Group #35320592 — subgroups

Subgroups — #35320592

Order 21248.

362 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 111 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×#19128119	C₂	1	1	yes	maximal, characteristic
2	#24909245	C₂	1	2	yes	maximal, complemented
3	#24907506	C₂	1	2	yes	maximal
4	#24905565	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₃×#1384	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14581798	C₂²	1	1	yes	characteristic
7	#19128119	—	2	2	no	—
8	#19128119	—	2	2	no	—
9	C₂×#14581731	C₄	1	2	yes	complemented
10	C₂²×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×D₈₃	C₄	1	2	yes	complemented
12	#19127739	C₂²	1	2	yes	—
13	#19127674	—	2	2	no	—
14	#19127528	—	2	2	no	—
15	#19127526	—	2	2	no	—
16	#19127457	—	2	2	no	—
17	C₂×M₃₂×C₈₃	C₂²	1	1	yes	characteristic
18	#427×C₈₃	C₂²	1	2	yes	—
19	C₂×#13050590	D₄	1	1	yes	characteristic
20	#14581798	—	2	2	no	—
21	#14581798	—	2	2	no	—
22	C₂×#13050574	D₄	1	1	yes	characteristic
23	C₂×#13050573	Q₈	1	2	yes	—
24	C₂×C₈×D₈₃	C₂×C₄	1	2	yes	—
25	#14581751	—	2	2	no	—
26	#14581731	—	2	4	no	—
27	C₂×C₈₃⋊C₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈₃⋊C₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈₃⋊C₁₆	D₄	1	2	yes	—
30	C₂×C₈₃⋊C₁₆	D₄	1	2	yes	—
31	C₁₆×D₈₃	—	2	2	no	—
32	C₁₆×D₈₃	—	2	2	no	—
33	#14581726	—	2	2	no	—
34	#14581724	—	2	2	no	—
35	#14581706	—	4	4	no	—
36	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
37	#117×C₈₃	—	2	2	no	—
38	M₃₂×C₈₃	—	2	2	no	—
39	C₂×C₁₆×C₈₃	C₂×C₄	1	2	yes	—
40	#107×C₈₃	—	2	2	no	—
41	#106×C₈₃	—	2	2	no	—
42	#13050590	—	2	2	no	—
43	#13050590	—	2	2	no	—
44	C₂×Dic₁₆₆	Q₁₆	1	1	yes	characteristic
45	C₂×D₃₃₂	Q₁₆	1	1	yes	characteristic
46	C₂×C₂²⋊D₈₃	D₈	1	2	yes	—
47	C₂×C₄×D₈₃	C₄⋊C₄	1	2	yes	—
48	#13050574	—	2	2	no	—
49	#13050573	—	2	4	no	—
50	C₂×C₈₃⋊C₈	C₄⋊C₄	1	2	yes	—
51	C₈×D₈₃	—	2	2	no	—
52	C₈×D₈₃	—	2	2	no	—
53	C₄⋊Dic₈₃	—	4	2	no	—
54	C₂²⋊Dic₈₃	—	4	2	no	—
55	C₄×Dic₈₃	—	4	2	no	—
56	C₈₃⋊C₁₆	C₂²⋊C₄	1	1	yes	characteristic
57	C₈₃⋊C₁₆	C₂²⋊C₄	1	1	yes	characteristic
58	C₈₃⋊C₁₆	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₈₃	#41	1	2	yes	—
64	C₄⋊C₄×C₈₃	—	4	2	no	—
65	C₂²⋊C₄×C₈₃	—	4	2	no	—
66	C₄²×C₈₃	—	4	2	no	—
67	C₁₆×C₈₃	—	2	2	no	—
68	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
69	C₂²×D₈₃	#107	1	2	yes	—
70	C₂×C₄×C₈₃	—	8	2	no	—
71	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₈₃	#121	1	2	yes	—
73	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
74	C₂×C₄×C₈₃	#135	1	2	yes	—
75	C₂²⋊D₈₃	—	2	4	no	—
76	C₂×Dic₈₃	—	8	2	no	—
77	C₂×Dic₈₃	#107	1	2	yes	—
78	D₃₃₂	—	2	2	no	—
79	C₄×D₈₃	—	2	2	no	—
80	C₄×D₈₃	—	2	2	no	—
81	Dic₁₆₆	—	2	2	no	—
82	C₈×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₈×C₈₃	#117	1	1	yes	characteristic
84	C₈×C₈₃	C₂²⋊Q₈	1	2	yes	—
85	C₈₃⋊C₈	—	2	2	no	—
86	C₂²×C₈₃	—	4	1	no	—
87	C₂²×C₈₃	C₂×#107	1	1	yes	characteristic
88	C₂²×C₈₃	#466	1	2	yes	—
89	C₂×D₈₃	—	2	2	no	—
90	C₂×D₈₃	—	4	2	no	—
91	C₄×C₈₃	—	16	2	no	—
92	C₄×C₈₃	#380	1	1	yes	characteristic
93	C₄×C₈₃	#427	1	1	yes	characteristic
