Group #36466924 — subgroups

Subgroups — #36466924

Order 22784.

330 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 59 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	#26031706	C₂	1	1	yes	maximal, characteristic
2	#26022186	C₂	1	2	yes	maximal
3	#26018269	C₂	1	2	yes	maximal
4	#26018194	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#2047	C₂	1	1	yes	maximal, characteristic
6	#19152210	—	2	1	no	—
7	C₂×#15667958	—	2	2	no	—
8	#19151906	—	2	2	no	—
9	#19151856	—	2	1	no	—
10	#19151839	C₂²	1	2	yes	—
11	#19151809	—	2	2	no	—
12	#19151402	—	2	2	no	—
13	C₂²⋊#13083549	C₂²	1	2	yes	—
14	C₂²×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	#424×C₈₉	C₂²	1	1	yes	characteristic
16	#419×C₈₉	—	2	1	no	—
17	C₂×#101×C₈₉	—	2	1	no	—
18	#341×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₃₅₆	—	2	1	no	—
20	C₂×Dic₃₅₆	D₄	1	2	yes	—
21	#15668012	—	2	1	no	—
22	#15668005	—	4	2	no	—
23	#15668004	—	2	1	no	—
24	#15668002	—	4	2	no	—
25	#15667991	—	2	2	no	—
26	C₄×Dic₁₇₈	—	2	2	no	—
27	#15667958	—	4	4	no	—
28	C₂×#13083549	—	2	2	no	—
29	#15667946	D₄	1	2	yes	—
30	#15667944	—	2	1	no	—
31	#15667937	—	2	2	no	—
32	#15667937	—	2	2	no	—
33	C₂²×Q₈×C₈₉	—	2	1	no	—
34	C₂×Q₁₆×C₈₉	—	2	2	no	—
35	C₂×Q₁₆×C₈₉	—	4	2	no	—
36	C₂×M₁₆×C₈₉	—	2	1	no	—
37	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
39	#117×C₈₉	D₄	1	1	yes	characteristic
40	C₄⋊C₈×C₈₉	—	2	2	no	—
41	#103×C₈₉	—	4	2	no	—
42	#101×C₈₉	—	4	2	no	—
43	C₂×Dic₁₇₈	—	2	2	no	—
44	Dic₃₅₆	—	4	2	no	—
45	Dic₃₅₆	—	8	1	no	—
46	C₄⋊Dic₈₉	—	2	2	no	—
47	C₄⋊Dic₈₉	—	4	2	no	—
48	#13083553	—	4	2	no	—
49	#13083553	—	4	2	no	—
50	#13083553	—	4	2	no	—
51	C₂²⋊Dic₈₉	—	4	2	no	—
52	C₄×Dic₈₉	—	4	2	no	—
53	#13083549	—	4	2	no	—
54	C₂×Q₈×C₈₉	—	2	2	no	—
55	C₂×Q₈×C₈₉	—	4	1	no	—
56	C₂×Q₈×C₈₉	—	4	1	no	—
57	C₂×Q₈×C₈₉	—	4	1	no	—
58	C₂×Q₈×C₈₉	—	4	1	no	—
59	C₂×Q₈×C₈₉	—	4	1	no	—
60	C₂²×C₄×C₈₉	—	4	1	no	—
61	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
62	Q₁₆×C₈₉	—	2	4	no	—
63	Q₁₆×C₈₉	—	4	2	no	—
64	M₁₆×C₈₉	—	4	1	no	—
65	M₁₆×C₈₉	—	4	1	no	—
66	M₁₆×C₈₉	—	4	1	no	—
67	C₂×C₈×C₈₉	—	2	1	no	—
68	C₂×C₈×C₈₉	—	4	1	no	—
69	C₂×C₈×C₈₉	—	4	1	no	—
70	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
71	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
72	C₂²⋊C₄×C₈₉	—	2	1	no	—
73	C₄²×C₈₉	—	2	1	no	—
74	C₂³×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
75	Q₈×C₈₉	—	4	2	no	—
76	Q₈×C₈₉	—	4	2	no	—
77	Q₈×C₈₉	—	8	1	no	—
78	Q₈×C₈₉	—	8	1	no	—
79	C₂×C₄×C₈₉	—	2	1	no	—
80	C₂×C₄×C₈₉	—	2	1	no	—
81	C₂×C₄×C₈₉	—	2	2	no	—
82	C₂×C₄×C₈₉	—	4	2	no	—
83	C₂×C₄×C₈₉	—	8	1	no	—
84	C₂×C₄×C₈₉	—	8	1	no	—
85	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
86	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
87	C₂×Dic₈₉	—	4	2	no	—
88	C₂×Dic₈₉	—	4	2	no	—
89	Dic₁₇₈	—	4	2	no	—
90	Dic₁₇₈	—	4	2	no	—
91	C₈×C₈₉	—	2	2	no	—
92	C₈×C₈₉	—	4	1	no	—
93	C₈×C₈₉	—	8	1	no	—
94	C₂²×C₈₉	—	2	1	no	—
95	C₂²×C₈₉	—	2	2	no	—
96	C₂²×C₈₉	#447	1	1	yes	characteristic
97	C₄×C₈₉	—	2	1	no	—
98	C₄×C₈₉	—	4	2	no	—
99	C₄×C₈₉	—	8	2	no	—
