Group #35324008 — subgroups

Subgroups — #35324008

Order 21248.

410 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 113 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×Dic₈₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127505	C₂	1	2	yes	maximal
3	#24908325	C₂	1	2	yes	maximal
4	#24905555	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₃₂×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14581726	C₂²	1	2	yes	—
8	C₂×#14581716	—	2	2	no	—
9	C₂×C₈×Dic₈₃	C₂²	1	1	yes	characteristic
10	#19127608	—	2	2	no	—
11	Dic₈₃⋊C₁₆	—	2	2	no	—
12	#19127505	—	2	4	no	—
13	C₂²×Q₁₆×C₈₃	C₂²	1	2	yes	—
14	C₂×Q₃₂×C₈₃	—	2	4	no	—
15	C₂×Q₃₂×C₈₃	—	2	2	no	—
16	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
17	C₂²×C₈₃⋊C₈	D₄	1	1	yes	characteristic
18	C₂×C₄⋊Dic₈₃	—	2	2	no	—
19	C₂×C₄×Dic₈₃	D₄	1	1	yes	characteristic
20	C₂×C₈₃⋊C₁₆	—	2	2	no	—
21	C₂×C₈₃⋊C₁₆	—	2	1	no	—
22	#14581726	—	2	2	no	—
23	#14581726	—	2	2	no	—
24	#14581716	—	4	4	no	—
25	C₈×Dic₈₃	—	2	1	no	—
26	C₈×Dic₈₃	—	2	1	no	—
27	C₈₃⋊#95	—	4	2	no	—
28	C₈₃⋊#95	—	4	2	no	—
29	C₂²×Q₈×C₈₃	—	2	2	no	—
30	C₂×Q₁₆×C₈₃	—	2	2	no	—
31	C₂×Q₁₆×C₈₃	—	2	2	no	—
32	C₂×Q₁₆×C₈₃	—	2	2	no	—
33	C₂×Q₁₆×C₈₃	—	2	2	no	—
34	C₂×Q₁₆×C₈₃	—	2	2	no	—
35	C₂×Q₁₆×C₈₃	D₄	1	4	yes	—
36	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
37	Q₃₂×C₈₃	—	2	8	no	—
38	C₂×C₁₆×C₈₃	—	2	2	no	—
39	C₂×C₁₆×C₈₃	D₄	1	2	yes	—
40	C₂²×Dic₈₃	—	4	2	no	—
41	C₂²×Dic₈₃	D₈	1	1	yes	characteristic
42	C₂²×Dic₈₃	D₈	1	1	yes	characteristic
43	C₂×C₈₃⋊C₈	—	2	1	no	—
44	C₂×C₈₃⋊C₈	—	2	2	no	—
45	C₄⋊Dic₈₃	—	4	2	no	—
46	C₄⋊Dic₈₃	—	4	2	no	—
47	C₄×Dic₈₃	—	2	1	no	—
48	C₄×Dic₈₃	—	2	1	no	—
49	C₈₃⋊C₁₆	—	4	1	no	—
50	C₂×Q₈×C₈₃	—	2	4	no	—
51	C₂×Q₈×C₈₃	—	4	2	no	—
52	C₂×Q₈×C₈₃	—	4	2	no	—
53	C₂×Q₈×C₈₃	—	4	2	no	—
54	C₂×Q₈×C₈₃	—	4	2	no	—
55	C₂×Q₈×C₈₃	—	4	2	no	—
56	C₂²×C₄×C₈₃	—	4	2	no	—
57	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
58	Q₁₆×C₈₃	—	2	4	no	—
59	Q₁₆×C₈₃	—	2	4	no	—
60	Q₁₆×C₈₃	—	4	2	no	—
61	Q₁₆×C₈₃	—	4	2	no	—
62	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
65	C₂×C₈×C₈₃	#41	1	2	yes	—
66	C₁₆×C₈₃	—	2	2	no	—
67	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
68	Q₈×C₈₃	—	4	4	no	—
