Group #22573291 — subgroups

Subgroups — #22573291

Order 8576.

356 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 113 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	#16864115	C₂	1	2	yes	maximal, complemented
2	#16863787	C₂	1	2	yes	maximal, complemented
3	#16863615	C₂	1	2	yes	maximal, complemented
4	#16863609	C₂	1	2	yes	maximal
5	#16863604	C₂	1	2	yes	maximal, complemented
6	#16863526	C₂	1	1	yes	maximal, characteristic, complemented
7	#16863497	C₂	1	2	yes	maximal, complemented
8	C₄×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic, complemented
9	#495×C₆₇	C₂	1	1	yes	maximal, characteristic
10	C₂×Dic₂₆₈	C₂²	1	1	yes	characteristic
11	C₂×#11761618	C₂²	1	2	yes	—
12	C₂×#11761616	C₂²	1	2	yes	—
13	#14524041	C₂²	1	2	yes	—
14	#14524036	—	2	2	no	—
15	#14524036	—	2	2	no	—
16	#14524030	C₂²	1	2	yes	—
17	#14524020	—	2	2	no	—
18	Q₈×Dic₆₇	C₂²	1	2	yes	—
19	C₄×Dic₁₃₄	C₂²	1	2	yes	—
20	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
21	D₄×Dic₆₇	C₂²	1	2	yes	—
22	#14523999	—	2	2	no	—
23	#14523977	C₂²	1	1	yes	characteristic
24	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
25	Dic₆₇⋊C₈	C₂²	1	2	yes	—
26	#14523968	C₂²	1	2	yes	—
27	#14523967	C₂²	1	2	yes	—
28	#14523964	C₂²	1	2	yes	—
29	#14523953	C₂²	1	2	yes	—
30	C₂×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×SD₁₆×C₆₇	C₂²	1	2	yes	—
32	C₂×D₈×C₆₇	C₂²	1	1	yes	characteristic
33	#124×C₆₇	C₂²	1	2	yes	—
34	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×Dic₁₃₄	D₄	1	1	yes	characteristic
36	C₂×Dic₁₃₄	C₂³	1	2	yes	—
37	C₂²×Dic₆₇	—	2	2	no	—
38	Dic₂₆₈	—	2	2	no	—
39	#11761618	—	4	2	no	—
40	#11761616	—	4	2	no	—
41	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
42	C₄⋊Dic₆₇	—	2	2	no	—
43	C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
44	C₄⋊Dic₆₇	C₂³	1	2	yes	—
45	Dic₆₇⋊C₄	—	2	2	no	—
46	Dic₆₇⋊C₄	—	2	2	no	—
47	Dic₆₇⋊C₄	—	2	2	no	—
48	Dic₆₇⋊C₄	D₄	1	2	yes	—
49	C₂²⋊Dic₆₇	—	2	2	no	—
50	C₂²⋊Dic₆₇	—	2	2	no	—
51	C₂²⋊Dic₆₇	—	2	2	no	—
52	C₄×Dic₆₇	—	2	2	no	—
53	C₄×Dic₆₇	—	2	2	no	—
54	C₄×Dic₆₇	C₂³	1	2	yes	—
55	C₂×Q₈×C₆₇	C₂³	1	2	yes	—
56	C₂×D₄×C₆₇	C₂³	1	2	yes	—
57	Q₁₆×C₆₇	—	4	1	no	—
58	SD₁₆×C₆₇	—	2	4	no	—
59	D₈×C₆₇	—	4	1	no	—
60	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₂²⋊C₄×C₆₇	—	2	2	no	—
63	C₂²⋊C₄×C₆₇	—	2	2	no	—
64	C₄²×C₆₇	C₂³	1	1	yes	characteristic
65	C₂³×C₆₇	—	2	2	no	—
66	Q₈×C₆₇	—	2	2	no	—
67	Q₈×C₆₇	—	2	2	no	—
68	D₄×C₆₇	—	2	2	no	—
69	D₄×C₆₇	—	2	2	no	—
70	C₂×C₄×C₆₇	—	2	2	no	—
71	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
74	C₂×Dic₆₇	—	2	2	no	—
75	C₂×Dic₆₇	—	2	2	no	—
76	C₂×Dic₆₇	—	4	4	no	—
77	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
78	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
79	Dic₁₃₄	—	2	2	no	—
80	Dic₁₃₄	—	2	2	no	—
81	Dic₁₃₄	—	4	1	no	—
82	C₈×C₆₇	—	2	1	no	—
83	C₈×C₆₇	#41	1	2	yes	—
84	C₆₇⋊C₈	—	2	2	no	—
85	C₂²×C₆₇	—	4	2	no	—
86	C₂²×C₆₇	—	4	2	no	—
87	C₂²×C₆₇	—	4	2	no	—
88	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
89	C₄×C₆₇	—	2	1	no	—
90	C₄×C₆₇	—	2	1	no	—
91	C₄×C₆₇	—	4	2	no	—
92	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
