Group #35307241 — subgroups

Subgroups — #35307241

Order 21248.

586 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 161 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×#19128097	C₂	1	2	yes	maximal
2	#24908266	C₂	1	2	yes	maximal
3	#24908265	C₂	1	1	yes	maximal, characteristic
4	C₈₃⋊#1273	C₂	1	1	yes	maximal, characteristic
5	C₂×#498×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581792	C₂²	1	1	yes	characteristic
7	#19128097	—	2	4	no	—
8	#19128097	—	2	4	no	—
9	C₂×#14581723	—	2	4	no	—
10	C₂×#14581717	—	2	2	no	—
11	C₂×#14581716	—	2	4	no	—
12	C₂×C₄×C₈₃⋊C₈	C₂²	1	2	yes	—
13	#19127571	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₈₃	C₂²	1	2	yes	—
15	#498×C₈₃	—	2	4	no	—
16	C₂×#106×C₈₃	—	2	2	no	—
17	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
18	C₂²×Dic₁₆₆	—	2	1	no	—
19	C₂²×C₈₃⋊C₈	D₄	1	4	yes	—
20	#14581792	—	2	2	no	—
21	#14581792	—	2	1	no	—
22	#14581792	D₄	1	2	yes	—
23	C₂×C₄⋊Dic₈₃	—	2	2	no	—
24	C₂×C₄⋊Dic₈₃	—	2	2	no	—
25	C₂×C₄⋊Dic₈₃	—	2	1	no	—
26	#14581723	—	2	16	no	—
27	#14581717	—	4	4	no	—
28	#14581716	—	4	8	no	—
29	C₄×C₈₃⋊C₈	—	2	2	no	—
30	C₄×C₈₃⋊C₈	—	2	2	no	—
31	C₂²×Q₈×C₈₃	—	2	2	no	—
32	C₂²×C₈×C₈₃	D₄	1	2	yes	—
33	C₄⋊Q₈×C₈₃	—	2	2	no	—
34	C₄⋊Q₈×C₈₃	—	2	2	no	—
35	C₄⋊Q₈×C₈₃	—	2	2	no	—
36	C₄⋊Q₈×C₈₃	D₄	1	4	yes	—
37	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
39	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
40	#106×C₈₃	—	2	8	no	—
41	C₄×C₈×C₈₃	—	2	2	no	—
42	C₂×Dic₁₆₆	—	2	2	no	—
43	C₂×Dic₁₆₆	—	4	2	no	—
44	C₂×Dic₁₆₆	—	4	2	no	—
45	C₂×Dic₁₆₆	—	4	1	no	—
46	C₂²×Dic₈₃	—	4	1	no	—
47	C₂²×Dic₈₃	—	4	1	no	—
48	C₂×C₈₃⋊C₈	—	2	4	no	—
49	C₂×C₈₃⋊C₈	—	2	4	no	—
50	C₂×C₈₃⋊C₈	SD₁₆	1	8	yes	—
51	C₄⋊Dic₈₃	—	2	4	no	—
52	C₄⋊Dic₈₃	—	4	2	no	—
53	C₄⋊Dic₈₃	—	4	4	no	—
54	C₄⋊Dic₈₃	—	4	2	no	—
55	C₂×Q₈×C₈₃	—	2	4	no	—
56	C₂×Q₈×C₈₃	—	4	2	no	—
57	C₂×Q₈×C₈₃	—	4	4	no	—
58	C₂×Q₈×C₈₃	—	4	4	no	—
59	C₂²×C₄×C₈₃	—	4	2	no	—
60	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₈₃	—	2	2	no	—
63	C₂×C₈×C₈₃	—	2	2	no	—
64	C₂×C₈×C₈₃	SD₁₆	1	4	yes	—
65	C₄⋊C₄×C₈₃	—	2	4	no	—
66	C₄⋊C₄×C₈₃	—	2	4	no	—
67	C₄⋊C₄×C₈₃	—	4	2	no	—
68	C₄⋊C₄×C₈₃	—	4	2	no	—
69	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₈₃	C₂×D₄	1	2	yes	—