69	Q₈×C₈₃	—	4	4	no	—
70	Q₈×C₈₃	—	8	2	no	—
71	Q₈×C₈₃	—	8	2	no	—
72	C₂×C₄×C₈₃	—	4	4	no	—
73	C₂×C₄×C₈₃	—	8	2	no	—
74	C₂×C₄×C₈₃	—	8	2	no	—
75	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
76	C₂×C₄×C₈₃	#121	1	2	yes	—
77	C₂×C₄×C₈₃	#135	1	1	yes	characteristic
78	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
79	C₂×Dic₈₃	—	2	1	no	—
80	C₂×Dic₈₃	—	2	1	no	—
81	C₂×Dic₈₃	—	2	1	no	—
82	C₂×Dic₈₃	—	2	2	no	—
83	C₂×Dic₈₃	—	8	2	no	—
84	C₂×Dic₈₃	—	8	2	no	—
85	C₂×Dic₈₃	—	8	2	no	—
86	C₂×Dic₈₃	Q₃₂	1	2	yes	complemented
87	C₈×C₈₃	#120	1	1	yes	characteristic
88	C₈×C₈₃	#123	1	2	yes	—
89	C₈×C₈₃	#124	1	1	yes	characteristic
90	C₈₃⋊C₈	—	4	1	no	—
91	C₂²×C₈₃	#459	1	2	yes	—
92	C₂²×C₈₃	#466	1	1	yes	characteristic
93	C₂²×C₈₃	C₂×Q₃₂	1	1	yes	characteristic
94	C₂²×C₈₃	#508	1	1	yes	characteristic
95	C₂²×C₈₃	#510	1	2	yes	—
96	C₄×C₈₃	—	8	4	no	—
97	C₄×C₈₃	#394	1	1	yes	characteristic
98	C₄×C₈₃	#463	1	2	yes	—
99	C₄×C₈₃	#469	1	1	yes	characteristic
100	Dic₈₃	—	2	2	no	—
101	Dic₈₃	—	4	1	no	—
102	Dic₈₃	—	16	2	no	—
103	#13450	—	83	1	no	maximal, Sylow
104	C₂×C₈₃	#2054	1	1	yes	characteristic
105	C₂×C₈₃	#2209	1	2	yes	—
106	C₂×C₈₃	#2211	1	2	yes	—
107	C₂×C₈₃	#2216	1	1	yes	characteristic
108	C₂×C₈₃	#2219	1	1	yes	characteristic
109	C₂²×Q₃₂	D₈₃	1	1	yes	characteristic
110	C₂×C₄⋊C₁₆	—	83	1	no	—
111	C₂×#358	—	83	2	no	—
112	#1969	—	83	2	no	—
113	#1380	—	83	1	no	—
114	C₈₃	#13450	1	1	yes	characteristic, complemented, Sylow
115	C₂²×Q₁₆	C₂×D₈₃	1	2	yes	—
116	C₂×Q₃₂	—	2	4	no	—
117	C₂×Q₃₂	—	2	2	no	—
118	C₂²×C₁₆	—	83	1	no	—
119	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
120	C₂×#107	—	83	2	no	—
121	C₂×#103	—	166	2	no	—
122	C₂×C₄×C₈	—	83	1	no	—
123	#391	—	166	2	no	—
124	C₄⋊C₁₆	—	166	2	no	—
125	#358	—	166	4	no	—
126	C₂²×Q₈	—	2	2	no	—
127	C₂×Q₁₆	—	2	2	no	—
128	C₂×Q₁₆	—	2	2	no	—
129	C₂×Q₁₆	—	2	2	no	—
130	C₂×Q₁₆	—	2	2	no	—
131	C₂×Q₁₆	—	2	2	no	—
132	C₂×Q₁₆	C₂²⋊D₈₃	1	4	yes	—
133	C₂²×C₈	—	83	1	no	—
134	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
135	C₂×C₄⋊C₄	—	166	2	no	—
136	C₂×C₄²	—	83	1	no	—
137	Q₃₂	—	2	8	no	—
138	C₂×C₁₆	—	2	2	no	—
139	C₂×C₁₆	—	166	2	no	—
140	C₂×C₁₆	—	166	1	no	—