94	C₄×C₈₃	#467	1	2	yes	—
95	Dic₈₃	—	2	2	no	—
96	Dic₈₃	—	16	2	no	—
97	#12839	—	83	1	no	maximal, Sylow
98	C₂×C₈₃	—	2	2	no	—
99	C₂×C₈₃	—	2	1	no	—
100	C₂×C₈₃	#1808	1	1	yes	characteristic
101	D₈₃	—	4	2	no	—
102	C₂×#504	—	83	1	no	—
103	#2158	—	83	2	no	—
104	#1812	—	83	2	no	—
105	#1384	—	83	1	no	—
106	#1384	D₈₃	1	1	yes	characteristic, complemented
107	C₈₃	#12839	1	1	yes	characteristic, complemented, Sylow
108	C₂×#131	—	83	1	no	—
109	#504	—	166	2	no	—
110	#504	—	166	2	no	—
111	C₂×M₃₂	—	83	2	no	—
112	C₂×M₃₂	C₂×D₈₃	1	1	yes	characteristic
113	C₂²×C₁₆	—	83	1	no	—
114	C₂²×C₁₆	—	83	2	no	—
115	#427	—	83	2	no	—
116	#427	C₂×D₈₃	1	2	yes	—
117	#410	—	166	2	no	—
118	#367	—	166	2	no	—
119	#366	—	166	2	no	—
120	H(3,ℤ/4)	—	166	2	no	—
121	C₂×#41	—	83	1	no	—
122	#131	—	166	2	no	—
123	#131	—	166	2	no	—
124	C₂×M₁₆	—	83	1	no	—
125	C₂×M₁₆	—	83	2	no	—
126	C₂²×C₈	—	83	2	no	—
127	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
128	#117	—	2	2	no	—
129	#117	—	166	2	no	—
130	M₃₂	—	2	2	no	—
131	M₃₂	—	166	4	no	—
132	C₂×C₁₆	—	83	1	no	—
133	C₂×C₁₆	—	83	1	no	—
134	C₂×C₁₆	—	83	2	no	—
135	C₂×C₁₆	—	83	2	no	—
136	C₂×C₁₆	—	166	2	no	—
137	C₂×C₁₆	—	166	2	no	—
138	C₂×C₁₆	C₄×D₈₃	1	2	yes	—
139	#107	—	2	2	no	—
140	#107	—	166	2	no	—
141	#106	—	2	2	no	—
142	#106	—	166	2	no	—
143	#99	—	332	4	no	—
144	#41	—	166	2	no	—
145	#41	—	166	2	no	—
146	C₂×Q₈	—	83	1	no	—
147	C₂×D₄	—	83	2	no	—
148	C₂×D₄	—	83	1	no	—
149	C₂²×C₄	—	83	2	no	—
150	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
151	M₁₆	—	166	2	no	—
152	M₁₆	—	166	4	no	—
153	C₂×C₈	—	83	2	no	—
154	C₂×C₈	—	166	2	no	—
155	C₂×C₈	—	166	2	no	—
156	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
157	C₂×C₈	D₄×D₈₃	1	2	yes	—
158	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
159	C₂×C₈	#13050590	1	2	yes	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	332	2	no	—
162	C₂²⋊C₄	—	4	2	no	—
163	C₂²⋊C₄	—	332	2	no	—
164	C₄²	—	4	2	no	—
165	C₄²	—	332	2	no	—
166	C₁₆	—	2	2	no	—
167	C₁₆	—	83	1	no	—
168	C₁₆	—	83	1	no	—
169	C₁₆	—	83	2	no	—
170	C₂³	—	83	2	no	—
171	C₂³	D₈×D₈₃	1	1	yes	characteristic
172	Q₈	—	166	2	no	—
173	D₄	—	166	4	no	—
174	D₄	—	166	2	no	—
175	C₂×C₄	—	8	2	no	—
176	C₂×C₄	—	83	2	no	—
177	C₂×C₄	—	664	2	no	—
178	C₂×C₄	—	166	2	no	—
179	C₂×C₄	—	166	2	no	—
180	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
181	C₂×C₄	#14581765	1	2	yes	—
182	C₂×C₄	Q₁₆×D₈₃	1	1	yes	characteristic
183	C₂×C₄	#14581813	1	2	yes	—
184	C₈	—	166	2	no	—
185	C₈	C₂²⋊C₄×D₈₃	1	1	yes	characteristic
186	C₈	#14581749	1	1	yes	characteristic
187	C₈	#14581770	1	2	yes	—
188	C₂²	—	4	1	no	—
189	C₂²	—	332	2	no	—
190	C₂²	—	166	2	no	—
191	C₂²	#107×D₈₃	1	1	yes	characteristic
192	C₂²	#19127936	1	2	yes	—
193	C₄	—	16	2	no	—
194	C₄	—	1328	2	no	—
195	C₄	—	166	2	no	—
196	C₄	#19127560	1	1	yes	characteristic
197	C₄	#19127736	1	1	yes	characteristic
198	C₄	#19127941	1	2	yes	—
199	C₂	—	2	2	no	—
200	C₂	—	2	1	no	—
201	C₂	—	332	2	no	—
202	C₂	#24907491	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19128119

