Group #35322641 — subgroups

Subgroups — #35322641

Order 21248.

486 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 167 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×#19127505	C₂	1	4	yes	maximal
2	C₂×#19127477	C₂	1	1	yes	maximal, characteristic
3	#24907627	C₂	1	1	yes	maximal, characteristic
4	C₂×#494×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈₃⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14581726	C₂²	1	2	yes	—
7	C₂×#14581724	—	2	1	no	—
8	#19127567	—	2	1	no	—
9	#19127505	—	2	8	no	—
10	#19127477	—	2	2	no	—
11	C₂²×Q₁₆×C₈₃	—	2	1	no	—
12	C₂²×Q₁₆×C₈₃	C₂²	1	2	yes	—
13	C₂×C₄⋊Q₈×C₈₃	—	2	1	no	—
14	#494×C₈₃	—	2	2	no	—
15	#494×C₈₃	C₄	1	4	yes	complemented
16	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₈₃	—	2	2	no	—
18	C₂×C₈₃⋊C₁₆	—	2	2	no	—
19	C₂×C₈₃⋊C₁₆	—	2	2	no	—
20	C₂×C₈₃⋊C₁₆	D₄	1	4	yes	—
21	#14581726	—	2	4	no	—
22	#14581724	—	4	2	no	—
23	C₈₃⋊#95	—	4	2	no	—
24	C₂²×Q₈×C₈₃	—	2	2	no	—
25	C₂×Q₁₆×C₈₃	—	2	4	no	—
26	C₂×Q₁₆×C₈₃	—	2	4	no	—
27	C₂×Q₁₆×C₈₃	—	2	2	no	—
28	C₂×Q₁₆×C₈₃	—	2	4	no	—
29	C₂×Q₁₆×C₈₃	—	4	2	no	—
30	C₂×Q₁₆×C₈₃	—	4	2	no	—
31	C₂×Q₁₆×C₈₃	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
34	C₄⋊Q₈×C₈₃	—	2	4	no	—
35	C₄⋊Q₈×C₈₃	—	4	2	no	—
36	C₂×C₄⋊C₄×C₈₃	—	4	2	no	—
37	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
38	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₈₃	D₄	1	2	yes	—
40	C₂²×Dic₈₃	—	4	2	no	—
41	C₄⋊Dic₈₃	—	4	2	no	—
42	C₄⋊Dic₈₃	—	4	2	no	—
43	C₈₃⋊C₁₆	—	2	4	no	—
44	C₂×Q₈×C₈₃	—	2	4	no	—
45	C₂×Q₈×C₈₃	—	4	2	no	—
46	C₂×Q₈×C₈₃	—	4	2	no	—
47	C₂×Q₈×C₈₃	—	4	2	no	—
48	C₂×Q₈×C₈₃	—	4	2	no	—
49	C₂×Q₈×C₈₃	—	4	2	no	—
50	C₂²×C₄×C₈₃	—	4	2	no	—
51	C₂²×C₄×C₈₃	D₈	1	2	yes	—
52	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
53	Q₁₆×C₈₃	—	2	8	no	—
54	Q₁₆×C₈₃	—	4	8	no	—
55	Q₁₆×C₈₃	—	4	4	no	—
56	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₈₃	D₈	1	2	yes	—
58	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
59	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₈₃	#41	1	2	yes	—
63	C₄⋊C₄×C₈₃	—	4	8	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	Q₈×C₈₃	—	4	4	no	—
68	Q₈×C₈₃	—	4	4	no	—
69	Q₈×C₈₃	—	8	2	no	—
70	Q₈×C₈₃	—	8	2	no	—
71	C₂×C₄×C₈₃	—	2	4	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	#102	1	4	yes	—
76	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
78	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
79	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
82	C₂×Dic₈₃	—	8	2	no	—
83	C₂×Dic₈₃	—	8	4	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	—	8	4	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	—	16	2	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	D₈₃	1	1	yes	characteristic
104	C₂×#358	—	83	4	no	—
105	C₂×#346	—	83	1	no	—
106	#1846	—	83	1	no	—
