Group #35303864 — subgroups

Subgroups — #35303864

Order 21248.

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

normal subgroups only

This group has 414 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	#24907635	C₂	1	1	yes	maximal, characteristic
2	#24906148	C₂	1	2	yes	maximal
3	#24906144	C₂	1	2	yes	maximal
4	#24905746	C₂	1	1	yes	maximal, characteristic
5	C₂×#500×C₈₃	C₂	1	1	yes	maximal, characteristic
6	#19127754	C₂²	1	1	yes	characteristic
7	#19127754	C₂²	1	1	yes	characteristic
8	C₂×#14581728	—	2	2	no	—
9	#19127725	C₂²	1	2	yes	—
10	#19127450	—	2	4	no	—
11	#19127447	—	2	4	no	—
12	#19127422	—	2	1	no	—
13	C₈×C₈₃⋊C₈	—	2	1	no	—
14	C₂×C₄⋊Q₈×C₈₃	C₂²	1	2	yes	—
15	#500×C₈₃	—	2	1	no	—
16	#500×C₈₃	—	2	1	no	—
17	#500×C₈₃	C₄	1	4	yes	—
18	C₂×#107×C₈₃	—	2	2	no	—
19	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
20	C₂×#13050574	—	2	2	no	—
21	#14581728	—	4	4	no	—
22	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
23	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
24	C₄⋊(C₈₃⋊C₈)	D₄	1	2	yes	—
25	C₄⋊(C₈₃⋊C₈)	D₄	1	2	yes	—
26	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
27	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
28	C₄×C₈₃⋊C₈	—	2	1	no	—
29	C₄×C₈₃⋊C₈	—	2	1	no	—
30	C₂²×Q₈×C₈₃	—	2	2	no	—
31	C₂²×C₈×C₈₃	D₄	1	2	yes	—
32	C₄⋊Q₈×C₈₃	—	2	2	no	—
33	C₄⋊Q₈×C₈₃	—	2	2	no	—
34	C₄⋊Q₈×C₈₃	—	2	2	no	—
35	C₄⋊Q₈×C₈₃	C₂×C₄	1	4	yes	—
36	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
38	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
39	#107×C₈₃	—	2	4	no	—
40	#107×C₈₃	—	2	4	no	—
41	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
44	#13050574	—	4	4	no	—
45	C₂×C₈₃⋊C₈	—	2	2	no	—
46	C₂×C₈₃⋊C₈	—	2	2	no	—
47	C₂×C₈₃⋊C₈	—	4	2	no	—
48	C₂×Q₈×C₈₃	—	2	4	no	—
49	C₂×Q₈×C₈₃	—	4	2	no	—
50	C₂×Q₈×C₈₃	—	8	4	no	—
51	C₂²×C₄×C₈₃	—	4	2	no	—
52	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
54	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
55	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₈₃	SD₁₆	1	4	yes	—
58	C₂×C₈×C₈₃	Q₁₆	1	4	yes	—
59	C₄⋊C₄×C₈₃	—	2	4	no	—
60	C₄⋊C₄×C₈₃	—	2	4	no	—
61	C₄⋊C₄×C₈₃	—	4	2	no	—
62	C₄⋊C₄×C₈₃	—	4	2	no	—
63	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
67	C₂³×C₈₃	#124	1	1	yes	characteristic
68	Q₈×C₈₃	—	8	4	no	—
69	Q₈×C₈₃	—	8	4	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	4	no	—
74	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
78	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
79	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
80	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
81	C₂×C₄×C₈₃	C₂×Q₁₆	1	2	yes	—
82	C₈×C₈₃	#103	1	4	yes	—
83	C₈×C₈₃	#103	1	4	yes	—
84	C₈₃⋊C₈	—	4	2	no	—
85	C₈₃⋊C₈	—	4	2	no	—
86	C₈₃⋊C₈	—	8	2	no	—
87	C₂²×C₈₃	—	2	1	no	—
88	C₂²×C₈₃	—	2	1	no	—
89	C₂²×C₈₃	#390	1	1	yes	characteristic
90	C₂²×C₈₃	#487	1	2	yes	—
91	C₄×C₈₃	—	2	1	no	—
92	C₄×C₈₃	—	2	1	no	—
93	C₄×C₈₃	—	8	4	no	—
94	C₄×C₈₃	C₂×#103	1	2	yes	—
95	C₄×C₈₃	#472	1	2	yes	—
96	C₄×C₈₃	#473	1	2	yes	—
97	C₄×C₈₃	#492	1	1	yes	characteristic
98	C₄×C₈₃	#494	1	1	yes	characteristic
99	#9771	—	83	1	no	maximal, Sylow
100	C₂×C₈₃	—	2	2	no	—
101	C₂×C₈₃	#1956	1	1	yes	characteristic
