Group #35316005 — subgroups

Subgroups — #35316005

Order 21248.

250 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 103 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	#24909265	C₂	1	2	yes	maximal
2	C₈²×D₈₃	C₂	1	1	yes	maximal, characteristic
3	#24905532	C₂	1	2	yes	maximal, complemented
4	#24905527	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₃×#1367	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₈×D₈₃	C₄	1	1	yes	characteristic
7	C₄×C₈×D₈₃	C₄	1	1	yes	characteristic
8	C₄×C₈×D₈₃	C₂²	1	1	yes	characteristic
9	C₄⋊(C₈₃⋊C₁₆)	C₂²	1	2	yes	—
10	(C₂×D₈₃)⋊C₁₆	—	2	2	no	—
11	#19127476	—	2	2	no	—
12	C₈×C₈₃⋊C₈	C₄	1	1	yes	characteristic
13	C₈×C₈₃⋊C₈	C₄	1	1	yes	characteristic
14	C₈×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₈₃	C₂²	1	2	yes	—
16	C₈²×C₈₃	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₈₃	—	2	1	no	—
18	C₂×C₈×D₈₃	C₈	1	1	yes	characteristic
19	C₂×C₈×D₈₃	C₈	1	1	yes	characteristic
20	C₂×C₈×D₈₃	D₄	1	1	yes	characteristic
21	C₂×C₈×D₈₃	Q₈	1	1	yes	characteristic
22	C₄²×D₈₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈₃⋊C₁₆	—	2	2	no	—
24	C₄×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₈₃	—	2	1	no	—
28	C₈×Dic₈₃	C₈	1	1	yes	characteristic
29	C₈×Dic₈₃	C₈	1	1	yes	characteristic
30	C₈×Dic₈₃	D₄	1	1	yes	characteristic
31	C₈×Dic₈₃	Q₈	1	1	yes	characteristic
32	C₂×C₁₆×C₈₃	—	2	2	no	—
33	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
36	C₂×C₄×D₈₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×D₈₃	C₂×C₈	1	1	yes	characteristic
38	C₂×C₄×D₈₃	M₁₆	1	1	yes	characteristic
39	C₂×C₈₃⋊C₈	—	2	1	no	—
40	C₂×C₈₃⋊C₈	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈₃⋊C₈	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈₃⋊C₈	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈₃⋊C₈	C₂×C₈	1	1	yes	characteristic
44	C₈×D₈₃	—	2	4	no	—
45	C₈×D₈₃	—	2	2	no	—
46	C₈×D₈₃	—	2	2	no	—
47	C₈×D₈₃	—	2	2	no	—
48	C₈×D₈₃	—	2	2	no	—
49	C₄×Dic₈₃	C₄⋊C₄	1	1	yes	characteristic
50	C₄×Dic₈₃	C₂×C₈	1	1	yes	characteristic
51	C₄×Dic₈₃	M₁₆	1	1	yes	characteristic
52	C₈₃⋊C₁₆	—	4	2	no	—
53	C₂×C₈×C₈₃	—	2	1	no	—
54	C₂×C₈×C₈₃	C₂×C₈	1	1	yes	characteristic
55	C₂×C₈×C₈₃	C₂×C₈	1	1	yes	characteristic
56	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
59	C₁₆×C₈₃	—	4	2	no	—
60	C₂²×D₈₃	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
63	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
64	C₂×Dic₈₃	C₄⋊C₈	1	1	yes	characteristic
65	C₂×Dic₈₃	C₄⋊C₈	1	1	yes	characteristic
66	C₂×Dic₈₃	C₄⋊C₈	1	1	yes	characteristic
67	C₄×D₈₃	—	2	2	no	—
68	C₄×D₈₃	—	2	1	no	—
69	C₄×D₈₃	—	2	1	no	—
70	C₄×D₈₃	—	2	1	no	—
71	C₄×D₈₃	—	2	1	no	—
72	C₈×C₈₃	—	2	2	no	—
73	C₈×C₈₃	—	2	1	no	—
74	C₈×C₈₃	—	2	1	no	—
75	C₈×C₈₃	M₃₂	1	2	yes	—
76	C₈×C₈₃	#135	1	2	yes	—
77	C₈₃⋊C₈	—	2	2	no	—
78	C₈₃⋊C₈	—	2	1	no	—
79	C₈₃⋊C₈	—	2	1	no	—
80	C₈₃⋊C₈	#108	1	2	yes	—
81	C₈₃⋊C₈	M₃₂	1	2	yes	—
82	C₂²×C₈₃	C₂×C₄⋊C₈	1	1	yes	characteristic
83	C₂×D₈₃	—	2	1	no	—
84	C₂×D₈₃	—	2	1	no	—
85	C₂×D₈₃	#364	1	2	yes	—
86	C₄×C₈₃	—	2	1	no	—
87	C₄×C₈₃	#427	1	1	yes	characteristic
88	C₄×C₈₃	C₂×#108	1	1	yes	characteristic
89	C₄×C₈₃	C₂×M₃₂	1	1	yes	characteristic
