Group #24990007 — subgroups

Subgroups — #24990007

Order 11008.

526 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 131 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	#19134357	C₂	1	2	yes	maximal
2	#19132476	C₂	1	4	yes	maximal, complemented
3	C₄₃×#2285	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
5	C₂×#13063236	C₂²	1	4	yes	—
6	#14582654	—	2	2	no	—
7	C₂×#117×C₄₃	C₄	1	2	yes	—
8	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
9	C₄₃×#384	C₄	1	1	yes	characteristic
10	C₄₃×#384	C₄	1	1	yes	characteristic
11	C₄₃×#382	C₂²	1	2	yes	—
12	C₄₃×#380	—	2	2	no	—
13	C₄×C₄⋊C₄×C₄₃	—	2	2	no	—
14	C₂²×C₄₃⋊C₈	—	2	4	no	—
15	C₂²⋊(C₄₃⋊C₈)	—	2	8	no	—
16	#13063236	—	2	8	no	—
17	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₄₃	D₄	1	2	yes	—
19	#117×C₄₃	—	4	4	no	—
20	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
21	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
22	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
23	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
24	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
25	C₂×C₄²×C₄₃	—	2	4	no	—
26	C₂×C₄²×C₄₃	—	2	2	no	—
27	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₄₃	D₄	1	2	yes	—
29	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
30	#95×C₄₃	—	2	4	no	—
31	C₂⁴×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₃	—	2	2	no	—
33	C₂²×C₄×C₄₃	—	2	2	no	—
34	C₂²×C₄×C₄₃	—	2	4	no	—
35	C₂²×C₄×C₄₃	—	2	4	no	—
36	C₂²×C₄×C₄₃	—	2	2	no	—
37	C₂²×C₄×C₄₃	—	2	2	no	—
38	C₂²×C₄×C₄₃	—	2	2	no	—
39	C₂²×C₄×C₄₃	—	4	2	no	—
40	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
44	C₄⋊C₄×C₄₃	—	4	2	no	—
45	C₄⋊C₄×C₄₃	—	4	2	no	—
46	C₄⋊C₄×C₄₃	—	4	4	no	—
47	C₄⋊C₄×C₄₃	—	4	4	no	—
48	C₂²⋊C₄×C₄₃	—	4	4	no	—
49	C₄²×C₄₃	—	2	2	no	—
50	C₄²×C₄₃	—	2	4	no	—
51	C₄²×C₄₃	—	4	4	no	—
52	C₄²×C₄₃	#41	1	4	yes	—
53	C₂×C₄₃⋊C₈	—	2	8	no	—
54	C₂×C₄₃⋊C₈	—	4	4	no	—
55	C₂×C₄₃⋊C₈	—	4	4	no	—
56	C₂³×C₄₃	—	2	1	no	—
57	C₂³×C₄₃	—	2	1	no	—
58	C₂³×C₄₃	—	2	2	no	—
59	C₂³×C₄₃	—	2	2	no	—
60	C₂³×C₄₃	#101	1	2	yes	—
61	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₄₃	—	2	8	no	—
63	C₂×C₄×C₄₃	—	2	1	no	—
64	C₂×C₄×C₄₃	—	2	2	no	—
65	C₂×C₄×C₄₃	—	2	1	no	—
66	C₂×C₄×C₄₃	—	2	1	no	—
67	C₂×C₄×C₄₃	—	2	1	no	—
68	C₂×C₄×C₄₃	—	2	2	no	—
69	C₂×C₄×C₄₃	—	2	2	no	—
70	C₂×C₄×C₄₃	—	2	2	no	—
71	C₂×C₄×C₄₃	—	4	8	no	—
72	C₂×C₄×C₄₃	—	4	2	no	—
73	C₂×C₄×C₄₃	—	4	2	no	—
74	C₂×C₄×C₄₃	—	4	4	no	—
75	C₂×C₄×C₄₃	—	4	4	no	—
76	C₂×C₄×C₄₃	#100	1	2	yes	—
77	C₂×C₄×C₄₃	#104	1	4	yes	—
78	C₂×C₄×C₄₃	#117	1	2	yes	—
79	C₂×C₄×C₄₃	#123	1	4	yes	—
80	C₄₃⋊C₈	—	4	8	no	—
81	#8721	—	43	1	no	maximal, Sylow
82	C₂²×C₄₃	—	2	2	no	—
83	C₂²×C₄₃	—	2	4	no	—
84	C₂²×C₄₃	—	4	1	no	—
85	C₂²×C₄₃	—	4	2	no	—
86	C₂²×C₄₃	—	4	1	no	—
87	C₂²×C₄₃	#381	1	1	yes	characteristic
88	C₂²×C₄₃	C₂×#100	1	1	yes	characteristic
89	C₂²×C₄₃	C₂×#101	1	1	yes	characteristic
90	C₂²×C₄₃	C₂×#104	1	2	yes	—
91	C₂²×C₄₃	#421	1	2	yes	—
92	C₄×C₄₃	—	2	2	no	—
93	C₄×C₄₃	—	2	2	no	—
94	C₄×C₄₃	—	2	4	no	—
95	C₄×C₄₃	—	2	4	no	—