107	C₈₃	#13207	1	1	yes	characteristic, complemented, Sylow
108	C₂²×Q₁₆	—	2	1	no	—
109	C₂²×Q₁₆	C₂×D₈₃	1	2	yes	—
110	C₂×C₄⋊Q₈	—	2	1	no	—
111	C₂²×C₁₆	—	83	2	no	—
112	#494	—	2	2	no	—
113	#494	Dic₈₃	1	4	yes	complemented
114	C₂×#107	—	83	2	no	—
115	C₂×#106	—	166	1	no	—
116	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
117	#382	—	166	1	no	—
118	#358	—	166	8	no	—
119	#346	—	166	2	no	—
120	C₂²×Q₈	—	2	2	no	—
121	C₂×Q₁₆	—	2	4	no	—
122	C₂×Q₁₆	—	2	4	no	—
123	C₂×Q₁₆	—	2	2	no	—
124	C₂×Q₁₆	—	2	4	no	—
125	C₂×Q₁₆	—	4	2	no	—
126	C₂×Q₁₆	—	4	2	no	—
127	C₂×Q₁₆	C₂×Dic₈₃	1	4	yes	—
128	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
129	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
130	C₄⋊Q₈	—	2	4	no	—
131	C₄⋊Q₈	—	4	2	no	—
132	C₂×C₄⋊C₄	—	4	2	no	—
133	C₂×C₄⋊C₄	—	166	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	—	166	4	no	—
139	#106	—	332	2	no	—
140	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
141	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
142	#95	—	332	2	no	—
143	C₂×Q₈	—	2	4	no	—
144	C₂×Q₈	—	4	2	no	—
145	C₂×Q₈	—	4	2	no	—
146	C₂×Q₈	—	4	2	no	—
147	C₂×Q₈	—	4	2	no	—
148	C₂×Q₈	—	4	2	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	—	2	8	no	—
154	Q₁₆	—	4	4	no	—
155	Q₁₆	—	4	8	no	—
156	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
157	C₂×C₈	#13050575	1	2	yes	—
158	C₂×C₈	#13050578	1	2	yes	—
159	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
160	C₂×C₈	D₄×D₈₃	1	2	yes	—
161	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
162	C₂×C₈	#13050589	1	2	yes	—
163	C₄⋊C₄	—	4	8	no	—
164	C₄⋊C₄	—	332	2	no	—
165	C₄⋊C₄	—	332	2	no	—
166	C₄²	C₂²⋊Dic₈₃	1	2	yes	—
167	C₄²	#13050578	1	2	yes	—
168	C₁₆	—	166	4	no	—
169	C₂³	C₂×#13050575	1	1	yes	characteristic
170	Q₈	—	4	4	no	—
171	Q₈	—	4	4	no	—
172	Q₈	—	8	2	no	—
173	Q₈	—	8	2	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	664	2	no	—
179	C₂×C₄	—	664	4	no	—
180	C₂×C₄	#14581713	1	4	yes	—
181	C₂×C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
182	C₂×C₄	#14581779	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×#13050578	1	1	yes	characteristic
186	C₂×C₄	C₂×#13050578	1	1	yes	characteristic
187	C₈	#14581713	1	4	yes	—
188	C₈	D₄×Dic₈₃	1	2	yes	—
189	C₈	#14581763	1	2	yes	—
190	C₂²	C₂×#14581713	1	1	yes	characteristic
191	C₂²	#19128047	1	1	yes	characteristic
192	C₂²	#19128077	1	1	yes	characteristic
193	C₂²	#19128142	1	4	yes	—
194	C₄	—	2	4	no	—
195	C₄	—	8	4	no	—
196	C₄	—	1328	2	no	—
197	C₄	#19127604	1	1	yes	characteristic
198	C₄	C₂×#14581713	1	1	yes	characteristic
199	C₄	#19128047	1	1	yes	characteristic
200	C₄	#19128073	1	1	yes	characteristic
201	C₂	#24908270	1	1	yes	characteristic
202	C₂	#24909183	1	2	yes	—
203	C₂	#24909708	1	2	yes	—
204	C₂	#24909732	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19127505

