Group #35323797 — subgroups

Subgroups — #35323797

Order 21248.

436 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 97 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×#19128081	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127518	C₂	1	2	yes	maximal
3	#24909184	C₂	1	2	yes	maximal
4	#24908211	C₂	1	1	yes	maximal, characteristic
5	C₈₃×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₃₂	—	2	1	no	—
7	C₂²×Dic₃₃₂	C₂²	1	2	yes	—
8	C₂×#14581792	—	2	1	no	—
9	#19128081	—	2	4	no	—
10	C₂×#14581728	C₂²	1	1	yes	characteristic
11	#19127739	C₂²	1	1	yes	characteristic
12	#19127727	—	2	1	no	—
13	#19127518	—	2	4	no	—
14	#19127506	—	2	4	no	—
15	#19127461	—	2	1	no	—
16	C₂×M₃₂×C₈₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
18	C₂²×Dic₁₆₆	—	2	2	no	—
19	C₂×Dic₃₃₂	—	2	4	no	—
20	C₂×Dic₃₃₂	—	2	2	no	—
21	C₂×Dic₃₃₂	—	2	2	no	—
22	C₂×Dic₃₃₂	—	2	2	no	—
23	C₂×Dic₃₃₂	—	2	2	no	—
24	C₂×Dic₃₃₂	—	2	2	no	—
25	C₂×Dic₃₃₂	—	4	2	no	—
26	C₂×Dic₃₃₂	—	4	2	no	—
27	C₂×Dic₃₃₂	D₄	1	4	yes	—
28	C₂×#13050574	—	2	1	no	—
29	#14581792	—	2	4	no	—
30	#14581792	—	4	2	no	—
31	#14581751	—	2	1	no	—
32	C₂×C₄⋊Dic₈₃	—	4	2	no	—
33	#14581728	—	2	2	no	—
34	#14581726	D₄	1	2	yes	—
35	#14581724	—	2	1	no	—
36	Dic₈₃⋊C₈	—	4	2	no	—
37	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
40	M₃₂×C₈₃	—	2	2	no	—
41	M₃₂×C₈₃	—	2	2	no	—
42	C₂×C₁₆×C₈₃	—	2	2	no	—
43	C₄×C₈×C₈₃	—	2	2	no	—
44	C₂×Dic₁₆₆	—	2	4	no	—
45	C₂×Dic₁₆₆	—	4	2	no	—
46	C₂×Dic₁₆₆	—	4	2	no	—
47	C₂×Dic₁₆₆	—	4	2	no	—
48	C₂×Dic₁₆₆	—	4	2	no	—
49	C₂×Dic₁₆₆	—	4	2	no	—
50	C₂²×Dic₈₃	—	4	2	no	—
51	Dic₃₃₂	—	2	4	no	—
52	Dic₃₃₂	—	2	4	no	—
53	Dic₃₃₂	—	4	8	no	—
54	Dic₃₃₂	—	4	2	no	—
55	Dic₃₃₂	—	4	2	no	—
56	#13050574	—	4	1	no	—
57	#13050574	—	4	1	no	—
58	C₂×C₈₃⋊C₈	—	4	1	no	—
59	C₄⋊Dic₈₃	—	2	2	no	—
60	C₄⋊Dic₈₃	—	4	8	no	—
61	C₂²⋊Dic₈₃	—	4	1	no	—
62	C₄×Dic₈₃	—	4	1	no	—
63	C₂²×C₄×C₈₃	—	2	1	no	—
64	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₈₃	—	2	1	no	—
66	C₂×C₈×C₈₃	—	2	1	no	—
67	C₂×C₈×C₈₃	D₈	1	2	yes	—
68	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₈₃	#41	1	2	yes	—
71	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
72	C₄²×C₈₃	—	2	1	no	—
73	C₄²×C₈₃	SD₁₆	1	2	yes	—
74	C₁₆×C₈₃	—	4	2	no	—
75	C₂³×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
76	C₂×C₄×C₈₃	—	2	4	no	—
77	C₂×C₄×C₈₃	—	2	2	no	—
78	C₂×C₄×C₈₃	#120	1	1	yes	characteristic
79	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
80	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
81	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
83	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
84	C₂×Dic₈₃	—	4	4	no	—
85	C₂×Dic₈₃	—	4	2	no	—
86	C₂×Dic₈₃	—	8	2	no	—
87	C₂×Dic₈₃	—	8	2	no	—
88	Dic₁₆₆	—	4	4	no	—
89	Dic₁₆₆	—	4	4	no	—
90	Dic₁₆₆	—	8	2	no	—
91	Dic₁₆₆	—	8	2	no	—
92	C₈×C₈₃	—	2	2	no	—
93	C₈×C₈₃	#121	1	2	yes	—
94	C₈×C₈₃	#123	1	2	yes	—
95	C₈₃⋊C₈	—	8	1	no	—
96	C₂²×C₈₃	—	2	2	no	—
97	C₂²×C₈₃	#450	1	1	yes	characteristic
98	C₂²×C₈₃	#459	1	1	yes	characteristic
99	C₂²×C₈₃	#490	1	1	yes	characteristic
100	C₄×C₈₃	—	2	4	no	—
101	C₄×C₈₃	#394	1	1	yes	characteristic
102	C₄×C₈₃	#447	1	1	yes	characteristic
103	C₄×C₈₃	#460	1	1	yes	characteristic
104	C₄×C₈₃	#486	1	1	yes	characteristic
105	Dic₈₃	—	8	2	no	—
106	Dic₈₃	—	8	4	no	—
107	#13415	—	83	1	no	maximal, Sylow
108	C₂×C₈₃	—	2	1	no	—
109	C₂×C₈₃	—	2	1	no	—
