Group #36467846 — subgroups

Subgroups — #36467846

Order 22784.

362 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 67 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	#26022333	C₂	1	2	yes	maximal, complemented
2	#26019921	C₂	1	1	yes	maximal, characteristic, complemented
3	#26018293	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018228	C₂	1	2	yes	maximal, complemented
5	C₈₉×#3291	C₂	1	1	yes	maximal, characteristic
6	#19152038	—	2	2	no	—
7	#19152001	—	2	1	no	—
8	#19151978	C₂²	1	1	yes	characteristic, complemented
9	#19151944	—	2	1	no	—
10	#19151419	C₂²	1	2	yes	complemented
11	#19151413	—	2	2	no	—
12	#19151319	C₂²	1	1	yes	characteristic, complemented
13	#577×C₈₉	—	2	2	no	—
14	#492×C₈₉	C₂²	1	1	yes	characteristic
15	#471×C₈₉	—	2	2	no	—
16	#456×C₈₉	—	2	1	no	—
17	#453×C₈₉	—	2	1	no	—
18	#443×C₈₉	C₂²	1	2	yes	—
19	C₂×Dic₃₅₆	—	2	2	no	—
20	#15668012	—	2	1	no	—
21	#15667958	—	4	2	no	—
22	#15667956	—	4	2	no	—
23	#15667952	—	2	2	no	—
24	#15667945	—	4	2	no	—
25	#15667933	—	4	2	no	—
26	C₄×#10623494	—	2	1	no	—
27	2₋¹⁺⁴×C₈₉	—	2	1	no	—
28	H(5,𝔽₂)×C₈₉	—	2	1	no	—
29	#137×C₈₉	—	2	2	no	—
30	#137×C₈₉	—	4	2	no	—
31	AGL(1,ℤ/8)×C₈₉	—	2	2	no	—
32	AGL(1,ℤ/8)×C₈₉	—	4	2	no	—
33	#135×C₈₉	—	4	2	no	—
34	#135×C₈₉	D₄	1	2	yes	—
35	C₂×SD₁₆×C₈₉	—	2	2	no	—
36	C₂×SD₁₆×C₈₉	—	4	2	no	—
37	#131×C₈₉	—	2	2	no	—
38	C₄⋊Q₈×C₈₉	D₄	1	1	yes	characteristic
39	#127×C₈₉	D₄	1	1	yes	characteristic
40	#108×C₈₉	D₄	1	2	yes	—
41	#104×C₈₉	—	2	2	no	—
42	#101×C₈₉	—	2	1	no	—
43	#100×C₈₉	—	2	1	no	—
44	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
45	C₂×Dic₁₇₈	—	4	1	no	—
46	Dic₃₅₆	—	4	2	no	—
47	Dic₃₅₆	—	4	2	no	—
48	C₂×#10623494	—	2	2	no	—
49	C₄⋊Dic₈₉	—	4	1	no	—
50	C₄⋊Dic₈₉	—	4	1	no	—
51	#13083549	—	4	2	no	—
52	#41×C₈₉	—	2	2	no	—
53	#41×C₈₉	—	4	2	no	—
54	#41×C₈₉	—	4	2	no	—
55	#41×C₈₉	—	8	1	no	—
56	C₂×Q₈×C₈₉	—	2	1	no	—
57	C₂×Q₈×C₈₉	—	4	2	no	—
58	C₂×D₄×C₈₉	—	2	1	no	—
59	C₂×D₄×C₈₉	—	4	1	no	—
60	C₂×D₄×C₈₉	—	4	2	no	—
61	C₂×D₄×C₈₉	—	8	1	no	—
62	Q₁₆×C₈₉	—	4	2	no	—
63	Q₁₆×C₈₉	D₈	1	2	yes	—
64	SD₁₆×C₈₉	—	2	2	no	—
65	SD₁₆×C₈₉	—	4	2	no	—
66	SD₁₆×C₈₉	—	4	2	no	—
67	SD₁₆×C₈₉	—	4	2	no	—
68	SD₁₆×C₈₉	—	4	2	no	—
69	D₈×C₈₉	—	4	2	no	—
70	D₈×C₈₉	D₈	1	2	yes	—
71	M₁₆×C₈₉	—	2	2	no	—
72	M₁₆×C₈₉	—	4	2	no	—
73	C₂×C₈×C₈₉	—	4	2	no	—
74	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
75	C₄⋊C₄×C₈₉	—	4	1	no	—
76	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
77	C₂³×C₈₉	—	4	1	no	—
78	C₂³×C₈₉	—	8	1	no	—
79	Q₈×C₈₉	—	2	2	no	—
80	Q₈×C₈₉	—	4	2	no	—
81	Q₈×C₈₉	—	8	1	no	—
82	D₄×C₈₉	—	2	2	no	—
83	D₄×C₈₉	—	4	2	no	—
84	D₄×C₈₉	—	4	2	no	—
85	D₄×C₈₉	—	8	2	no	—
86	D₄×C₈₉	—	8	2	no	—
87	D₄×C₈₉	—	8	1	no	—
88	C₂×C₄×C₈₉	—	2	1	no	—
89	C₂×C₄×C₈₉	—	4	2	no	—
90	C₂×C₄×C₈₉	—	4	1	no	—
91	C₂×C₄×C₈₉	—	8	1	no	—
92	C₂×C₄×C₈₉	—	8	2	no	—
93	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
94	C₂×Dic₈₉	—	8	1	no	—
95	Dic₁₇₈	—	8	2	no	—
96	C₈×C₈₉	—	4	2	no	—
97	C₈×C₈₉	C₂×D₈	1	2	yes	—
98	C₈×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
99	#10623494	—	4	2	no	—
100	C₂²×C₈₉	—	4	2	no	—
101	C₂²×C₈₉	—	8	1	no	—
102	C₂²×C₈₉	—	16	1	no	—
103	C₂²×C₈₉	—	32	1	no	—
