Group #35304086 — subgroups

Subgroups — #35304086

Order 21248.

402 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 127 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×#19128106	C₂	1	1	yes	maximal, characteristic
2	#24908451	C₂	1	2	yes	maximal
3	#24906149	C₂	1	2	yes	maximal
4	#24905752	C₂	1	1	yes	maximal, characteristic
5	C₈₃×#1566	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581792	C₂²	1	2	yes	—
7	#19128106	—	2	1	no	—
8	#19128106	—	2	1	no	—
9	#19128106	—	2	2	no	—
10	#19127754	C₂²	1	1	yes	characteristic
11	C₂×#14581726	C₄	1	4	yes	—
12	C₄×#13050574	C₂²	1	1	yes	characteristic
13	#19127469	—	2	4	no	—
14	#19127446	—	2	4	no	—
15	#19127423	—	2	1	no	—
16	#19127422	—	2	1	no	—
17	#423×C₈₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
19	#334×C₈₃	—	2	2	no	—
20	C₂²×Dic₁₆₆	—	2	2	no	—
21	C₂×#13050574	D₄	1	2	yes	—
22	#14581792	—	2	2	no	—
23	#14581792	—	2	2	no	—
24	#14581792	—	2	2	no	—
25	#14581792	—	2	2	no	—
26	C₂×C₄⋊Dic₈₃	—	2	2	no	—
27	C₂×C₄⋊Dic₈₃	C₂×C₄	1	4	yes	—
28	#14581726	—	2	4	no	—
29	#14581726	—	2	4	no	—
30	C₄⋊(C₈₃⋊C₈)	D₄	1	2	yes	—
31	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
32	#14581703	—	2	1	no	—
33	C₄×C₈₃⋊C₈	—	2	1	no	—
34	C₄×C₈₃⋊C₈	—	2	1	no	—
35	C₂×M₁₆×C₈₃	—	2	2	no	—
36	C₂²×C₈×C₈₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₈₃	—	2	2	no	—
39	C₄⋊C₈×C₈₃	—	2	2	no	—
40	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
42	C₄×C₈×C₈₃	D₄	1	2	yes	—
43	C₂×Dic₁₆₆	—	4	2	no	—
44	C₂×Dic₁₆₆	—	4	2	no	—
45	C₂×Dic₁₆₆	—	8	4	no	—
46	C₂²×Dic₈₃	—	2	4	no	—
47	#13050574	—	2	4	no	—
48	C₂×C₈₃⋊C₈	—	2	2	no	—
49	C₂×C₈₃⋊C₈	—	2	2	no	—
50	C₄⋊Dic₈₃	—	2	4	no	—
51	C₄⋊Dic₈₃	—	2	4	no	—
52	C₄⋊Dic₈₃	—	4	2	no	—
53	C₄⋊Dic₈₃	—	4	2	no	—
54	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
57	M₁₆×C₈₃	—	4	4	no	—
58	C₂×C₈×C₈₃	—	2	4	no	—
59	C₂×C₈×C₈₃	—	4	2	no	—
60	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
62	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₈₃	#41	1	1	yes	characteristic
66	C₂³×C₈₃	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₈₃	—	2	2	no	—
68	C₂×C₄×C₈₃	—	2	2	no	—
69	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
72	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
73	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
74	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
75	C₂×C₄×C₈₃	#137	1	4	yes	—
76	C₂×Dic₈₃	—	4	4	no	—
77	C₂×Dic₈₃	—	8	4	no	—
78	Dic₁₆₆	—	8	4	no	—
79	Dic₁₆₆	—	8	4	no	—
80	C₈×C₈₃	—	2	2	no	—
81	C₈×C₈₃	—	8	2	no	—
82	C₈×C₈₃	#101	1	4	yes	—
83	C₈₃⋊C₈	—	4	2	no	—
84	C₈₃⋊C₈	—	4	2	no	—
85	C₂²×C₈₃	—	2	1	no	—
86	C₂²×C₈₃	—	2	1	no	—
87	C₂²×C₈₃	#390	1	1	yes	characteristic
88	C₂²×C₈₃	#441	1	2	yes	—
89	C₄×C₈₃	—	2	1	no	—
90	C₄×C₈₃	—	2	1	no	—
91	C₄×C₈₃	C₂×#101	1	2	yes	—
92	C₄×C₈₃	#470	1	2	yes	—
93	C₄×C₈₃	#472	1	2	yes	—
94	C₄×C₈₃	#497	1	2	yes	—
95	Dic₈₃	—	8	4	no	—
96	#9810	—	83	1	no	maximal, Sylow
97	C₂×C₈₃	—	2	2	no	—
98	C₂×C₈₃	#1966	1	1	yes	characteristic
99	C₂×C₈₃	#1971	1	1	yes	characteristic
100	C₂×C₈₃	#1978	1	1	yes	characteristic