110	C₂×C₈₃	#2041	1	1	yes	characteristic
111	C₂×C₈₃	#2214	1	2	yes	—
112	C₂×#494	—	83	1	no	—
113	C₂×#362	—	83	2	no	—
114	#2141	—	83	2	no	—
115	#1945	—	83	1	no	—
116	#1376	D₈₃	1	1	yes	characteristic
117	C₈₃	#13415	1	1	yes	characteristic, complemented, Sylow
118	C₂²×Q₁₆	—	83	2	no	—
119	C₂²×Q₁₆	—	166	1	no	—
120	C₂×C₄⋊Q₈	—	166	1	no	—
121	C₂×M₃₂	C₂×D₈₃	1	2	yes	—
122	#494	—	166	4	no	—
123	C₂×#108	—	83	1	no	—
124	#427	—	83	1	no	—
125	#424	—	166	1	no	—
126	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
127	#362	—	166	4	no	—
128	#358	—	166	4	no	—
129	#339	—	166	1	no	—
130	C₂²×Q₈	—	166	2	no	—
131	C₂×Q₁₆	—	332	2	no	—
132	C₂×Q₁₆	—	332	2	no	—
133	C₂×Q₁₆	—	83	4	no	—
134	C₂×Q₁₆	—	166	4	no	—
135	C₂×Q₁₆	—	166	2	no	—
136	C₂×Q₁₆	—	166	2	no	—
137	C₂×Q₁₆	—	166	2	no	—
138	C₂×Q₁₆	—	166	2	no	—
139	C₂×Q₁₆	—	166	2	no	—
140	C₂×M₁₆	—	166	1	no	—
141	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
142	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
143	C₄⋊Q₈	—	332	2	no	—
144	C₄⋊Q₈	—	166	4	no	—
145	#117	—	166	1	no	—
146	C₂×C₄⋊C₄	—	332	2	no	—
147	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
148	M₃₂	—	2	2	no	—
149	M₃₂	—	2	2	no	—
150	C₂×C₁₆	—	2	2	no	—
151	#108	—	166	2	no	—
152	#107	—	83	2	no	—
153	#106	—	166	1	no	—
154	C₄⋊C₈	—	332	2	no	—
155	C₄×C₈	—	2	2	no	—
156	C₂×Q₈	—	332	2	no	—
157	C₂×Q₈	—	332	2	no	—
158	C₂×Q₈	—	332	2	no	—
159	C₂×Q₈	—	332	2	no	—
160	C₂×Q₈	—	332	2	no	—
161	C₂×Q₈	—	166	4	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	332	2	no	—
164	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
165	Q₁₆	—	332	8	no	—
166	Q₁₆	—	332	2	no	—
167	Q₁₆	—	332	2	no	—
168	Q₁₆	—	166	4	no	—
169	Q₁₆	—	166	4	no	—
170	M₁₆	—	332	1	no	—
171	M₁₆	—	332	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	332	1	no	—
175	C₂×C₈	D₆₆₄	1	2	yes	—
176	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
177	C₂×C₈	D₄×D₈₃	1	2	yes	—
178	C₂×C₈	#13050590	1	2	yes	—
179	C₂×C₈	#13050591	1	1	yes	characteristic
180	C₄⋊C₄	—	332	8	no	—
181	C₄⋊C₄	—	166	2	no	—
182	C₂²⋊C₄	—	332	1	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	332	1	no	—
185	C₄²	#13050579	1	2	yes	—
186	C₁₆	—	4	2	no	—
187	C₂³	#14581820	1	1	yes	characteristic
188	Q₈	—	332	4	no	—
189	Q₈	—	332	4	no	—
190	Q₈	—	664	2	no	—
191	Q₈	—	664	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	332	4	no	—
195	C₂×C₄	—	332	2	no	—
196	C₂×C₄	—	664	2	no	—
197	C₂×C₄	—	664	2	no	—
198	C₂×C₄	#14581762	1	1	yes	characteristic
199	C₂×C₄	#14581767	1	1	yes	characteristic
200	C₂×C₄	#14581780	1	1	yes	characteristic
201	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
202	C₂×C₄	C₂×#13050579	1	1	yes	characteristic
203	C₂×C₄	#14581820	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	664	1	no	—
206	C₈	#14581765	1	2	yes	—
207	C₈	#14581777	1	2	yes	—
208	C₂²	—	2	2	no	—
209	C₂²	#19127854	1	1	yes	characteristic
210	C₂²	#19127897	1	1	yes	characteristic
211	C₂²	#19128066	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	664	4	no	—
214	C₄	—	664	2	no	—
215	C₄	#19127619	1	1	yes	characteristic
216	C₄	#19127837	1	1	yes	characteristic
217	C₄	#19127902	1	1	yes	characteristic
218	C₄	#19128048	1	1	yes	characteristic
219	C₂	—	2	1	no	—
220	C₂	—	2	1	no	—
221	C₂	#24908698	1	1	yes	characteristic
222	C₂	#24909545	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19128081