102	C₂×C₈₃	#1978	1	1	yes	characteristic
103	C₂×C₈₃	#1979	1	1	yes	characteristic
104	C₂×#500	D₈₃	1	1	yes	characteristic
105	#1848	—	83	1	no	—
106	#1549	—	83	2	no	—
107	#1548	—	83	2	no	—
108	#1450	—	83	1	no	—
109	C₈₃	#9771	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₄⋊Q₈	C₂×D₈₃	1	2	yes	—
111	#500	—	2	1	no	—
112	#500	—	2	1	no	—
113	#500	Dic₈₃	1	4	yes	—
114	#431	—	83	1	no	—
115	#431	—	83	1	no	—
116	C₂×#108	—	166	2	no	—
117	C₂×#107	—	2	2	no	—
118	#423	—	83	2	no	—
119	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
120	#333	—	166	4	no	—
121	#332	—	166	4	no	—
122	#322	—	166	1	no	—
123	C₈²	—	166	1	no	—
124	C₂²×Q₈	—	2	2	no	—
125	C₂×M₁₆	—	166	2	no	—
126	C₂²×C₈	C₂²⋊D₈₃	1	2	yes	—
127	C₄⋊Q₈	—	2	2	no	—
128	C₄⋊Q₈	—	2	2	no	—
129	C₄⋊Q₈	—	2	2	no	—
130	C₄⋊Q₈	C₂×Dic₈₃	1	4	yes	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
134	#108	—	332	4	no	—
135	#107	—	2	4	no	—
136	#107	—	2	4	no	—
137	C₄⋊C₈	—	83	2	no	—
138	C₄⋊C₈	—	83	2	no	—
139	C₄⋊C₈	—	166	2	no	—
140	C₄⋊C₈	—	166	2	no	—
141	C₂²⋊C₈	—	166	1	no	—
142	C₂²⋊C₈	—	166	1	no	—
143	C₄×C₈	—	166	1	no	—
144	C₄×C₈	—	166	1	no	—
145	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
146	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
147	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
148	C₂×Q₈	—	2	4	no	—
149	C₂×Q₈	—	4	2	no	—
150	C₂×Q₈	—	8	4	no	—
151	C₂²×C₄	—	4	2	no	—
152	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
153	C₂²×C₄	#13050589	1	1	yes	characteristic
154	C₂²×C₄	#13050589	1	1	yes	characteristic
155	M₁₆	—	332	4	no	—
156	C₂×C₈	—	332	2	no	—
157	C₂×C₈	—	166	2	no	—
158	C₂×C₈	—	166	2	no	—
159	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
160	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
161	C₂×C₈	#13050577	1	4	yes	—
162	C₂×C₈	#13050580	1	4	yes	—
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
168	C₄²	D₄×D₈₃	1	1	yes	characteristic
169	C₄²	D₄×D₈₃	1	1	yes	characteristic
170	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
171	C₂³	#14581779	1	1	yes	characteristic
172	Q₈	—	8	4	no	—
173	Q₈	—	8	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	4	no	—
177	C₂×C₄	—	8	4	no	—
178	C₂×C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
179	C₂×C₄	D₄×Dic₈₃	1	1	yes	characteristic
180	C₂×C₄	D₄×Dic₈₃	1	1	yes	characteristic
181	C₂×C₄	#14581763	1	1	yes	characteristic
182	C₂×C₄	#14581763	1	1	yes	characteristic
183	C₂×C₄	#14581788	1	1	yes	characteristic
184	C₂×C₄	C₂×#13050577	1	2	yes	—
185	C₂×C₄	C₂×#13050580	1	2	yes	—
186	C₈	—	332	2	no	—
187	C₈	—	332	2	no	—
188	C₈	—	664	2	no	—
189	C₈	#14581716	1	4	yes	—
190	C₈	#14581716	1	4	yes	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	#19127604	1	1	yes	characteristic
194	C₂²	#19128051	1	2	yes	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	8	4	no	—
198	C₄	C₂×#14581716	1	2	yes	—
199	C₄	#19127968	1	2	yes	—
200	C₄	#19127973	1	2	yes	—
201	C₄	#19128075	1	1	yes	characteristic
202	C₄	#19128080	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	#24908266	1	1	yes	characteristic
205	C₂	#24908374	1	1	yes	characteristic
206	C₂	#24908379	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24907635

