Group #35303490 — subgroups

Subgroups — #35303490

Order 21248.

538 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 175 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	#24907580	C₂	1	1	yes	maximal, characteristic
2	#24907316	C₂	1	1	yes	maximal, characteristic
3	C₂×#19127447	C₂	1	4	yes	maximal
4	C₈₃×#2868	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
6	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
7	C₂×#14581703	—	2	1	no	—
8	#19127456	—	2	1	no	—
9	#19127456	—	2	2	no	—
10	#19127447	—	2	8	no	—
11	C₂×C₄⋊Q₈×C₈₃	—	2	2	no	—
12	C₂×C₄⋊Q₈×C₈₃	C₂²	1	2	yes	—
13	#384×C₈₃	—	2	1	no	—
14	#384×C₈₃	—	2	1	no	—
15	#384×C₈₃	C₄	1	4	yes	—
16	C₄³×C₈₃	C₂²	1	1	yes	characteristic
17	C₂²×C₈₃⋊C₈	—	2	2	no	—
18	C₂²×C₈₃⋊C₈	—	2	2	no	—
19	C₄⋊(C₈₃⋊C₈)	—	2	4	no	—
20	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
21	C₄⋊(C₈₃⋊C₈)	D₄	1	4	yes	—
22	#14581703	—	4	2	no	—
23	C₂²×Q₈×C₈₃	—	2	2	no	—
24	C₄⋊Q₈×C₈₃	—	2	4	no	—
25	C₄⋊Q₈×C₈₃	—	2	4	no	—
26	C₄⋊Q₈×C₈₃	—	4	8	no	—
27	C₂×C₄⋊C₄×C₈₃	—	2	4	no	—
28	C₂×C₄⋊C₄×C₈₃	—	2	4	no	—
29	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₈₃	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₈₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₈₃	D₄	1	2	yes	—
33	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
36	C₂×C₈₃⋊C₈	—	2	4	no	—
37	C₂×C₈₃⋊C₈	—	4	2	no	—
38	C₂×C₈₃⋊C₈	—	4	2	no	—
39	C₂×C₈₃⋊C₈	—	4	2	no	—
40	C₂×C₈₃⋊C₈	—	4	4	no	—
41	C₂×Q₈×C₈₃	—	4	2	no	—
42	C₂×Q₈×C₈₃	—	4	2	no	—
43	C₂×Q₈×C₈₃	—	4	8	no	—
44	C₂²×C₄×C₈₃	—	2	4	no	—
45	C₂²×C₄×C₈₃	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₈₃	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
50	C₄⋊C₄×C₈₃	—	2	8	no	—
51	C₄⋊C₄×C₈₃	—	4	8	no	—
52	C₄⋊C₄×C₈₃	—	4	8	no	—
53	C₄⋊C₄×C₈₃	—	4	4	no	—
54	C₄²×C₈₃	—	2	4	no	—
55	C₄²×C₈₃	—	2	4	no	—
56	C₄²×C₈₃	—	2	2	no	—
57	C₄²×C₈₃	SD₁₆	1	2	yes	—
58	C₄²×C₈₃	Q₁₆	1	2	yes	—
59	C₄²×C₈₃	C₂×D₄	1	2	yes	—
60	C₄²×C₈₃	#41	1	2	yes	—
61	C₂³×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	Q₈×C₈₃	—	8	4	no	—
63	Q₈×C₈₃	—	8	4	no	—
64	C₂×C₄×C₈₃	—	2	2	no	—
65	C₂×C₄×C₈₃	—	2	2	no	—
66	C₂×C₄×C₈₃	—	2	2	no	—
67	C₂×C₄×C₈₃	—	2	2	no	—
68	C₂×C₄×C₈₃	—	2	2	no	—
69	C₂×C₄×C₈₃	—	4	4	no	—
70	C₂×C₄×C₈₃	—	4	8	no	—
71	C₂×C₄×C₈₃	#101	1	4	yes	—
72	C₂×C₄×C₈₃	#103	1	4	yes	—
73	C₂×C₄×C₈₃	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₈₃	#121	1	2	yes	—
75	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
76	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
77	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	4	yes	—
80	C₂×C₄×C₈₃	#137	1	2	yes	—
81	C₈₃⋊C₈	—	4	4	no	—
82	C₈₃⋊C₈	—	8	2	no	—
83	C₂²×C₈₃	#390	1	1	yes	characteristic
84	C₂²×C₈₃	C₂×#101	1	2	yes	—
85	C₂²×C₈₃	C₂×#103	1	1	yes	characteristic
86	C₂²×C₈₃	#418	1	2	yes	—
87	C₂²×C₈₃	#419	1	1	yes	characteristic
88	C₄×C₈₃	—	2	4	no	—
89	C₄×C₈₃	—	2	4	no	—
90	C₄×C₈₃	—	2	2	no	—
91	C₄×C₈₃	—	8	4	no	—
92	C₄×C₈₃	#461	1	2	yes	—
93	C₄×C₈₃	#462	1	2	yes	—
94	C₄×C₈₃	#490	1	2	yes	—
95	C₄×C₈₃	#496	1	2	yes	—
96	#9703	—	83	1	no	maximal, Sylow
