C₄⋊C₄	—	67	2	no	—
108	C₂×C₄⋊C₄	—	67	8	no	—
109	C₂×C₄⋊C₄	—	67	2	no	—
110	C₂×C₄⋊C₄	—	67	2	no	—
111	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
112	C₂×C₂²⋊C₄	—	67	8	no	—
113	C₂×C₂²⋊C₄	C₂×D₆₇	1	2	yes	—
114	C₂×C₄²	—	67	2	no	—
115	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
116	#95	—	67	2	no	—
117	C₂⁴	C₂²×D₆₇	1	2	yes	—
118	C₂×Q₈	—	134	8	no	—
119	C₂×Q₈	—	134	4	no	—
120	C₂×D₄	—	2	2	no	—
121	C₂×D₄	—	2	2	no	—
122	C₂×D₄	Dic₁₃₄	1	4	yes	complemented
123	C₂²×C₄	—	2	8	no	—
124	C₂²×C₄	—	2	4	no	—
125	C₂²×C₄	—	67	4	no	—
126	C₂²×C₄	—	67	4	no	—
127	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
128	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
129	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
130	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	67	8	no	—
133	C₄⋊C₄	—	134	16	no	—
134	C₄⋊C₄	—	134	4	no	—
135	C₄⋊C₄	—	134	4	no	—
136	C₂²⋊C₄	—	2	4	no	—
137	C₂²⋊C₄	—	134	16	no	—
138	C₄²	—	134	4	no	—
139	C₄²	C₂²⋊D₆₇	1	4	yes	—
140	C₂³	—	2	4	no	—
141	C₂³	—	2	2	no	—
142	C₂³	—	2	2	no	—
143	C₂³	—	2	4	no	—
144	C₂³	C₂×Dic₁₃₄	1	4	yes	—
145	C₂³	C₂³×D₆₇	1	1	yes	characteristic
146	Q₈	—	268	8	no	—
147	D₄	—	2	8	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	8	no	—
152	C₂×C₄	—	67	8	no	—
153	C₂×C₄	—	134	8	no	—
154	C₂×C₄	—	134	4	no	—
155	C₂×C₄	—	134	4	no	—
156	C₂×C₄	—	134	4	no	—
157	C₂×C₄	D₄×D₆₇	1	4	yes	—
158	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
159	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
160	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
161	C₂×C₄	C₂×Dic₁₃₄	1	2	yes	—
162	C₂×C₄	#11761628	1	2	yes	—
163	C₂²	—	2	8	no	—
164	C₂²	—	2	4	no	—
165	C₂²	—	4	4	no	—
166	C₂²	—	4	2	no	—
167	C₂²	—	4	2	no	—
168	C₂²	C₂×D₄×D₆₇	1	2	yes	—
169	C₂²	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
170	C₂²	C₂²×Dic₁₃₄	1	1	yes	characteristic
171	C₂²	C₂×#11761628	1	1	yes	characteristic
172	C₂²	#14524090	1	2	yes	—
173	C₄	—	2	2	no	—
174	C₄	—	4	2	no	—
175	C₄	—	268	4	no	—
176	C₄	—	134	8	no	—
177	C₄	#14524022	1	4	yes	complemented
178	C₄	#14524039	1	4	yes	—
179	C₂	—	4	4	no	—
180	C₂	C₂×#14524022	1	1	yes	characteristic
181	C₂	C₂×#14524039	1	1	yes	characteristic
182	C₂	#16863967	1	1	yes	characteristic
183	C₂	#16864051	1	2	yes	—
184	C₂	D₄×Dic₁₃₄	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524043

C₂

yes

maximal, characteristic, complemented

C₂×#14524022

C₂

yes

maximal, complemented

#16863332

C₂

yes

maximal, complemented

#16863299

C₂

yes

maximal

#16863265

C₂

yes

maximal, complemented

C₄×C₄⋊Dic₆₇

C₂

yes

maximal, characteristic, complemented

C₂×C₄×D₄×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

C₂²

yes

—

#14524043

—

#14524022

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×Dic₆₇⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₆₇

