Group #35324236 — subgroups

Subgroups — #35324236

Order 21248.

302 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 93 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×#19127518	C₂	1	2	yes	maximal
2	#24908334	C₂	1	1	yes	maximal, characteristic
3	#24907638	C₂	1	1	yes	maximal, characteristic
4	#24905562	C₂	1	2	yes	maximal
5	C₈₃×#1377	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₃₂	C₂²	1	1	yes	characteristic
7	C₂×#14581728	—	2	1	no	—
8	C₂×#14581728	C₂²	1	2	yes	—
9	C₂×#14581724	C₂²	1	1	yes	characteristic
10	#19127725	—	2	1	no	—
11	C₂×#14581717	—	2	1	no	—
12	#19127607	—	2	1	no	—
13	#19127518	—	2	4	no	—
14	C₂×M₃₂×C₈₃	C₂²	1	2	yes	—
15	C₂×#97×C₈₃	C₂²	1	1	yes	characteristic
16	C₂²×Dic₁₆₆	—	2	1	no	—
17	C₂×Dic₃₃₂	—	2	2	no	—
18	C₂×Dic₃₃₂	—	2	2	no	—
19	C₂×Dic₃₃₂	—	2	1	no	—
20	C₂×Dic₃₃₂	D₄	1	2	yes	—
21	C₂×#13050574	—	2	2	no	—
22	C₂×C₄⋊Dic₈₃	—	2	1	no	—
23	#14581728	—	2	2	no	—
24	#14581728	—	2	2	no	—
25	#14581728	—	4	2	no	—
26	#14581724	—	2	2	no	—
27	C₄⋊(C₈₃⋊C₈)	—	4	2	no	—
28	#14581717	—	4	2	no	—
29	C₈₃⋊#95	—	4	2	no	—
30	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
33	M₃₂×C₈₃	—	2	2	no	—
34	M₃₂×C₈₃	—	2	2	no	—
35	C₂×C₁₆×C₈₃	—	2	2	no	—
36	#97×C₈₃	—	2	2	no	—
37	C₂×Dic₁₆₆	—	2	2	no	—
38	C₂×Dic₁₆₆	—	4	1	no	—
39	C₂×Dic₁₆₆	—	4	1	no	—
40	C₂×Dic₁₆₆	—	4	1	no	—
41	C₂×Dic₁₆₆	—	4	1	no	—
42	C₂×Dic₁₆₆	—	4	1	no	—
43	C₂²×Dic₈₃	—	4	1	no	—
44	C₂²×Dic₈₃	—	4	1	no	—
45	Dic₃₃₂	—	2	4	no	—
46	Dic₃₃₂	—	4	2	no	—
47	#13050574	—	4	2	no	—
48	#13050574	—	4	2	no	—
49	C₂×C₈₃⋊C₈	—	4	2	no	—
50	C₄⋊Dic₈₃	—	4	1	no	—
51	C₄⋊Dic₈₃	—	4	1	no	—
52	C₂²×C₄×C₈₃	SD₁₆	1	2	yes	—
53	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₈₃	—	2	1	no	—
55	C₂×C₈×C₈₃	—	2	1	no	—
56	C₂×C₈×C₈₃	D₈	1	2	yes	—
57	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₈₃	#41	1	2	yes	—
59	C₂×C₈×C₈₃	#41	1	2	yes	—
60	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
61	C₄²×C₈₃	—	2	1	no	—
62	C₄²×C₈₃	—	2	1	no	—
63	C₁₆×C₈₃	—	4	2	no	—
64	C₂³×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₈₃	—	2	2	no	—
66	C₂×C₄×C₈₃	—	4	2	no	—
67	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
68	C₂×C₄×C₈₃	#123	1	1	yes	characteristic
69	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
70	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
73	C₂×Dic₈₃	—	4	2	no	—
74	C₂×Dic₈₃	—	8	1	no	—
75	C₂×Dic₈₃	—	8	1	no	—
76	C₂×Dic₈₃	—	8	2	no	—
77	C₂×Dic₈₃	—	8	1	no	—
78	Dic₁₆₆	—	4	2	no	—
79	Dic₁₆₆	—	4	2	no	—
80	Dic₁₆₆	—	8	1	no	—
81	Dic₁₆₆	—	8	1	no	—
82	C₈×C₈₃	—	2	2	no	—
83	C₈×C₈₃	#123	1	2	yes	—
84	C₈×C₈₃	#126	1	2	yes	—
85	C₈₃⋊C₈	—	8	2	no	—
86	C₂²×C₈₃	—	2	2	no	—
87	C₂²×C₈₃	#461	1	1	yes	characteristic
88	C₂²×C₈₃	#481	1	1	yes	characteristic
89	C₂²×C₈₃	#486	1	1	yes	characteristic
90	C₄×C₈₃	—	4	2	no	—
91	C₄×C₈₃	#397	1	1	yes	characteristic
92	C₄×C₈₃	#459	1	1	yes	characteristic
93	C₄×C₈₃	#480	1	1	yes	characteristic
94	C₄×C₈₃	#489	1	1	yes	characteristic
95	Dic₈₃	—	8	2	no	—
96	Dic₈₃	—	16	1	no	—
97	#13488	—	83	1	no	maximal, Sylow
98	C₂×C₈₃	—	2	1	no	—
99	C₂×C₈₃	—	2	1	no	—
