Group #35323480 — subgroups

Subgroups — #35323480

Order 21248.

382 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 117 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×#19127504	C₂	1	2	yes	maximal
2	#24908324	C₂	1	1	yes	maximal, characteristic
3	#24905554	C₂	1	2	yes	maximal
4	#24905541	C₂	1	1	yes	maximal, characteristic
5	C₂×#499×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₃₂	C₂²	1	1	yes	characteristic
7	C₂²×C₈₃⋊C₁₆	C₂²	1	2	yes	—
8	C₂×#14581726	C₂²	1	1	yes	characteristic
9	C₂×#14581717	—	2	1	no	—
10	#19127607	—	2	1	no	—
11	#19127504	—	2	4	no	—
12	C₂×#125×C₈₃	—	2	1	no	—
13	#499×C₈₃	—	2	4	no	—
14	C₂×#107×C₈₃	C₂²	1	2	yes	—
15	C₂×#106×C₈₃	—	2	1	no	—
16	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₆₆	—	2	1	no	—
18	C₂×Dic₃₃₂	—	2	2	no	—
19	C₂×Dic₃₃₂	—	2	2	no	—
20	C₂×Dic₃₃₂	—	2	1	no	—
21	C₂×Dic₃₃₂	Q₈	1	2	yes	—
22	C₂×C₄⋊Dic₈₃	—	2	1	no	—
23	C₂×C₈₃⋊C₁₆	—	2	2	no	—
24	C₂×C₈₃⋊C₁₆	—	2	2	no	—
25	C₂×C₈₃⋊C₁₆	—	2	2	no	—
26	#14581726	—	2	2	no	—
27	#14581717	—	4	2	no	—
28	C₈₃⋊#95	—	4	2	no	—
29	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
31	#125×C₈₃	—	2	4	no	—
32	#125×C₈₃	—	4	2	no	—
33	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₈₃	—	4	2	no	—
35	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
36	#107×C₈₃	—	2	2	no	—
37	#107×C₈₃	D₄	1	4	yes	—
38	#106×C₈₃	—	2	4	no	—
39	C₄×C₈×C₈₃	—	2	2	no	—
40	C₂×Dic₁₆₆	—	2	2	no	—
41	C₂×Dic₁₆₆	—	4	1	no	—
42	C₂×Dic₁₆₆	—	4	1	no	—
43	C₂×Dic₁₆₆	—	4	1	no	—
44	C₂×Dic₁₆₆	—	4	1	no	—
45	C₂×Dic₁₆₆	—	4	1	no	—
46	C₂²×Dic₈₃	—	4	1	no	—
47	C₂²×Dic₈₃	—	4	1	no	—
48	Dic₃₃₂	—	2	4	no	—
49	Dic₃₃₂	—	4	2	no	—
50	C₄⋊Dic₈₃	—	4	1	no	—
51	C₄⋊Dic₈₃	—	4	1	no	—
52	C₈₃⋊C₁₆	—	4	2	no	—
53	C₂²×C₄×C₈₃	—	4	2	no	—
54	C₂²×C₄×C₈₃	D₈	1	2	yes	—
55	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₈₃	—	2	1	no	—
57	C₂×C₈×C₈₃	—	2	1	no	—
58	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
59	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₈₃	#41	1	2	yes	—
62	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
63	C₄⋊C₄×C₈₃	—	2	4	no	—
64	C₄⋊C₄×C₈₃	—	4	8	no	—
65	C₄⋊C₄×C₈₃	—	4	2	no	—
66	C₄²×C₈₃	—	2	1	no	—
67	C₄²×C₈₃	—	2	1	no	—
68	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₈₃	—	2	2	no	—
70	C₂×C₄×C₈₃	—	2	2	no	—
71	C₂×C₄×C₈₃	—	2	2	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	#120	1	1	yes	characteristic
76	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
78	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
81	C₂×Dic₈₃	—	4	2	no	—
82	C₂×Dic₈₃	—	8	1	no	—
83	C₂×Dic₈₃	—	8	1	no	—
84	C₂×Dic₈₃	—	8	1	no	—
85	C₂×Dic₈₃	—	8	2	no	—
86	Dic₁₆₆	—	4	2	no	—
87	Dic₁₆₆	—	4	2	no	—
88	Dic₁₆₆	—	8	1	no	—
89	Dic₁₆₆	—	8	1	no	—
90	C₈×C₈₃	—	2	2	no	—
91	C₈×C₈₃	C₂²⋊Q₈	1	2	yes	—
92	C₈×C₈₃	#124	1	2	yes	—
93	C₂²×C₈₃	#447	1	1	yes	characteristic
94	C₂²×C₈₃	#474	1	1	yes	characteristic
95	C₂²×C₈₃	#486	1	1	yes	characteristic
96	C₂²×C₈₃	#508	1	2	yes	—
97	C₂²×C₈₃	#510	1	2	yes	—
98	C₄×C₈₃	—	4	2	no	—
99	C₄×C₈₃	—	8	4	no	—
100	C₄×C₈₃	#393	1	1	yes	characteristic
101	C₄×C₈₃	#448	1	1	yes	characteristic
102	C₄×C₈₃	#476	1	1	yes	characteristic
103	C₄×C₈₃	#489	1	1	yes	characteristic
104	Dic₈₃	—	8	2	no	—
105	Dic₈₃	—	16	1	no	—
106	#13366	—	83	1	no	maximal, Sylow
107	C₂×C₈₃	#2023	1	1	yes	characteristic
108	C₂×C₈₃	#2189	1	2	yes	—
109	C₂×C₈₃	#2229	1	2	yes	—
