Group #35304546 — subgroups

Subgroups — #35304546

Order 21248.

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

normal subgroups only

This group has 438 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	#24915826	C₂	1	1	yes	maximal, characteristic
2	#24907316	C₂	1	1	yes	maximal, characteristic
3	#24905347	C₂	1	4	yes	maximal
4	C₈₃×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×#14581784	—	2	1	no	—
6	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
7	C₂×#14581703	—	2	1	no	—
8	#19127599	—	2	2	no	—
9	#19127599	C₄	1	4	yes	—
10	#19127571	C₂²	1	2	yes	—
11	#19127567	—	2	1	no	—
12	#19127456	—	2	1	no	—
13	C₂×C₄⋊C₈×C₈₃	—	2	2	no	—
14	C₂×C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
15	C₄³×C₈₃	C₂²	1	1	yes	characteristic
16	C₂²×C₈₃⋊C₈	—	2	2	no	—
17	#14581784	—	4	4	no	—
18	C₂×C₄⋊Dic₈₃	—	2	2	no	—
19	C₂×C₄⋊Dic₈₃	—	2	2	no	—
20	C₂×C₄⋊Dic₈₃	C₂×C₄	1	4	yes	—
21	C₂×Dic₈₃⋊C₄	—	2	4	no	—
22	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
23	C₄⋊(C₈₃⋊C₈)	Q₈	1	4	yes	—
24	#14581703	—	4	2	no	—
25	C₈₃⋊#95	—	2	4	no	—
26	C₂²×C₈×C₈₃	—	2	2	no	—
27	C₂×C₄²×C₈₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₈₃	D₄	1	2	yes	—
31	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₈₃	—	2	4	no	—
33	C₄⋊C₈×C₈₃	—	4	4	no	—
34	C₂²×Dic₈₃	—	2	4	no	—
35	C₂×C₈₃⋊C₈	—	2	4	no	—
36	C₂×C₈₃⋊C₈	—	4	2	no	—
37	C₂×C₈₃⋊C₈	—	4	2	no	—
38	C₄⋊Dic₈₃	—	4	4	no	—
39	C₄⋊Dic₈₃	—	4	2	no	—
40	C₄⋊Dic₈₃	—	4	2	no	—
41	C₄⋊Dic₈₃	—	4	4	no	—
42	Dic₈₃⋊C₄	—	4	8	no	—
43	C₂²×C₄×C₈₃	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₈₃	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₈₃	—	4	2	no	—
49	C₂×C₈×C₈₃	—	4	4	no	—
50	C₄²×C₈₃	—	2	2	no	—
51	C₄²×C₈₃	—	2	2	no	—
52	C₄²×C₈₃	—	2	2	no	—
53	C₄²×C₈₃	—	4	2	no	—
54	C₄²×C₈₃	SD₁₆	1	2	yes	—
55	C₄²×C₈₃	Q₁₆	1	2	yes	—
56	C₄²×C₈₃	C₂×Q₈	1	2	yes	—
57	C₄²×C₈₃	#41	1	2	yes	—
58	C₂³×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₈₃	—	2	2	no	—
60	C₂×C₄×C₈₃	—	2	2	no	—
61	C₂×C₄×C₈₃	—	2	2	no	—
62	C₂×C₄×C₈₃	—	2	2	no	—
63	C₂×C₄×C₈₃	—	2	2	no	—
64	C₂×C₄×C₈₃	—	2	2	no	—
65	C₂×C₄×C₈₃	#103	1	4	yes	—
66	C₂×C₄×C₈₃	C₄×Q₈	1	2	yes	—
67	C₂×C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
68	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
69	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
73	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×C₄×C₈₃	#137	1	2	yes	—
75	C₂×Dic₈₃	—	4	4	no	—
76	C₂×Dic₈₃	—	4	4	no	—
77	C₂×Dic₈₃	—	4	4	no	—
78	C₈×C₈₃	—	8	2	no	—
79	C₈₃⋊C₈	—	4	4	no	—
80	C₂²×C₈₃	#391	1	1	yes	characteristic
81	C₂²×C₈₃	#413	1	2	yes	—
82	C₂²×C₈₃	C₂×#103	1	1	yes	characteristic
83	C₂²×C₈₃	#417	1	2	yes	—
84	C₂²×C₈₃	#418	1	1	yes	characteristic
85	C₄×C₈₃	—	2	4	no	—
86	C₄×C₈₃	—	2	2	no	—
87	C₄×C₈₃	—	4	2	no	—
88	C₄×C₈₃	#476	1	2	yes	—
89	C₄×C₈₃	#477	1	2	yes	—
90	C₄×C₈₃	#490	1	1	yes	characteristic
91	C₄×C₈₃	#491	1	1	yes	characteristic
92	C₄×C₈₃	#501	1	2	yes	—
93	Dic₈₃	—	8	4	no	—
94	#9902	—	83	1	no	maximal, Sylow