100	C₂×C₈₃	#2071	1	1	yes	characteristic
101	C₂×C₈₃	#2239	1	2	yes	—
102	C₂×#362	—	83	2	no	—
103	#1971	—	83	1	no	—
104	#1849	—	83	1	no	—
105	#1383	—	83	2	no	—
106	#1377	D₈₃	1	1	yes	characteristic
107	C₈₃	#13488	1	1	yes	characteristic, complemented, Sylow
108	C₂²×Q₁₆	—	83	1	no	—
109	C₂×M₃₂	C₂×D₈₃	1	2	yes	—
110	C₂×#108	—	83	2	no	—
111	C₂×#108	—	166	1	no	—
112	C₂×#106	—	83	1	no	—
113	#423	—	166	1	no	—
114	C₂×#103	—	166	1	no	—
115	C₂×#97	C₂×D₈₃	1	1	yes	characteristic
116	#391	—	166	1	no	—
117	#362	—	166	4	no	—
118	C₂²×Q₈	—	166	1	no	—
119	C₂×Q₁₆	—	83	2	no	—
120	C₂×Q₁₆	—	166	2	no	—
121	C₂×Q₁₆	—	166	2	no	—
122	C₂×Q₁₆	—	166	1	no	—
123	C₂×M₁₆	—	166	2	no	—
124	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
126	C₂×C₄⋊C₄	—	166	1	no	—
127	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
128	M₃₂	—	2	2	no	—
129	M₃₂	—	2	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	#108	—	332	2	no	—
132	#108	—	166	2	no	—
133	#108	—	166	2	no	—
134	#106	—	166	2	no	—
135	C₄⋊C₈	—	332	2	no	—
136	#103	—	332	2	no	—
137	#97	—	2	2	no	—
138	#95	—	332	2	no	—
139	C₂×Q₈	—	332	1	no	—
140	C₂×Q₈	—	332	1	no	—
141	C₂×Q₈	—	332	1	no	—
142	C₂×Q₈	—	332	1	no	—
143	C₂×Q₈	—	332	1	no	—
144	C₂×Q₈	—	166	2	no	—
145	C₂²×C₄	—	332	1	no	—
146	C₂²×C₄	—	332	1	no	—
147	C₂²×C₄	#13050579	1	2	yes	—
148	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
149	Q₁₆	—	332	2	no	—
150	Q₁₆	—	166	4	no	—
151	M₁₆	—	332	2	no	—
152	M₁₆	—	332	2	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	332	2	no	—
156	C₂×C₈	D₆₆₄	1	2	yes	—
157	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
158	C₂×C₈	#13050589	1	2	yes	—
159	C₂×C₈	#13050590	1	2	yes	—
160	C₂×C₈	#13050591	1	1	yes	characteristic
161	C₄⋊C₄	—	332	1	no	—
162	C₄⋊C₄	—	332	1	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	2	1	no	—
165	C₁₆	—	4	2	no	—
166	C₂³	C₂×#13050579	1	1	yes	characteristic
167	Q₈	—	332	2	no	—
168	Q₈	—	332	2	no	—
169	Q₈	—	664	1	no	—
170	Q₈	—	664	1	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	332	2	no	—
174	C₂×C₄	—	664	2	no	—
175	C₂×C₄	—	664	1	no	—
176	C₂×C₄	—	664	1	no	—
177	C₂×C₄	—	664	1	no	—
178	C₂×C₄	#14581767	1	1	yes	characteristic
179	C₂×C₄	#14581776	1	1	yes	characteristic
180	C₂×C₄	#14581780	1	1	yes	characteristic
181	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
182	C₂×C₄	#14581820	1	1	yes	characteristic
183	C₂×C₄	#14581825	1	1	yes	characteristic
184	C₈	—	2	2	no	—
185	C₈	—	664	2	no	—
186	C₈	#14581774	1	2	yes	—
187	C₈	#14581787	1	2	yes	—
188	C₂²	—	2	2	no	—
189	C₂²	#19127907	1	1	yes	characteristic
190	C₂²	#19128020	1	1	yes	characteristic
191	C₂²	#19128048	1	1	yes	characteristic
192	C₄	—	4	2	no	—
193	C₄	—	1328	1	no	—
194	C₄	—	664	2	no	—
195	C₄	#19127628	1	1	yes	characteristic
196	C₄	#19127897	1	1	yes	characteristic
197	C₄	#19128015	1	1	yes	characteristic
198	C₄	#19128061	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#24908868	1	1	yes	characteristic
202	C₂	#24909683	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19127518