97	C₂×C₈₃	#1549	1	2	yes	—
98	C₂×C₈₃	#1556	1	2	yes	—
99	C₂×C₈₃	#1961	1	1	yes	characteristic
100	C₂×C₈₃	#1966	1	1	yes	characteristic
101	C₂×C₈₃	#1981	1	1	yes	characteristic
102	#2868	D₈₃	1	1	yes	characteristic
103	#1831	—	83	1	no	—
104	#1767	—	83	1	no	—
105	C₂×#332	—	83	4	no	—
106	C₈₃	#9703	1	1	yes	characteristic, complemented, Sylow
107	C₂×C₄⋊Q₈	—	2	2	no	—
108	C₂×C₄⋊Q₈	C₂×D₈₃	1	2	yes	—
109	C₂×C₄⋊C₈	—	83	2	no	—
110	C₂×C₄⋊C₈	—	83	2	no	—
111	C₂×#97	—	166	1	no	—
112	#384	—	2	1	no	—
113	#384	—	2	1	no	—
114	#384	Dic₈₃	1	4	yes	—
115	C₄³	C₂×D₈₃	1	1	yes	characteristic
116	#336	—	166	2	no	—
117	#336	—	166	1	no	—
118	#332	—	166	8	no	—
119	C₂²×Q₈	—	2	2	no	—
120	C₂²×C₈	—	166	2	no	—
121	C₂²×C₈	—	166	2	no	—
122	C₄⋊Q₈	—	2	4	no	—
123	C₄⋊Q₈	—	2	4	no	—
124	C₄⋊Q₈	—	4	8	no	—
125	C₂×C₄⋊C₄	—	2	4	no	—
126	C₂×C₄⋊C₄	—	2	4	no	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	C₂×Dic₈₃	1	4	yes	—
129	C₂×C₄²	C₂×Dic₈₃	1	2	yes	—
130	C₂×C₄²	C₂²⋊D₈₃	1	2	yes	—
131	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
132	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
133	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
134	C₄⋊C₈	—	83	4	no	—
135	C₄⋊C₈	—	166	4	no	—
136	C₄⋊C₈	—	166	2	no	—
137	#97	—	332	2	no	—
138	C₂×Q₈	—	4	2	no	—
139	C₂×Q₈	—	4	2	no	—
140	C₂×Q₈	—	4	8	no	—
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	C₂²⋊Dic₈₃	1	2	yes	—
143	C₂²×C₄	C₂²⋊Dic₈₃	1	2	yes	—
144	C₂²×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
147	C₂×C₈	—	332	2	no	—
148	C₂×C₈	—	332	2	no	—
149	C₂×C₈	—	332	2	no	—
150	C₂×C₈	—	332	4	no	—
151	C₂×C₈	—	166	4	no	—
152	C₄⋊C₄	—	2	8	no	—
153	C₄⋊C₄	—	4	8	no	—
154	C₄⋊C₄	—	4	8	no	—
155	C₄⋊C₄	—	4	4	no	—
156	C₄²	—	2	4	no	—
157	C₄²	—	2	2	no	—
158	C₄²	—	2	4	no	—
159	C₄²	#13050577	1	2	yes	—
160	C₄²	#13050580	1	2	yes	—
161	C₄²	D₄×D₈₃	1	2	yes	—
162	C₄²	#13050589	1	2	yes	—
163	C₂³	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
164	Q₈	—	8	4	no	—
165	Q₈	—	8	4	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	4	8	no	—
173	C₂×C₄	#14581711	1	4	yes	—
174	C₂×C₄	#14581716	1	4	yes	—
175	C₂×C₄	D₄×Dic₈₃	1	2	yes	—
176	C₂×C₄	#14581763	1	2	yes	—
177	C₂×C₄	#14581779	1	1	yes	characteristic
178	C₂×C₄	#14581788	1	1	yes	characteristic
179	C₂×C₄	C₂×#13050577	1	1	yes	characteristic
180	C₂×C₄	C₂×#13050580	1	1	yes	characteristic
181	C₂×C₄	#14581818	1	4	yes	—
182	C₂×C₄	#14581823	1	2	yes	—
183	C₈	—	332	4	no	—
184	C₈	—	664	2	no	—
185	C₂²	#19127604	1	1	yes	characteristic
186	C₂²	C₂×#14581711	1	2	yes	—
187	C₂²	C₂×#14581716	1	1	yes	characteristic
188	C₂²	#19127706	1	2	yes	—
189	C₂²	#19127710	1	1	yes	characteristic
190	C₄	—	2	2	no	—
191	C₄	—	2	4	no	—
192	C₄	—	2	4	no	—
193	C₄	—	8	4	no	—
194	C₄	#19127903	1	2	yes	—
195	C₄	#19127908	1	2	yes	—
196	C₄	#19128065	1	2	yes	—
197	C₄	#19128088	1	2	yes	—
198	C₂	#24906148	1	2	yes	—
199	C₂	#24906180	1	2	yes	—
200	C₂	#24908288	1	1	yes	characteristic
201	C₂	#24908308	1	1	yes	characteristic
202	C₂	#24908387	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24907580