96	C₄×C₄₃	—	8	2	no	—
97	C₄×C₄₃	—	8	2	no	—
98	#2285	D₄₃	1	1	yes	characteristic
99	#1749	—	43	2	no	—
100	#1295	—	43	4	no	—
101	C₂×C₄₃	—	4	2	no	—
102	C₂×C₄₃	#1525	1	1	yes	characteristic
103	C₂×C₄₃	#1531	1	2	yes	—
104	C₂×C₄₃	#1536	1	1	yes	characteristic
105	C₂×C₄₃	#1821	1	1	yes	characteristic
106	C₂×C₄₃	#1826	1	2	yes	—
107	C₂×#117	Dic₄₃	1	2	yes	—
108	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
109	C₂×C₂²⋊C₈	—	86	2	no	—
110	C₂×#97	—	43	4	no	—
111	#384	Dic₄₃	1	1	yes	characteristic
112	#384	Dic₄₃	1	1	yes	characteristic
113	#382	C₂×D₄₃	1	2	yes	—
114	#380	—	2	2	no	—
115	C₄×C₄⋊C₄	—	2	2	no	—
116	#336	—	86	2	no	—
117	C₄₃	#8721	1	1	yes	characteristic, complemented, Sylow
118	C₂³×C₄	C₂×Dic₄₃	1	1	yes	characteristic
119	C₂³×C₄	C₂²⋊D₄₃	1	2	yes	—
120	C₂²×C₈	—	86	4	no	—
121	#117	—	4	4	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
125	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
126	C₂×C₂²⋊C₄	—	2	2	no	—
127	C₂×C₄²	—	2	4	no	—
128	C₂×C₄²	—	2	2	no	—
129	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
130	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
131	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
132	C₂²⋊C₈	—	86	8	no	—
133	#97	—	86	8	no	—
134	#95	—	2	4	no	—
135	C₂⁴	C₂²⋊Dic₄₃	1	1	yes	characteristic
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	—	2	2	no	—
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	—	2	4	no	—
140	C₂²×C₄	—	2	4	no	—
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
145	C₂²×C₄	C₂²⋊Dic₄₃	1	2	yes	—
146	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
148	C₂×C₈	—	86	8	no	—
149	C₂×C₈	—	172	4	no	—
150	C₂×C₈	—	172	4	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	4	no	—
154	C₄⋊C₄	—	4	4	no	—
155	C₂²⋊C₄	—	4	4	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	2	4	no	—
158	C₄²	—	4	4	no	—
159	C₄²	#10621898	1	4	yes	—
160	C₂³	—	2	1	no	—
161	C₂³	—	2	2	no	—
162	C₂³	—	2	2	no	—
163	C₂³	—	2	1	no	—
164	C₂³	#13063244	1	2	yes	—
165	C₂³	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
166	C₂×C₄	—	2	8	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	4	8	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	#13063242	1	2	yes	—
181	C₂×C₄	#13063252	1	4	yes	—
182	C₂×C₄	#13063284	1	2	yes	—
183	C₂×C₄	#13063308	1	4	yes	—
184	C₈	—	172	8	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	2	2	no	—
187	C₂²	—	4	1	no	—
188	C₂²	—	4	1	no	—
189	C₂²	—	4	2	no	—
190	C₂²	#14582762	1	1	yes	characteristic
191	C₂²	C₂×#13063242	1	1	yes	characteristic
192	C₂²	C₂×#13063244	1	1	yes	characteristic
193	C₂²	C₂×#13063252	1	2	yes	—
194	C₂²	#14582916	1	2	yes	—
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	4	no	—
198	C₄	—	2	4	no	—
199	C₄	—	8	2	no	—
200	C₄	—	8	2	no	—
201	C₂	—	4	2	no	—
202	C₂	#19133187	1	1	yes	characteristic
203	C₂	#19133210	1	2	yes	—
204	C₂	#19133230	1	1	yes	characteristic
205	C₂	#19134675	1	1	yes	characteristic
206	C₂	#19134694	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19134357