100	C₄×C₈₉	#454	1	1	yes	characteristic
101	C₄×C₈₉	B(4,𝔽₂)	1	1	yes	characteristic
102	Dic₈₉	—	8	2	no	—
103	Dic₈₉	—	8	2	no	—
104	#12168	—	89	1	no	maximal, Sylow
105	C₂×C₈₉	—	4	1	no	—
106	C₂×C₈₉	#1598	1	1	yes	characteristic
107	C₂×C₈₉	#2195	1	2	yes	—
108	#3261	—	89	1	no	—
109	#2188	—	89	2	no	—
110	#2047	D₈₉	1	1	yes	characteristic
111	#1350	—	89	2	no	—
112	#1323	—	89	1	no	—
113	C₈₉	#12168	1	1	yes	characteristic, complemented, Sylow
114	C₂²×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
115	#541	—	178	1	no	—
116	C₂×Q₃₂	—	178	2	no	—
117	#479	—	178	2	no	—
118	#470	—	178	1	no	—
119	#467	—	89	2	no	—
120	#462	—	178	2	no	—
121	#424	C₂×D₈₉	1	1	yes	characteristic
122	#419	—	2	1	no	—
123	C₂×#101	—	2	1	no	—
124	#358	—	178	2	no	—
125	C₂²⋊C₁₆	—	89	2	no	—
126	#341	C₂×D₈₉	1	1	yes	characteristic
127	C₂²×Q₈	—	2	1	no	—
128	C₂×Q₁₆	—	2	2	no	—
129	C₂×Q₁₆	—	4	2	no	—
130	C₂×Q₁₆	—	89	2	no	—
131	C₂×Q₁₆	—	178	1	no	—
132	C₂×M₁₆	—	2	1	no	—
133	C₂×M₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
134	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
135	C₄⋊Q₈	—	178	1	no	—
136	#126	—	356	2	no	—
137	#125	—	356	2	no	—
138	#125	—	178	1	no	—
139	C₂²⋊Q₈	—	178	2	no	—
140	C₄×Q₈	—	178	2	no	—
141	#117	C₂²⋊D₈₉	1	1	yes	characteristic
142	Q₃₂	—	356	4	no	—
143	C₂×C₁₆	—	178	2	no	—
144	#107	—	89	2	no	—
145	#106	—	178	1	no	—
146	C₄⋊C₈	—	2	2	no	—
147	#103	—	4	2	no	—
148	#103	—	178	2	no	—
149	#103	—	178	2	no	—
150	#101	—	4	2	no	—
151	C₂×Q₈	—	2	2	no	—
152	C₂×Q₈	—	4	1	no	—
153	C₂×Q₈	—	4	1	no	—
154	C₂×Q₈	—	4	1	no	—
155	C₂×Q₈	—	4	1	no	—
156	C₂×Q₈	—	4	1	no	—
157	C₂×Q₈	—	178	2	no	—
158	C₂²×C₄	—	4	1	no	—
159	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
160	Q₁₆	—	2	4	no	—
161	Q₁₆	—	4	2	no	—
162	Q₁₆	—	356	2	no	—
163	Q₁₆	—	712	1	no	—
164	M₁₆	—	4	1	no	—
165	M₁₆	—	4	1	no	—
166	M₁₆	—	4	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	4	1	no	—
169	C₂×C₈	—	4	1	no	—
170	C₂×C₈	D₄×D₈₉	1	2	yes	—
171	C₄⋊C₄	—	356	2	no	—
172	C₄⋊C₄	—	356	2	no	—
173	C₄⋊C₄	—	356	2	no	—
174	C₄⋊C₄	—	356	2	no	—
175	C₄⋊C₄	—	178	2	no	—
176	C₄⋊C₄	#13083559	1	2	yes	—
177	C₂²⋊C₄	—	2	1	no	—
178	C₂²⋊C₄	—	356	2	no	—
179	C₄²	—	2	1	no	—
180	C₄²	—	356	2	no	—
181	C₁₆	—	356	2	no	—
182	C₂³	#15668037	1	1	yes	characteristic
183	Q₈	—	4	2	no	—
184	Q₈	—	4	2	no	—
185	Q₈	—	8	1	no	—
186	Q₈	—	8	1	no	—
187	Q₈	—	356	2	no	—
188	Q₈	—	356	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	8	1	no	—
194	C₂×C₄	—	8	1	no	—
195	C₂×C₄	—	356	2	no	—
196	C₂×C₄	—	356	2	no	—
197	C₂×C₄	#15667980	1	1	yes	characteristic
198	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
199	C₈	—	2	2	no	—
200	C₈	—	4	1	no	—
201	C₈	—	8	1	no	—
202	C₂²	—	2	2	no	—
203	C₂²	—	2	1	no	—
204	C₂²	#19151731	1	1	yes	characteristic
205	C₄	—	2	1	no	—
206	C₄	—	4	2	no	—
207	C₄	—	8	2	no	—
208	C₄	—	712	2	no	—
209	C₄	—	712	2	no	—
210	C₄	#19151770	1	1	yes	characteristic
211	C₄	#19151784	1	1	yes	characteristic
212	C₂	—	4	1	no	—
213	C₂	#26019186	1	1	yes	characteristic
214	C₂	#26022228	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26031706