C₂

yes

maximal, characteristic

#24907316

C₂

yes

maximal, characteristic

C₂×#19127447

C₂

yes

maximal

C₈₃×#2868

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

C₂×C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Q₈×C₈₃

—

C₂×C₄⋊Q₈×C₈₃

C₂²

yes

—

#384×C₈₃

—

#384×C₈₃

—

#384×C₈₃

C₄

yes

—

C₄³×C₈₃

C₂²

yes

characteristic

C₂²×C₈₃⋊C₈

—

C₂²×C₈₃⋊C₈

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

D₄

yes

—

#14581703

—

C₂²×Q₈×C₈₃

—

C₄⋊Q₈×C₈₃

—

C₄⋊Q₈×C₈₃

—

C₄⋊Q₈×C₈₃

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄⋊C₄×C₈₃

C₂×C₄

yes

—

C₂×C₄²×C₈₃

C₂×C₄

yes

—

C₂×C₄²×C₈₃

D₄

yes

—

C₂×C₄²×C₈₃

D₄

yes

characteristic

C₂×C₄²×C₈₃

D₄

yes

characteristic

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂²×C₄×C₈₃

—

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

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

SD₁₆

yes

—

C₄²×C₈₃

Q₁₆

yes

—

C₄²×C₈₃

C₂×D₄

yes

—

C₄²×C₈₃

#41

yes

—

C₂³×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

#101

yes

—

C₂×C₄×C₈₃

#103

yes

—

C₂×C₄×C₈₃

C₄×D₄

yes

—

C₂×C₄×C₈₃

#121

yes

—

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

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

#137

yes

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

#390

yes

characteristic

C₂²×C₈₃

C₂×#101

yes

—

C₂²×C₈₃

C₂×#103

yes

characteristic

C₂²×C₈₃

#418

yes

—

C₂²×C₈₃

#419

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#461

yes

—

C₄×C₈₃

#462

yes

—

C₄×C₈₃

#490

yes

—

C₄×C₈₃

#496

yes

—

#9703

—

maximal, Sylow

C₂×C₈₃

#1549

yes

—

C₂×C₈₃

#1556

yes

—

C₂×C₈₃

#1961

yes

characteristic

100

C₂×C₈₃

#1966

yes

characteristic

101

C₂×C₈₃

#1981

yes

characteristic

102

#2868

D₈₃

yes

characteristic

103

#1831

—

104

#1767

—

105

C₂×#332

—

106

C₈₃

#9703

yes

characteristic, complemented, Sylow

107

C₂×C₄⋊Q₈

—

108

C₂×C₄⋊Q₈

C₂×D₈₃

yes

—

109

C₂×C₄⋊C₈

—

110

C₂×C₄⋊C₈

—

111

C₂×#97

—

166

—

112

#384

—

113

#384

—

114

#384

Dic₈₃

yes

—

115

C₄³

C₂×D₈₃

yes

characteristic

116

#336

—

166

—

117

#336

—

166

—

118

#332

—

166

—

119

C₂²×Q₈

—

120

C₂²×C₈

—

166

—

121

C₂²×C₈

—

166

—

122

C₄⋊Q₈

—

123

C₄⋊Q₈

—

124

C₄⋊Q₈

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

C₂×Dic₈₃

yes

—

129

C₂×C₄²

C₂×Dic₈₃

yes

—

130

C₂×C₄²

C₂²⋊D₈₃

yes

—

131

C₂×C₄²

C₂²⋊D₈₃

yes

characteristic

132

C₂×C₄²

C₂²⋊D₈₃

yes

characteristic

133

C₂×C₄²

C₂²×D₈₃