110	C₂×C₈₃	#2242	1	1	yes	characteristic
111	C₂×C₈₃	#2245	1	1	yes	characteristic
112	C₂×#499	D₈₃	1	1	yes	characteristic
113	C₂×#358	—	83	2	no	—
114	#1969	—	83	1	no	—
115	#1380	—	83	2	no	—
116	#1374	—	83	1	no	—
117	C₈₃	#13366	1	1	yes	characteristic, complemented, Sylow
118	C₂²×Q₁₆	—	83	1	no	—
119	C₂×#125	—	2	1	no	—
120	C₂²×C₁₆	—	83	2	no	—
121	#499	—	2	4	no	—
122	C₂×#107	—	83	1	no	—
123	C₂×#107	C₂×D₈₃	1	2	yes	—
124	C₂×#106	—	2	1	no	—
125	C₂×#103	—	166	1	no	—
126	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
127	#391	—	166	1	no	—
128	#358	—	166	4	no	—
129	C₂²×Q₈	—	166	1	no	—
130	C₂×Q₁₆	—	83	2	no	—
131	C₂×Q₁₆	—	166	2	no	—
132	C₂×Q₁₆	—	166	1	no	—
133	C₂×Q₁₆	—	166	2	no	—
134	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
136	#125	—	2	4	no	—
137	#125	—	4	2	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	4	2	no	—
140	C₂×C₄⋊C₄	—	166	1	no	—
141	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
142	C₂×C₁₆	—	166	2	no	—
143	C₂×C₁₆	—	166	2	no	—
144	C₂×C₁₆	—	166	2	no	—
145	#107	—	2	2	no	—
146	#107	—	166	2	no	—
147	#107	C₂²⋊D₈₃	1	4	yes	—
148	#106	—	2	4	no	—
149	#103	—	332	2	no	—
150	C₄×C₈	—	2	2	no	—
151	#95	—	332	2	no	—
152	C₂×Q₈	—	332	1	no	—
153	C₂×Q₈	—	332	1	no	—
154	C₂×Q₈	—	332	1	no	—
155	C₂×Q₈	—	332	1	no	—
156	C₂×Q₈	—	332	1	no	—
157	C₂×Q₈	—	166	2	no	—
158	C₂²×C₄	—	4	2	no	—
159	C₂²×C₄	—	332	1	no	—
160	C₂²×C₄	—	332	1	no	—
161	C₂²×C₄	#13050575	1	2	yes	—
162	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
163	Q₁₆	—	332	2	no	—
164	Q₁₆	—	166	4	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	#13050577	1	2	yes	—
168	C₂×C₈	C₂×C₂²⋊D₈₃	1	2	yes	—
169	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
170	C₂×C₈	#13050591	1	2	yes	—
171	C₂×C₈	#13050591	1	1	yes	characteristic
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	4	8	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	332	1	no	—
176	C₄⋊C₄	—	332	1	no	—
177	C₄²	—	2	1	no	—
178	C₄²	—	2	1	no	—
179	C₁₆	—	332	2	no	—
180	C₂³	C₂×#13050575	1	1	yes	characteristic
181	Q₈	—	332	2	no	—
182	Q₈	—	332	2	no	—
183	Q₈	—	664	1	no	—
184	Q₈	—	664	1	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	4	no	—
189	C₂×C₄	—	8	2	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	332	2	no	—
192	C₂×C₄	—	664	2	no	—
193	C₂×C₄	—	664	1	no	—
194	C₂×C₄	—	664	1	no	—
195	C₂×C₄	—	664	1	no	—
196	C₂×C₄	#14581761	1	1	yes	characteristic
197	C₂×C₄	#14581768	1	1	yes	characteristic
198	C₂×C₄	#14581781	1	1	yes	characteristic
199	C₂×C₄	C₂×#13050577	1	1	yes	characteristic
200	C₂×C₄	#14581818	1	1	yes	characteristic
201	C₂×C₄	#14581823	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	#14581768	1	2	yes	—
204	C₈	#14581781	1	2	yes	—
205	C₂²	#19127835	1	1	yes	characteristic
206	C₂²	#19127978	1	1	yes	characteristic
207	C₂²	#19128049	1	1	yes	characteristic
208	C₂²	#19128131	1	2	yes	—
209	C₂²	#19128141	1	2	yes	—
210	C₄	—	4	2	no	—
211	C₄	—	8	4	no	—
212	C₄	—	1328	1	no	—
213	C₄	—	664	2	no	—
214	C₄	#19127612	1	1	yes	characteristic
215	C₄	#19127840	1	1	yes	characteristic
216	C₄	#19127988	1	1	yes	characteristic
217	C₄	#19128063	1	1	yes	characteristic
218	C₂	#24908596	1	1	yes	characteristic
219	C₂	#24909403	1	2	yes	—
220	C₂	#24909624	1	2	yes	—
221	C₂	#24909697	1	1	yes	characteristic
222	C₂	#24909710	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19127504