141	C₂×C₁₆	C₂²⋊D₈₃	1	2	yes	—
142	#107	—	166	2	no	—
143	#107	—	166	2	no	—
144	#103	—	332	4	no	—
145	C₄×C₈	—	166	1	no	—
146	C₄×C₈	—	166	1	no	—
147	#95	—	332	2	no	—
148	#95	—	332	2	no	—
149	C₂×Q₈	—	2	4	no	—
150	C₂×Q₈	—	4	2	no	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	4	2	no	—
153	C₂×Q₈	—	4	2	no	—
154	C₂×Q₈	—	4	2	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	—	332	2	no	—
157	C₂²×C₄	—	83	1	no	—
158	C₂²×C₄	—	83	1	no	—
159	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
160	Q₁₆	—	2	4	no	—
161	Q₁₆	—	2	4	no	—
162	Q₁₆	—	4	2	no	—
163	Q₁₆	—	4	2	no	—
164	C₂×C₈	—	166	1	no	—
165	C₂×C₈	—	166	2	no	—
166	C₂×C₈	C₂×C₂²⋊D₈₃	1	2	yes	—
167	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
168	C₂×C₈	#13050589	1	1	yes	characteristic
169	C₂×C₈	#13050589	1	2	yes	—
170	C₄⋊C₄	—	332	2	no	—
171	C₄⋊C₄	—	332	2	no	—
172	C₄²	—	166	1	no	—
173	C₄²	—	166	1	no	—
174	C₁₆	—	2	2	no	—
175	C₁₆	—	332	1	no	—
176	C₂³	D₈×D₈₃	1	1	yes	characteristic
177	Q₈	—	4	4	no	—
178	Q₈	—	4	4	no	—
179	Q₈	—	8	2	no	—
180	Q₈	—	8	2	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	—	8	2	no	—
184	C₂×C₄	—	83	2	no	—
185	C₂×C₄	—	664	2	no	—
186	C₂×C₄	—	664	2	no	—
187	C₂×C₄	—	664	2	no	—
188	C₂×C₄	—	166	2	no	—
189	C₂×C₄	—	166	1	no	—
190	C₂×C₄	—	166	1	no	—
191	C₂×C₄	—	166	1	no	—
192	C₂×C₄	#14581763	1	1	yes	characteristic
193	C₂×C₄	#14581764	1	2	yes	—
194	C₂×C₄	#14581811	1	1	yes	characteristic
195	C₂×C₄	#14581816	1	2	yes	—
196	C₈	—	332	1	no	—
197	C₈	#14581761	1	1	yes	characteristic
198	C₈	#14581773	1	2	yes	—
199	C₈	#14581779	1	1	yes	characteristic
200	C₂²	#19127894	1	2	yes	—
201	C₂²	#19127934	1	1	yes	characteristic
202	C₂²	Q₃₂×D₈₃	1	1	yes	characteristic
203	C₂²	#19128134	1	1	yes	characteristic
204	C₂²	#19128143	1	2	yes	—
205	C₄	—	8	4	no	—
206	C₄	—	1328	2	no	—
207	C₄	—	332	1	no	—
208	C₄	—	166	2	no	—
209	C₄	#19127615	1	1	yes	characteristic
210	C₄	#19127914	1	2	yes	—
211	C₄	#19127951	1	1	yes	characteristic
212	C₂	#24908771	1	1	yes	characteristic
213	C₂	#24909511	1	2	yes	—
214	C₂	#24909527	1	2	yes	—
215	C₂	#24909551	1	1	yes	characteristic
216	C₂	#24909568	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₈₃⋊C₁₆

