Group #35322639 — subgroups

Subgroups — #35322639

Order 21248.

486 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 159 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×#19128081	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127504	C₂	1	4	yes	maximal
3	C₂×#19127477	C₂	1	1	yes	maximal, characteristic
4	C₈₃×#1846	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₃₂	—	2	1	no	—
6	C₂²×Dic₃₃₂	C₂²	1	2	yes	—
7	C₂×#14581792	—	2	1	no	—
8	C₂²×C₈₃⋊C₁₆	C₂²	1	2	yes	—
9	#19128081	—	2	2	no	—
10	#19128081	C₄	1	4	yes	complemented
11	#19127504	—	2	8	no	—
12	#19127477	—	2	2	no	—
13	C₂×#107×C₈₃	C₂²	1	2	yes	—
14	C₂×#106×C₈₃	—	2	1	no	—
15	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
16	#382×C₈₃	—	2	1	no	—
17	C₂²×Dic₁₆₆	—	2	2	no	—
18	C₂×Dic₃₃₂	—	2	4	no	—
19	C₂×Dic₃₃₂	—	2	4	no	—
20	C₂×Dic₃₃₂	—	2	2	no	—
21	C₂×Dic₃₃₂	—	2	4	no	—
22	C₂×Dic₃₃₂	—	4	2	no	—
23	C₂×Dic₃₃₂	—	4	2	no	—
24	C₂×Dic₃₃₂	C₂×C₄	1	4	yes	—
25	#14581792	—	2	4	no	—
26	#14581792	—	4	2	no	—
27	C₂×C₄⋊Dic₈₃	—	4	2	no	—
28	C₂×C₈₃⋊C₁₆	—	2	2	no	—
29	C₂×C₈₃⋊C₁₆	—	2	2	no	—
30	C₂×C₈₃⋊C₁₆	D₄	1	4	yes	—
31	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
34	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
35	#107×C₈₃	—	2	4	no	—
36	#106×C₈₃	—	4	2	no	—
37	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₈₃	D₄	1	2	yes	—
39	#95×C₈₃	—	4	2	no	—
40	C₂×Dic₁₆₆	—	2	4	no	—
41	C₂×Dic₁₆₆	—	4	2	no	—
42	C₂×Dic₁₆₆	—	4	2	no	—
43	C₂×Dic₁₆₆	—	4	2	no	—
44	C₂×Dic₁₆₆	—	4	2	no	—
45	C₂×Dic₁₆₆	—	4	2	no	—
46	C₂²×Dic₈₃	—	4	2	no	—
47	Dic₃₃₂	—	2	8	no	—
48	Dic₃₃₂	—	4	8	no	—
49	Dic₃₃₂	—	4	4	no	—
50	C₄⋊Dic₈₃	—	4	8	no	—
51	C₈₃⋊C₁₆	—	2	4	no	—
52	C₂²×C₄×C₈₃	—	4	2	no	—
53	C₂²×C₄×C₈₃	D₈	1	2	yes	—
54	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₈₃	D₈	1	2	yes	—
57	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
58	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₈₃	#41	1	2	yes	—
62	C₄⋊C₄×C₈₃	—	4	2	no	—
63	C₄⋊C₄×C₈₃	—	4	2	no	—
64	C₄²×C₈₃	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₈₃	SD₁₆	1	2	yes	—
66	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₈₃	—	2	4	no	—
68	C₂×C₄×C₈₃	—	8	2	no	—
69	C₂×C₄×C₈₃	—	8	4	no	—
70	C₂×C₄×C₈₃	#102	1	4	yes	—
71	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
73	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
74	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×Dic₈₃	—	4	4	no	—
78	C₂×Dic₈₃	—	8	2	no	—
79	C₂×Dic₈₃	—	8	2	no	—
80	Dic₁₆₆	—	4	4	no	—
81	Dic₁₆₆	—	4	4	no	—
82	Dic₁₆₆	—	8	2	no	—
83	Dic₁₆₆	—	8	2	no	—
84	C₈×C₈₃	#102	1	4	yes	—
85	C₈×C₈₃	C₄×D₄	1	2	yes	—
86	C₈×C₈₃	#121	1	2	yes	—
87	C₂²×C₈₃	C₂×#102	1	1	yes	characteristic
88	C₂²×C₈₃	#486	1	1	yes	characteristic
89	C₂²×C₈₃	#493	1	1	yes	characteristic
90	C₂²×C₈₃	#510	1	4	yes	—
91	C₄×C₈₃	—	2	4	no	—
92	C₄×C₈₃	—	16	2	no	—
93	C₄×C₈₃	#390	1	1	yes	characteristic
94	C₄×C₈₃	C₂×#102	1	1	yes	characteristic
95	C₄×C₈₃	#486	1	1	yes	characteristic
96	C₄×C₈₃	#492	1	1	yes	characteristic
97	Dic₈₃	—	8	4	no	—
98	#13207	—	83	1	no	maximal, Sylow
99	C₂×C₈₃	#1957	1	1	yes	characteristic
100	C₂×C₈₃	#2141	1	2	yes	—
101	C₂×C₈₃	#2245	1	2	yes	—
102	C₂×C₈₃	#2251	1	2	yes	—
103	C₂×#494	—	83	1	no	—