C₂

yes

maximal, characteristic

#24909245

C₂

yes

maximal, complemented

#24907506

C₂

yes

maximal

#24905565

C₂

yes

maximal, characteristic, complemented

C₈₃×#1384

C₂

yes

maximal, characteristic, complemented

C₂×#14581798

C₂²

yes

—

—

C₄

yes

complemented

C₂²×C₈₃⋊C₁₆

C₂²

yes

characteristic

C₂×C₁₆×D₈₃

C₄

yes

complemented

#19127739

C₂²

yes

—

—

—

—

—

C₂×M₃₂×C₈₃

C₂²

yes

characteristic

#427×C₈₃

C₂²

yes

—

C₂×#13050590

D₄

yes

—

—

D₄

yes

characteristic

C₂×#13050573

Q₈

yes

—

C₂×C₈×D₈₃

C₂×C₄

yes

—

#14581751

—

#14581731

—

C₂×C₈₃⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₈₃⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₈₃⋊C₁₆

D₄

yes

—

C₂×C₈₃⋊C₁₆

D₄

yes

—

C₁₆×D₈₃

—

C₁₆×D₈₃

—

#14581726

—

#14581724

—

#14581706

—

C₂²×C₈×C₈₃

C₂³

yes

characteristic

#117×C₈₃

—

M₃₂×C₈₃

—

C₂×C₁₆×C₈₃

C₂×C₄

yes

—

#107×C₈₃

—

#106×C₈₃

—

#13050590

—

#13050590

—

C₂×Dic₁₆₆

Q₁₆

yes

characteristic

C₂×D₃₃₂

Q₁₆

yes

characteristic

C₂×C₂²⋊D₈₃

D₈

yes

—

C₂×C₄×D₈₃

C₄⋊C₄

yes

—

#13050574

—

#13050573

—

C₂×C₈₃⋊C₈

C₄⋊C₄

yes

—

C₈×D₈₃

—

C₈×D₈₃

—

C₄⋊Dic₈₃

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₈₃⋊C₁₆

C₂²⋊C₄

yes

characteristic

C₈₃⋊C₁₆

C₂²⋊C₄

yes

characteristic

C₈₃⋊C₁₆

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂×D₄

yes

—

C₂×C₈×C₈₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₃

#41

yes

—

C₄⋊C₄×C₈₃

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₁₆×C₈₃

—

C₂³×C₈₃

C₂×D₈

yes

characteristic

C₂²×D₈₃

#107

yes

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

#121

yes

—

C₂×C₄×C₈₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₈₃

#135

yes

—

C₂²⋊D₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

#107

yes

—

D₃₃₂

—

C₄×D₈₃

—

C₄×D₈₃

—

Dic₁₆₆

—

C₈×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₈₃

#117

yes

characteristic

C₈×C₈₃

C₂²⋊Q₈

yes

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×#107

yes

characteristic

C₂²×C₈₃

#466

yes

—

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

—

Dic₈₃

—

Dic₈₃

—

#12839

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

—

100

C₂×C₈₃

#1808

yes

characteristic

101

D₈₃

—

102

C₂×#504

—

103

#2158

—

104

#1812

—

105

#1384

—

106

#1384

D₈₃

yes

characteristic, complemented

107

C₈₃

#12839

yes

characteristic, complemented, Sylow

108

C₂×#131

—

109

#504

—

166

—

110

#504

—

166

—

111

C₂×M₃₂

—

112

C₂×M₃₂

C₂×D₈₃

yes

characteristic

113

C₂²×C₁₆

—

114

C₂²×C₁₆

—

115

#427

—

116

#427

C₂×D₈₃

yes

—

117

#410

—

166

—

118

#367

—

166

—

119

#366

—

166

—

120

H(3,ℤ/4)