C₂

yes

maximal

#24908324

C₂

yes

maximal, characteristic

#24905554

C₂

yes

maximal

#24905541

C₂

yes

maximal, characteristic

C₂×#499×C₈₃

C₂

yes

maximal, characteristic

C₂²×Dic₃₃₂

C₂²

yes

characteristic

C₂²×C₈₃⋊C₁₆

C₂²

yes

—

C₂×#14581726

C₂²

yes

—

—

—

C₂×#125×C₈₃

—

#499×C₈₃

—

C₂×#107×C₈₃

C₂²

yes

—

C₂×#106×C₈₃

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂²×Dic₁₆₆

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

Q₈

yes

—

C₂×C₄⋊Dic₈₃

—

C₂×C₈₃⋊C₁₆

—

C₂×C₈₃⋊C₁₆

—

C₂×C₈₃⋊C₁₆

—

#14581726

—

#14581717

—

C₈₃⋊#95

—

C₂²×C₈×C₈₃

D₄

yes

characteristic

C₂²×C₈×C₈₃

C₂³

yes

characteristic

#125×C₈₃

—

#125×C₈₃

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄²×C₈₃

D₄

yes

characteristic

#107×C₈₃

—

#107×C₈₃

D₄

yes

—

#106×C₈₃

—

C₄×C₈×C₈₃

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂²×Dic₈₃

—

C₂²×Dic₈₃

—

Dic₃₃₂

—

Dic₃₃₂

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₈₃⋊C₁₆

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

D₈

yes

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

SD₁₆

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

—

C₂×C₈×C₈₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₃

#41

yes

—

C₂×C₈×C₈₃

#41

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

—

C₂³×C₈₃

C₂×D₈

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

#120

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₈₃

#124

yes

characteristic

C₂×C₄×C₈₃

C₂×SD₁₆

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

C₂²⋊Q₈

yes

—

C₈×C₈₃

#124

yes

—

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

—

C₂²×C₈₃

#510

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

100

C₄×C₈₃

#393

yes

characteristic

101

C₄×C₈₃

#448

yes

characteristic

102

C₄×C₈₃

#476

yes

characteristic

103

C₄×C₈₃

#489

yes

characteristic

104

Dic₈₃

—

105

Dic₈₃

—

106

#13366

—

maximal, Sylow

107

C₂×C₈₃

#2023

yes

characteristic

108

C₂×C₈₃

#2189

yes

—

109

C₂×C₈₃

#2229

yes

—

110

C₂×C₈₃

#2242

yes

characteristic

111

C₂×C₈₃

#2245

yes

characteristic

112

C₂×#499

D₈₃

yes

characteristic

113

C₂×#358

—

114

#1969

—

115

#1380

—

116

#1374

—

117

C₈₃

#13366

yes

characteristic, complemented, Sylow

118

C₂²×Q₁₆

—

119

C₂×#125

—

120

C₂²×C₁₆

—

121

#499

—

122

C₂×#107

—

123

C₂×#107

C₂×D₈₃

yes

—

124

C₂×#106

—

125

C₂×#103

—

166

—

126

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

127

#391

—

166

—

128

#358

—

166

—

129

C₂²×Q₈

—

166

—

130

C₂×Q₁₆

—

131

C₂×Q₁₆

—

166

—

132

C₂×Q₁₆

—

166

—

133

C₂×Q₁₆

—

166

—

134

C₂²×C₈

C₂²⋊D₈₃

yes

characteristic

135

C₂²×C₈

C₂²×D₈₃

yes

characteristic

136

#125

—

137

#125

—

138

C₂×C₄⋊C₄

—

139

C₂×C₄⋊C₄

—

140

C₂×C₄⋊C₄

—

166

—

141

C₂×C₄²

C₂²⋊D₈₃

yes

characteristic

142

C₂×C₁₆

—

166

—

143

C₂×C₁₆

—

166

—

144

C₂×C₁₆

—