104	C₂×#358	—	83	4	no	—
105	C₂×#346	—	83	1	no	—
106	#1846	D₈₃	1	1	yes	characteristic
107	C₈₃	#13207	1	1	yes	characteristic, complemented, Sylow
108	C₂²×Q₁₆	—	83	2	no	—
109	C₂²×Q₁₆	—	166	1	no	—
110	C₂×C₄⋊Q₈	—	166	1	no	—
111	C₂²×C₁₆	—	83	2	no	—
112	#494	—	83	4	no	—
113	#494	—	166	2	no	—
114	C₂×#107	C₂×D₈₃	1	2	yes	—
115	C₂×#106	—	2	1	no	—
116	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
117	#382	—	2	1	no	—
118	#358	—	166	8	no	—
119	#346	—	166	2	no	—
120	C₂²×Q₈	—	166	2	no	—
121	C₂×Q₁₆	—	332	2	no	—
122	C₂×Q₁₆	—	332	2	no	—
123	C₂×Q₁₆	—	83	4	no	—
124	C₂×Q₁₆	—	166	4	no	—
125	C₂×Q₁₆	—	166	4	no	—
126	C₂×Q₁₆	—	166	2	no	—
127	C₂×Q₁₆	—	166	4	no	—
128	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
129	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
130	C₄⋊Q₈	—	332	2	no	—
131	C₄⋊Q₈	—	166	4	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	—	332	2	no	—
134	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
135	C₂×C₁₆	—	83	4	no	—
136	C₂×C₁₆	—	166	2	no	—
137	C₂×C₁₆	—	166	2	no	—
138	#107	—	2	4	no	—
139	#106	—	4	2	no	—
140	C₄×C₈	C₄×D₈₃	1	2	yes	—
141	C₄×C₈	D₃₃₂	1	2	yes	—
142	#95	—	4	2	no	—
143	C₂×Q₈	—	332	2	no	—
144	C₂×Q₈	—	332	2	no	—
145	C₂×Q₈	—	332	2	no	—
146	C₂×Q₈	—	332	2	no	—
147	C₂×Q₈	—	332	2	no	—
148	C₂×Q₈	—	166	4	no	—
149	C₂²×C₄	—	4	2	no	—
150	C₂²×C₄	—	332	2	no	—
151	C₂²×C₄	#13050575	1	2	yes	—
152	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
153	Q₁₆	—	332	4	no	—
154	Q₁₆	—	332	8	no	—
155	Q₁₆	—	166	8	no	—
156	C₂×C₈	(C₂×D₈₃)⋊C₄	1	2	yes	—
157	C₂×C₈	#13050575	1	2	yes	—
158	C₂×C₈	#13050577	1	2	yes	—
159	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
160	C₂×C₈	D₄×D₈₃	1	2	yes	—
161	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
162	C₂×C₈	#13050591	1	2	yes	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	332	8	no	—
166	C₄²	(C₂×D₈₃)⋊C₄	1	2	yes	—
167	C₄²	#13050577	1	2	yes	—
168	C₁₆	—	166	4	no	—
169	C₂³	C₂×#13050575	1	1	yes	characteristic
170	Q₈	—	332	4	no	—
171	Q₈	—	332	4	no	—
172	Q₈	—	664	2	no	—
173	Q₈	—	664	2	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	8	2	no	—
176	C₂×C₄	—	8	4	no	—
177	C₂×C₄	—	332	4	no	—
178	C₂×C₄	—	664	2	no	—
179	C₂×C₄	—	664	2	no	—
180	C₂×C₄	#14581712	1	4	yes	—
181	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
182	C₂×C₄	#14581781	1	1	yes	characteristic
183	C₂×C₄	#14581788	1	1	yes	characteristic
184	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
185	C₂×C₄	C₂×#13050577	1	1	yes	characteristic
186	C₂×C₄	C₂×#13050577	1	1	yes	characteristic
187	C₈	#14581712	1	4	yes	—
188	C₈	#14581754	1	2	yes	—
189	C₈	#14581767	1	2	yes	—
190	C₂²	C₂×#14581712	1	1	yes	characteristic
191	C₂²	#19128049	1	1	yes	characteristic
192	C₂²	#19128077	1	1	yes	characteristic
193	C₂²	#19128141	1	4	yes	—
194	C₄	—	2	4	no	—
195	C₄	—	16	2	no	—
196	C₄	—	664	4	no	—
197	C₄	#19127602	1	1	yes	characteristic
198	C₄	C₂×#14581712	1	1	yes	characteristic
199	C₄	#19128049	1	1	yes	characteristic
200	C₄	#19128075	1	1	yes	characteristic
201	C₂	#24908268	1	1	yes	characteristic
202	C₂	#24909182	1	2	yes	—
203	C₂	#24909710	1	2	yes	—
204	C₂	#24909735	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19128081