90	C₄×C₈₃	C₂×M₃₂	1	1	yes	characteristic
91	Dic₈₃	—	2	1	no	—
92	Dic₈₃	—	2	1	no	—
93	Dic₈₃	—	2	1	no	—
94	#12013	—	83	1	no	maximal, Sylow
95	C₂×C₈₃	#1564	1	1	yes	characteristic
96	C₂×C₈₃	#2150	1	1	yes	characteristic
97	C₂×C₈₃	C₂×#364	1	1	yes	characteristic
98	D₈₃	—	2	2	no	—
99	#2163	—	83	2	no	—
100	C₂×C₈²	—	83	1	no	—
101	#1370	—	83	2	no	—
102	#1367	—	83	1	no	—
103	#1367	D₈₃	1	1	yes	characteristic, complemented
104	C₈₃	#12013	1	1	yes	characteristic, complemented, Sylow
105	C₂×C₄×C₈	—	83	1	no	—
106	C₂×C₄×C₈	—	83	1	no	—
107	C₂×C₄×C₈	—	83	1	no	—
108	C₄⋊C₁₆	—	83	2	no	—
109	C₄⋊C₁₆	C₂×D₈₃	1	2	yes	—
110	C₂²⋊C₁₆	—	166	2	no	—
111	#346	—	166	2	no	—
112	C₈²	—	83	1	no	—
113	C₈²	—	83	1	no	—
114	C₈²	—	83	1	no	—
115	C₈²	C₂×D₈₃	1	1	yes	characteristic
116	C₂²×C₈	—	83	1	no	—
117	C₂²×C₈	—	83	1	no	—
118	C₂²×C₈	—	83	1	no	—
119	C₂²×C₈	—	83	1	no	—
120	C₂²×C₈	—	166	1	no	—
121	C₂×C₄²	—	83	1	no	—
122	C₂×C₁₆	—	2	2	no	—
123	C₂×C₁₆	—	166	2	no	—
124	C₄×C₈	—	83	1	no	—
125	C₄×C₈	—	83	1	no	—
126	C₄×C₈	—	83	1	no	—
127	C₄×C₈	—	83	1	no	—
128	C₄×C₈	—	83	1	no	—
129	C₄×C₈	—	83	1	no	—
130	C₄×C₈	—	83	1	no	—
131	C₄×C₈	—	166	1	no	—
132	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
133	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
134	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
135	C₂²×C₄	—	83	1	no	—
136	C₂²×C₄	—	83	1	no	—
137	C₂²×C₄	—	83	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	83	1	no	—
140	C₂×C₈	—	83	1	no	—
141	C₂×C₈	—	83	1	no	—
142	C₂×C₈	—	83	1	no	—
143	C₂×C₈	—	166	2	no	—
144	C₂×C₈	—	166	2	no	—
145	C₂×C₈	—	166	4	no	—
146	C₂×C₈	—	166	1	no	—
147	C₂×C₈	—	166	2	no	—
148	C₂×C₈	—	166	2	no	—
149	C₂×C₈	C₈×D₈₃	1	1	yes	characteristic
150	C₂×C₈	C₈×D₈₃	1	1	yes	characteristic
151	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
152	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
153	C₄²	—	83	1	no	—
154	C₄²	—	83	1	no	—
155	C₄²	—	83	1	no	—
156	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
157	C₁₆	—	4	2	no	—
158	C₁₆	—	332	2	no	—
159	C₂³	—	83	1	no	—
160	C₂×C₄	—	83	1	no	—
161	C₂×C₄	—	83	1	no	—
162	C₂×C₄	—	83	1	no	—
163	C₂×C₄	—	166	2	no	—
164	C₂×C₄	—	166	1	no	—
165	C₂×C₄	—	166	1	no	—
166	C₂×C₄	—	166	1	no	—
167	C₂×C₄	—	166	1	no	—
168	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
169	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
170	C₂×C₄	M₁₆×D₈₃	1	1	yes	characteristic
171	C₈	—	2	2	no	—
172	C₈	—	2	1	no	—
173	C₈	—	2	1	no	—
174	C₈	—	83	2	no	—
175	C₈	—	83	2	no	—
176	C₈	—	166	2	no	—
177	C₈	—	166	1	no	—
178	C₈	—	166	1	no	—
179	C₈	#14581731	1	2	yes	—
180	C₈	#14581813	1	2	yes	—
181	C₂²	—	83	2	no	—
182	C₂²	—	166	1	no	—
183	C₂²	—	166	1	no	—
184	C₂²	C₄⋊C₈×D₈₃	1	1	yes	characteristic
185	C₄	—	2	1	no	—
186	C₄	—	166	1	no	—
187	C₄	—	166	1	no	—
188	C₄	—	166	1	no	—
189	C₄	#19127736	1	1	yes	characteristic
190	C₄	#108×D₈₃	1	1	yes	characteristic
191	C₄	M₃₂×D₈₃	1	1	yes	characteristic
192	C₄	C₂×#14581731	1	1	yes	characteristic
193	C₂	—	166	2	no	—
194	C₂	#24906208	1	1	yes	characteristic
195	C₂	#24909217	1	1	yes	characteristic
196	C₂	#364×D₈₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24909265