71	C₄²×C₈₃	#41	1	1	yes	characteristic
72	C₂³×C₈₃	#127	1	1	yes	characteristic
73	Q₈×C₈₃	—	4	8	no	—
74	Q₈×C₈₃	—	8	4	no	—
75	C₂×C₄×C₈₃	—	4	4	no	—
76	C₂×C₄×C₈₃	—	8	2	no	—
77	C₂×C₄×C₈₃	—	8	2	no	—
78	C₂×C₄×C₈₃	#120	1	2	yes	—
79	C₂×C₄×C₈₃	#120	1	1	yes	characteristic
80	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
81	C₂×C₄×C₈₃	#121	1	2	yes	—
82	C₂×C₄×C₈₃	C₂×SD₁₆	1	4	yes	—
83	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
84	C₂×C₄×C₈₃	#137	1	4	yes	—
85	C₂×C₄×C₈₃	#137	1	2	yes	—
86	C₂×Dic₈₃	—	4	2	no	—
87	C₂×Dic₈₃	—	8	1	no	—
88	C₂×Dic₈₃	—	8	1	no	—
89	C₂×Dic₈₃	—	8	2	no	—
90	C₂×Dic₈₃	—	8	1	no	—
91	Dic₁₆₆	—	4	4	no	—
92	Dic₁₆₆	—	8	2	no	—
93	C₈×C₈₃	—	2	4	no	—
94	C₈₃⋊C₈	—	2	8	no	—
95	C₂²×C₈₃	#392	1	1	yes	characteristic
96	C₂²×C₈₃	#492	1	2	yes	—
97	C₂²×C₈₃	#492	1	1	yes	characteristic
98	C₂²×C₈₃	#497	1	2	yes	—
99	C₂²×C₈₃	#497	1	1	yes	characteristic
100	C₄×C₈₃	—	8	4	no	—
101	C₄×C₈₃	#448	1	4	yes	—
102	C₄×C₈₃	#448	1	2	yes	—
103	C₄×C₈₃	#461	1	2	yes	—
104	C₄×C₈₃	#461	1	4	yes	—
105	Dic₈₃	—	8	2	no	—
106	Dic₈₃	—	16	1	no	—
107	#10316	—	83	1	no	maximal, Sylow
108	C₂×C₈₃	#1705	1	1	yes	characteristic
109	C₂×C₈₃	#1716	1	2	yes	—
110	C₂×C₈₃	#1716	1	1	yes	characteristic
111	C₂×C₈₃	#2013	1	2	yes	—
112	C₂×C₈₃	#2013	1	1	yes	characteristic
113	C₂×#498	—	83	2	no	—
114	C₂×#498	D₈₃	1	1	yes	characteristic
115	#1956	—	83	1	no	—
116	#1956	—	83	2	no	—
117	#1273	—	83	1	no	—
118	C₈₃	#10316	1	1	yes	characteristic, complemented, Sylow
119	C₂×C₄⋊Q₈	—	83	1	no	—
120	C₂×C₄⋊Q₈	C₂×D₈₃	1	2	yes	—
121	#498	—	2	4	no	—
122	#498	—	166	4	no	—
123	#498	—	166	4	no	—
124	C₂×#106	—	2	2	no	—
125	C₂×#106	—	166	4	no	—
126	C₂×#103	—	166	4	no	—
127	C₂×#103	—	166	2	no	—
128	C₂×C₄×C₈	—	83	2	no	—
129	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
130	#384	—	83	1	no	—
131	C₂²×Q₈	—	2	2	no	—
132	C₂²×Q₈	—	166	1	no	—
133	C₂²×C₈	—	83	4	no	—
134	C₂²×C₈	C₂²⋊D₈₃	1	2	yes	—
135	C₄⋊Q₈	—	2	2	no	—
136	C₄⋊Q₈	—	2	2	no	—
137	C₄⋊Q₈	—	2	2	no	—
138	C₄⋊Q₈	—	83	2	no	—
139	C₄⋊Q₈	—	166	2	no	—
140	C₄⋊Q₈	—	166	1	no	—
141	C₄⋊Q₈	C₂²⋊D₈₃	1	4	yes	—
142	C₂×C₄⋊C₄	—	2	2	no	—
143	C₂×C₄⋊C₄	—	2	2	no	—
144	C₂×C₄⋊C₄	—	166	2	no	—
145	C₂×C₄⋊C₄	—	166	1	no	—