C₂

yes

maximal, characteristic

C₂×#19127518

C₂

yes

maximal

#24909184

C₂

yes

maximal

#24908211

C₂

yes

maximal, characteristic

C₈₃×#1376

C₂

yes

maximal, characteristic

C₂²×Dic₃₃₂

—

C₂²×Dic₃₃₂

C₂²

yes

—

C₂×#14581792

—

#19128081

—

C₂×#14581728

C₂²

yes

characteristic

#19127739

C₂²

yes

—

—

—

—

C₂×M₃₂×C₈₃

C₂²

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂²×Dic₁₆₆

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

D₄

yes

—

—

—

—

—

C₂×C₄⋊Dic₈₃

—

#14581728

—

#14581726

D₄

yes

—

#14581724

—

Dic₈₃⋊C₈

—

C₂²×C₈×C₈₃

D₄

yes

characteristic

C₂²×C₈×C₈₃

C₂³

yes

characteristic

C₂×C₄²×C₈₃

D₄

yes

characteristic

M₃₂×C₈₃

—

M₃₂×C₈₃

—

C₂×C₁₆×C₈₃

—

C₄×C₈×C₈₃

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂²×Dic₈₃

—

Dic₃₃₂

—

Dic₃₃₂

—

Dic₃₃₂

—

Dic₃₃₂

—

Dic₃₃₂

—

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

D₈

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

C₂×D₄

yes

—

C₂×C₈×C₈₃

#41

yes

—

C₂×C₈×C₈₃

#41

yes

characteristic

C₄²×C₈₃

—

C₄²×C₈₃

SD₁₆

yes

—

C₁₆×C₈₃

—

C₂³×C₈₃

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

C₂×D₈

yes

characteristic

C₂×C₄×C₈₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₃

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

#121

yes

—

C₈×C₈₃

#123

yes

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

100

C₄×C₈₃

—

101

C₄×C₈₃

#394

yes

characteristic

102

C₄×C₈₃

#447

yes

characteristic

103

C₄×C₈₃

#460

yes

characteristic

104

C₄×C₈₃

#486

yes

characteristic

105

Dic₈₃

—

106

Dic₈₃

—

107

#13415

—

maximal, Sylow

108

C₂×C₈₃

—

109

C₂×C₈₃

—

110

C₂×C₈₃

#2041

yes

characteristic

111

C₂×C₈₃

#2214

yes

—

112

C₂×#494

—

113

C₂×#362

—

114

#2141

—

115

#1945

—

116

#1376

D₈₃

yes

characteristic

117

C₈₃

#13415

yes

characteristic, complemented, Sylow

118

C₂²×Q₁₆

—

119

C₂²×Q₁₆

—

166

—

120

C₂×C₄⋊Q₈

—

166

—

121

C₂×M₃₂

C₂×D₈₃

yes

—

122

#494

—

166

—

123

C₂×#108

—

124

#427

—

125

#424

—

166

—

126

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

127

#362

—

166

—

128

#358

—

166

—

129

#339

—

166

—

130

C₂²×Q₈

—

166

—

131

C₂×Q₁₆

—

332

—

132

C₂×Q₁₆

—

332

—

133

C₂×Q₁₆

—

134

C₂×Q₁₆

—

166

—

135

C₂×Q₁₆

—

166

—

136

C₂×Q₁₆

—

166

—

137

C₂×Q₁₆

—

166

—

138

C₂×Q₁₆

—

166

—

139

C₂×Q₁₆

—

166

—

140

C₂×M₁₆

—

166

—

141

C₂²×C₈

D₃₃₂

yes

characteristic

142

C₂²×C₈

C₂²×D₈₃

yes

characteristic

143

C₄⋊Q₈

—

332

—

144

C₄⋊Q₈

—

166

—

145

#117

—

166

—

146

C₂×C₄⋊C₄

—

332

—

147

C₂×C₄²

D₃₃₂

yes

characteristic

148

M₃₂

—

149

M₃₂

—

150

C₂×C₁₆

—

151

#108

—

166

—

152