95	C₂×C₈₃	#1550	1	2	yes	—
96	C₂×C₈₃	#1557	1	2	yes	—
97	C₂×C₈₃	#1987	1	1	yes	characteristic
98	C₂×C₈₃	#1992	1	1	yes	characteristic
99	C₂×C₈₃	#1998	1	1	yes	characteristic
100	#2859	—	83	1	no	—
101	#1842	D₈₃	1	1	yes	characteristic
102	#1767	—	83	1	no	—
103	#1296	—	83	4	no	—
104	C₈₃	#9902	1	1	yes	characteristic, complemented, Sylow
105	C₂×#125	—	166	1	no	—
106	C₂×C₄⋊C₈	—	2	2	no	—
107	C₂×C₄⋊C₈	—	83	2	no	—
108	C₂×C₄⋊C₈	C₂×D₈₃	1	2	yes	—
109	C₂×#97	—	166	1	no	—
110	#389	—	83	4	no	—
111	#389	—	166	2	no	—
112	#384	—	83	2	no	—
113	#382	—	166	1	no	—
114	C₄³	C₂×D₈₃	1	1	yes	characteristic
115	#336	—	166	1	no	—
116	C₂²×C₈	—	2	2	no	—
117	C₂²×C₈	—	166	2	no	—
118	#125	—	332	4	no	—
119	C₂×C₄⋊C₄	—	83	4	no	—
120	C₂×C₄⋊C₄	—	166	4	no	—
121	C₂×C₄⋊C₄	—	166	2	no	—
122	C₂×C₄⋊C₄	—	166	2	no	—
123	C₂×C₄²	C₄×D₈₃	1	2	yes	—
124	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
125	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
126	C₂×C₄²	C₂²⋊D₈₃	1	2	yes	—
127	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
128	C₄⋊C₈	—	2	4	no	—
129	C₄⋊C₈	—	4	4	no	—
130	C₄⋊C₈	—	83	4	no	—
131	C₄⋊C₈	—	166	2	no	—
132	#97	—	332	2	no	—
133	#95	—	166	4	no	—
134	C₂²×C₄	—	166	4	no	—
135	C₂²×C₄	(C₂×D₈₃)⋊C₄	1	2	yes	—
136	C₂²×C₄	(C₂×D₈₃)⋊C₄	1	2	yes	—
137	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
139	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	4	4	no	—
142	C₂×C₈	—	332	2	no	—
143	C₂×C₈	—	332	2	no	—
144	C₂×C₈	—	166	4	no	—
145	C₄⋊C₄	—	332	4	no	—
146	C₄⋊C₄	—	332	8	no	—
147	C₄⋊C₄	—	332	2	no	—
148	C₄⋊C₄	—	332	4	no	—
149	C₄⋊C₄	—	332	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	—	4	2	no	—
154	C₄²	#13050579	1	2	yes	—
155	C₄²	Dic₃₃₂	1	2	yes	—
156	C₄²	Q₈×D₈₃	1	2	yes	—
157	C₄²	#13050589	1	2	yes	—
158	C₂³	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	332	4	no	—
166	C₂×C₄	—	332	4	no	—
167	C₂×C₄	—	332	4	no	—
168	C₂×C₄	#14581717	1	4	yes	—
169	C₂×C₄	#14581757	1	2	yes	—
170	C₂×C₄	#14581772	1	2	yes	—
171	C₂×C₄	#14581779	1	1	yes	characteristic
172	C₂×C₄	#14581790	1	1	yes	characteristic
173	C₂×C₄	C₂×#13050579	1	1	yes	characteristic
174	C₂×C₄	C₂×Dic₃₃₂	1	1	yes	characteristic
175	C₂×C₄	#14581818	1	2	yes	—
176	C₂×C₄	#14581820	1	2	yes	—
177	C₂×C₄	#14581823	1	2	yes	—
178	C₈	—	8	2	no	—
179	C₈	—	332	4	no	—
180	C₂²	#19127606	1	1	yes	characteristic
181	C₂²	#19127683	1	2	yes	—
182	C₂²	C₂×#14581717	1	1	yes	characteristic
183	C₂²	#19127700	1	2	yes	—
184	C₂²	#19127707	1	1	yes	characteristic
185	C₄	—	2	4	no	—
186	C₄	—	2	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	664	4	no	—
189	C₄	#19127992	1	2	yes	—
190	C₄	#19127997	1	2	yes	—
191	C₄	#19128065	1	1	yes	characteristic
192	C₄	#19128069	1	1	yes	characteristic
193	C₄	#19128107	1	2	yes	—
194	C₂	#24906151	1	2	yes	—
195	C₂	#24906183	1	2	yes	—
196	C₂	#24908410	1	1	yes	characteristic
197	C₂	#24908431	1	1	yes	characteristic
198	C₂	#24908455	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24915826