93	C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
94	Dic₆₇	—	2	2	no	—
95	Dic₆₇	—	2	2	no	—
96	Dic₆₇	—	4	2	no	—
97	Dic₆₇	—	4	2	no	—
98	C₂×C₆₇	—	8	2	no	—
99	C₂×C₆₇	#550	1	1	yes	characteristic
100	C₂×C₆₇	#575	1	1	yes	characteristic
101	C₂×C₆₇	#578	1	1	yes	characteristic
102	#3402	—	67	1	no	maximal, Sylow
103	C₆₇	#3402	1	1	yes	characteristic, complemented, Sylow
104	#556	—	67	2	no	—
105	#497	—	67	2	no	—
106	#495	D₆₇	1	1	yes	characteristic
107	#464	—	67	2	no	—
108	#463	—	67	2	no	—
109	#462	—	67	2	no	—
110	#446	—	67	1	no	—
111	#440	—	67	2	no	—
112	C₄×Q₁₆	—	67	1	no	—
113	C₂×Q₁₆	—	67	2	no	—
114	C₂×Q₁₆	—	67	1	no	—
115	C₂×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
116	C₂×SD₁₆	—	67	2	no	—
117	C₂×SD₁₆	C₂×D₆₇	1	2	yes	—
118	C₂×D₈	C₂×D₆₇	1	1	yes	characteristic
119	C₄⋊Q₈	—	67	2	no	—
120	#126	—	134	2	no	—
121	#126	—	134	2	no	—
122	#124	—	67	2	no	—
123	#124	C₂×D₆₇	1	2	yes	—
124	C₂²⋊Q₈	—	134	2	no	—
125	C₄×Q₈	—	67	1	no	—
126	C₄×Q₈	—	67	2	no	—
127	C₄×Q₈	—	67	2	no	—
128	C₄×D₄	—	67	2	no	—
129	#117	—	134	2	no	—
130	#107	—	67	1	no	—
131	C₄⋊C₈	—	67	2	no	—
132	C₄⋊C₈	—	67	1	no	—
133	#103	—	67	2	no	—
134	#103	—	67	2	no	—
135	#102	—	67	2	no	—
136	#97	—	67	2	no	—
137	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
138	C₂×Q₈	—	67	1	no	—
139	C₂×Q₈	—	67	2	no	—
140	C₂×Q₈	C₂²×D₆₇	1	2	yes	—
141	C₂×D₄	C₂²×D₆₇	1	2	yes	—
142	C₂²×C₄	—	134	2	no	—
143	Q₁₆	—	4	1	no	—
144	Q₁₆	—	268	2	no	—
145	Q₁₆	—	134	2	no	—
146	SD₁₆	—	2	4	no	—
147	SD₁₆	—	268	2	no	—
148	D₈	—	4	1	no	—
149	C₂×C₈	—	67	2	no	—
150	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
151	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
152	C₄⋊C₄	—	67	2	no	—
153	C₄⋊C₄	—	67	1	no	—
154	C₄⋊C₄	—	67	2	no	—
155	C₄⋊C₄	—	134	2	no	—
156	C₄⋊C₄	—	134	2	no	—
157	C₄⋊C₄	—	134	2	no	—
158	C₄⋊C₄	—	134	2	no	—
159	C₂²⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	2	no	—
161	C₂²⋊C₄	—	134	2	no	—
162	C₂²⋊C₄	—	134	2	no	—
163	C₂²⋊C₄	—	134	2	no	—
164	C₄²	—	67	2	no	—
165	C₄²	—	134	2	no	—
166	C₄²	—	134	2	no	—
167	C₄²	C₂²×D₆₇	1	1	yes	characteristic
168	C₂³	—	2	2	no	—
169	Q₈	—	2	2	no	—
170	Q₈	—	2	2	no	—
171	Q₈	—	268	1	no	—
172	Q₈	—	134	2	no	—
173	Q₈	—	134	2	no	—
174	D₄	—	2	2	no	—
175	D₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	268	4	no	—
178	C₂×C₄	—	67	2	no	—
179	C₂×C₄	—	67	2	no	—
180	C₂×C₄	—	134	2	no	—
181	C₂×C₄	—	134	2	no	—
182	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
183	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
184	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
185	C₈	—	2	1	no	—
186	C₈	—	134	2	no	—
187	C₈	#11761627	1	2	yes	—
188	C₂²	—	4	2	no	—
189	C₂²	—	4	2	no	—
190	C₂²	—	4	2	no	—
191	C₂²	C₂×D₄×D₆₇	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	—	4	2	no	—
195	C₄	—	268	2	no	—
196	C₄	—	268	2	no	—
197	C₄	—	134	2	no	—
198	C₄	—	134	2	no	—
199	C₄	C₂×#11761627	1	1	yes	characteristic
200	C₄	#14524088	1	1	yes	characteristic
201	C₂	—	8	2	no	—
202	C₂	#16864066	1	1	yes	characteristic
203	C₂	#16864193	1	1	yes	characteristic
204	C₂	#16864210	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16864115