C₂

yes

maximal, characteristic

C₂×#19127504

C₂

yes

maximal

C₂×#19127477

C₂

yes

maximal, characteristic

C₈₃×#1846

C₂

yes

maximal, characteristic

C₂²×Dic₃₃₂

—

C₂²×Dic₃₃₂

C₂²

yes

—

C₂×#14581792

—

C₂²×C₈₃⋊C₁₆

C₂²

yes

—

#19128081

—

#19128081

C₄

yes

complemented

#19127504

—

#19127477

—

C₂×#107×C₈₃

C₂²

yes

—

C₂×#106×C₈₃

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

#382×C₈₃

—

C₂²×Dic₁₆₆

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

C₂×C₄

yes

—

#14581792

—

#14581792

—

C₂×C₄⋊Dic₈₃

—

C₂×C₈₃⋊C₁₆

—

C₂×C₈₃⋊C₁₆

—

C₂×C₈₃⋊C₁₆

D₄

yes

—

C₂²×C₈×C₈₃

D₄

yes

characteristic

C₂²×C₈×C₈₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄²×C₈₃

D₄

yes

characteristic

#107×C₈₃

—

#106×C₈₃

—

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

D₄

yes

—

#95×C₈₃

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂²×Dic₈₃

—

Dic₃₃₂

—

Dic₃₃₂

—

Dic₃₃₂

—

C₄⋊Dic₈₃

—

C₈₃⋊C₁₆

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

D₈

yes

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

C₂²⋊C₄

yes

—

C₂×C₈×C₈₃

D₈

yes

—

C₂×C₈×C₈₃

SD₁₆

yes

—

C₂×C₈×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂×D₄

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

#41

yes

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

C₂²⋊C₄

yes

—

C₄²×C₈₃

SD₁₆

yes

—

C₂³×C₈₃

C₂×D₈

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

#102

yes

—

C₂×C₄×C₈₃

C₂×C₂²⋊C₄

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

C₂×D₈

yes

characteristic

C₂×C₄×C₈₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₃

C₂×SD₁₆

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

#102

yes

—

C₈×C₈₃

C₄×D₄

yes

—

C₈×C₈₃

#121

yes

—

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

—

C₂×C₈₃

yes

100

C₂×C₈₃

#2141

yes

—

101

C₂×C₈₃

#2245

yes

—

102

C₂×C₈₃

#2251

yes

—

103

C₂×#494

—

104

C₂×#358

—

105

C₂×#346

—

106

#1846

D₈₃

yes

characteristic

107

C₈₃

#13207

yes

characteristic, complemented, Sylow

108

C₂²×Q₁₆

—

109

C₂²×Q₁₆

—

166

—

110

C₂×C₄⋊Q₈

—

166

—

111

C₂²×C₁₆

—

112

#494

—

113

#494

—

166

—

114

C₂×#107

C₂×D₈₃

yes

—

115

C₂×#106

—

116

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

117

#382

—

118

#358

—

166

—

119

#346

—

166

—

120

C₂²×Q₈

—

166

—

121

C₂×Q₁₆

—

332

—

122

C₂×Q₁₆

—

332

—

123

C₂×Q₁₆

—

124

C₂×Q₁₆

—

166

—

125

C₂×Q₁₆

—

166

—

126

C₂×Q₁₆

—

166

—

127

C₂×Q₁₆

—

166

—

128

C₂²×C₈

C₂²⋊D₈₃

yes

characteristic

129

C₂²×C₈

C₂²×D₈₃

yes

characteristic

130

C₄⋊Q₈

—

332

—

131

C₄⋊Q₈

—

166

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

332

—

134

C₂×C₄²

C₂²⋊D₈₃

yes

characteristic

135

C₂×C₁₆

—

136

C₂×C₁₆

—

166

—

137

C₂×C₁₆

—

166

—

138

#107

—

139