—

166

—

121

C₂×#41

—

122

#131

—

166

—

123

#131

—

166

—

124

C₂×M₁₆

—

125

C₂×M₁₆

—

126

C₂²×C₈

—

127

C₂²×C₈

C₂²×D₈₃

yes

characteristic

128

#117

—

129

#117

—

166

—

130

M₃₂

—

131

M₃₂

—

166

—

132

C₂×C₁₆

—

133

C₂×C₁₆

—

134

C₂×C₁₆

—

135

C₂×C₁₆

—

136

C₂×C₁₆

—

166

—

137

C₂×C₁₆

—

166

—

138

C₂×C₁₆

C₄×D₈₃

yes

—

139

#107

—

140

#107

—

166

—

141

#106

—

142

#106

—

166

—

143

#99

—

332

—

144

#41

—

166

—

145

#41

—

166

—

146

C₂×Q₈

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂²×C₄

—

150

C₂²×C₄

D₄×D₈₃

yes

characteristic

151

M₁₆

—

166

—

152

M₁₆

—

166

—

153

C₂×C₈

—

154

C₂×C₈

—

166

—

155

C₂×C₈

—

166

—

156

C₂×C₈

C₂×C₄×D₈₃

yes

characteristic

157

C₂×C₈

D₄×D₈₃

yes

—

158

C₂×C₈

Q₈×D₈₃

yes

characteristic

159

C₂×C₈

#13050590

yes

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

332

—

162

C₂²⋊C₄

—

163

C₂²⋊C₄

—

332

—

164

C₄²

—

165

C₄²

—

332

—

166

C₁₆

—

167

C₁₆

—

168

C₁₆

—

169

C₁₆

—

170

C₂³

—

171

C₂³

D₈×D₈₃

yes

characteristic

172

Q₈

—

166

—

173

D₄

—

166

—

174

D₄

—

166

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

664

—

178

C₂×C₄

—

166

—

179

C₂×C₄

—

166

—

180

C₂×C₄

C₄⋊C₄×D₈₃

yes

characteristic

181

C₂×C₄

#14581765

yes

—

182

C₂×C₄

Q₁₆×D₈₃

yes

characteristic

183

C₂×C₄

#14581813

yes

—

184

C₈

—

166

—

185

C₈

C₂²⋊C₄×D₈₃

yes

characteristic

186

C₈

#14581749

yes

characteristic

187

C₈

#14581770

yes

—

188

C₂²

—

189

C₂²

—

332

—

190

C₂²

—

166

—

191

C₂²

#107×D₈₃

yes

characteristic

192

C₂²

#19127936

yes

—

193

C₄

—

194

C₄

—

1328

—

195

C₄

—

166

—

196

C₄

#19127560

yes

characteristic

197

C₄

#19127736

yes

characteristic

198

C₄

#19127941

yes

—

199

C₂

—

200

C₂

—

201

C₂

—

332

—

202

C₂

#24907491

yes

characteristic

	subgroup	G/H
center	C₄	#19127736
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#1384	C₂
socle	C₂×C₈₃	#1808
prime radical	C₂×#13050590	D₄
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	#1384	D₈₃
hypocenter	C₈₃	#12839
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1384

subgroup

G/H

center

C₄

#19127736

commutator

C₂×C₈×C₈₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₃

Fitting

C₈₃×#1384

C₂

C₂×C₈₃

D₄

C₂²×C₈×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1384

N	H ≅ G/N	type
2. #24909245	C₂	semidirect
4. #24905565	C₂	semidirect
5. C₈₃×#1384	C₂	semidirect
9. C₂×#14581731	C₄	semidirect
11. C₂×C₁₆×D₈₃	C₄	semidirect
106. #1384	D₈₃	semidirect
107. C₈₃	#12839	semidirect

H ≅ G/N

type

2. #24909245

C₂

semidirect

4. #24905565

C₂

semidirect

5. C₈₃×#1384

C₂

semidirect

9. C₂×#14581731

C₄

semidirect

11. C₂×C₁₆×D₈₃

C₄

semidirect

106. #1384

D₈₃

semidirect

107. C₈₃

#12839

semidirect

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

subgroup	index
C₂×#19128119	2
#24909245	2
#24907506	2
#24905565	2
C₈₃×#1384	2
#12839	83

subgroup

index

C₈₃×#1384

Subgroups — #35320592

Conjugacy classes of subgroups

Complements

Maximal subgroups