101	C₂×#500	—	83	1	no	—
102	#1997	—	83	2	no	—
103	#1566	D₈₃	1	1	yes	characteristic
104	#1549	—	83	2	no	—
105	#1451	—	83	1	no	—
106	C₈₃	#9810	1	1	yes	characteristic, complemented, Sylow
107	C₂×C₄⋊Q₈	—	83	2	no	—
108	#500	—	166	1	no	—
109	#500	—	166	1	no	—
110	#500	—	166	2	no	—
111	#431	—	83	1	no	—
112	C₂×#107	—	83	4	no	—
113	#423	C₂×D₈₃	1	2	yes	—
114	C₄×M₁₆	—	83	1	no	—
115	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
116	#343	—	166	4	no	—
117	#334	—	2	2	no	—
118	#332	—	166	4	no	—
119	#322	—	166	1	no	—
120	#322	—	166	1	no	—
121	C₂²×Q₈	—	166	2	no	—
122	C₂×M₁₆	—	2	2	no	—
123	C₂×M₁₆	—	83	2	no	—
124	C₂²×C₈	C₄×D₈₃	1	2	yes	—
125	C₄⋊Q₈	—	166	2	no	—
126	C₄⋊Q₈	—	166	2	no	—
127	C₄⋊Q₈	—	166	2	no	—
128	C₄⋊Q₈	—	166	2	no	—
129	C₂×C₄⋊C₄	—	83	4	no	—
130	C₂×C₄⋊C₄	—	166	2	no	—
131	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
132	#107	—	166	4	no	—
133	#107	—	166	4	no	—
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	83	2	no	—
137	C₂²⋊C₈	—	166	1	no	—
138	#97	—	166	1	no	—
139	C₄×C₈	—	166	1	no	—
140	C₄×C₈	—	166	1	no	—
141	C₄×C₈	D₃₃₂	1	1	yes	characteristic
142	C₄×C₈	D₃₃₂	1	1	yes	characteristic
143	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
144	C₂×Q₈	—	332	2	no	—
145	C₂×Q₈	—	332	2	no	—
146	C₂×Q₈	—	664	4	no	—
147	C₂²×C₄	—	166	4	no	—
148	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
149	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
150	C₂²×C₄	#13050591	1	1	yes	characteristic
151	M₁₆	—	4	4	no	—
152	M₁₆	—	166	4	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	—	166	2	no	—
156	C₂×C₈	—	166	2	no	—
157	C₂×C₈	(C₂×D₈₃)⋊C₄	1	2	yes	—
158	C₂×C₈	(C₂×D₈₃)⋊C₄	1	2	yes	—
159	C₄⋊C₄	—	332	2	no	—
160	C₄⋊C₄	—	332	2	no	—
161	C₄⋊C₄	—	166	4	no	—
162	C₄⋊C₄	—	166	4	no	—
163	C₄²	D₄×D₈₃	1	1	yes	characteristic
164	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
165	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
166	C₄²	#13050591	1	1	yes	characteristic
167	C₂³	#14581754	1	1	yes	characteristic
168	Q₈	—	664	4	no	—
169	Q₈	—	664	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	332	4	no	—
173	C₂×C₄	—	664	4	no	—
174	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
175	C₂×C₄	#14581754	1	1	yes	characteristic
176	C₂×C₄	#14581767	1	1	yes	characteristic
177	C₂×C₄	#14581767	1	1	yes	characteristic
178	C₂×C₄	#14581781	1	1	yes	characteristic
179	C₂×C₄	#14581788	1	1	yes	characteristic
180	C₂×C₄	#14581821	1	4	yes	—
181	C₈	—	2	2	no	—
182	C₈	—	8	2	no	—
183	C₈	—	332	2	no	—
184	C₈	—	332	2	no	—
185	C₈	#14581710	1	4	yes	—
186	C₂²	—	2	1	no	—
187	C₂²	—	2	1	no	—
188	C₂²	#19127602	1	1	yes	characteristic
189	C₂²	#19127807	1	2	yes	—
190	C₄	—	2	1	no	—
191	C₄	—	2	1	no	—
192	C₄	—	664	4	no	—
193	C₄	C₂×#14581710	1	2	yes	—
194	C₄	#19127960	1	2	yes	—
195	C₄	#19127972	1	2	yes	—
196	C₄	#19128092	1	2	yes	—
197	C₂	—	2	2	no	—
198	C₂	#24908306	1	1	yes	characteristic
199	C₂	#24908332	1	1	yes	characteristic
200	C₂	#24908371	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19128106

