Group #35322437 — subgroups

Subgroups — #35322437

Order 21248.

326 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 135 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×#19127529	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₈₃	C₂	1	1	yes	maximal, characteristic
3	#24908194	C₂	1	2	yes	maximal
4	#24905554	C₂	1	2	yes	maximal
5	C₂×#367×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×C₈×Dic₈₃	C₂²	1	1	yes	characteristic
9	#19127597	—	2	2	no	—
10	#19127529	C₄	1	4	yes	complemented
11	C₁₆×Dic₈₃	—	2	2	no	—
12	#19127463	—	2	2	no	—
13	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
14	C₂×#107×C₈₃	C₂²	1	2	yes	—
15	#367×C₈₃	—	2	2	no	—
16	C₂²×C₈₃⋊C₈	D₄	1	1	yes	characteristic
17	C₂×C₄⋊Dic₈₃	—	2	2	no	—
18	C₂×Dic₈₃⋊C₄	—	4	2	no	—
19	C₂×C₄×Dic₈₃	D₄	1	1	yes	characteristic
20	C₂×C₈₃⋊C₁₆	—	2	1	no	—
21	C₂×C₈₃⋊C₁₆	—	2	1	no	—
22	C₂×C₈₃⋊C₁₆	Q₈	1	2	yes	—
23	#14581726	—	2	2	no	—
24	#14581726	C₂×C₄	1	4	yes	—
25	C₈×Dic₈₃	—	2	1	no	—
26	C₈×Dic₈₃	—	2	1	no	—
27	C₈₃⋊#95	—	4	2	no	—
28	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
30	C₂×C₁₆×C₈₃	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₈₃	D₄	1	2	yes	—
32	C₂×C₁₆×C₈₃	Q₈	1	2	yes	—
33	#107×C₈₃	—	2	2	no	—
34	#107×C₈₃	—	2	2	no	—
35	C₂²×Dic₈₃	—	4	2	no	—
36	C₂²×Dic₈₃	D₈	1	2	yes	—
37	C₂×C₈₃⋊C₈	—	2	1	no	—
38	C₂×C₈₃⋊C₈	—	2	1	no	—
39	C₂×C₈₃⋊C₈	Q₁₆	1	2	yes	—
40	C₄⋊Dic₈₃	—	2	4	no	—
41	C₄⋊Dic₈₃	—	4	2	no	—
42	Dic₈₃⋊C₄	—	8	4	no	—
43	C₄×Dic₈₃	—	2	1	no	—
44	C₄×Dic₈₃	—	2	1	no	—
45	C₈₃⋊C₁₆	—	2	2	no	—
46	C₂²×C₄×C₈₃	—	4	2	no	—
47	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
53	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
54	C₄⋊C₄×C₈₃	—	4	2	no	—
55	C₄⋊C₄×C₈₃	—	4	2	no	—
56	C₁₆×C₈₃	C₄⋊C₄	1	4	yes	complemented
57	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₈₃	—	8	2	no	—
59	C₂×C₄×C₈₃	—	8	2	no	—
60	C₂×C₄×C₈₃	—	8	2	no	—
61	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
62	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
63	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
65	C₂×C₄×C₈₃	#135	1	1	yes	characteristic
66	C₂×C₄×C₈₃	#135	1	1	yes	characteristic
67	C₂×Dic₈₃	—	2	2	no	—
68	C₂×Dic₈₃	—	2	2	no	—
69	C₂×Dic₈₃	—	4	4	no	—
70	C₂×Dic₈₃	—	8	2	no	—
71	C₂×Dic₈₃	—	8	2	no	—
72	C₂×Dic₈₃	SD₃₂	1	4	yes	—
73	C₈×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₈×C₈₃	C₄×Q₈	1	1	yes	characteristic
75	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
76	C₈×C₈₃	#125	1	1	yes	characteristic
77	C₈₃⋊C₈	—	2	2	no	—
78	C₂²×C₈₃	C₄×D₈	1	1	yes	characteristic
79	C₂²×C₈₃	#466	1	1	yes	characteristic
80	C₂²×C₈₃	#500	1	1	yes	characteristic
81	C₂²×C₈₃	C₂×SD₃₂	1	2	yes	—
82	C₂²×C₈₃	#508	1	2	yes	—
83	C₄×C₈₃	—	16	2	no	—
84	C₄×C₈₃	#389	1	1	yes	characteristic
85	C₄×C₈₃	C₄×Q₁₆	1	1	yes	characteristic
86	C₄×C₈₃	#467	1	1	yes	characteristic
87	C₄×C₈₃	#499	1	1	yes	characteristic
88	Dic₈₃	—	2	4	no	—
89	Dic₈₃	—	8	4	no	—
90	#13166	—	83	1	no	maximal, Sylow
91	C₂×C₈₃	#1942	1	1	yes	characteristic
92	C₂×C₈₃	C₄×SD₃₂	1	2	yes	—
93	C₂×C₈₃	#2217	1	2	yes	—
94	C₂×C₈₃	#2253	1	1	yes	characteristic