146	C₂×C₄⋊C₄	—	166	2	no	—
147	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
148	#106	—	2	8	no	—
149	#106	—	166	16	no	—
150	#103	—	332	8	no	—
151	#103	—	332	4	no	—
152	C₄×C₈	—	2	2	no	—
153	C₄×C₈	—	166	2	no	—
154	C₄×C₈	—	166	2	no	—
155	C₂×Q₈	—	2	4	no	—
156	C₂×Q₈	—	4	2	no	—
157	C₂×Q₈	—	4	4	no	—
158	C₂×Q₈	—	4	4	no	—
159	C₂×Q₈	—	332	1	no	—
160	C₂×Q₈	—	332	2	no	—
161	C₂×Q₈	—	332	2	no	—
162	C₂×Q₈	—	166	2	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	—	332	1	no	—
165	C₂²×C₄	—	332	1	no	—
166	C₂²×C₄	D₄×D₈₃	1	2	yes	—
167	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	83	8	no	—
171	C₂×C₈	—	166	4	no	—
172	C₂×C₈	—	166	4	no	—
173	C₂×C₈	#13050577	1	4	yes	—
174	C₄⋊C₄	—	2	4	no	—
175	C₄⋊C₄	—	2	4	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	332	2	no	—
179	C₄⋊C₄	—	332	4	no	—
180	C₄⋊C₄	—	332	2	no	—
181	C₄⋊C₄	—	166	4	no	—
182	C₄²	D₄×D₈₃	1	1	yes	characteristic
183	C₄²	C₂×C₂²⋊D₈₃	1	2	yes	—
184	C₄²	#13050589	1	1	yes	characteristic
185	C₂³	#14581788	1	1	yes	characteristic
186	Q₈	—	4	8	no	—
187	Q₈	—	8	4	no	—
188	Q₈	—	332	4	no	—
189	Q₈	—	664	2	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	8	2	no	—
193	C₂×C₄	—	332	2	no	—
194	C₂×C₄	—	664	2	no	—
195	C₂×C₄	—	664	1	no	—
196	C₂×C₄	—	664	1	no	—
197	C₂×C₄	—	664	1	no	—
198	C₂×C₄	#14581760	1	2	yes	—
199	C₂×C₄	#14581761	1	1	yes	characteristic
200	C₂×C₄	#14581763	1	1	yes	characteristic
201	C₂×C₄	#14581764	1	2	yes	—
202	C₂×C₄	SD₁₆×D₈₃	1	4	yes	—
203	C₂×C₄	C₂×#13050577	1	2	yes	—
204	C₂×C₄	#14581821	1	4	yes	—
205	C₂×C₄	#14581823	1	2	yes	—
206	C₈	—	2	4	no	—
207	C₈	—	166	8	no	—
208	C₂²	#19127609	1	1	yes	characteristic
209	C₂²	#19128072	1	2	yes	—
210	C₂²	#19128075	1	1	yes	characteristic
211	C₂²	#19128092	1	2	yes	—
212	C₂²	#19128093	1	1	yes	characteristic
213	C₄	—	8	4	no	—
214	C₄	—	1328	1	no	—
215	C₄	—	664	2	no	—
216	C₄	#19127838	1	4	yes	—
217	C₄	#19127840	1	2	yes	—
218	C₄	#19127903	1	2	yes	—
219	C₄	#19127904	1	4	yes	—
220	C₂	#24907051	1	1	yes	characteristic
221	C₂	#24907095	1	2	yes	—
222	C₂	#24907097	1	1	yes	characteristic
223	C₂	#24908544	1	2	yes	—
224	C₂	#24908546	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19128097