C₂

yes

maximal

C₈²×D₈₃

C₂

yes

maximal, characteristic

#24905532

C₂

yes

maximal, complemented

#24905527

C₂

yes

maximal, characteristic, complemented

C₈₃×#1367

C₂

yes

maximal, characteristic, complemented

C₄×C₈×D₈₃

C₄

yes

characteristic

C₄×C₈×D₈₃

C₄

yes

characteristic

C₄×C₈×D₈₃

C₂²

yes

characteristic

C₄⋊(C₈₃⋊C₁₆)

C₂²

yes

—

(C₂×D₈₃)⋊C₁₆

—

#19127476

—

C₈×C₈₃⋊C₈

C₄

yes

characteristic

C₈×C₈₃⋊C₈

C₄

yes

characteristic

C₈×C₈₃⋊C₈

C₂²

yes

characteristic

C₄⋊C₁₆×C₈₃

C₂²

yes

—

C₈²×C₈₃

C₂²

yes

characteristic

C₂×C₈×D₈₃

—

C₂×C₈×D₈₃

C₈

yes

characteristic

C₂×C₈×D₈₃

C₈

yes

characteristic

C₂×C₈×D₈₃

D₄

yes

characteristic

C₂×C₈×D₈₃

Q₈

yes

characteristic

C₄²×D₈₃

C₂×C₄

yes

characteristic

C₂×C₈₃⋊C₁₆

—

C₄×C₈₃⋊C₈

C₂×C₄

yes

characteristic

C₄×C₈₃⋊C₈

C₂×C₄

yes

characteristic

C₄×C₈₃⋊C₈

C₂×C₄

yes

characteristic

C₈×Dic₈₃

—

C₈×Dic₈₃

C₈

yes

characteristic

C₈×Dic₈₃

C₈

yes

characteristic

C₈×Dic₈₃

D₄

yes

characteristic

C₈×Dic₈₃

Q₈

yes

characteristic

C₂×C₁₆×C₈₃

—

C₄×C₈×C₈₃

C₂×C₄

yes

characteristic

C₄×C₈×C₈₃

C₂×C₄

yes

characteristic

C₄×C₈×C₈₃

C₂³

yes

characteristic

C₂×C₄×D₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×D₈₃

C₂×C₈

yes

characteristic

C₂×C₄×D₈₃

M₁₆

yes

characteristic

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₈₃⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₈₃⋊C₈

