Group #35304140 — subgroups

Subgroups — #35304140

Order 21248.

526 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 207 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	#24908425	C₂	1	2	yes	maximal
2	#24908266	C₂	1	2	yes	maximal
3	#24907620	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×#498×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581724	—	2	2	no	—
7	C₂×#14581716	—	2	4	no	—
8	C₂×C₄×C₈₃⋊C₈	C₂²	1	2	yes	—
9	#19127571	C₂²	1	1	yes	characteristic
10	#19127571	C₂²	1	1	yes	characteristic
11	#19127463	—	2	4	no	—
12	C₈×C₈₃⋊C₈	—	2	2	no	—
13	C₂×C₄⋊Q₈×C₈₃	C₂²	1	2	yes	—
14	#498×C₈₃	—	2	2	no	—
15	#498×C₈₃	C₄	1	4	yes	complemented
16	C₂×#106×C₈₃	—	2	2	no	—
17	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
18	C₂²×C₈₃⋊C₈	D₄	1	4	yes	—
19	C₂×C₄⋊Dic₈₃	—	2	2	no	—
20	C₂×C₄⋊Dic₈₃	—	2	2	no	—
21	C₂×C₄⋊Dic₈₃	—	2	1	no	—
22	C₂×C₄⋊Dic₈₃	—	2	1	no	—
23	#14581724	—	4	4	no	—
24	#14581716	—	4	8	no	—
25	C₄×C₈₃⋊C₈	—	2	2	no	—
26	C₄×C₈₃⋊C₈	—	2	2	no	—
27	C₂²×Q₈×C₈₃	—	2	2	no	—
28	C₂²×C₈×C₈₃	D₄	1	2	yes	—
29	C₄⋊Q₈×C₈₃	—	2	2	no	—
30	C₄⋊Q₈×C₈₃	—	2	2	no	—
31	C₄⋊Q₈×C₈₃	—	2	2	no	—
32	C₄⋊Q₈×C₈₃	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
35	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
36	#106×C₈₃	—	2	8	no	—
37	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₈₃	D₄	1	2	yes	—
39	C₂²×Dic₈₃	—	4	1	no	—
40	C₂²×Dic₈₃	—	4	1	no	—
41	C₂×C₈₃⋊C₈	—	2	4	no	—
42	C₂×C₈₃⋊C₈	—	2	4	no	—
43	C₂×C₈₃⋊C₈	SD₁₆	1	8	yes	—
44	C₄⋊Dic₈₃	—	4	4	no	—
45	C₄⋊Dic₈₃	—	4	2	no	—
46	C₄⋊Dic₈₃	—	4	2	no	—
47	C₄⋊Dic₈₃	—	4	4	no	—
48	C₂×Q₈×C₈₃	—	2	4	no	—
49	C₂×Q₈×C₈₃	—	4	2	no	—
50	C₂×Q₈×C₈₃	—	4	4	no	—
51	C₂×Q₈×C₈₃	—	4	4	no	—
52	C₂²×C₄×C₈₃	—	4	2	no	—
53	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₈₃	C₂²⋊C₄	1	4	yes	—
56	C₂×C₈×C₈₃	SD₁₆	1	4	yes	—
57	C₂×C₈×C₈₃	Q₁₆	1	4	yes	—
58	C₄⋊C₄×C₈₃	—	2	4	no	—
59	C₄⋊C₄×C₈₃	—	2	4	no	—
60	C₄⋊C₄×C₈₃	—	4	2	no	—
61	C₄⋊C₄×C₈₃	—	4	2	no	—
62	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₈₃	#41	1	2	yes	—
65	C₂³×C₈₃	#127	1	1	yes	characteristic
66	Q₈×C₈₃	—	4	8	no	—
67	Q₈×C₈₃	—	8	4	no	—
68	C₂×C₄×C₈₃	—	4	4	no	—
69	C₂×C₄×C₈₃	—	8	2	no	—
70	C₂×C₄×C₈₃	—	8	2	no	—
71	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₈₃	C₄×D₄	1	2	yes	—
73	C₂×C₄×C₈₃	#121	1	2	yes	—
74	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
75	C₂×C₄×C₈₃	C₂×SD₁₆	1	4	yes	—
76	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
77	C₂×C₄×C₈₃	C₂×Q₁₆	1	2	yes	—
78	C₂×C₄×C₈₃	#135	1	4	yes	—
79	C₂×Dic₈₃	—	8	2	no	—
80	C₂×Dic₈₃	—	8	1	no	—
81	C₂×Dic₈₃	—	8	1	no	—
82	C₂×Dic₈₃	—	8	2	no	—
83	C₈×C₈₃	#103	1	8	yes	complemented
84	C₈₃⋊C₈	—	2	8	no	—
85	C₂²×C₈₃	#390	1	1	yes	characteristic
86	C₂²×C₈₃	#492	1	2	yes	—
87	C₂²×C₈₃	#492	1	1	yes	characteristic
88	C₂²×C₈₃	#494	1	1	yes	characteristic
89	C₂²×C₈₃	#495	1	2	yes	—
90	C₄×C₈₃	—	8	4	no	—
91	C₄×C₈₃	C₂×#103	1	2	yes	—
92	C₄×C₈₃	C₄×SD₁₆	1	4	yes	—
93	C₄×C₈₃	#465	1	4	yes	—
94	C₄×C₈₃	#487	1	2	yes	—
95	Dic₈₃	—	16	1	no	—
96	Dic₈₃	—	16	1	no	—
97	#9819	—	83	1	no	maximal, Sylow
98	C₂×C₈₃	#1705	1	1	yes	characteristic
99	C₂×C₈₃	#1707	1	2	yes	—
100	C₂×C₈₃	#1708	1	1	yes	characteristic