C₂

yes

maximal, characteristic

C₂×#19127505

C₂

yes

maximal

#24908325

C₂

yes

maximal

#24905555

C₂

yes

maximal, characteristic

C₂²×Q₃₂×C₈₃

C₂

yes

maximal, characteristic

C₂²×C₈₃⋊C₁₆

C₂²

yes

characteristic

C₂×#14581726

C₂²

yes

—

C₂×#14581716

—

C₂×C₈×Dic₈₃

C₂²

yes

characteristic

#19127608

—

Dic₈₃⋊C₁₆

—

#19127505

—

C₂²×Q₁₆×C₈₃

C₂²

yes

—

C₂×Q₃₂×C₈₃

—

C₂×Q₃₂×C₈₃

—

C₂²×C₁₆×C₈₃

C₂²

yes

characteristic

C₂²×C₈₃⋊C₈

D₄

yes

characteristic

C₂×C₄⋊Dic₈₃

—

C₂×C₄×Dic₈₃

D₄

yes

characteristic

C₂×C₈₃⋊C₁₆

—

C₂×C₈₃⋊C₁₆

—

#14581726

—

#14581726

—

#14581716

—

C₈×Dic₈₃

—

C₈×Dic₈₃

—

C₈₃⋊#95

—

C₈₃⋊#95

—

C₂²×Q₈×C₈₃

—

C₂×Q₁₆×C₈₃

—

C₂×Q₁₆×C₈₃

—

C₂×Q₁₆×C₈₃

—

C₂×Q₁₆×C₈₃

—

C₂×Q₁₆×C₈₃

—

C₂×Q₁₆×C₈₃

D₄

yes

—

C₂²×C₈×C₈₃

C₂³

yes

characteristic

Q₃₂×C₈₃

—

C₂×C₁₆×C₈₃

—

C₂×C₁₆×C₈₃

D₄

yes

—

C₂²×Dic₈₃

—

C₂²×Dic₈₃

D₈

yes

characteristic

C₂²×Dic₈₃

D₈

yes

characteristic

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄×Dic₈₃

—

C₄×Dic₈₃

—

C₈₃⋊C₁₆

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

Q₁₆×C₈₃

—

Q₁₆×C₈₃

—

Q₁₆×C₈₃

—

Q₁₆×C₈₃

—

C₂×C₈×C₈₃

C₂×D₄

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

C₂×C₈×C₈₃

yes

C₂×C₈×C₈₃

yes

—

C₁₆×C₈₃

—

C₂³×C₈₃

C₂×D₈

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

#121

yes

characteristic

C₂×C₄×C₈₃

#121

yes

—

C₂×C₄×C₈₃

#135

yes

characteristic

C₂×C₄×C₈₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

Q₃₂

yes

C₈×C₈₃

yes

C₈×C₈₃

yes

—

C₈×C₈₃

#124

yes

characteristic

C₈₃⋊C₈

—

C₂²×C₈₃

#459

yes

—

C₂²×C₈₃

#466

yes

characteristic

C₂²×C₈₃

C₂×Q₃₂

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

—

C₄×C₈₃

—

C₄×C₈₃

#394

yes

characteristic

C₄×C₈₃

#463

yes

—

C₄×C₈₃

#469

yes

characteristic

100

Dic₈₃

—

101

Dic₈₃

—

102

Dic₈₃

—

103

#13450

—

maximal, Sylow

104

C₂×C₈₃

#2054

yes

characteristic

105

C₂×C₈₃

#2209

yes

—

106

C₂×C₈₃

#2211

yes

—

107

C₂×C₈₃

#2216

yes

characteristic

108

C₂×C₈₃

#2219

yes

characteristic

109

C₂²×Q₃₂

D₈₃

yes

characteristic

110

C₂×C₄⋊C₁₆

—

111

C₂×#358

—

112

#1969

—

113

#1380

—

114

C₈₃

#13450

yes

characteristic, complemented, Sylow

115

C₂²×Q₁₆

C₂×D₈₃

yes

—

116

C₂×Q₃₂

—

117

C₂×Q₃₂

—

118

C₂²×C₁₆

—

119

C₂²×C₁₆

C₂×D₈₃

yes

characteristic

120

C₂×#107

—

121

C₂×#103

—

166

—

122

C₂×C₄×C₈

—

123

#391

—

166

—

124

C₄⋊C₁₆

—

166

—

125

#358

—

166

—

126

C₂²×Q₈

—

127

C₂×Q₁₆

—

128

C₂×Q₁₆

—

129

C₂×Q₁₆

—

130

C₂×Q₁₆

—

131

C₂×Q₁₆

—

132

C₂×Q₁₆

C₂²⋊D₈₃

yes

—

133

C₂²×C₈

—

134

C₂²×C₈

C₂²×D₈₃

yes

characteristic

135

C₂×C₄⋊C₄

—

166

—

136

C₂×C₄²

—

137

Q₃₂

—

138

C₂×C₁₆

—

139

C₂×C₁₆

—

166

—

140

C₂×C₁₆

—

166

—

141

C₂×C₁₆

C₂²⋊D₈₃

yes

—

142

#107

—

166

—

143

#107

—

166