C₂

yes

maximal

#24908334

C₂

yes

maximal, characteristic

#24907638

C₂

yes

maximal, characteristic

#24905562

C₂

yes

maximal

C₈₃×#1377

C₂

yes

maximal, characteristic

C₂²×Dic₃₃₂

C₂²

yes

characteristic

C₂×#14581728

—

C₂×#14581728

C₂²

yes

—

C₂×#14581724

C₂²

yes

—

—

—

—

C₂×M₃₂×C₈₃

C₂²

yes

—

C₂×#97×C₈₃

C₂²

yes

characteristic

C₂²×Dic₁₆₆

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

D₄

yes

—

C₂×#13050574

—

C₂×C₄⋊Dic₈₃

—

—

—

—

—

C₄⋊(C₈₃⋊C₈)

—

#14581717

—

C₈₃⋊#95

—

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

—

#97×C₈₃

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂²×Dic₈₃

—

C₂²×Dic₈₃

—

Dic₃₃₂

—

Dic₃₃₂

—

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂²×C₄×C₈₃

SD₁₆

yes

—

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

#41

yes

—

C₂×C₈×C₈₃

#41

yes

—

C₂×C₈×C₈₃

#41

yes

characteristic

C₄²×C₈₃

—

C₄²×C₈₃

—

C₁₆×C₈₃

—

C₂³×C₈₃

C₂×SD₁₆

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

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

#123

yes

—

C₈×C₈₃

#126

yes

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

Dic₈₃

—

#13488

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

—

100

C₂×C₈₃

#2071

yes

characteristic

101

C₂×C₈₃

#2239

yes

—

102

C₂×#362

—

103

#1971

—

104

#1849

—

105

#1383

—

106

#1377

D₈₃

yes

characteristic

107

C₈₃

#13488

yes

characteristic, complemented, Sylow

108

C₂²×Q₁₆

—

109

C₂×M₃₂

C₂×D₈₃

yes

—

110

C₂×#108

—

111

C₂×#108

—

166

—

112

C₂×#106

—

113

#423

—

166

—

114

C₂×#103

—

166

—

115

C₂×#97

C₂×D₈₃

yes

characteristic

116

#391

—

166

—

117

#362

—

166

—

118

C₂²×Q₈

—

166

—

119

C₂×Q₁₆

—

120

C₂×Q₁₆

—

166

—

121

C₂×Q₁₆

—

166

—

122

C₂×Q₁₆

—

166

—

123

C₂×M₁₆

—

166

—

124

C₂²×C₈

D₃₃₂

yes

characteristic

125

C₂²×C₈

C₂²×D₈₃

yes

characteristic

126

C₂×C₄⋊C₄

—

166

—

127

C₂×C₄²

D₃₃₂

yes

characteristic

128

M₃₂

—

129

M₃₂

—

130

C₂×C₁₆

—

131

#108

—

332

—

132

#108

—

166

—

133

#108

—

166

—

134

#106

—

166

—

135

C₄⋊C₈

—

332

—

136

#103

—

332

—

137

#97

—

138

#95

—

332

—

139

C₂×Q₈

—

332

—

140

C₂×Q₈

—

332

—

141

C₂×Q₈

—

332

—

142

C₂×Q₈

—

332

—

143

C₂×Q₈

—

332

—

144

C₂×Q₈

—

166

—

145

C₂²×C₄

—

332

—

146

C₂²×C₄

—

332

—

147

C₂²×C₄

#13050579

yes

—