yes

characteristic

134

C₄⋊C₈

—

135

C₄⋊C₈

—

166

—

136

C₄⋊C₈

—

166

—

137

#97

—

332

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂²×C₄

—

142

C₂²×C₄

C₂²⋊Dic₈₃

yes

—

143

C₂²×C₄

C₂²⋊Dic₈₃

yes

—

144

C₂²×C₄

C₂²×Dic₈₃

yes

characteristic

145

C₂²×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

146

C₂²×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

147

C₂×C₈

—

332

—

148

C₂×C₈

—

332

—

149

C₂×C₈

—

332

—

150

C₂×C₈

—

332

—

151

C₂×C₈

—

166

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄²

—

157

C₄²

—

158

C₄²

—

159

C₄²

#13050577

yes

—

160

C₄²

#13050580

yes

—

161

C₄²

D₄×D₈₃

yes

—

162

C₄²

#13050589

yes

—

163

C₂³

C₂×C₂²⋊Dic₈₃

yes

characteristic

164

Q₈

—

165

Q₈

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

#14581711

yes

—

174

C₂×C₄

#14581716

yes

—

175

C₂×C₄

D₄×Dic₈₃

yes

—

176

C₂×C₄

#14581763

yes

—

177

C₂×C₄

#14581779

yes

characteristic

178

C₂×C₄

#14581788

yes

characteristic

179

C₂×C₄

C₂×#13050577

yes

characteristic

180

C₂×C₄

C₂×#13050580

yes

characteristic

181

C₂×C₄

#14581818

yes

—

182

C₂×C₄

#14581823

yes

—

183

C₈

—

332

—

184

C₈

—

664

—

185

C₂²

#19127604

yes

characteristic

186

C₂²

C₂×#14581711

yes

—

187

C₂²

C₂×#14581716

yes

characteristic

188

C₂²

#19127706

yes

—

189

C₂²

#19127710

yes

characteristic

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

#19127903

yes

—

195

C₄

#19127908

yes

—

196

C₄

#19128065

yes

—

197

C₄

#19128088

yes

—

198

C₂

#24906148

yes

—

199

C₂

#24906180

yes

—

200

C₂

#24908288

yes

characteristic

201

C₂

#24908308

yes

characteristic

202

C₂

#24908387

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₈₃
commutator	C₂²×C₄×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#2868	C₂
socle, prime radical	C₂³×C₈₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#2868	D₈₃
hypocenter	C₈₃	#9703
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₈₃

commutator

C₂²×C₄×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₈₃×#2868

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 ◁ #2868

N	H ≅ G/N	type
106. C₈₃	#9703	semidirect

H ≅ G/N

type

106. C₈₃

#9703

semidirect

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

subgroup	index
#24907580	2
#24907316	2
C₂×#19127447	2
C₈₃×#2868	2
#9703	83

subgroup

index

C₈₃×#2868

Subgroups — #35303490

Conjugacy classes of subgroups

Complements

Maximal subgroups