104	C₂²×C₈₉	#453	1	1	yes	characteristic
105	C₄×C₈₉	—	2	2	no	—
106	C₄×C₈₉	—	4	2	no	—
107	C₄×C₈₉	—	8	1	no	—
108	C₄×C₈₉	#447	1	2	yes	—
109	Dic₈₉	—	16	1	no	—
110	#12315	—	89	1	no	maximal, Sylow
111	C₂×C₈₉	—	2	1	no	—
112	C₂×C₈₉	—	8	2	no	—
113	C₂×C₈₉	—	16	1	no	—
114	C₂×C₈₉	#1619	1	1	yes	characteristic
115	#3291	D₈₉	1	1	yes	characteristic
116	#2214	—	89	2	no	—
117	#1722	—	89	1	no	—
118	#1359	—	89	1	no	—
119	#1336	—	89	2	no	—
120	C₈₉	#12315	1	1	yes	characteristic, complemented, Sylow
121	#577	—	2	2	no	—
122	#510	—	178	2	no	—
123	#500	—	178	1	no	—
124	#494	—	89	1	no	—
125	#492	C₂×D₈₉	1	1	yes	characteristic
126	#486	—	178	1	no	—
127	#471	—	2	2	no	—
128	#456	—	2	1	no	—
129	#453	—	2	1	no	—
130	#443	C₂×D₈₉	1	2	yes	—
131	#364	—	89	2	no	—
132	#362	—	178	2	no	—
133	#322	—	89	1	no	—
134	2₋¹⁺⁴	—	2	1	no	—
135	H(5,𝔽₂)	—	2	1	no	—
136	#137	—	2	2	no	—
137	#137	—	4	2	no	—
138	AGL(1,ℤ/8)	—	2	2	no	—
139	AGL(1,ℤ/8)	—	4	2	no	—
140	#135	—	4	2	no	—
141	#135	C₂²⋊D₈₉	1	2	yes	—
142	C₂×Q₁₆	—	178	2	no	—
143	C₂×SD₁₆	—	2	2	no	—
144	C₂×SD₁₆	—	4	2	no	—
145	#131	—	2	2	no	—
146	C₄⋊Q₈	—	178	1	no	—
147	C₄⋊Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
148	#127	C₂²⋊D₈₉	1	1	yes	characteristic
149	Q₃₂	—	356	2	no	—
150	SD₃₂	—	356	2	no	—
151	M₃₂	—	178	2	no	—
152	#108	C₂²⋊D₈₉	1	2	yes	—
153	#107	—	356	2	no	—
154	#104	—	2	2	no	—
155	#102	—	356	2	no	—
156	#101	—	2	1	no	—
157	#100	—	2	1	no	—
158	C₄×C₈	—	178	1	no	—
159	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
160	#41	—	2	2	no	—
161	#41	—	4	2	no	—
162	#41	—	4	2	no	—
163	#41	—	8	1	no	—
164	C₂×Q₈	—	2	1	no	—
165	C₂×Q₈	—	4	2	no	—
166	C₂×Q₈	—	356	1	no	—
167	C₂×D₄	—	2	1	no	—
168	C₂×D₄	—	4	1	no	—
169	C₂×D₄	—	4	2	no	—
170	C₂×D₄	—	8	1	no	—
171	Q₁₆	—	4	2	no	—
172	Q₁₆	—	356	2	no	—
173	Q₁₆	—	356	2	no	—
174	Q₁₆	#13083559	1	2	yes	—
175	SD₁₆	—	2	2	no	—
176	SD₁₆	—	4	2	no	—
177	SD₁₆	—	4	2	no	—
178	SD₁₆	—	4	2	no	—
179	SD₁₆	—	4	2	no	—
180	D₈	—	4	2	no	—
181	D₈	#13083559	1	2	yes	—
182	M₁₆	—	2	2	no	—
183	M₁₆	—	4	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	178	2	no	—
186	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
187	C₄⋊C₄	—	4	1	no	—
188	C₄⋊C₄	—	356	1	no	—
189	C₄⋊C₄	—	356	1	no	—
190	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
191	C₁₆	—	356	2	no	—
192	C₂³	—	4	1	no	—
193	C₂³	—	8	1	no	—
194	Q₈	—	2	2	no	—
195	Q₈	—	4	2	no	—
196	Q₈	—	8	1	no	—
197	Q₈	—	712	2	no	—
198	D₄	—	2	2	no	—
199	D₄	—	4	2	no	—
200	D₄	—	4	2	no	—
201	D₄	—	8	2	no	—
202	D₄	—	8	2	no	—
203	D₄	—	8	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	8	1	no	—
208	C₂×C₄	—	8	2	no	—
209	C₂×C₄	—	712	1	no	—
210	C₂×C₄	#15667981	1	1	yes	characteristic
211	C₈	—	4	2	no	—
212	C₈	—	356	2	no	—
213	C₈	C₂×#13083559	1	2	yes	—
214	C₈	#15668038	1	2	yes	—
215	C₂²	—	4	2	no	—
216	C₂²	—	8	1	no	—
217	C₂²	—	16	1	no	—
218	C₂²	—	32	1	no	—
219	C₂²	#19151766	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	4	2	no	—
222	C₄	—	8	1	no	—
223	C₄	—	1424	1	no	—
224	C₄	#19151732	1	2	yes	—
225	C₂	—	2	1	no	—
226	C₂	—	8	2	no	—
227	C₂	—	16	1	no	—
228	C₂	#26019306	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022333