C₂

yes

maximal, complemented

#16863787

C₂

yes

maximal, complemented

#16863615

C₂

yes

maximal, complemented

#16863609

C₂

yes

maximal

#16863604

C₂

yes

maximal, complemented

#16863526

C₂

yes

maximal, characteristic, complemented

#16863497

C₂

yes

maximal, complemented

C₄×Dic₂₆₈

C₂

yes

maximal, characteristic, complemented

#495×C₆₇

C₂

yes

maximal, characteristic

C₂×Dic₂₆₈

C₂²

yes

characteristic

C₂×#11761618

C₂²

yes

—

C₂×#11761616

C₂²

yes

—

#14524041

C₂²

yes

—

#14524036

—

#14524036

—

#14524030

C₂²

yes

—

#14524020

—

Q₈×Dic₆₇

C₂²

yes

—

C₄×Dic₁₃₄

C₂²

yes

—

C₄×Dic₁₃₄

C₂²

yes

characteristic

D₄×Dic₆₇

C₂²

yes

—

#14523999

—

#14523977

C₂²

yes

characteristic

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

Dic₆₇⋊C₈

C₂²

yes

—

#14523968

C₂²

yes

—

#14523967

C₂²

yes

—

#14523964

C₂²

yes

—

#14523953

C₂²

yes

—

C₂×Q₁₆×C₆₇

C₂²

yes

characteristic

C₂×SD₁₆×C₆₇

C₂²

yes

—

C₂×D₈×C₆₇

C₂²

yes

characteristic

#124×C₆₇

C₂²

yes

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

D₄

yes

characteristic

C₂×Dic₁₃₄

C₂³

yes

—

C₂²×Dic₆₇

—

Dic₂₆₈

—

#11761618

—

#11761616

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₄

yes

characteristic

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

D₄

yes

—

C₂²⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

C₂³

yes

—

C₂×Q₈×C₆₇

C₂³

yes

—

C₂×D₄×C₆₇

C₂³

yes

—

Q₁₆×C₆₇

—

SD₁₆×C₆₇

—

D₈×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₂²⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂×D₄

yes

—

C₂×Dic₆₇

C₂×D₄

yes

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

#41

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×#41

yes

characteristic

C₄×C₆₇

H(5,𝔽₂)