C₂

yes

maximal

#24908266

C₂

yes

maximal

#24908265

C₂

yes

maximal, characteristic

C₈₃⋊#1273

C₂

yes

maximal, characteristic

C₂×#498×C₈₃

C₂

yes

maximal, characteristic

C₂×#14581792

C₂²

yes

—

—

—

—

—

C₂×C₄×C₈₃⋊C₈

C₂²

yes

—

#19127571

C₂²

yes

characteristic

C₂×C₄⋊Q₈×C₈₃

C₂²

yes

—

#498×C₈₃

—

C₂×#106×C₈₃

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂²×Dic₁₆₆

—

C₂²×C₈₃⋊C₈

D₄

yes

—

#14581792

—

#14581792

—

#14581792

D₄

yes

—

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

—

#14581723

—

#14581717

—

#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

D₄

yes

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

#106×C₈₃

—

C₄×C₈×C₈₃

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

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

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

SD₁₆

yes

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₄²×C₈₃

C₂×D₄

yes

—

C₄²×C₈₃

yes

C₂³×C₈₃

yes

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

#120

yes

—

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

—

C₂×C₄×C₈₃

C₂×SD₁₆

yes

—

C₂×C₄×C₈₃

C₂×SD₁₆

yes

—

C₂×C₄×C₈₃

#137

yes

—

C₂×C₄×C₈₃

#137

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

#392

yes

characteristic

C₂²×C₈₃

#492

yes

—

C₂²×C₈₃

#492

yes

characteristic

C₂²×C₈₃

#497

yes

—

C₂²×C₈₃

#497

yes

characteristic

100

C₄×C₈₃

—

101

C₄×C₈₃

#448

yes

—

102

C₄×C₈₃

#448

yes

—

103

C₄×C₈₃

#461

yes

—

104

C₄×C₈₃

#461

yes

—

105

Dic₈₃

—

106

Dic₈₃

—

107

#10316

—

maximal, Sylow

108

C₂×C₈₃

#1705

yes

characteristic

109

C₂×C₈₃

#1716

yes

—

110

C₂×C₈₃

#1716

yes

characteristic

111

C₂×C₈₃

#2013

yes

—

112

C₂×C₈₃

#2013

yes

characteristic

113

C₂×#498

—

114

C₂×#498

D₈₃

yes

characteristic

115

#1956

—

116

#1956

—

117

#1273

—

118

C₈₃

#10316

yes

characteristic, complemented, Sylow

119

C₂×C₄⋊Q₈

—

120

C₂×C₄⋊Q₈

C₂×D₈₃

yes

—

121

#498

—

122

#498

—

166

—

123

#498

—

166

—

124

C₂×#106

—

125

C₂×#106

—

166

—

126

C₂×#103

—

166

—

127

C₂×#103

—

166

—

128

C₂×C₄×C₈

—

129

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

130

#384

—

131

C₂²×Q₈

—

132

C₂²×Q₈

—

166

—

133

C₂²×C₈

—

134

C₂²×C₈

C₂²⋊D₈₃

yes

—

135

C₄⋊Q₈

—

136

C₄⋊Q₈

—

137

C₄⋊Q₈

—

138

C₄⋊Q₈

—

139

C₄⋊Q₈

—

166

—

140

C₄⋊Q₈

—

166

—

141

C₄⋊Q₈

C₂²⋊D₈₃

yes

—

142

C₂×C₄⋊C₄

—

143

C₂×C₄⋊C₄

—

144

C₂×C₄⋊C₄

—

166

—

145

C₂×C₄⋊C₄

—

166

—

146

C₂×C₄⋊C₄

—

166

—

147

C₂×C₄²

C₂²×D₈₃

yes

characteristic

148

#106

—

149

#106

—

166

—

150

#103

—

332

—

151

#103

—

332

—

152

C₄×C₈

—

153

C₄×C₈