166

—

145

#107

—

146

#107

—

166

—

147

#107

C₂²⋊D₈₃

yes

—

148

#106

—

149

#103

—

332

—

150

C₄×C₈

—

151

#95

—

332

—

152

C₂×Q₈

—

332

—

153

C₂×Q₈

—

332

—

154

C₂×Q₈

—

332

—

155

C₂×Q₈

—

332

—

156

C₂×Q₈

—

332

—

157

C₂×Q₈

—

166

—

158

C₂²×C₄

—

159

C₂²×C₄

—

332

—

160

C₂²×C₄

—

332

—

161

C₂²×C₄

#13050575

yes

—

162

C₂²×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

163

Q₁₆

—

332

—

164

Q₁₆

—

166

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

#13050577

yes

—

168

C₂×C₈

C₂×C₂²⋊D₈₃

yes

—

169

C₂×C₈

Q₈×D₈₃

yes

characteristic

170

C₂×C₈

#13050591

yes

—

171

C₂×C₈

#13050591

yes

characteristic

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

332

—

176

C₄⋊C₄

—

332

—

177

C₄²

—

178

C₄²

—

179

C₁₆

—

332

—

180

C₂³

C₂×#13050575

yes

characteristic

181

Q₈

—

332

—

182

Q₈

—

332

—

183

Q₈

—

664

—

184

Q₈

—

664

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

332

—

192

C₂×C₄

—

664

—

193

C₂×C₄

—

664

—

194

C₂×C₄

—

664

—

195

C₂×C₄

—

664

—

196

C₂×C₄

#14581761

yes

characteristic

197

C₂×C₄

#14581768

yes

characteristic

198

C₂×C₄

#14581781

yes

characteristic

199

C₂×C₄

C₂×#13050577

yes

characteristic

200

C₂×C₄

#14581818

yes

characteristic

201

C₂×C₄

#14581823

yes

characteristic

202

C₈

—

203

C₈

#14581768

yes

—

204

C₈

#14581781

yes

—

205

C₂²

#19127835

yes

characteristic

206

C₂²

#19127978

yes

characteristic

207

C₂²

#19128049

yes

characteristic

208

C₂²

#19128131

yes

—

209

C₂²

#19128141

yes

—

210

C₄

—

211

C₄

—

212

C₄

—

1328

—

213

C₄

—

664

—

214

C₄

#19127612

yes

characteristic

215

C₄

#19127840

yes

characteristic

216

C₄

#19127988

yes

characteristic

217

C₄

#19128063

yes

characteristic

218

C₂

#24908596

yes

characteristic

219

C₂

#24909403

yes

—

220

C₂

#24909624

yes

—

221

C₂

#24909697

yes

characteristic

222

C₂

#24909710

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#13050575
commutator, cosocle	C₂²×C₈×C₈₃	C₂³
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₂×#499×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×D₈
hypercenter	C₂×#499	D₈₃
hypocenter	C₈₃	#13366
LCS	G ▷ C₂²×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

subgroup

G/H

center

C₂³

C₂×#13050575

commutator, cosocle

C₂²×C₈×C₈₃

C₂³

Frattini

C₂²×C₈

C₂²×D₈₃

Fitting

C₂×#499×C₈₃

C₂

socle, prime radical

C₂³×C₈₃

C₂×D₈

D₈₃

C₈₃

G ▷ C₂²×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

N	H ≅ G/N	type
117. C₈₃	#13366	semidirect

H ≅ G/N

type

117. C₈₃

#13366

semidirect

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

subgroup	index
C₂×#19127504	2
#24908324	2
#24905554	2
#24905541	2
C₂×#499×C₈₃	2
#13366	83

subgroup

index

C₂×#499×C₈₃

Subgroups — #35323480

Conjugacy classes of subgroups

Complements

Maximal subgroups