Group #35322889 — subgroups

Subgroups — #35322889

Order 21248.

462 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 143 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	#24909219	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127517	C₂	1	4	yes	maximal
3	C₄×#14581728	C₂	1	1	yes	maximal, characteristic
4	C₂×#494×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14581732	C₂²	1	2	yes	—
6	C₂×#14581728	C₂²	1	2	yes	—
7	C₄×#13050574	—	2	1	no	—
8	C₄⋊(C₈₃⋊C₁₆)	—	2	2	no	—
9	#19127517	—	2	8	no	—
10	#19127466	—	2	1	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	—
16	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	C₂×#13050574	—	2	2	no	—
18	#14581732	—	2	4	no	—
19	C₂×C₈₃⋊C₁₆	—	2	2	no	—
20	#14581728	—	2	2	no	—
21	#14581728	D₄	1	4	yes	—
22	#14581703	—	4	1	no	—
23	C₄×C₈₃⋊C₈	—	4	1	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	4	no	—
30	C₂×Q₁₆×C₈₃	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈×C₈₃	—	2	4	no	—
34	C₄⋊Q₈×C₈₃	—	4	1	no	—
35	C₄⋊Q₈×C₈₃	—	4	1	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	#13050574	—	2	4	no	—
41	#13050574	—	4	2	no	—
42	C₂×C₈₃⋊C₈	—	4	2	no	—
43	C₈₃⋊C₁₆	—	4	2	no	—
44	C₂×Q₈×C₈₃	—	2	4	no	—
45	C₂×Q₈×C₈₃	—	4	2	no	—
46	C₂×Q₈×C₈₃	—	4	4	no	—
47	C₂×Q₈×C₈₃	—	4	4	no	—
48	C₂²×C₄×C₈₃	—	4	2	no	—
49	C₂²×C₄×C₈₃	SD₁₆	1	2	yes	—
50	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
51	Q₁₆×C₈₃	—	2	8	no	—
52	Q₁₆×C₈₃	—	4	8	no	—
53	Q₁₆×C₈₃	—	4	2	no	—
54	Q₁₆×C₈₃	—	4	2	no	—
55	C₂×C₈×C₈₃	—	2	2	no	—
56	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₈₃	#41	1	2	yes	—
61	C₄⋊C₄×C₈₃	—	4	4	no	—
62	C₄⋊C₄×C₈₃	—	4	4	no	—
63	C₄²×C₈₃	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₈₃	D₈	1	1	yes	characteristic
65	C₄²×C₈₃	D₈	1	1	yes	characteristic
66	C₂³×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
67	Q₈×C₈₃	—	4	4	no	—
68	Q₈×C₈₃	—	4	4	no	—
69	Q₈×C₈₃	—	8	4	no	—
70	C₂×C₄×C₈₃	—	2	4	no	—
71	C₂×C₄×C₈₃	—	4	4	no	—
72	C₂×C₄×C₈₃	—	8	4	no	—
73	C₂×C₄×C₈₃	#102	1	2	yes	—
74	C₂×C₄×C₈₃	#102	1	2	yes	—
75	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₈₃	#121	1	2	yes	—
77	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
79	C₈×C₈₃	—	2	2	no	—
80	C₈×C₈₃	C₄×D₄	1	2	yes	—
81	C₈×C₈₃	#124	1	1	yes	characteristic
82	C₈×C₈₃	#127	1	1	yes	characteristic
83	C₈₃⋊C₈	—	4	4	no	—
84	C₂²×C₈₃	—	2	2	no	—
85	C₂²×C₈₃	C₂×#102	1	1	yes	characteristic
86	C₂²×C₈₃	#460	1	2	yes	—
87	C₄×C₈₃	—	2	2	no	—
88	C₄×C₈₃	—	8	4	no	—
89	C₄×C₈₃	#362	1	4	yes	complemented
90	C₄×C₈₃	#390	1	1	yes	characteristic
91	C₄×C₈₃	#418	1	1	yes	characteristic
92	C₄×C₈₃	#459	1	2	yes	—
93	#13256	—	83	1	no	maximal, Sylow
94	C₂×C₈₃	—	2	2	no	—
95	C₂×C₈₃	#1982	1	1	yes	characteristic
96	C₂×C₈₃	C₂×#362	1	2	yes	—
97	C₂×#494	D₈₃	1	1	yes	characteristic
98	#2150	—	83	1	no	—
99	C₂×#362	—	83	4	no	—
100	C₄×#108	—	83	1	no	—
101	C₈₃	#13256	1	1	yes	characteristic, complemented, Sylow
102	C₂²×Q₁₆	—	2	1	no	—
103	C₂²×Q₁₆	C₂×D₈₃	1	2	yes	—
104	C₂×C₄⋊Q₈	—	2	1	no	—
105	C₂×M₃₂	—	83	2	no	—
106	#494	—	2	2	no	—
107	#494	Dic₈₃	1	4	yes	—
108	C₂×#108	—	83	2	no	—
109	C₄×M₁₆	—	166	1	no	—
110	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
111	C₄⋊C₁₆	—	166	2	no	—
112	#362	—	166	8	no	—
113	#341	—	166	1	no	—
114	C₂²×Q₈	—	2	2	no	—
115	C₂×Q₁₆	—	2	4	no	—
116	C₂×Q₁₆	—	2	4	no	—
117	C₂×Q₁₆	—	2	2	no	—
118	C₂×Q₁₆	—	2	4	no	—
119	C₂×Q₁₆	—	4	4	no	—
120	C₂×Q₁₆	C₂×Dic₈₃	1	4	yes	—
121	C₂×M₁₆	—	166	2	no	—
122	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
124	C₄⋊Q₈	—	2	4	no	—
125	C₄⋊Q₈	—	4	1	no	—
126	C₄⋊Q₈	—	4	1	no	—
127	C₂×C₄⋊C₄	—	4	2	no	—
128	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
129	M₃₂	—	166	4	no	—
130	C₂×C₁₆	—	166	2	no	—
131	#108	—	83	4	no	—
132	#108	—	166	2	no	—
133	#97	—	332	1	no	—
134	C₄×C₈	—	332	1	no	—
135	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
136	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
137	C₂×Q₈	—	2	4	no	—
138	C₂×Q₈	—	4	2	no	—
139	C₂×Q₈	—	4	4	no	—
140	C₂×Q₈	—	4	4	no	—
141	C₂²×C₄	—	4	2	no	—
142	C₂²×C₄	#13050578	1	2	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
144	Q₁₆	—	2	8	no	—
145	Q₁₆	—	4	8	no	—
146	Q₁₆	—	4	2	no	—
147	Q₁₆	—	4	2	no	—
148	M₁₆	—	332	2	no	—
149	M₁₆	—	166	4	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	332	2	no	—
152	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
153	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
154	C₂×C₈	D₄×D₈₃	1	2	yes	—
155	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
156	C₂×C₈	#13050589	1	2	yes	—
157	C₄⋊C₄	—	4	4	no	—
158	C₄⋊C₄	—	4	4	no	—
159	C₄²	C₂²⋊Dic₈₃	1	2	yes	—
160	C₄²	#13050575	1	1	yes	characteristic
161	C₄²	#13050575	1	1	yes	characteristic
162	C₁₆	—	332	2	no	—
163	C₂³	C₂×#13050578	1	1	yes	characteristic
164	Q₈	—	4	4	no	—
165	Q₈	—	4	4	no	—
166	Q₈	—	8	4	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	—	8	4	no	—
170	C₂×C₄	#14581713	1	2	yes	—
171	C₂×C₄	#14581713	1	2	yes	—
172	C₂×C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
173	C₂×C₄	#14581763	1	2	yes	—
174	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
175	C₂×C₄	#14581818	1	2	yes	—
176	C₈	—	2	2	no	—
177	C₈	—	332	4	no	—
178	C₈	D₄×Dic₈₃	1	2	yes	—
179	C₈	#14581779	1	1	yes	characteristic
180	C₈	#14581788	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	C₂×#14581713	1	1	yes	characteristic
183	C₂²	#19127898	1	2	yes	—
184	C₄	—	2	2	no	—
185	C₄	—	8	4	no	—
186	C₄	#19127517	1	4	yes	complemented
187	C₄	#19127604	1	1	yes	characteristic
188	C₄	#19127706	1	1	yes	characteristic
189	C₄	#19127893	1	2	yes	—
190	C₂	—	2	2	no	—
191	C₂	#24908391	1	1	yes	characteristic
192	C₂	C₂×#19127517	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24909219