yes

characteristic

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#550

yes

characteristic

100

C₂×C₆₇

#575

yes

characteristic

101

C₂×C₆₇

yes

102

—

103

C₆₇

#3402

yes

characteristic, complemented, Sylow

104

#556

—

105

#497

—

106

#495

D₆₇

yes

characteristic

107

#464

—

108

#463

—

109

#462

—

110

#446

—

111

#440

—

112

C₄×Q₁₆

—

113

C₂×Q₁₆

—

114

C₂×Q₁₆

—

115

C₂×Q₁₆

C₂×D₆₇

yes

characteristic

116

C₂×SD₁₆

—

117

C₂×SD₁₆

C₂×D₆₇

yes

—

118

C₂×D₈

C₂×D₆₇

yes

characteristic

119

C₄⋊Q₈

—

120

#126

—

134

—

121

#126

—

134

—

122

#124

—

123

#124

C₂×D₆₇

yes

—

124

C₂²⋊Q₈

—

134

—

125

C₄×Q₈

—

126

C₄×Q₈

—

127

C₄×Q₈

—

128

C₄×D₄

—

129

#117

—

134

—

130

#107

—

131

C₄⋊C₈

—

132

C₄⋊C₈

—

133

#103

—

134

#103

—

135

#102

—

136

#97

—

137

C₄×C₈

C₂×D₆₇

yes

characteristic

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

C₂²×D₆₇

yes

—

141

C₂×D₄

C₂²×D₆₇

yes

—

142

C₂²×C₄

—

134

—

143

Q₁₆

—

144

Q₁₆

—

268

—

145

Q₁₆

—

134

—

146

SD₁₆

—

147

SD₁₆

—

268

—

148

D₈

—

149

C₂×C₈

—

150

C₂×C₈

C₂²×D₆₇

yes

characteristic

151

C₂×C₈

C₂²×D₆₇

yes

characteristic

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

134

—

156

C₄⋊C₄

—

134

—

157

C₄⋊C₄

—

134

—

158

C₄⋊C₄

—

134

—

159

C₂²⋊C₄

—

160

C₂²⋊C₄

—

161

C₂²⋊C₄

—

134

—

162

C₂²⋊C₄

—

134

—

163

C₂²⋊C₄

—

134

—

164

C₄²

—

165

C₄²

—

134

—

166

C₄²

—

134

—

167

C₄²

C₂²×D₆₇

yes

characteristic

168

C₂³

—

169

Q₈

—

170

Q₈

—

171

Q₈

—

268

—

172

Q₈

—

134

—

173

Q₈

—

134

—

174

D₄

—

175

D₄

—

176

C₂×C₄

—

177

C₂×C₄

—

268

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

134

—

181

C₂×C₄

—

134

—

182

C₂×C₄

D₄×D₆₇

yes

characteristic

183

C₂×C₄

D₄×D₆₇

yes

characteristic

184

C₂×C₄

C₂³×D₆₇

yes

characteristic

185

C₈

—

186

C₈

—

134

—

187

C₈

#11761627

yes

—

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

C₂×D₄×D₆₇

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

268

—

196

C₄

—

268

—

197

C₄

—

134

—

198

C₄

—

134

—

199

C₄

C₂×#11761627

yes

characteristic

200

C₄

#14524088

yes

characteristic

201

C₂

—

202

C₂

#16864066

yes

characteristic

203

C₂

#16864193

yes

characteristic

204

C₂

#16864210

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	#495×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	#495	D₆₇
hypocenter	C₆₇	#3402
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #495

subgroup

G/H

center

C₂²

C₂×D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

#495×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #495

N	H ≅ G/N	type
1. #16864115	C₂	semidirect
2. #16863787	C₂	semidirect
3. #16863615	C₂	semidirect
5. #16863604	C₂	semidirect
6. #16863526	C₂	semidirect
7. #16863497	C₂	semidirect
8. C₄×Dic₂₆₈	C₂	semidirect
103. C₆₇	#3402	semidirect

H ≅ G/N

type

1. #16864115

C₂

semidirect

2. #16863787

C₂

semidirect

3. #16863615

C₂

semidirect

5. #16863604

C₂

semidirect

6. #16863526

C₂

semidirect

7. #16863497

C₂

semidirect

8. C₄×Dic₂₆₈

C₂

semidirect

103. C₆₇

#3402

semidirect

Subgroups — #22573291

Conjugacy classes of subgroups

Complements