101	C₂×C₈₃	#1956	1	1	yes	characteristic
102	C₂×C₈₃	#1968	1	2	yes	—
103	C₂×#498	D₈₃	1	1	yes	characteristic
104	#1990	—	83	2	no	—
105	#1956	—	83	2	no	—
106	#1844	—	83	1	no	—
107	C₂×C₈²	—	83	1	no	—
108	C₈₃	#9819	1	1	yes	characteristic, complemented, Sylow
109	C₂×C₄⋊Q₈	C₂×D₈₃	1	2	yes	—
110	#498	—	2	2	no	—
111	#498	Dic₈₃	1	4	yes	complemented
112	C₂×#106	—	2	2	no	—
113	C₂×#106	—	166	2	no	—
114	C₂×#103	—	166	4	no	—
115	C₂×C₄×C₈	—	83	2	no	—
116	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
117	#384	—	83	1	no	—
118	#384	—	83	1	no	—
119	#340	—	166	4	no	—
120	C₈²	—	166	2	no	—
121	C₂²×Q₈	—	2	2	no	—
122	C₂²×C₈	—	83	4	no	—
123	C₂²×C₈	C₂²⋊D₈₃	1	2	yes	—
124	C₄⋊Q₈	—	2	2	no	—
125	C₄⋊Q₈	—	2	2	no	—
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	C₂×Dic₈₃	1	4	yes	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	166	2	no	—
131	C₂×C₄⋊C₄	—	166	2	no	—
132	C₂×C₄⋊C₄	—	166	1	no	—
133	C₂×C₄⋊C₄	—	166	1	no	—
134	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
135	#106	—	2	8	no	—
136	#106	—	332	4	no	—
137	#103	—	332	8	no	—
138	C₄×C₈	—	166	2	no	—
139	C₄×C₈	—	166	2	no	—
140	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
141	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
142	C₂×Q₈	—	2	4	no	—
143	C₂×Q₈	—	4	2	no	—
144	C₂×Q₈	—	4	4	no	—
145	C₂×Q₈	—	4	4	no	—
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	—	332	1	no	—
148	C₂²×C₄	—	332	1	no	—
149	C₂²×C₄	D₄×D₈₃	1	2	yes	—
150	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
151	C₂×C₈	—	83	8	no	—
152	C₂×C₈	—	166	4	no	—
153	C₂×C₈	—	166	4	no	—
154	C₂×C₈	C₂²⋊Dic₈₃	1	4	yes	—
155	C₂×C₈	#13050577	1	4	yes	—
156	C₂×C₈	#13050580	1	4	yes	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	332	4	no	—
162	C₄⋊C₄	—	332	2	no	—
163	C₄⋊C₄	—	332	2	no	—
164	C₄⋊C₄	—	332	4	no	—
165	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
166	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
167	C₄²	#13050589	1	2	yes	—
168	C₂³	#14581788	1	1	yes	characteristic
169	Q₈	—	4	8	no	—
170	Q₈	—	8	4	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	664	2	no	—
175	C₂×C₄	—	664	1	no	—
176	C₂×C₄	—	664	1	no	—
177	C₂×C₄	—	664	2	no	—
178	C₂×C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
179	C₂×C₄	D₄×Dic₈₃	1	2	yes	—
180	C₂×C₄	#14581763	1	2	yes	—
181	C₂×C₄	#14581779	1	1	yes	characteristic
182	C₂×C₄	SD₁₆×D₈₃	1	4	yes	—
183	C₂×C₄	C₂×#13050577	1	2	yes	—
184	C₂×C₄	C₂×#13050580	1	2	yes	—
185	C₂×C₄	#14581813	1	4	yes	—
186	C₈	—	166	8	no	—
187	C₈	#14581716	1	8	yes	complemented
188	C₂²	#19127604	1	1	yes	characteristic
189	C₂²	#19128072	1	2	yes	—
190	C₂²	#19128075	1	1	yes	characteristic
191	C₂²	#19128080	1	1	yes	characteristic
192	C₂²	#19128082	1	2	yes	—
193	C₄	—	8	4	no	—
194	C₄	—	1328	1	no	—
195	C₄	—	1328	1	no	—
196	C₄	C₂×#14581716	1	2	yes	—
197	C₄	SD₁₆×Dic₈₃	1	4	yes	—
198	C₄	#19127927	1	4	yes	—
199	C₄	#19128051	1	2	yes	—
200	C₂	#24907051	1	1	yes	characteristic
201	C₂	#24907058	1	2	yes	—
202	C₂	#24907065	1	1	yes	characteristic
203	C₂	#24908266	1	1	yes	characteristic
204	C₂	#24908319	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24908425