C₂

yes

maximal, characteristic

#24906148

C₂

yes

maximal

#24906144

C₂

yes

maximal

#24905746

C₂

yes

maximal, characteristic

C₂×#500×C₈₃

C₂

yes

maximal, characteristic

#19127754

C₂²

yes

characteristic

#19127754

C₂²

yes

characteristic

C₂×#14581728

—

#19127725

C₂²

yes

—

#19127450

—

#19127447

—

#19127422

—

C₈×C₈₃⋊C₈

—

C₂×C₄⋊Q₈×C₈₃

C₂²

yes

—

#500×C₈₃

—

#500×C₈₃

—

#500×C₈₃

C₄

yes

—

C₂×#107×C₈₃

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×#13050574

—

#14581728

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

D₄

yes

—

C₄⋊(C₈₃⋊C₈)

D₄

yes

—

C₂²⋊(C₈₃⋊C₈)

—

C₂²⋊(C₈₃⋊C₈)

—

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

#107×C₈₃

—

#107×C₈₃

—

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

D₄

yes

characteristic

C₄×C₈×C₈₃

D₄

yes

characteristic

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂×D₄

yes

C₂²×C₄×C₈₃

yes

C₂²×C₄×C₈₃

yes

C₂×C₈×C₈₃

C₂²⋊C₄

yes

—

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

C₂×D₄

yes

characteristic

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₂³×C₈₃

#124

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄×D₄

yes

characteristic

C₂×C₄×C₈₃

C₄×D₄

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

C₂×SD₁₆

yes

—

C₂×C₄×C₈₃

C₂×Q₁₆

yes

—

C₈×C₈₃

#103

yes

—

C₈×C₈₃

#103

yes

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#390

yes

characteristic

C₂²×C₈₃

#487

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×#103

yes

—

C₄×C₈₃

#472

yes

—

C₄×C₈₃

#473

yes

—

C₄×C₈₃

yes

C₄×C₈₃

yes

—

100

C₂×C₈₃

—

101

C₂×C₈₃

#1956

yes

characteristic

102

C₂×C₈₃

#1978

yes

characteristic

103

C₂×C₈₃

#1979

yes

characteristic

104

C₂×#500

D₈₃

yes

characteristic

105

#1848

—

106

#1549

—

107

#1548

—

108

#1450

—

109

C₈₃

#9771

yes

characteristic, complemented, Sylow

110

C₂×C₄⋊Q₈

C₂×D₈₃

yes

—

111

#500

—

112

#500

—

113

#500

Dic₈₃

yes

—

114

#431

—

115

#431

—

116

C₂×#108

—

166

—

117

C₂×#107

—

118

#423

—

119

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

120

#333

—

166

—

121

#332

—

166

—

122

#322

—

166

—

123

C₈²

—

166

—

124

C₂²×Q₈

—

125

C₂×M₁₆

—

166

—

126

C₂²×C₈

C₂²⋊D₈₃

yes

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

—

129

C₄⋊Q₈

—

130

C₄⋊Q₈

C₂×Dic₈₃

yes

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄²

C₂²×D₈₃

yes

characteristic

134

#108

—

332

—

135

#107

—

136

#107

—

137

C₄⋊C₈

—

138

C₄⋊C₈

—

139

C₄⋊C₈

—

166

—

140

C₄⋊C₈

—

166

—

141

C₂²⋊C₈

—

166

—

142

C₂²⋊C₈

—

166