C₂

yes

maximal, characteristic

#24907316

C₂

yes

maximal, characteristic

#24905347

C₂

yes

maximal

C₈₃×#1842

C₂

yes

maximal, characteristic

C₂×#14581784

—

C₂×C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

C₂×#14581703

—

#19127599

—

#19127599

C₄

yes

—

#19127571

C₂²

yes

—

#19127567

—

#19127456

—

C₂×C₄⋊C₈×C₈₃

—

C₂×C₄⋊C₈×C₈₃

C₂²

yes

—

C₄³×C₈₃

C₂²

yes

characteristic

C₂²×C₈₃⋊C₈

—

#14581784

—

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

C₂×C₄

yes

—

C₂×Dic₈₃⋊C₄

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

Q₈

yes

—

#14581703

—

C₈₃⋊#95

—

C₂²×C₈×C₈₃

—

C₂×C₄²×C₈₃

C₂×C₄

yes

—

C₂×C₄²×C₈₃

D₄

yes

characteristic

C₂×C₄²×C₈₃

D₄

yes

characteristic

C₂×C₄²×C₈₃

D₄

yes

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄⋊C₈×C₈₃

—

C₄⋊C₈×C₈₃

—

C₂²×Dic₈₃

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

Dic₈₃⋊C₄

—

C₂²×C₄×C₈₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

SD₁₆

yes

—

C₄²×C₈₃

Q₁₆

yes

—

C₄²×C₈₃

C₂×Q₈

yes

—

C₄²×C₈₃

#41

yes

—

C₂³×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

#103

yes

—

C₂×C₄×C₈₃

C₄×Q₈

yes

—

C₂×C₄×C₈₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₃

#124

yes

characteristic

C₂×C₄×C₈₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₈₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₈₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₈₃