95	C₂×C₈₃	#2254	1	1	yes	characteristic
96	C₂×#367	—	83	1	no	—
97	C₂×#367	D₈₃	1	1	yes	characteristic
98	C₂×C₄×C₁₆	—	83	1	no	—
99	#1942	—	83	2	no	—
100	#1380	—	83	2	no	—
101	C₈₃	#13166	1	1	yes	characteristic, complemented, Sylow
102	C₂²×C₁₆	—	83	1	no	—
103	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
104	C₂×#107	—	83	2	no	—
105	C₂×#107	C₂×D₈₃	1	2	yes	—
106	C₂×C₄×C₈	—	83	1	no	—
107	#389	—	166	2	no	—
108	#367	—	2	2	no	—
109	#367	—	83	4	no	—
110	C₄×C₁₆	—	166	2	no	—
111	#340	—	166	2	no	—
112	C₂²×C₈	—	83	1	no	—
113	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	332	2	no	—
116	C₂×C₄⋊C₄	—	166	2	no	—
117	C₂×C₄²	—	83	1	no	—
118	C₂×C₁₆	—	83	2	no	—
119	C₂×C₁₆	—	166	1	no	—
120	C₂×C₁₆	—	166	1	no	—
121	C₂×C₁₆	Dic₁₆₆	1	2	yes	—
122	C₂×C₁₆	C₄×D₈₃	1	2	yes	—
123	C₂×C₁₆	C₂²⋊D₈₃	1	2	yes	—
124	#107	—	2	2	no	—
125	#107	—	2	2	no	—
126	#107	—	83	4	no	—
127	#107	—	166	2	no	—
128	C₄×C₈	—	166	1	no	—
129	C₄×C₈	—	166	1	no	—
130	#95	—	332	2	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	332	2	no	—
133	C₂²×C₄	—	83	2	no	—
134	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
135	C₂×C₈	—	83	2	no	—
136	C₂×C₈	—	166	1	no	—
137	C₂×C₈	—	166	1	no	—
138	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
139	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
140	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
141	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
142	C₂×C₈	#13050589	1	1	yes	characteristic
143	C₂×C₈	#13050591	1	1	yes	characteristic
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	332	2	no	—
147	C₄⋊C₄	—	664	4	no	—
148	C₄⋊C₄	—	166	4	no	—
149	C₄²	—	166	1	no	—
150	C₄²	—	166	1	no	—
151	C₁₆	—	166	2	no	—
152	C₁₆	Dic₈₃⋊C₄	1	4	yes	complemented
153	C₂³	D₈×D₈₃	1	1	yes	characteristic
154	C₂×C₄	—	8	2	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	332	4	no	—
158	C₂×C₄	—	83	4	no	—
159	C₂×C₄	—	664	2	no	—
160	C₂×C₄	—	664	2	no	—
161	C₂×C₄	—	166	2	no	—
162	C₂×C₄	—	166	2	no	—
163	C₂×C₄	#14581754	1	1	yes	characteristic
164	C₂×C₄	#14581763	1	1	yes	characteristic
165	C₂×C₄	#14581791	1	1	yes	characteristic
166	C₂×C₄	Q₁₆×D₈₃	1	1	yes	characteristic
167	C₂×C₄	#14581811	1	1	yes	characteristic
168	C₂×C₄	#14581815	1	1	yes	characteristic
169	C₈	—	166	2	no	—
170	C₈	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
171	C₈	#14581757	1	1	yes	characteristic
172	C₈	#14581768	1	1	yes	characteristic
173	C₈	#14581783	1	1	yes	characteristic
174	C₂²	#19127783	1	1	yes	characteristic
175	C₂²	#19127934	1	1	yes	characteristic
176	C₂²	#19128105	1	1	yes	characteristic
177	C₂²	SD₃₂×D₈₃	1	2	yes	—
178	C₂²	#19128132	1	2	yes	—
179	C₄	—	16	2	no	—
180	C₄	—	664	4	no	—
181	C₄	—	166	4	no	—
182	C₄	#19127597	1	1	yes	characteristic
183	C₄	#19127795	1	1	yes	characteristic
184	C₄	#19127939	1	1	yes	characteristic
185	C₄	#19128102	1	1	yes	characteristic
186	C₂	#24908194	1	1	yes	characteristic
187	C₂	#24909314	1	2	yes	—
188	C₂	#24909556	1	2	yes	—
189	C₂	#24909741	1	1	yes	characteristic
190	C₂	#24909744	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19127529