C₂

yes

maximal

#19132476

C₂

yes

maximal, complemented

C₄₃×#2285

C₂

yes

maximal, characteristic

C₂×C₂²⋊(C₄₃⋊C₈)

—

C₂×#13063236

C₂²

yes

—

#14582654

—

C₂×#117×C₄₃

C₄

yes

—

C₂²×C₄²×C₄₃

C₂²

yes

characteristic

C₄₃×#384

C₄

yes

characteristic

C₄₃×#384

C₄

yes

characteristic

C₄₃×#382

C₂²

yes

—

C₄₃×#380

—

C₄×C₄⋊C₄×C₄₃

—

C₂²×C₄₃⋊C₈

—

C₂²⋊(C₄₃⋊C₈)

—

#13063236

—

C₂³×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂³×C₄×C₄₃

D₄

yes

—

#117×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₄₃

C₂×C₄

yes

—

C₂×C₂²⋊C₄×C₄₃

—

C₂×C₄²×C₄₃

—

C₂×C₄²×C₄₃

—

C₂×C₄²×C₄₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

#95×C₄₃

—

C₂⁴×C₄₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

#41

yes

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

#101

yes

—

C₂³×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#100

yes

—

C₂×C₄×C₄₃

#104

yes

—

C₂×C₄×C₄₃

#117

yes

—

C₂×C₄×C₄₃

#123

yes

—

C₄₃⋊C₈

—

#8721

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

#421

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

#2285

D₄₃

yes

characteristic

#1749

—

100

#1295

—

101

C₂×C₄₃

—

102

C₂×C₄₃

#1525

yes

characteristic

103

C₂×C₄₃

#1531

yes

—

104

C₂×C₄₃

#1536

yes

characteristic

105

C₂×C₄₃

#1821

yes

characteristic

106

C₂×C₄₃

#1826

yes

—

107

C₂×#117

Dic₄₃

yes

—

108

C₂²×C₄²

C₂×D₄₃

yes

characteristic

109

C₂×C₂²⋊C₈

—

110

C₂×#97

—

111

#384

Dic₄₃

yes

characteristic

112

#384

Dic₄₃

yes

characteristic

113

#382

C₂×D₄₃

yes

—

114

#380

—

115

C₄×C₄⋊C₄

—

116

#336

—

117

C₄₃

#8721

yes

characteristic, complemented, Sylow

118

C₂³×C₄

C₂×Dic₄₃

yes

characteristic

119

C₂³×C₄

C₂²⋊D₄₃

yes

—

120

C₂²×C₈

—

121

#117

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

C₂×Dic₄₃

yes

—

125

C₂×C₄⋊C₄

C₂×Dic₄₃

yes

—

126

C₂×C₂²⋊C₄

—

127

C₂×C₄²

—

128

C₂×C₄²

—

129

C₂×C₄²

C₂×Dic₄₃

yes

characteristic

130

C₂×C₄²

C₂²⋊D₄₃

yes

—

131

C₂×C₄²

C₂²×D₄₃

yes

characteristic

132

C₂²⋊C₈

—

133

#97

—

134

#95

—

135

C₂⁴

C₂²⋊Dic₄₃

yes

characteristic

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

C₂²⋊Dic₄₃

yes

characteristic

145

C₂²×C₄

C₂²⋊Dic₄₃

yes

—

146

C₂²×C₄

C₂²×Dic₄₃

yes

characteristic

147

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

—

148

C₂×C₈

—

149

C₂×C₈

—

172

—

150

C₂×C₈

—

172

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₂²⋊C₄

—

156

C₄²

—

157

C₄²

—

158

C₄²

—

159

C₄²

#10621898

yes

—

160

C₂³

—

161

C₂³

—

162

C₂³

—

163

C₂³

—

164

C₂³

#13063244

yes

—

165

C₂³

C₂×C₂²⋊Dic₄₃

yes

characteristic

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

#13063242

yes

—

181

C₂×C₄

#13063252

yes

—

182

C₂×C₄

#13063284

yes

—

183

C₂×C₄

#13063308

yes

—

184

C₈

—

172

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

#14582762

yes

characteristic

191

C₂²

C₂×#13063242

yes

characteristic

192

C₂²

C₂×#13063244

yes

characteristic

193

C₂²

C₂×#13063252

yes

—

194

C₂²

#14582916

yes

—

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

201

C₂

—

202

C₂

#19133187

yes

characteristic

203

C₂

#19133210

yes

—

204

C₂

#19133230

yes

characteristic

205

C₂

#19134675

yes

characteristic

206

C₂

#19134694

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₃
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#2285	C₂
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#2285	D₄₃
hypocenter	C₄₃	#8721
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2285

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₄₃

commutator

C₂²×C₄×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#2285

C₂

socle

C₂³×C₄₃

C₂×C₂²⋊C₄

prime radical

C₂⁴×C₄₃

C₂²⋊C₄

cosocle

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2285

N	H ≅ G/N	type
2. #19132476	C₂	semidirect
117. C₄₃	#8721	semidirect

H ≅ G/N

type

2. #19132476

C₂

semidirect

117. C₄₃

#8721

semidirect

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

subgroup	index
#19134357	2
#19132476	2
C₄₃×#2285	2
#8721	43

subgroup

index

#19134357

#19132476

C₄₃×#2285

#8721

Subgroups — #24990007

Conjugacy classes of subgroups

Complements

Maximal subgroups