C₂

yes

maximal

C₂×#19127477

C₂

yes

maximal, characteristic

#24907627

C₂

yes

maximal, characteristic

C₂×#494×C₈₃

C₂

yes

maximal, characteristic

C₂²×C₈₃⋊C₁₆

C₂²

yes

—

C₂×#14581726

C₂²

yes

—

—

—

—

—

C₂²×Q₁₆×C₈₃

—

C₂²×Q₁₆×C₈₃

C₂²

yes

—

C₂×C₄⋊Q₈×C₈₃

—

#494×C₈₃

—

#494×C₈₃

C₄

yes

complemented

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₃

—

C₂×C₈₃⋊C₁₆

—

C₂×C₈₃⋊C₁₆

—

C₂×C₈₃⋊C₁₆

D₄

yes

—

#14581726

—

#14581724

—

C₈₃⋊#95

—

C₂²×Q₈×C₈₃

—

C₂×Q₁₆×C₈₃

—

C₂×Q₁₆×C₈₃

—

C₂×Q₁₆×C₈₃

—

C₂×Q₁₆×C₈₃

—

C₂×Q₁₆×C₈₃

—

C₂×Q₁₆×C₈₃

—

C₂×Q₁₆×C₈₃

C₂×C₄

yes

—

C₂²×C₈×C₈₃

D₄

yes

characteristic

C₂²×C₈×C₈₃

C₂³

yes

characteristic

C₄⋊Q₈×C₈₃

—

C₄⋊Q₈×C₈₃

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄²×C₈₃

D₄

yes

characteristic

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

D₄

yes

—

C₂²×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

D₈

yes

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

Q₁₆×C₈₃

—

Q₁₆×C₈₃

—

Q₁₆×C₈₃

—

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

—

C₄²×C₈₃

C₂²⋊C₄

yes

—

C₄²×C₈₃

SD₁₆

yes

—

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

—

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

—

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

D₈₃

yes

characteristic

104

C₂×#358

—

105

C₂×#346

—

106

#1846

—

107

C₈₃

#13207

yes

characteristic, complemented, Sylow

108

C₂²×Q₁₆

—

109

C₂²×Q₁₆

C₂×D₈₃

yes

—

110

C₂×C₄⋊Q₈

—

111

C₂²×C₁₆

—

112

#494

—

113

#494

Dic₈₃

yes

complemented

114

C₂×#107

—

115

C₂×#106

—

166

—

116

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

117

#382

—

166

—

118

#358

—

166

—

119

#346

—

166

—

120

C₂²×Q₈

—

121

C₂×Q₁₆

—

122

C₂×Q₁₆

—

123

C₂×Q₁₆

—

124

C₂×Q₁₆

—

125

C₂×Q₁₆

—

126

C₂×Q₁₆

—

127

C₂×Q₁₆

C₂×Dic₈₃

yes

—

128

C₂²×C₈

C₂²⋊D₈₃

yes

characteristic

129

C₂²×C₈

C₂²×D₈₃

yes

characteristic

130

C₄⋊Q₈

—

131

C₄⋊Q₈

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

166

—

134

C₂×C₄²

C₂²⋊D₈₃

yes

characteristic

135

C₂×C₁₆

—

136

C₂×C₁₆

—

166

—

137

C₂×C₁₆

—

166

—

138

#107

—

166

—

139

#106

—

332

—

140

C₄×C₈

C₂×Dic₈₃

yes

—

141

C₄×C₈

C₂²⋊D₈₃

yes

—

142

#95

—

332

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂×Q₈

—

147

C₂×Q₈

—

148

C₂×Q₈

—

149

C₂²×C₄

—

150

C₂²×C₄

—

332

—

151

C₂²×C₄

#13050575

yes

—

152

C₂²×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

153

Q₁₆

—

154

Q₁₆

—

155

Q₁₆

—

156

C₂×C₈

C₂²⋊Dic₈₃

yes

—

157

C₂×C₈

#13050575

yes

—

158

C₂×C₈

#13050578

yes

—

159

C₂×C₈

C₂²×Dic₈₃

yes

characteristic

160

C₂×C₈

D₄×D₈₃

yes

—

161

C₂×C₈

C₂×C₂²⋊D₈₃

yes

characteristic

162

C₂×C₈

#13050589

yes

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

332

—

165

C₄⋊C₄

—

332

—

166

C₄²

C₂²⋊Dic₈₃

yes

—

167

C₄²

#13050578

yes

—

168

C₁₆

—

166

—

169

C₂³

C₂×#13050575

yes

characteristic

170

Q₈

—

171

Q₈

—

172

Q₈

—

173

Q₈

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

664

—

179

C₂×C₄

—

664

—

180

C₂×C₄

#14581713

yes

—

181

C₂×C₄

C₂×C₂²⋊Dic₈₃

yes

characteristic

182

C₂×C₄

#14581779

yes

characteristic

183

C₂×C₄

#14581788

yes

characteristic

184

C₂×C₄

C₂×#13050575

yes

characteristic

185

C₂×C₄

C₂×#13050578

yes

characteristic

186

C₂×C₄

C₂×#13050578

yes

characteristic

187

C₈

#14581713

yes

—

188

C₈

D₄×Dic₈₃

yes

—

189

C₈

#14581763

yes

—

190

C₂²

C₂×#14581713

yes

characteristic

191

C₂²

#19128047

yes

characteristic

192

C₂²

#19128077

yes

characteristic

193

C₂²

#19128142

yes

—

194

C₄

—

195

C₄

—

196

C₄

—

1328

—

197

C₄

#19127604

yes

characteristic

198

C₄

C₂×#14581713

yes

characteristic

199

C₄

#19128047

yes

characteristic

200

C₄

#19128073

yes

characteristic

201

C₂

#24908270

yes

characteristic

202

C₂

#24909183

yes

—

203

C₂

#24909708

yes

—

204

C₂

#24909732

yes

—

	subgroup	G/H
center	C₂³	C₂×#13050575
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₂×#494×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×D₈
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₂×#494	D₈₃
hypocenter	C₈₃	#13207
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

subgroup

G/H

center

C₂³

C₂×#13050575

commutator

C₂×C₈×C₈₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₃

Fitting

C₂×#494×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 ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

N	H ≅ G/N	type
15. #494×C₈₃	C₄	semidirect
107. C₈₃	#13207	semidirect
113. #494	Dic₈₃	semidirect

H ≅ G/N

type

15. #494×C₈₃

C₄

semidirect

107. C₈₃

#13207

semidirect

113. #494

Dic₈₃

semidirect

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

subgroup	index
C₂×#19127505	2
C₂×#19127477	2
#24907627	2
C₂×#494×C₈₃	2
#13207	83

subgroup

index

C₂×#494×C₈₃

Subgroups — #35322641

Conjugacy classes of subgroups

Complements

Maximal subgroups