—

144

#103

—

332

—

145

C₄×C₈

—

166

—

146

C₄×C₈

—

166

—

147

#95

—

332

—

148

#95

—

332

—

149

C₂×Q₈

—

150

C₂×Q₈

—

151

C₂×Q₈

—

152

C₂×Q₈

—

153

C₂×Q₈

—

154

C₂×Q₈

—

155

C₂²×C₄

—

156

C₂²×C₄

—

332

—

157

C₂²×C₄

—

158

C₂²×C₄

—

159

C₂²×C₄

D₄×D₈₃

yes

characteristic

160

Q₁₆

—

161

Q₁₆

—

162

Q₁₆

—

163

Q₁₆

—

164

C₂×C₈

—

166

—

165

C₂×C₈

—

166

—

166

C₂×C₈

C₂×C₂²⋊D₈₃

yes

—

167

C₂×C₈

C₂×C₂²⋊D₈₃

yes

characteristic

168

C₂×C₈

#13050589

yes

characteristic

169

C₂×C₈

#13050589

yes

—

170

C₄⋊C₄

—

332

—

171

C₄⋊C₄

—

332

—

172

C₄²

—

166

—

173

C₄²

—

166

—

174

C₁₆

—

175

C₁₆

—

332

—

176

C₂³

D₈×D₈₃

yes

characteristic

177

Q₈

—

178

Q₈

—

179

Q₈

—

180

Q₈

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

664

—

186

C₂×C₄

—

664

—

187

C₂×C₄

—

664

—

188

C₂×C₄

—

166

—

189

C₂×C₄

—

166

—

190

C₂×C₄

—

166

—

191

C₂×C₄

—

166

—

192

C₂×C₄

#14581763

yes

characteristic

193

C₂×C₄

#14581764

yes

—

194

C₂×C₄

#14581811

yes

characteristic

195

C₂×C₄

#14581816

yes

—

196

C₈

—

332

—

197

C₈

#14581761

yes

characteristic

198

C₈

#14581773

yes

—

199

C₈

#14581779

yes

characteristic

200

C₂²

#19127894

yes

—

201

C₂²

#19127934

yes

characteristic

202

C₂²

Q₃₂×D₈₃

yes

characteristic

203

C₂²

#19128134

yes

characteristic

204

C₂²

#19128143

yes

—

205

C₄

—

206

C₄

—

1328

—

207

C₄

—

332

—

208

C₄

—

166

—

209

C₄

#19127615

yes

characteristic

210

C₄

#19127914

yes

—

211

C₄

#19127951

yes

characteristic

212

C₂

#24908771

yes

characteristic

213

C₂

#24909511

yes

—

214

C₂

#24909527

yes

—

215

C₂

#24909551

yes

characteristic

216

C₂

#24909568

yes

characteristic

	subgroup	G/H
center	C₂³	D₈×D₈₃
commutator, cosocle	C₂²×C₈×C₈₃	C₂³
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₂²×Q₃₂×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×D₈
hypercenter	C₂²×Q₃₂	D₈₃
hypocenter	C₈₃	#13450
LCS	G ▷ C₂²×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

subgroup

G/H

center

C₂³

D₈×D₈₃

commutator, cosocle

C₂²×C₈×C₈₃

C₂³

Frattini

C₂²×C₈

C₂²×D₈₃

Fitting

C₂²×Q₃₂×C₈₃

C₂

socle, prime radical

C₂³×C₈₃

C₂×D₈

hypercenter

C₂²×Q₃₂

D₈₃

hypocenter

C₈₃

#13450

LCS

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

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

N	H ≅ G/N	type
86. C₂×Dic₈₃	Q₃₂	semidirect
114. C₈₃	#13450	semidirect

H ≅ G/N

type

86. C₂×Dic₈₃

Q₃₂

semidirect

114. C₈₃

#13450

semidirect

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

subgroup	index
C₂×Dic₈₃⋊C₁₆	2
C₂×#19127505	2
#24908325	2
#24905555	2
C₂²×Q₃₂×C₈₃	2
#13450	83

subgroup

index

C₂×Dic₈₃⋊C₁₆

C₂²×Q₃₂×C₈₃

Subgroups — #35324008

Conjugacy classes of subgroups

Complements

Maximal subgroups