—

166

—

154

C₄×C₈

—

166

—

155

C₂×Q₈

—

156

C₂×Q₈

—

157

C₂×Q₈

—

158

C₂×Q₈

—

159

C₂×Q₈

—

332

—

160

C₂×Q₈

—

332

—

161

C₂×Q₈

—

332

—

162

C₂×Q₈

—

166

—

163

C₂²×C₄

—

164

C₂²×C₄

—

332

—

165

C₂²×C₄

—

332

—

166

C₂²×C₄

D₄×D₈₃

yes

—

167

C₂²×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

166

—

172

C₂×C₈

—

166

—

173

C₂×C₈

#13050577

yes

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

176

C₄⋊C₄

—

177

C₄⋊C₄

—

178

C₄⋊C₄

—

332

—

179

C₄⋊C₄

—

332

—

180

C₄⋊C₄

—

332

—

181

C₄⋊C₄

—

166

—

182

C₄²

D₄×D₈₃

yes

characteristic

183

C₄²

C₂×C₂²⋊D₈₃

yes

—

184

C₄²

#13050589

yes

characteristic

185

C₂³

#14581788

yes

characteristic

186

Q₈

—

187

Q₈

—

188

Q₈

—

332

—

189

Q₈

—

664

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

332

—

194

C₂×C₄

—

664

—

195

C₂×C₄

—

664

—

196

C₂×C₄

—

664

—

197

C₂×C₄

—

664

—

198

C₂×C₄

#14581760

yes

—

199

C₂×C₄

#14581761

yes

characteristic

200

C₂×C₄

#14581763

yes

characteristic

201

C₂×C₄

#14581764

yes

—

202

C₂×C₄

SD₁₆×D₈₃

yes

—

203

C₂×C₄

C₂×#13050577

yes

—

204

C₂×C₄

#14581821

yes

—

205

C₂×C₄

#14581823

yes

—

206

C₈

—

207

C₈

—

166

—

208

C₂²

#19127609

yes

characteristic

209

C₂²

#19128072

yes

—

210

C₂²

#19128075

yes

characteristic

211

C₂²

#19128092

yes

—

212

C₂²

#19128093

yes

characteristic

213

C₄

—

214

C₄

—

1328

—

215

C₄

—

664

—

216

C₄

#19127838

yes

—

217

C₄

#19127840

yes

—

218

C₄

#19127903

yes

—

219

C₄

#19127904

yes

—

220

C₂

#24907051

yes

characteristic

221

C₂

#24907095

yes

—

222

C₂

#24907097

yes

characteristic

223

C₂

#24908544

yes

—

224

C₂

#24908546

yes

characteristic

	subgroup	G/H
center	C₂³	#14581788
commutator, cosocle	C₂×C₄²×C₈₃	C₂³
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₂×#498×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	#127
hypercenter	C₂×#498	D₈₃
hypocenter	C₈₃	#10316
LCS	G ▷ C₂×C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

subgroup

G/H

center

C₂³

#14581788

commutator, cosocle

C₂×C₄²×C₈₃

C₂³

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₂×#498×C₈₃

C₂

C₂³×C₈₃

D₈₃

C₈₃

G ▷ C₂×C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

N	H ≅ G/N	type
118. C₈₃	#10316	semidirect

H ≅ G/N

type

118. C₈₃

#10316

semidirect

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

subgroup	index
C₂×#19128097	2
#24908266	2
#24908265	2
C₈₃⋊#1273	2
C₂×#498×C₈₃	2
#10316	83

subgroup

index

C₂×#19128097

#24908266

#24908265

C₈₃⋊#1273

C₂×#498×C₈₃

#10316

Subgroups — #35307241

Conjugacy classes of subgroups

Complements

Maximal subgroups