—

143

C₄×C₈

—

166

—

144

C₄×C₈

—

166

—

145

C₄×C₈

C₂×Dic₈₃

yes

—

146

C₄×C₈

C₂²⋊D₈₃

yes

characteristic

147

C₄×C₈

C₂²⋊D₈₃

yes

characteristic

148

C₂×Q₈

—

149

C₂×Q₈

—

150

C₂×Q₈

—

151

C₂²×C₄

—

152

C₂²×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

153

C₂²×C₄

#13050589

yes

characteristic

154

C₂²×C₄

#13050589

yes

characteristic

155

M₁₆

—

332

—

156

C₂×C₈

—

332

—

157

C₂×C₈

—

166

—

158

C₂×C₈

—

166

—

159

C₂×C₈

C₂²⋊Dic₈₃

yes

—

160

C₂×C₈

C₂²⋊Dic₈₃

yes

—

161

C₂×C₈

#13050577

yes

—

162

C₂×C₈

#13050580

yes

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄²

C₂²×Dic₈₃

yes

characteristic

168

C₄²

D₄×D₈₃

yes

characteristic

169

C₄²

D₄×D₈₃

yes

characteristic

170

C₄²

C₂×C₂²⋊D₈₃

yes

characteristic

171

C₂³

#14581779

yes

characteristic

172

Q₈

—

173

Q₈

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

C₂×C₂²⋊Dic₈₃

yes

characteristic

179

C₂×C₄

D₄×Dic₈₃

yes

characteristic

180

C₂×C₄

D₄×Dic₈₃

yes

characteristic

181

C₂×C₄

#14581763

yes

characteristic

182

C₂×C₄

#14581763

yes

characteristic

183

C₂×C₄

#14581788

yes

characteristic

184

C₂×C₄

C₂×#13050577

yes

—

185

C₂×C₄

C₂×#13050580

yes

—

186

C₈

—

332

—

187

C₈

—

332

—

188

C₈

—

664

—

189

C₈

#14581716

yes

—

190

C₈

#14581716

yes

—

191

C₂²

—

192

C₂²

—

193

C₂²

#19127604

yes

characteristic

194

C₂²

#19128051

yes

—

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

C₂×#14581716

yes

—

199

C₄

#19127968

yes

—

200

C₄

#19127973

yes

—

201

C₄

#19128075

yes

characteristic

202

C₄

#19128080

yes

characteristic

203

C₂

—

204

C₂

#24908266

yes

characteristic

205

C₂

#24908374

yes

characteristic

206

C₂

#24908379

yes

characteristic

	subgroup	G/H
center	C₂²	#19127604
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₂×#500×C₈₃	C₂
socle	C₂²×C₈₃	#390
prime radical	C₂³×C₈₃	#124
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₂×#500	D₈₃
hypocenter	C₈₃	#9771
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

subgroup

G/H

center

C₂²

#19127604

commutator

C₄²×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₂×#500×C₈₃

C₂

C₂²×C₈₃

C₂³×C₈₃

C₂×C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

N	H ≅ G/N	type
109. C₈₃	#9771	semidirect

H ≅ G/N

type

109. C₈₃

#9771

semidirect

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

subgroup	index
#24907635	2
#24906148	2
#24906144	2
#24905746	2
C₂×#500×C₈₃	2
#9771	83

subgroup

index

C₂×#500×C₈₃

Subgroups — #35303864

Conjugacy classes of subgroups

Complements

Maximal subgroups