C₂

yes

maximal, characteristic

#26022186

C₂

yes

maximal

#26018269

C₂

yes

maximal

#26018194

C₂

yes

maximal, characteristic

C₈₉×#2047

C₂

yes

maximal, characteristic

—

—

—

—

C₂²

yes

—

#19151809

—

#19151402

—

C₂²⋊#13083549

C₂²

yes

—

C₂²×Q₁₆×C₈₉

C₂²

yes

characteristic

#424×C₈₉

C₂²

yes

characteristic

#419×C₈₉

—

C₂×#101×C₈₉

—

#341×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

D₄

yes

—

—

—

—

—

—

C₄×Dic₁₇₈

—

#15667958

—

C₂×#13083549

—

#15667946

D₄

yes

—

#15667944

—

#15667937

—

#15667937

—

C₂²×Q₈×C₈₉

—

C₂×Q₁₆×C₈₉

—

C₂×Q₁₆×C₈₉

—

C₂×M₁₆×C₈₉

—

C₂×M₁₆×C₈₉

D₄

yes

characteristic

C₂²×C₈×C₈₉

C₂³

yes

characteristic

#117×C₈₉

D₄

yes

characteristic

C₄⋊C₈×C₈₉

—

#103×C₈₉

—

#101×C₈₉

—

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

#13083553

—

#13083553

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

#13083549

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

M₁₆×C₈₉

—

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₄⋊C₄×C₈₉

D₈

yes

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₂³×C₈₉

AGL(1,ℤ/8)

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#120

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#447

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

100

C₄×C₈₉

#454

yes

characteristic

101

C₄×C₈₉

B(4,𝔽₂)