#137

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

#391

yes

characteristic

C₂²×C₈₃

#413

yes

—

C₂²×C₈₃

C₂×#103

yes

characteristic

C₂²×C₈₃

#417

yes

—

C₂²×C₈₃

#418

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#476

yes

—

C₄×C₈₃

#477

yes

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

—

Dic₈₃

—

#9902

—

maximal, Sylow

C₂×C₈₃

#1550

yes

—

C₂×C₈₃

#1557

yes

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

100

—

101

#1842

D₈₃

yes

characteristic

102

#1767

—

103

#1296

—

104

C₈₃

#9902

yes

characteristic, complemented, Sylow

105

C₂×#125

—

166

—

106

C₂×C₄⋊C₈

—

107

C₂×C₄⋊C₈

—

108

C₂×C₄⋊C₈

C₂×D₈₃

yes

—

109

C₂×#97

—

166

—

110

#389

—

111

#389

—

166

—

112

#384

—

113

#382

—

166

—

114

C₄³

C₂×D₈₃

yes

characteristic

115

#336

—

166

—

116

C₂²×C₈

—

117

C₂²×C₈

—

166

—

118

#125

—

332

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

166

—

121

C₂×C₄⋊C₄

—

166

—

122

C₂×C₄⋊C₄

—

166

—

123

C₂×C₄²

C₄×D₈₃

yes

—

124

C₂×C₄²

D₃₃₂

yes

characteristic

125

C₂×C₄²

D₃₃₂

yes

characteristic

126

C₂×C₄²

C₂²⋊D₈₃

yes

—

127

C₂×C₄²

C₂²×D₈₃

yes

characteristic

128

C₄⋊C₈

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

131

C₄⋊C₈

—

166

—

132

#97

—

332

—

133

#95

—

166

—

134

C₂²×C₄

—

166

—

135

C₂²×C₄

(C₂×D₈₃)⋊C₄

yes

—

136

C₂²×C₄

(C₂×D₈₃)⋊C₄

yes

—

137

C₂²×C₄

C₂×C₄×D₈₃

yes

characteristic

138

C₂²×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

139

C₂²×C₄

C₂×D₃₃₂

yes

characteristic

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

332

—

143

C₂×C₈

—

332

—

144

C₂×C₈

—

166

—

145

C₄⋊C₄

—

332

—

146

C₄⋊C₄

—

332

—

147

C₄⋊C₄

—

332

—

148

C₄⋊C₄

—

332

—

149

C₄⋊C₄

—

332

—

150

C₄²

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

#13050579

yes

—

155

C₄²

Dic₃₃₂

yes

—

156

C₄²

Q₈×D₈₃

yes

—

157

C₄²

#13050589

yes

—

158

C₂³

C₂×(C₂×D₈₃)⋊C₄

yes

characteristic

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

332

—

166

C₂×C₄

—

332

—

167

C₂×C₄

—

332

—

168

C₂×C₄

#14581717

yes

—

169

C₂×C₄

#14581757

yes

—

170

C₂×C₄

#14581772

yes

—

171

C₂×C₄

#14581779

yes

characteristic

172

C₂×C₄

#14581790

yes

characteristic

173

C₂×C₄

C₂×#13050579

yes

characteristic

174

C₂×C₄

C₂×Dic₃₃₂

yes

characteristic

175

C₂×C₄

#14581818

yes

—

176

C₂×C₄

#14581820

yes

—

177

C₂×C₄

#14581823

yes

—

178

C₈

—

179

C₈

—

332

—

180

C₂²

#19127606

yes

characteristic

181

C₂²

#19127683

yes

—

182

C₂²

C₂×#14581717

yes

characteristic

183

C₂²

#19127700

yes

—

184

C₂²

#19127707

yes

characteristic

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

664

—

189

C₄

#19127992

yes

—

190

C₄

#19127997

yes

—

191

C₄

#19128065

yes

characteristic

192

C₄

#19128069

yes

characteristic

193

C₄

#19128107

yes

—

194

C₂

#24906151

yes

—

195

C₂

#24906183

yes

—

196

C₂

#24908410

yes

characteristic

197

C₂

#24908431

yes

characteristic

198

C₂

#24908455

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₈₃)⋊C₄
commutator	C₂²×C₄×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1842	C₂
socle, prime radical	C₂³×C₈₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1842	D₈₃
hypocenter	C₈₃	#9902
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

subgroup

G/H

center

C₂³

C₂×(C₂×D₈₃)⋊C₄

commutator

C₂²×C₄×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₈₃×#1842

C₂

socle, prime radical

C₂³×C₈₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂²×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

N	H ≅ G/N	type
104. C₈₃	#9902	semidirect

H ≅ G/N

type

104. C₈₃

#9902

semidirect

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

subgroup	index
#24915826	2
#24907316	2
#24905347	2
C₈₃×#1842	2
#9902	83

subgroup

index

C₈₃×#1842

Subgroups — #35304546

Conjugacy classes of subgroups

Complements

Maximal subgroups