148

C₂²×C₄

C₂×D₃₃₂

yes

characteristic

149

Q₁₆

—

332

—

150

Q₁₆

—

166

—

151

M₁₆

—

332

—

152

M₁₆

—

332

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

332

—

156

C₂×C₈

D₆₆₄

yes

—

157

C₂×C₈

D₄×D₈₃

yes

characteristic

158

C₂×C₈

#13050589

yes

—

159

C₂×C₈

#13050590

yes

—

160

C₂×C₈

#13050591

yes

characteristic

161

C₄⋊C₄

—

332

—

162

C₄⋊C₄

—

332

—

163

C₄²

—

164

C₄²

—

165

C₁₆

—

166

C₂³

C₂×#13050579

yes

characteristic

167

Q₈

—

332

—

168

Q₈

—

332

—

169

Q₈

—

664

—

170

Q₈

—

664

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

332

—

174

C₂×C₄

—

664

—

175

C₂×C₄

—

664

—

176

C₂×C₄

—

664

—

177

C₂×C₄

—

664

—

178

C₂×C₄

#14581767

yes

characteristic

179

C₂×C₄

#14581776

yes

characteristic

180

C₂×C₄

#14581780

yes

characteristic

181

C₂×C₄

C₂×D₆₆₄

yes

characteristic

182

C₂×C₄

#14581820

yes

characteristic

183

C₂×C₄

#14581825

yes

characteristic

184

C₈

—

185

C₈

—

664

—

186

C₈

#14581774

yes

—

187

C₈

#14581787

yes

—

188

C₂²

—

189

C₂²

#19127907

yes

characteristic

190

C₂²

#19128020

yes

characteristic

191

C₂²

#19128048

yes

characteristic

192

C₄

—

193

C₄

—

1328

—

194

C₄

—

664

—

195

C₄

#19127628

yes

characteristic

196

C₄

#19127897

yes

characteristic

197

C₄

#19128015

yes

characteristic

198

C₄

#19128061

yes

characteristic

199

C₂

—

200

C₂

—

201

C₂

#24908868

yes

characteristic

202

C₂

#24909683

yes

—

	subgroup	G/H
center	C₂²	#19128020
commutator, cosocle	C₂²×C₈×C₈₃	C₂³
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#1377	C₂
socle	C₂²×C₈₃	#481
prime radical	C₂³×C₈₃	C₂×SD₁₆
hypercenter	#1377	D₈₃
hypocenter	C₈₃	#13488
LCS	G ▷ C₂²×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1377

subgroup

G/H

center

C₂²

#19128020

commutator, cosocle

C₂²×C₈×C₈₃

C₂³

Frattini

C₂²×C₈

C₂²×D₈₃

Fitting

C₈₃×#1377

C₂

socle

C₂²×C₈₃

#481

prime radical

C₂³×C₈₃

C₂×SD₁₆

D₈₃

C₈₃

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1377

N	H ≅ G/N	type
107. C₈₃	#13488	semidirect

H ≅ G/N

type

107. C₈₃

#13488

semidirect

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

subgroup	index
C₂×#19127518	2
#24908334	2
#24907638	2
#24905562	2
C₈₃×#1377	2
#13488	83

subgroup

index

C₈₃×#1377

Subgroups — #35324236

Conjugacy classes of subgroups

Complements

Maximal subgroups