C₂

yes

maximal

#24908266

C₂

yes

maximal

#24907620

C₂

yes

maximal, characteristic

C₂×C₈×C₈₃⋊C₈

C₂

yes

maximal, characteristic

C₂×#498×C₈₃

C₂

yes

maximal, characteristic

C₂×#14581724

—

C₂×#14581716

—

C₂×C₄×C₈₃⋊C₈

C₂²

yes

—

#19127571

C₂²

yes

characteristic

#19127571

C₂²

yes

characteristic

#19127463

—

C₈×C₈₃⋊C₈

—

C₂×C₄⋊Q₈×C₈₃

C₂²

yes

—

#498×C₈₃

—

#498×C₈₃

C₄

yes

complemented

C₂×#106×C₈₃

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂²×C₈₃⋊C₈

D₄

yes

—

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

—

#14581724

—

#14581716

—

C₄×C₈₃⋊C₈

—

C₄×C₈₃⋊C₈

—

C₂²×Q₈×C₈₃

—

C₂²×C₈×C₈₃

D₄

yes

—

C₄⋊Q₈×C₈₃

—

C₄⋊Q₈×C₈₃

—

C₄⋊Q₈×C₈₃

—

C₄⋊Q₈×C₈₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

#106×C₈₃

—

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

D₄

yes

—

C₂²×Dic₈₃

—

C₂²×Dic₈₃

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

SD₁₆

yes

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂×D₄

yes

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

C₂²⋊C₄

yes

—

C₂×C₈×C₈₃

SD₁₆

yes

—

C₂×C₈×C₈₃

Q₁₆

yes

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

C₂²×C₄

yes

characteristic

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₄²×C₈₃

#41

yes

—

C₂³×C₈₃

#127

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄×D₄

yes

—

C₂×C₄×C₈₃

#121

yes

—

C₂×C₄×C₈₃

#124

yes

characteristic

C₂×C₄×C₈₃

C₂×SD₁₆

yes

—

C₂×C₄×C₈₃

C₂×SD₁₆

yes

—

C₂×C₄×C₈₃

C₂×Q₁₆

yes

—

C₂×C₄×C₈₃

#135

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈×C₈₃

#103

yes

complemented

C₈₃⋊C₈

—

C₂²×C₈₃

#390

yes

characteristic

C₂²×C₈₃

#492

yes

—

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

—

C₄×C₈₃

—

C₄×C₈₃

C₂×#103

yes

—

C₄×C₈₃

C₄×SD₁₆

yes

—

C₄×C₈₃

#465

yes

—

C₄×C₈₃

#487

yes

—

Dic₈₃

—

Dic₈₃

—

—

C₂×C₈₃

yes

C₂×C₈₃

yes

—

100

C₂×C₈₃

#1708

yes

characteristic

101

C₂×C₈₃

#1956

yes

characteristic

102

C₂×C₈₃

#1968

yes

—

103

C₂×#498

D₈₃

yes

characteristic

104

#1990

—

105

#1956

—

106

#1844

—

107

C₂×C₈²

—

108

C₈₃

#9819

yes

characteristic, complemented, Sylow

109

C₂×C₄⋊Q₈

C₂×D₈₃

yes

—

110

#498

—

111

#498

Dic₈₃

yes

complemented

112

C₂×#106

—

113

C₂×#106

—

166

—

114

C₂×#103

—

166

—

115

C₂×C₄×C₈

—

116

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

117

#384

—

118

#384

—

119

#340

—

166

—

120

C₈²

—

166

—

121

C₂²×Q₈

—

122

C₂²×C₈

—

123

C₂²×C₈

C₂²⋊D₈₃

yes

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

C₂×Dic₈₃

yes

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

166

—

131

C₂×C₄⋊C₄

—

166

—

132

C₂×C₄⋊C₄

—

166

—

133

C₂×C₄⋊C₄

—

166

—

134

C₂×C₄²

C₂²×D₈₃

yes

characteristic

135

#106

—

136

#106

—

332

—

137

#103

—

332

—

138

C₄×C₈

—

166

—

139

C₄×C₈

—

166

—