C₂

yes

maximal, complemented

#26019921

C₂

yes

maximal, characteristic, complemented

#26018293

C₂

yes

maximal, characteristic, complemented

#26018228

C₂

yes

maximal, complemented

C₈₉×#3291

C₂

yes

maximal, characteristic

#19152038

—

#19152001

—

#19151978

C₂²

yes

characteristic, complemented

#19151944

—

#19151419

C₂²

yes

complemented

#19151413

—

#19151319

C₂²

yes

characteristic, complemented

#577×C₈₉

—

#492×C₈₉

C₂²

yes

characteristic

#471×C₈₉

—

#456×C₈₉

—

#453×C₈₉

—

#443×C₈₉

C₂²

yes

—

C₂×Dic₃₅₆

—

—

—

—

—

—

—

—

2₋¹⁺⁴×C₈₉

—

H(5,𝔽₂)×C₈₉

—

#137×C₈₉

—

#137×C₈₉

—

AGL(1,ℤ/8)×C₈₉

—

AGL(1,ℤ/8)×C₈₉

—

#135×C₈₉

—

#135×C₈₉

D₄

yes

—

C₂×SD₁₆×C₈₉

—

C₂×SD₁₆×C₈₉

—

#131×C₈₉

—

C₄⋊Q₈×C₈₉

D₄

yes

characteristic

#127×C₈₉

D₄

yes

characteristic

#108×C₈₉

D₄

yes

—

#104×C₈₉

—

#101×C₈₉

—

#100×C₈₉

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083549

—

#41×C₈₉

—

#41×C₈₉

—

#41×C₈₉

—

#41×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

D₈

yes

—

SD₁₆×C₈₉

—

SD₁₆×C₈₉

—

SD₁₆×C₈₉

—

SD₁₆×C₈₉

—

SD₁₆×C₈₉

—

D₈×C₈₉

—

D₈×C₈₉

D₈

yes

—

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₂³×C₈₉

—

C₂³×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#120

yes

characteristic

C₂×Dic₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

C₂×D₈

yes

—

C₈×C₈₉

AGL(1,ℤ/8)

yes

—

#10623494

—

100

C₂²×C₈₉

—

101

C₂²×C₈₉

—

102

C₂²×C₈₉

—

103

C₂²×C₈₉

—

104

C₂²×C₈₉

#453

yes

characteristic

105

C₄×C₈₉

—

106

C₄×C₈₉

—

107

C₄×C₈₉

—

108

C₄×C₈₉

#447

yes

—

109

Dic₈₉

—

110

#12315

—

maximal, Sylow

111

C₂×C₈₉

—

112

C₂×C₈₉

—

113

C₂×C₈₉

—

114

C₂×C₈₉

#1619

yes

characteristic

115

#3291

D₈₉

yes

characteristic

116

#2214

—

117

#1722

—

118

#1359

—

119

#1336

—

120

C₈₉

#12315

yes

characteristic, complemented, Sylow

121

#577

—

122

#510

—

178

—

123

#500

—

178

—

124

#494

—

125

#492

C₂×D₈₉

yes

characteristic

126

#486

—

178

—

127

#471

—

128

#456

—

129

#453

—

130

#443

C₂×D₈₉

yes

—

131

#364

—

132

#362

—

178

—

133

#322

—

134

2₋¹⁺⁴

—

135

H(5,𝔽₂)