C₂

yes

maximal, characteristic

C₂×#19127517

C₂

yes

maximal

C₄×#14581728

C₂

yes

maximal, characteristic

C₂×#494×C₈₃

C₂

yes

maximal, characteristic

C₂×#14581732

C₂²

yes

—

C₂×#14581728

C₂²

yes

—

C₄×#13050574

—

C₄⋊(C₈₃⋊C₁₆)

—

#19127517

—

#19127466

—

C₂²×Q₁₆×C₈₃

—

C₂²×Q₁₆×C₈₃

C₂²

yes

—

C₂×C₄⋊Q₈×C₈₃

—

#494×C₈₃

—

#494×C₈₃

C₄

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×#13050574

—

#14581732

—

C₂×C₈₃⋊C₁₆

—

#14581728

—

#14581728

D₄

yes

—

#14581703

—

C₄×C₈₃⋊C₈

—

C₂²×Q₈×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

—

C₄⋊Q₈×C₈₃

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄²×C₈₃

D₄

yes

characteristic

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

D₄

yes

—

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₈₃⋊C₁₆

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

SD₁₆

yes

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

Q₁₆×C₈₃

—

Q₁₆×C₈₃

—

Q₁₆×C₈₃

—

Q₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂²⋊C₄

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

D₈

yes

characteristic

C₄²×C₈₃

D₈

yes

characteristic

C₂³×C₈₃

C₂×SD₁₆

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

#102

yes

—

C₂×C₄×C₈₃

#102

yes

—

C₂×C₄×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

#121

yes

—

C₂×C₄×C₈₃

C₂×D₈

yes

characteristic

C₂×C₄×C₈₃

AGL(1,ℤ/8)