C₂

yes

maximal, characteristic

#24908451

C₂

yes

maximal

#24906149

C₂

yes

maximal

#24905752

C₂

yes

maximal, characteristic

C₈₃×#1566

C₂

yes

maximal, characteristic

C₂×#14581792

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×#14581726

C₄

yes

—

C₄×#13050574

C₂²

yes

—

—

—

—

#423×C₈₃

C₂²

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

#334×C₈₃

—

C₂²×Dic₁₆₆

—

C₂×#13050574

D₄

yes

—

—

—

—

—

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

C₂×C₄

yes

—

#14581726

—

#14581726

—

C₄⋊(C₈₃⋊C₈)

D₄

yes

—

C₂²⋊(C₈₃⋊C₈)

—

#14581703

—

C₄×C₈₃⋊C₈

—

C₄×C₈₃⋊C₈

—

C₂×M₁₆×C₈₃

—

C₂²×C₈×C₈₃

C₂×C₄

yes

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄⋊C₈×C₈₃

—

C₄⋊C₈×C₈₃

—

C₄×C₈×C₈₃

D₄

yes

characteristic

C₄×C₈×C₈₃

D₄

yes

characteristic

C₄×C₈×C₈₃

D₄

yes

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂²×Dic₈₃

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂²×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₃

#41

yes

characteristic

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂²⋊C₄

yes

—

C₂×C₈×C₈₃

C₂²⋊C₄

yes

—

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₄²×C₈₃

#41

yes

characteristic

C₂³×C₈₃

C₄×D₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄×D₄

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

#101

yes

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#390

yes

characteristic

C₂²×C₈₃

#441

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×#101

yes

—

C₄×C₈₃

#470

yes

—

C₄×C₈₃

#472

yes

—

C₄×C₈₃

#497

yes

—

Dic₈₃

—

#9810

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

100

C₂×C₈₃

#1978

yes

characteristic

101

C₂×#500

—

102

#1997

—

103

#1566

D₈₃

yes

characteristic

104

#1549

—

105

#1451

—

106

C₈₃

#9810

yes

characteristic, complemented, Sylow

107

C₂×C₄⋊Q₈

—

108

#500

—

166

—

109

#500

—

166

—

110

#500

—

166

—

111

#431

—

112

C₂×#107

—

113

#423

C₂×D₈₃

yes

—

114

C₄×M₁₆

—

115

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

116

#343

—

166

—

117

#334

—

118

#332

—

166

—

119

#322

—

166

—

120

#322

—

166

—

121

C₂²×Q₈

—

166

—

122

C₂×M₁₆

—

123

C₂×M₁₆

—

124

C₂²×C₈

C₄×D₈₃

yes

—

125

C₄⋊Q₈

—

166

—

126

C₄⋊Q₈

—

166

—

127

C₄⋊Q₈

—

166

—

128

C₄⋊Q₈

—

166

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

166

—

131

C₂×C₄²

C₂²×D₈₃

yes

characteristic

132

#107

—

166

—

133

#107

—

166

—

134

C₄⋊C₈

—

135

C₄⋊C₈

—

136

C₄⋊C₈

—

137

C₂²⋊C₈

—

166

—

138

#97

—

166

—

139

C₄×C₈

—

166

—

140

C₄×C₈

—

166

—

141

C₄×C₈