C₂

yes

maximal, characteristic

C₂×C₁₆×Dic₈₃

C₂

yes

maximal, characteristic

#24908194

C₂

yes

maximal

#24905554

C₂

yes

maximal

C₂×#367×C₈₃

C₂

yes

maximal, characteristic

C₂²×C₈₃⋊C₁₆

C₂²

yes

characteristic

C₂×#14581726

C₂²

yes

—

C₂×C₈×Dic₈₃

C₂²

yes

characteristic

#19127597

—

#19127529

C₄

yes

complemented

C₁₆×Dic₈₃

—

#19127463

—

C₂²×C₁₆×C₈₃

C₂²

yes

characteristic

C₂×#107×C₈₃

C₂²

yes

—

#367×C₈₃

—

C₂²×C₈₃⋊C₈

D₄

yes

characteristic

C₂×C₄⋊Dic₈₃

—

C₂×Dic₈₃⋊C₄

—

C₂×C₄×Dic₈₃

D₄

yes

characteristic

C₂×C₈₃⋊C₁₆

—

C₂×C₈₃⋊C₁₆

—

C₂×C₈₃⋊C₁₆

Q₈

yes

—

#14581726

—

#14581726

C₂×C₄

yes

—

C₈×Dic₈₃

—

C₈×Dic₈₃

—

C₈₃⋊#95

—

C₂²×C₈×C₈₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₈₃

—

C₂×C₁₆×C₈₃

C₂×C₄

yes

—

C₂×C₁₆×C₈₃

D₄

yes

—

C₂×C₁₆×C₈₃

Q₈

yes

—

#107×C₈₃

—

#107×C₈₃

—

C₂²×Dic₈₃

—

C₂²×Dic₈₃

D₈

yes

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

Q₁₆

yes

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

Dic₈₃⋊C₄

—

C₄×Dic₈₃

—

C₄×Dic₈₃

—

C₈₃⋊C₁₆

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₃

C₂×Q₈

yes

C₂×C₈×C₈₃

yes

C₂×C₈×C₈₃

yes

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₁₆×C₈₃

C₄⋊C₄

yes

complemented

C₂³×C₈₃

C₂×D₈

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₄×D₄

yes

characteristic

C₂×C₄×C₈₃

#121

yes

characteristic

C₂×C₄×C₈₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₁₆

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

SD₃₂

yes

—

C₈×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₈×C₈₃

C₄×Q₈

yes

characteristic

C₈×C₈₃

C₂²⋊Q₈

yes

characteristic

C₈×C₈₃

#125

yes

characteristic

C₈₃⋊C₈

—

C₂²×C₈₃

C₄×D₈

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

C₂×SD₃₂

yes

—

C₂²×C₈₃

#508

yes

—

C₄×C₈₃

—

C₄×C₈₃

#389

yes

characteristic

C₄×C₈₃

C₄×Q₁₆

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

Dic₈₃

—

—

C₂×C₈₃

yes

C₂×C₈₃

C₄×SD₃₂

yes

—

C₂×C₈₃

#2217

yes

—

C₂×C₈₃

yes

C₂×C₈₃

yes

—

D₈₃

yes

characteristic

C₂×C₄×C₁₆

—

#1942

—

100

#1380

—

101

C₈₃

#13166

yes

characteristic, complemented, Sylow

102

C₂²×C₁₆

—

103

C₂²×C₁₆

C₂×D₈₃

yes

characteristic

104

C₂×#107

—

105

C₂×#107

C₂×D₈₃

yes

—

106

C₂×C₄×C₈

—

107

#389

—

166

—

108

#367

—

109

#367

—

110

C₄×C₁₆

—

166

—

111

#340

—

166

—

112

C₂²×C₈

—

113

C₂²×C₈

C₂²×D₈₃

yes

characteristic

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

332

—

116

C₂×C₄⋊C₄

—

166

—

117

C₂×C₄²

—

118

C₂×C₁₆

—

119

C₂×C₁₆

—

166

—

120

C₂×C₁₆

—

166

—

121

C₂×C₁₆

Dic₁₆₆

yes

—

122

C₂×C₁₆

C₄×D₈₃

yes

—

123

C₂×C₁₆

C₂²⋊D₈₃

yes

—

124

#107

—

125

#107

—

126

#107

—

127

#107

—

166

—

128

C₄×C₈

—

166

—

129

C₄×C₈

—

166

—

130

#95

—

332

—

131

C₂²×C₄

—

132

C₂²×C₄

—

332

—

133

C₂²×C₄

—

134

C₂²×C₄