yes

characteristic

102

Dic₈₉

—

103

Dic₈₉

—

104

#12168

—

maximal, Sylow

105

C₂×C₈₉

—

106

C₂×C₈₉

#1598

yes

characteristic

107

C₂×C₈₉

#2195

yes

—

108

#3261

—

109

#2188

—

110

#2047

D₈₉

yes

characteristic

111

#1350

—

112

#1323

—

113

C₈₉

#12168

yes

characteristic, complemented, Sylow

114

C₂²×Q₁₆

C₂×D₈₉

yes

characteristic

115

#541

—

178

—

116

C₂×Q₃₂

—

178

—

117

#479

—

178

—

118

#470

—

178

—

119

#467

—

120

#462

—

178

—

121

#424

C₂×D₈₉

yes

characteristic

122

#419

—

123

C₂×#101

—

124

#358

—

178

—

125

C₂²⋊C₁₆

—

126

#341

C₂×D₈₉

yes

characteristic

127

C₂²×Q₈

—

128

C₂×Q₁₆

—

129

C₂×Q₁₆

—

130

C₂×Q₁₆

—

131

C₂×Q₁₆

—

178

—

132

C₂×M₁₆

—

133

C₂×M₁₆

C₂²⋊D₈₉

yes

characteristic

134

C₂²×C₈

C₂²×D₈₉

yes

characteristic

135

C₄⋊Q₈

—

178

—

136

#126

—

356

—

137

#125

—

356

—

138

#125

—

178

—

139

C₂²⋊Q₈

—

178

—

140

C₄×Q₈

—

178

—

141

#117

C₂²⋊D₈₉

yes

characteristic

142

Q₃₂

—

356

—

143

C₂×C₁₆

—

178

—

144

#107

—

145

#106

—

178

—

146

C₄⋊C₈

—

147

#103

—

148

#103

—

178

—

149

#103

—

178

—

150

#101

—

151

C₂×Q₈

—

152

C₂×Q₈

—

153

C₂×Q₈

—

154

C₂×Q₈

—

155

C₂×Q₈

—

156

C₂×Q₈

—

157

C₂×Q₈

—

178

—

158

C₂²×C₄

—

159

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

160

Q₁₆

—

161

Q₁₆

—

162

Q₁₆

—

356

—

163

Q₁₆

—

712

—

164

M₁₆

—

165

M₁₆

—

166

M₁₆

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

D₄×D₈₉

yes

—

171

C₄⋊C₄

—

356

—

172

C₄⋊C₄

—

356

—

173

C₄⋊C₄

—

356

—

174

C₄⋊C₄

—

356

—

175

C₄⋊C₄

—

178

—

176

C₄⋊C₄

#13083559

yes

—

177

C₂²⋊C₄

—

178

C₂²⋊C₄

—

356

—

179

C₄²

—

180

C₄²

—

356

—

181

C₁₆

—

356

—

182

C₂³

#15668037

yes

characteristic

183

Q₈

—

184

Q₈

—

185

Q₈

—

186

Q₈

—

187

Q₈

—

356

—

188

Q₈

—

356

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

356

—

196

C₂×C₄

—

356

—

197

C₂×C₄

#15667980

yes

characteristic

198

C₂×C₄

C₂×#13083559

yes

characteristic

199

C₈

—

200

C₈

—

201

C₈

—

202

C₂²

—

203

C₂²

—

204

C₂²

#19151731

yes

characteristic

205

C₄

—

206

C₄

—

207

C₄

—

208

C₄

—

712

—

209

C₄

—

712

—

210

C₄

#19151770

yes

characteristic

211

C₄

#19151784

yes

characteristic

212

C₂

—

213

C₂

#26019186

yes

characteristic

214

C₂

#26022228

yes

—

	subgroup	G/H
center	C₂²	#19151731
commutator, cosocle	C₂²×C₈×C₈₉	C₂³
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#2047	C₂
socle	C₂²×C₈₉	#447
prime radical	C₂³×C₈₉	AGL(1,ℤ/8)
hypercenter	#2047	D₈₉
hypocenter	C₈₉	#12168
LCS	G ▷ C₂²×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424 ◁ #2047

subgroup

G/H

center

C₂²

#19151731

commutator, cosocle

C₂²×C₈×C₈₉

C₂³

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₈₉×#2047

C₂

C₂²×C₈₉

C₂³×C₈₉

D₈₉

C₈₉

G ▷ C₂²×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424 ◁ #2047

N	H ≅ G/N	type
113. C₈₉	#12168	semidirect

H ≅ G/N

type

113. C₈₉

#12168

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#26031706	2
#26022186	2
#26018269	2
#26018194	2
C₈₉×#2047	2
#12168	89

subgroup

index

C₈₉×#2047

Subgroups — #36466924

Conjugacy classes of subgroups

Complements

Maximal subgroups