C₂×C₈

yes

characteristic

C₂×C₈₃⋊C₈

C₂×C₈

yes

characteristic

C₈×D₈₃

—

C₈×D₈₃

—

C₈×D₈₃

—

C₈×D₈₃

—

C₈×D₈₃

—

C₄×Dic₈₃

C₄⋊C₄

yes

characteristic

C₄×Dic₈₃

C₂×C₈

yes

characteristic

C₄×Dic₈₃

M₁₆

yes

characteristic

C₈₃⋊C₁₆

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×C₈

yes

characteristic

C₂×C₈×C₈₃

C₂×C₈

yes

characteristic

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

C₂×Q₈

yes

characteristic

C₄²×C₈₃

C₂²×C₄

yes

characteristic

C₁₆×C₈₃

—

C₂²×D₈₃

C₄⋊C₈

yes

characteristic

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₈₃

C₂×M₁₆

yes

characteristic

C₂×Dic₈₃

C₄⋊C₈

yes

characteristic

C₂×Dic₈₃

C₄⋊C₈

yes

characteristic

C₂×Dic₈₃

C₄⋊C₈

yes

characteristic

C₄×D₈₃

—

C₄×D₈₃

—

C₄×D₈₃

—

C₄×D₈₃

—

C₄×D₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

M₃₂

yes

—

C₈×C₈₃

#135

yes

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

#108

yes

—

C₈₃⋊C₈

M₃₂

yes

—

C₂²×C₈₃

C₂×C₄⋊C₈

yes

characteristic

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

#364

yes

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

C₂×M₃₂

yes

characteristic

C₄×C₈₃

C₂×M₃₂

yes

characteristic

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

D₈₃

—

#2163

—

100

C₂×C₈²

—

101

#1370

—

102

#1367

—

103

#1367

D₈₃

yes

characteristic, complemented

104

C₈₃

#12013

yes

characteristic, complemented, Sylow

105

C₂×C₄×C₈

—

106

C₂×C₄×C₈

—

107

C₂×C₄×C₈

—

108

C₄⋊C₁₆

—

109

C₄⋊C₁₆

C₂×D₈₃

yes

—

110

C₂²⋊C₁₆

—

166

—

111

#346

—

166

—

112

C₈²

—

113

C₈²

—

114

C₈²

—

115

C₈²

C₂×D₈₃

yes

characteristic

116

C₂²×C₈

—

117

C₂²×C₈

—

118

C₂²×C₈

—

119

C₂²×C₈

—

120

C₂²×C₈

—

166

—

121

C₂×C₄²

—

122

C₂×C₁₆

—

123

C₂×C₁₆

—

166

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₄×C₈

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

—

166

—

132

C₄×C₈

C₄×D₈₃

yes

characteristic

133

C₄×C₈

C₄×D₈₃

yes

characteristic

134

C₄×C₈

C₂²×D₈₃

yes

characteristic

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

166

—

144

C₂×C₈

—

166

—

145

C₂×C₈

—

166

—

146

C₂×C₈

—

166

—

147

C₂×C₈

—

166

—

148

C₂×C₈

—

166

—

149

C₂×C₈

C₈×D₈₃

yes

characteristic

150

C₂×C₈

C₈×D₈₃

yes

characteristic

151

C₂×C₈

D₄×D₈₃

yes

characteristic

152

C₂×C₈

Q₈×D₈₃

yes

characteristic

153

C₄²

—

154

C₄²

—

155

C₄²

—

156

C₄²

C₂×C₄×D₈₃

yes

characteristic

157

C₁₆

—

158

C₁₆

—

332

—

159

C₂³

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

166

—

164

C₂×C₄

—

166

—

165

C₂×C₄

—

166

—

166

C₂×C₄

—

166

—

167

C₂×C₄

—

166

—

168

C₂×C₄

C₄⋊C₄×D₈₃

yes

characteristic

169

C₂×C₄

C₂×C₈×D₈₃

yes

characteristic

170

C₂×C₄

M₁₆×D₈₃

yes

characteristic

171

C₈

—

172

C₈

—

173

C₈

—

174

C₈

—

175

C₈

—

176

C₈

—

166

—

177

C₈

—

166

—

178

C₈

—

166

—

179

C₈

#14581731

yes

—

180

C₈

#14581813

yes

—

181

C₂²

—

182

C₂²

—

166

—

183

C₂²

—

166

—

184

C₂²

C₄⋊C₈×D₈₃

yes

characteristic

185

C₄

—

186

C₄

—

166

—

187

C₄

—

166

—

188

C₄

—

166

—

189

C₄

#19127736

yes

characteristic

190

C₄

#108×D₈₃

yes

characteristic

191

C₄

M₃₂×D₈₃

yes

characteristic

192

C₄

C₂×#14581731

yes

characteristic

193

C₂

—

166

—

194

C₂

#24906208

yes

characteristic

195

C₂

#24909217

yes

characteristic

196

C₂

#364×D₈₃

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₈₃×#1367	C₂
socle	C₂²×C₈₃	C₂×C₄⋊C₈
prime radical	C₂²×D₈₃	C₄⋊C₈
cosocle	C₄×C₈×C₈₃	C₂³
hypercenter	#1367	D₈₃
hypocenter	C₈₃	#12013
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1367

subgroup

G/H

center

C₂×C₈

D₄×D₈₃

commutator

C₂×C₄×C₈₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₈₃

Fitting

C₈₃×#1367

C₂

socle

C₂²×C₈₃

C₂×C₄⋊C₈

prime radical

C₂²×D₈₃

C₄⋊C₈

cosocle

C₄×C₈×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1367

N	H ≅ G/N	type
3. #24905532	C₂	semidirect
4. #24905527	C₂	semidirect
5. C₈₃×#1367	C₂	semidirect
103. #1367	D₈₃	semidirect
104. C₈₃	#12013	semidirect

H ≅ G/N

type

3. #24905532

C₂

semidirect

4. #24905527

C₂

semidirect

5. C₈₃×#1367

C₂

semidirect

103. #1367

D₈₃

semidirect

104. C₈₃

#12013

semidirect

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

subgroup	index
#24909265	2
C₈²×D₈₃	2
#24905532	2
#24905527	2
C₈₃×#1367	2
#12013	83

subgroup

index

C₈²×D₈₃

C₈₃×#1367

Subgroups — #35316005

Conjugacy classes of subgroups

Complements

Maximal subgroups