—

136

#137

—

137

#137

—

138

AGL(1,ℤ/8)

—

139

AGL(1,ℤ/8)

—

140

#135

—

141

#135

C₂²⋊D₈₉

yes

—

142

C₂×Q₁₆

—

178

—

143

C₂×SD₁₆

—

144

C₂×SD₁₆

—

145

#131

—

146

C₄⋊Q₈

—

178

—

147

C₄⋊Q₈

C₂²⋊D₈₉

yes

characteristic

148

#127

C₂²⋊D₈₉

yes

characteristic

149

Q₃₂

—

356

—

150

SD₃₂

—

356

—

151

M₃₂

—

178

—

152

#108

C₂²⋊D₈₉

yes

—

153

#107

—

356

—

154

#104

—

155

#102

—

356

—

156

#101

—

157

#100

—

158

C₄×C₈

—

178

—

159

C₄×C₈

C₂²×D₈₉

yes

characteristic

160

#41

—

161

#41

—

162

#41

—

163

#41

—

164

C₂×Q₈

—

165

C₂×Q₈

—

166

C₂×Q₈

—

356

—

167

C₂×D₄

—

168

C₂×D₄

—

169

C₂×D₄

—

170

C₂×D₄

—

171

Q₁₆

—

172

Q₁₆

—

356

—

173

Q₁₆

—

356

—

174

Q₁₆

#13083559

yes

—

175

SD₁₆

—

176

SD₁₆

—

177

SD₁₆

—

178

SD₁₆

—

179

SD₁₆

—

180

D₈

—

181

D₈

#13083559

yes

—

182

M₁₆

—

183

M₁₆

—

184

C₂×C₈

—

185

C₂×C₈

—

178

—

186

C₂×C₈

C₂×C₂²⋊D₈₉

yes

—

187

C₄⋊C₄

—

188

C₄⋊C₄

—

356

—

189

C₄⋊C₄

—

356

—

190

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

191

C₁₆

—

356

—

192

C₂³

—

193

C₂³

—

194

Q₈

—

195

Q₈

—

196

Q₈

—

197

Q₈

—

712

—

198

D₄

—

199

D₄

—

200

D₄

—

201

D₄

—

202

D₄

—

203

D₄

—

204

C₂×C₄

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

—

208

C₂×C₄

—

209

C₂×C₄

—

712

—

210

C₂×C₄

#15667981

yes

characteristic

211

C₈

—

212

C₈

—

356

—

213

C₈

C₂×#13083559

yes

—

214

C₈

#15668038

yes

—

215

C₂²

—

216

C₂²

—

217

C₂²

—

218

C₂²

—

219

C₂²

#19151766

yes

characteristic

220

C₄

—

221

C₄

—

222

C₄

—

223

C₄

—

1424

—

224

C₄

#19151732

yes

—

225

C₂

—

226

C₂

—

227

C₂

—

228

C₂

#26019306

yes

characteristic

	subgroup	G/H
center	C₂	#26019306
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting, prime radical	C₈₉×#3291	C₂
socle	C₂×C₈₉	#1619
hypercenter	#3291	D₈₉
hypocenter	C₈₉	#12315
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #3291

subgroup

G/H

center

C₂

#26019306

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting, prime radical

C₈₉×#3291

C₂

C₂×C₈₉

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #3291

N	H ≅ G/N	type
1. #26022333	C₂	semidirect
2. #26019921	C₂	semidirect
3. #26018293	C₂	semidirect
4. #26018228	C₂	semidirect
8. #19151978	C₂²	semidirect
10. #19151419	C₂²	semidirect
12. #19151319	C₂²	semidirect
120. C₈₉	#12315	semidirect

H ≅ G/N

type

1. #26022333

C₂

semidirect

2. #26019921

C₂

semidirect

3. #26018293

C₂

semidirect

4. #26018228

C₂

semidirect

8. #19151978

C₂²

semidirect

10. #19151419

C₂²

semidirect

12. #19151319

C₂²

semidirect

120. C₈₉

#12315

semidirect

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

subgroup	index
#26022333	2
#26019921	2
#26018293	2
#26018228	2
C₈₉×#3291	2
#12315	89

subgroup

index

C₈₉×#3291

Subgroups — #36467846

Conjugacy classes of subgroups

Complements

Maximal subgroups