#107

—

153

#106

—

166

—

154

C₄⋊C₈

—

332

—

155

C₄×C₈

—

156

C₂×Q₈

—

332

—

157

C₂×Q₈

—

332

—

158

C₂×Q₈

—

332

—

159

C₂×Q₈

—

332

—

160

C₂×Q₈

—

332

—

161

C₂×Q₈

—

166

—

162

C₂²×C₄

—

163

C₂²×C₄

—

332

—

164

C₂²×C₄

C₂×D₃₃₂

yes

characteristic

165

Q₁₆

—

332

—

166

Q₁₆

—

332

—

167

Q₁₆

—

332

—

168

Q₁₆

—

166

—

169

Q₁₆

—

166

—

170

M₁₆

—

332

—

171

M₁₆

—

332

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

332

—

175

C₂×C₈

D₆₆₄

yes

—

176

C₂×C₈

D₄×D₈₃

yes

characteristic

177

C₂×C₈

D₄×D₈₃

yes

—

178

C₂×C₈

#13050590

yes

—

179

C₂×C₈

#13050591

yes

characteristic

180

C₄⋊C₄

—

332

—

181

C₄⋊C₄

—

166

—

182

C₂²⋊C₄

—

332

—

183

C₄²

—

184

C₄²

—

332

—

185

C₄²

#13050579

yes

—

186

C₁₆

—

187

C₂³

#14581820

yes

characteristic

188

Q₈

—

332

—

189

Q₈

—

332

—

190

Q₈

—

664

—

191

Q₈

—

664

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

332

—

195

C₂×C₄

—

332

—

196

C₂×C₄

—

664

—

197

C₂×C₄

—

664

—

198

C₂×C₄

#14581762

yes

characteristic

199

C₂×C₄

#14581767

yes

characteristic

200

C₂×C₄

#14581780

yes

characteristic

201

C₂×C₄

C₂×D₆₆₄

yes

characteristic

202

C₂×C₄

C₂×#13050579

yes

characteristic

203

C₂×C₄

#14581820

yes

characteristic

204

C₈

—

205

C₈

—

664

—

206

C₈

#14581765

yes

—

207

C₈

#14581777

yes

—

208

C₂²

—

209

C₂²

#19127854

yes

characteristic

210

C₂²

#19127897

yes

characteristic

211

C₂²

#19128066

yes

characteristic

212

C₄

—

213

C₄

—

664

—

214

C₄

—

664

—

215

C₄

#19127619

yes

characteristic

216

C₄

#19127837

yes

characteristic

217

C₄

#19127902

yes

characteristic

218

C₄

#19128048

yes

characteristic

219

C₂

—

220

C₂

—

221

C₂

#24908698

yes

characteristic

222

C₂

#24909545

yes

—

	subgroup	G/H
center	C₂²	#19127897
commutator, cosocle	C₂²×C₈×C₈₃	C₂³
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#1376	C₂
socle	C₂²×C₈₃	#459
prime radical	C₂³×C₈₃	AGL(1,ℤ/8)
hypercenter	#1376	D₈₃
hypocenter	C₈₃	#13415
LCS	G ▷ C₂²×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

subgroup

G/H

center

C₂²

#19127897

commutator, cosocle

C₂²×C₈×C₈₃

C₂³

Frattini

C₂²×C₈

C₂²×D₈₃

Fitting

C₈₃×#1376

C₂

C₂²×C₈₃

C₂³×C₈₃

D₈₃

C₈₃

G ▷ C₂²×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

N	H ≅ G/N	type
117. C₈₃	#13415	semidirect

H ≅ G/N

type

117. C₈₃

#13415

semidirect

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

subgroup	index
C₂×#19128081	2
C₂×#19127518	2
#24909184	2
#24908211	2
C₈₃×#1376	2
#13415	83

subgroup

index

C₈₃×#1376

Subgroups — #35323797

Conjugacy classes of subgroups

Complements

Maximal subgroups