#106

—

140

C₄×C₈

C₄×D₈₃

yes

—

141

C₄×C₈

D₃₃₂

yes

—

142

#95

—

143

C₂×Q₈

—

332

—

144

C₂×Q₈

—

332

—

145

C₂×Q₈

—

332

—

146

C₂×Q₈

—

332

—

147

C₂×Q₈

—

332

—

148

C₂×Q₈

—

166

—

149

C₂²×C₄

—

150

C₂²×C₄

—

332

—

151

C₂²×C₄

#13050575

yes

—

152

C₂²×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

153

Q₁₆

—

332

—

154

Q₁₆

—

332

—

155

Q₁₆

—

166

—

156

C₂×C₈

(C₂×D₈₃)⋊C₄

yes

—

157

C₂×C₈

#13050575

yes

—

158

C₂×C₈

#13050577

yes

—

159

C₂×C₈

C₂×C₄×D₈₃

yes

characteristic

160

C₂×C₈

D₄×D₈₃

yes

—

161

C₂×C₈

C₂×D₃₃₂

yes

characteristic

162

C₂×C₈

#13050591

yes

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

332

—

166

C₄²

(C₂×D₈₃)⋊C₄

yes

—

167

C₄²

#13050577

yes

—

168

C₁₆

—

166

—

169

C₂³

C₂×#13050575

yes

characteristic

170

Q₈

—

332

—

171

Q₈

—

332

—

172

Q₈

—

664

—

173

Q₈

—

664

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

332

—

178

C₂×C₄

—

664

—

179

C₂×C₄

—

664

—

180

C₂×C₄

#14581712

yes

—

181

C₂×C₄

C₂×(C₂×D₈₃)⋊C₄

yes

characteristic

182

C₂×C₄

#14581781

yes

characteristic

183

C₂×C₄

#14581788

yes

characteristic

184

C₂×C₄

C₂×#13050575

yes

characteristic

185

C₂×C₄

C₂×#13050577

yes

characteristic

186

C₂×C₄

C₂×#13050577

yes

characteristic

187

C₈

#14581712

yes

—

188

C₈

#14581754

yes

—

189

C₈

#14581767

yes

—

190

C₂²

C₂×#14581712

yes

characteristic

191

C₂²

#19128049

yes

characteristic

192

C₂²

#19128077

yes

characteristic

193

C₂²

#19128141

yes

—

194

C₄

—

195

C₄

—

196

C₄

—

664

—

197

C₄

#19127602

yes

characteristic

198

C₄

C₂×#14581712

yes

characteristic

199

C₄

#19128049

yes

characteristic

200

C₄

#19128075

yes

characteristic

201

C₂

#24908268

yes

characteristic

202

C₂

#24909182

yes

—

203

C₂

#24909710

yes

—

204

C₂

#24909735

yes

—

	subgroup	G/H
center	C₂³	C₂×#13050575
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#1846	C₂
socle, prime radical	C₂³×C₈₃	C₂×D₈
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	#1846	D₈₃
hypocenter	C₈₃	#13207
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

subgroup

G/H

center

C₂³

C₂×#13050575

commutator

C₂×C₈×C₈₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₃

Fitting

C₈₃×#1846

C₂

socle, prime radical

C₂³×C₈₃

C₂×D₈

cosocle

C₂²×C₈×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

N	H ≅ G/N	type
10. #19128081	C₄	semidirect
107. C₈₃	#13207	semidirect

H ≅ G/N

type

10. #19128081

C₄

semidirect

107. C₈₃

#13207

semidirect

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

subgroup	index
C₂×#19128081	2
C₂×#19127504	2
C₂×#19127477	2
C₈₃×#1846	2
#13207	83

subgroup

index

C₈₃×#1846

Subgroups — #35322639

Conjugacy classes of subgroups

Complements

Maximal subgroups