D₃₃₂

yes

characteristic

142

C₄×C₈

D₃₃₂

yes

characteristic

143

C₄×C₈

C₂²⋊D₈₃

yes

—

144

C₂×Q₈

—

332

—

145

C₂×Q₈

—

332

—

146

C₂×Q₈

—

664

—

147

C₂²×C₄

—

166

—

148

C₂²×C₄

C₂×C₄×D₈₃

yes

characteristic

149

C₂²×C₄

D₄×D₈₃

yes

characteristic

150

C₂²×C₄

#13050591

yes

characteristic

151

M₁₆

—

152

M₁₆

—

166

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

166

—

156

C₂×C₈

—

166

—

157

C₂×C₈

(C₂×D₈₃)⋊C₄

yes

—

158

C₂×C₈

(C₂×D₈₃)⋊C₄

yes

—

159

C₄⋊C₄

—

332

—

160

C₄⋊C₄

—

332

—

161

C₄⋊C₄

—

166

—

162

C₄⋊C₄

—

166

—

163

C₄²

D₄×D₈₃

yes

characteristic

164

C₄²

C₂×C₂²⋊D₈₃

yes

characteristic

165

C₄²

C₂×D₃₃₂

yes

characteristic

166

C₄²

#13050591

yes

characteristic

167

C₂³

#14581754

yes

characteristic

168

Q₈

—

664

—

169

Q₈

—

664

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

332

—

173

C₂×C₄

—

664

—

174

C₂×C₄

C₂×(C₂×D₈₃)⋊C₄

yes

characteristic

175

C₂×C₄

#14581754

yes

characteristic

176

C₂×C₄

#14581767

yes

characteristic

177

C₂×C₄

#14581767

yes

characteristic

178

C₂×C₄

#14581781

yes

characteristic

179

C₂×C₄

#14581788

yes

characteristic

180

C₂×C₄

#14581821

yes

—

181

C₈

—

182

C₈

—

183

C₈

—

332

—

184

C₈

—

332

—

185

C₈

#14581710

yes

—

186

C₂²

—

187

C₂²

—

188

C₂²

#19127602

yes

characteristic

189

C₂²

#19127807

yes

—

190

C₄

—

191

C₄

—

192

C₄

—

664

—

193

C₄

C₂×#14581710

yes

—

194

C₄

#19127960

yes

—

195

C₄

#19127972

yes

—

196

C₄

#19128092

yes

—

197

C₂

—

198

C₂

#24908306

yes

characteristic

199

C₂

#24908332

yes

characteristic

200

C₂

#24908371

yes

characteristic

	subgroup	G/H
center	C₂²	#19127602
commutator	C₂²×C₄×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1566	C₂
socle	C₂²×C₈₃	#390
prime radical	C₂³×C₈₃	C₄×D₄
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1566	D₈₃
hypocenter	C₈₃	#9810
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1566

subgroup

G/H

center

C₂²

#19127602

commutator

C₂²×C₄×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₈₃×#1566

C₂

socle

C₂²×C₈₃

#390

prime radical

C₂³×C₈₃

C₄×D₄

cosocle

C₂×C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂²×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1566

N	H ≅ G/N	type
106. C₈₃	#9810	semidirect

H ≅ G/N

type

106. C₈₃

#9810

semidirect

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

subgroup	index
C₂×#19128106	2
#24908451	2
#24906149	2
#24905752	2
C₈₃×#1566	2
#9810	83

subgroup

index

C₈₃×#1566

Subgroups — #35304086

Conjugacy classes of subgroups

Complements

Maximal subgroups