140

C₄×C₈

C₂×Dic₈₃

yes

—

141

C₄×C₈

C₂²⋊D₈₃

yes

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂²×C₄

—

147

C₂²×C₄

—

332

—

148

C₂²×C₄

—

332

—

149

C₂²×C₄

D₄×D₈₃

yes

—

150

C₂²×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

151

C₂×C₈

—

152

C₂×C₈

—

166

—

153

C₂×C₈

—

166

—

154

C₂×C₈

C₂²⋊Dic₈₃

yes

—

155

C₂×C₈

#13050577

yes

—

156

C₂×C₈

#13050580

yes

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

332

—

162

C₄⋊C₄

—

332

—

163

C₄⋊C₄

—

332

—

164

C₄⋊C₄

—

332

—

165

C₄²

C₂²×Dic₈₃

yes

characteristic

166

C₄²

C₂×C₂²⋊D₈₃

yes

characteristic

167

C₄²

#13050589

yes

—

168

C₂³

#14581788

yes

characteristic

169

Q₈

—

170

Q₈

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

664

—

175

C₂×C₄

—

664

—

176

C₂×C₄

—

664

—

177

C₂×C₄

—

664

—

178

C₂×C₄

C₂×C₂²⋊Dic₈₃

yes

characteristic

179

C₂×C₄

D₄×Dic₈₃

yes

—

180

C₂×C₄

#14581763

yes

—

181

C₂×C₄

#14581779

yes

characteristic

182

C₂×C₄

SD₁₆×D₈₃

yes

—

183

C₂×C₄

C₂×#13050577

yes

—

184

C₂×C₄

C₂×#13050580

yes

—

185

C₂×C₄

#14581813

yes

—

186

C₈

—

166

—

187

C₈

#14581716

yes

complemented

188

C₂²

#19127604

yes

characteristic

189

C₂²

#19128072

yes

—

190

C₂²

#19128075

yes

characteristic

191

C₂²

#19128080

yes

characteristic

192

C₂²

#19128082

yes

—

193

C₄

—

194

C₄

—

1328

—

195

C₄

—

1328

—

196

C₄

C₂×#14581716

yes

—

197

C₄

SD₁₆×Dic₈₃

yes

—

198

C₄

#19127927

yes

—

199

C₄

#19128051

yes

—

200

C₂

#24907051

yes

characteristic

201

C₂

#24907058

yes

—

202

C₂

#24907065

yes

characteristic

203

C₂

#24908266

yes

characteristic

204

C₂

#24908319

yes

—

	subgroup	G/H
center	C₂³	#14581788
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₂×#498×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	#127
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₂×#498	D₈₃
hypocenter	C₈₃	#9819
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

subgroup

G/H

center

C₂³

#14581788

commutator

C₄²×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₂×#498×C₈₃

C₂

socle, prime radical

C₂³×C₈₃

#127

cosocle

C₂×C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

N	H ≅ G/N	type
15. #498×C₈₃	C₄	semidirect
83. C₈×C₈₃	#103	semidirect
108. C₈₃	#9819	semidirect
111. #498	Dic₈₃	semidirect
187. C₈	#14581716	semidirect

H ≅ G/N

type

15. #498×C₈₃

C₄

semidirect

83. C₈×C₈₃

#103

semidirect

108. C₈₃

#9819

semidirect

111. #498

Dic₈₃

semidirect

187. C₈

#14581716

semidirect

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

subgroup	index
#24908425	2
#24908266	2
#24907620	2
C₂×C₈×C₈₃⋊C₈	2
C₂×#498×C₈₃	2
#9819	83

subgroup

index

#24908425

#24908266

#24907620

C₂×C₈×C₈₃⋊C₈

C₂×#498×C₈₃

#9819

Subgroups — #35304140

Conjugacy classes of subgroups

Complements

Maximal subgroups