C₂²

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₂²×D₄×C₆₇

C₂²

yes

characteristic

C₂³×C₄×C₆₇

C₂²

yes

—

C₄×D₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

C₂³

yes

—

C₂²×Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₄

yes

complemented

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂⁴×C₆₇

C₂³

yes

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

Q₈

yes

complemented

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

D₄

yes

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

C₂×Q₈

yes

—

C₂³×C₆₇

C₂⁴

yes

characteristic

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂×D₄

yes

—

Dic₁₃₄

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂²×D₄

yes

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₂²×C₆₇

C₂²×Q₈

yes

characteristic

C₂²×C₆₇

C₂×#41

yes

characteristic

C₂²×C₆₇

2₋¹⁺⁴

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂²⋊Q₈

yes

complemented

C₄×C₆₇

#127

yes

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

C₂×C₂²⋊Q₈

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×C₆₇

D₄×Q₈

yes

—

#2542

—

maximal, Sylow

C₆₇

#2542

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×C₂²⋊Q₈

—

C₂×C₄×D₄

D₆₇

yes

—

—

—

100

C₄×C₄⋊C₄

—

101

C₂²×Q₈

—

102

C₂²×D₄

C₂×D₆₇

yes

characteristic

103

C₂³×C₄

C₂×D₆₇

yes

—

104

C₄⋊Q₈

—

134

—

105

C₂²⋊Q₈

—

134

—

106

C₄×D₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

C₂×D₆₇

yes

characteristic

112

C₂×C₂²⋊C₄

—

113

C₂×C₂²⋊C₄

C₂×D₆₇

yes

—

114

C₂×C₄²

—

115

C₂×C₄²

C₂×D₆₇

yes

characteristic

116

#95

—

117

C₂⁴

C₂²×D₆₇

yes

—

118

C₂×Q₈

—

134

—

119

C₂×Q₈

—

134

—

120

C₂×D₄

—

121

C₂×D₄

—

122

C₂×D₄

Dic₁₃₄

yes

complemented

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

C₂²×D₆₇

yes

—

128

C₂²×C₄

C₂²×D₆₇

yes

characteristic

129

C₂²×C₄

C₂²×D₆₇

yes

characteristic

130

C₂²×C₄

C₂²×D₆₇

yes

characteristic

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

—

134

C₄⋊C₄

—

134

—

135

C₄⋊C₄

—

134

—

136

C₂²⋊C₄

—

137

C₂²⋊C₄

—

134

—

138

C₄²

—

134

—

139

C₄²

C₂²⋊D₆₇

yes

—

140

C₂³

—

141

C₂³

—

142

C₂³

—

143

C₂³

—

144

C₂³

C₂×Dic₁₃₄

yes

—

145

C₂³

C₂³×D₆₇

yes

characteristic

146

Q₈

—

268

—

147

D₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

134

—

154

C₂×C₄

—

134

—

155

C₂×C₄

—

134

—

156

C₂×C₄

—

134

—

157

C₂×C₄

D₄×D₆₇

yes

—

158

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

159

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

160

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

161

C₂×C₄

C₂×Dic₁₃₄

yes

—

162

C₂×C₄

#11761628

yes

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

C₂×D₄×D₆₇

yes

—

169

C₂²

C₂²×C₂²⋊D₆₇

yes

characteristic

170

C₂²

C₂²×Dic₁₃₄

yes

characteristic

171

C₂²

C₂×#11761628

yes

characteristic

172

C₂²

#14524090

yes

—

173

C₄

—

174

C₄

—

175

C₄

—

268

—

176

C₄

—

134

—

177

C₄

#14524022

yes

complemented

178

C₄

#14524039

yes

—

179

C₂

—

180

C₂

C₂×#14524022

yes

characteristic

181

C₂

C₂×#14524039

yes

characteristic

182

C₂

#16863967

yes

characteristic

183

C₂

#16864051

yes

—

184

C₂

D₄×Dic₁₃₄

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂⁴
Fitting	C₂×C₄×D₄×C₆₇	C₂
hypercenter	C₂×C₄×D₄	D₆₇
hypocenter	C₆₇	#2542
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₇

commutator, socle

C₂³×C₆₇

C₂⁴

Fitting

C₂×C₄×D₄×C₆₇

C₂

hypercenter

C₂×C₄×D₄

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₆₇

C₁ ◁ C₂³ ◁ C₂×C₄×D₄

N	H ≅ G/N	type
1. C₂×#14524043	C₂	semidirect
2. C₂×#14524022	C₂	semidirect
3. #16863332	C₂	semidirect
5. #16863265	C₂	semidirect
6. C₄×C₄⋊Dic₆₇	C₂	semidirect
30. C₄⋊Dic₆₇	D₄	semidirect
37. C₂×D₄×C₆₇	Q₈	semidirect
82. C₄×C₆₇	C₂²⋊Q₈	semidirect
93. C₆₇	#2542	semidirect
122. C₂×D₄	Dic₁₃₄	semidirect
177. C₄	#14524022	semidirect

H ≅ G/N

type

1. C₂×#14524043

C₂

semidirect

2. C₂×#14524022

C₂

semidirect

3. #16863332

C₂

semidirect

5. #16863265

C₂

semidirect

6. C₄×C₄⋊Dic₆₇

C₂

semidirect

30. C₄⋊Dic₆₇

D₄

semidirect

37. C₂×D₄×C₆₇

Q₈

semidirect

82. C₄×C₆₇

C₂²⋊Q₈

semidirect

93. C₆₇

#2542

semidirect

122. C₂×D₄

Dic₁₃₄

semidirect

177. C₄

#14524022

semidirect

Subgroups — #22565162

Conjugacy classes of subgroups

Complements