D₄×D₈₃

yes

characteristic

135

C₂×C₈

—

136

C₂×C₈

—

166

—

137

C₂×C₈

—

166

—

138

C₂×C₈

C₂×C₄×D₈₃

yes

characteristic

139

C₂×C₈

C₂×C₂²⋊D₈₃

yes

characteristic

140

C₂×C₈

Q₈×D₈₃

yes

characteristic

141

C₂×C₈

C₂×Dic₁₆₆

yes

characteristic

142

C₂×C₈

#13050589

yes

characteristic

143

C₂×C₈

#13050591

yes

characteristic

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

332

—

147

C₄⋊C₄

—

664

—

148

C₄⋊C₄

—

166

—

149

C₄²

—

166

—

150

C₄²

—

166

—

151

C₁₆

—

166

—

152

C₁₆

Dic₈₃⋊C₄

yes

complemented

153

C₂³

D₈×D₈₃

yes

characteristic

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

332

—

158

C₂×C₄

—

159

C₂×C₄

—

664

—

160

C₂×C₄

—

664

—

161

C₂×C₄

—

166

—

162

C₂×C₄

—

166

—

163

C₂×C₄

#14581754

yes

characteristic

164

C₂×C₄

#14581763

yes

characteristic

165

C₂×C₄

#14581791

yes

characteristic

166

C₂×C₄

Q₁₆×D₈₃

yes

characteristic

167

C₂×C₄

#14581811

yes

characteristic

168

C₂×C₄

#14581815

yes

characteristic

169

C₈

—

166

—

170

C₈

C₂×Dic₈₃⋊C₄

yes

characteristic

171

C₈

#14581757

yes

characteristic

172

C₈

#14581768

yes

characteristic

173

C₈

#14581783

yes

characteristic

174

C₂²

#19127783

yes

characteristic

175

C₂²

#19127934

yes

characteristic

176

C₂²

#19128105

yes

characteristic

177

C₂²

SD₃₂×D₈₃

yes

—

178

C₂²

#19128132

yes

—

179

C₄

—

180

C₄

—

664

—

181

C₄

—

166

—

182

C₄

#19127597

yes

characteristic

183

C₄

#19127795

yes

characteristic

184

C₄

#19127939

yes

characteristic

185

C₄

#19128102

yes

characteristic

186

C₂

#24908194

yes

characteristic

187

C₂

#24909314

yes

—

188

C₂

#24909556

yes

—

189

C₂

#24909741

yes

characteristic

190

C₂

#24909744

yes

characteristic

	subgroup	G/H
center	C₂³	D₈×D₈₃
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₂×#367×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×D₈
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₂×#367	D₈₃
hypocenter	C₈₃	#13166
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

subgroup

G/H

center

C₂³

D₈×D₈₃

commutator

C₂×C₈×C₈₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₃

Fitting

C₂×#367×C₈₃

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 ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

N	H ≅ G/N	type
10. #19127529	C₄	semidirect
56. C₁₆×C₈₃	C₄⋊C₄	semidirect
101. C₈₃	#13166	semidirect
152. C₁₆	Dic₈₃⋊C₄	semidirect

H ≅ G/N

type

10. #19127529

C₄

semidirect

56. C₁₆×C₈₃

C₄⋊C₄

semidirect

101. C₈₃

#13166

semidirect

152. C₁₆

Dic₈₃⋊C₄

semidirect

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

subgroup	index
C₂×#19127529	2
C₂×C₁₆×Dic₈₃	2
#24908194	2
#24905554	2
C₂×#367×C₈₃	2
#13166	83

subgroup

index

C₂×C₁₆×Dic₈₃

C₂×#367×C₈₃

Subgroups — #35322437

Conjugacy classes of subgroups

Complements

Maximal subgroups