yes

—

C₈×C₈₃

—

C₈×C₈₃

C₄×D₄

yes

—

C₈×C₈₃

yes

C₈×C₈₃

yes

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×#102

yes

characteristic

C₂²×C₈₃

#460

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

—

#13256

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

#1982

yes

characteristic

C₂×C₈₃

C₂×#362

yes

—

C₂×#494

D₈₃

yes

characteristic

#2150

—

C₂×#362

—

100

C₄×#108

—

101

C₈₃

#13256

yes

characteristic, complemented, Sylow

102

C₂²×Q₁₆

—

103

C₂²×Q₁₆

C₂×D₈₃

yes

—

104

C₂×C₄⋊Q₈

—

105

C₂×M₃₂

—

106

#494

—

107

#494

Dic₈₃

yes

—

108

C₂×#108

—

109

C₄×M₁₆

—

166

—

110

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

111

C₄⋊C₁₆

—

166

—

112

#362

—

166

—

113

#341

—

166

—

114

C₂²×Q₈

—

115

C₂×Q₁₆

—

116

C₂×Q₁₆

—

117

C₂×Q₁₆

—

118

C₂×Q₁₆

—

119

C₂×Q₁₆

—

120

C₂×Q₁₆

C₂×Dic₈₃

yes

—

121

C₂×M₁₆

—

166

—

122

C₂²×C₈

C₂²⋊D₈₃

yes

characteristic

123

C₂²×C₈

C₂²×D₈₃

yes

characteristic

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄²

C₂²⋊D₈₃

yes

characteristic

129

M₃₂

—

166

—

130

C₂×C₁₆

—

166

—

131

#108

—

132

#108

—

166

—

133

#97

—

332

—

134

C₄×C₈

—

332

—

135

C₄×C₈

C₂×Dic₈₃

yes

—

136

C₄×C₈

C₂²⋊D₈₃

yes

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂²×C₄

—

142

C₂²×C₄

#13050578

yes

—

143

C₂²×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

144

Q₁₆

—

145

Q₁₆

—

146

Q₁₆

—

147

Q₁₆

—

148

M₁₆

—

332

—

149

M₁₆

—

166

—

150

C₂×C₈

—

151

C₂×C₈

—

332

—

152

C₂×C₈

C₂²⋊Dic₈₃

yes

—

153

C₂×C₈

C₂²×Dic₈₃

yes

characteristic

154

C₂×C₈

D₄×D₈₃

yes

—

155

C₂×C₈

C₂×C₂²⋊D₈₃

yes

characteristic

156

C₂×C₈

#13050589

yes

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄²

C₂²⋊Dic₈₃

yes

—

160

C₄²

#13050575

yes

characteristic

161

C₄²

#13050575

yes

characteristic

162

C₁₆

—

332

—

163

C₂³

C₂×#13050578

yes

characteristic

164

Q₈

—

165

Q₈

—

166

Q₈

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

#14581713

yes

—

171

C₂×C₄

#14581713

yes

—

172

C₂×C₄

C₂×C₂²⋊Dic₈₃

yes

characteristic

173

C₂×C₄

#14581763

yes

—

174

C₂×C₄

C₂×#13050575

yes

characteristic

175

C₂×C₄

#14581818

yes

—

176

C₈

—

177

C₈

—

332

—

178

C₈

D₄×Dic₈₃

yes

—

179

C₈

#14581779

yes

characteristic

180

C₈

#14581788

yes

characteristic

181

C₂²

—

182

C₂²

C₂×#14581713

yes

characteristic

183

C₂²

#19127898

yes

—

184

C₄

—

185

C₄

—

186

C₄

#19127517

yes

complemented

187

C₄

#19127604

yes

characteristic

188

C₄

#19127706

yes

characteristic

189

C₄

#19127893

yes

—

190

C₂

—

191

C₂

#24908391

yes

characteristic

192

C₂

C₂×#19127517

yes

—

	subgroup	G/H
center	C₂²	C₂×#14581713
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₂×#494×C₈₃	C₂
socle	C₂²×C₈₃	C₂×#102
prime radical	C₂³×C₈₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₂×#494	D₈₃
hypocenter	C₈₃	#13256
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

subgroup

G/H

center

C₂²

C₂×#14581713

commutator

C₂×C₈×C₈₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₃

Fitting

C₂×#494×C₈₃

C₂

socle

C₂²×C₈₃

C₂×#102

prime radical

C₂³×C₈₃

C₂×SD₁₆

cosocle

C₂²×C₈×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

N	H ≅ G/N	type
89. C₄×C₈₃	#362	semidirect
101. C₈₃	#13256	semidirect
186. C₄	#19127517	semidirect

H ≅ G/N

type

89. C₄×C₈₃

#362

semidirect

101. C₈₃

#13256

semidirect

186. C₄

#19127517

semidirect

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

subgroup	index
#24909219	2
C₂×#19127517	2
C₄×#14581728	2
C₂×#494×C₈₃	2
#13256	83

subgroup

index

C₂×#494×C₈₃

Subgroups — #35322889

Conjugacy classes of subgroups

Complements

Maximal subgroups