Group #24989554 — subgroups

Subgroups — #24989554

Order 11008.

362 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 101 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×#14583023	C₂	1	1	yes	maximal, characteristic
2	#19133356	C₂	1	2	yes	maximal
3	#19132938	C₂	1	2	yes	maximal
4	#19132465	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#1290	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
7	#14583023	—	2	2	no	—
8	#14583023	—	2	2	no	—
9	#14582956	C₂²	1	2	yes	—
10	C₂×Dic₄₃⋊C₈	C₄	1	2	yes	—
11	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₄₃	C₄	1	2	yes	—
13	#14582651	—	2	2	no	—
14	#14582649	—	2	2	no	—
15	#14582624	—	2	4	no	—
16	C₄₃×#432	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
18	C₂²×Dic₈₆	Q₈	1	1	yes	characteristic
19	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₈₆	—	2	2	no	—
22	C₄×Dic₈₆	—	2	2	no	—
23	C₂×C₄⋊Dic₄₃	Q₈	1	1	yes	characteristic
24	C₂×Dic₄₃⋊C₄	D₄	1	2	yes	—
25	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
26	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
27	Dic₄₃⋊C₈	—	2	4	no	—
28	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
29	#13063236	—	2	2	no	—
30	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
31	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
32	C₈×Dic₄₃	—	2	2	no	—
33	C₈×Dic₄₃	—	2	2	no	—
34	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₄₃	—	2	2	no	—
37	C₄⋊C₈×C₄₃	—	2	2	no	—
38	C₂²⋊C₈×C₄₃	—	2	2	no	—
39	#97×C₄₃	—	2	2	no	—
40	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
43	C₂²×Dic₄₃	C₄⋊C₄	1	2	yes	—
44	C₂²×Dic₄₃	C₄⋊C₄	1	2	yes	—
45	C₂×Dic₈₆	—	2	2	no	—
46	C₂×Dic₈₆	—	2	4	no	—
47	C₂×C₈×C₄₃	—	2	2	no	—
48	C₂×C₈×C₄₃	—	4	2	no	—
49	C₂×C₈×C₄₃	—	4	2	no	—
50	C₄²×C₄₃	C₂×D₄	1	2	yes	—
51	C₄²×C₄₃	#41	1	2	yes	—
52	C₄⋊Dic₄₃	—	2	2	no	—
53	Dic₄₃⋊C₄	—	2	4	no	—
54	C₄×Dic₄₃	—	2	2	no	—
55	C₄×Dic₄₃	—	2	2	no	—
56	C₂×C₄₃⋊C₈	—	2	2	no	—
57	C₂×C₄₃⋊C₈	—	4	1	no	—
58	C₂×C₄₃⋊C₈	—	4	2	no	—
59	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
61	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
62	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₄₃	—	2	2	no	—
64	C₂×C₄×C₄₃	—	2	1	no	—
65	C₂×C₄×C₄₃	—	2	1	no	—
66	C₂×C₄×C₄₃	—	4	1	no	—
67	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
69	C₂×C₄×C₄₃	#121	1	2	yes	—
70	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
71	C₂×Dic₄₃	—	2	2	no	—
72	C₂×Dic₄₃	—	2	2	no	—
73	C₂×Dic₄₃	—	2	2	no	—
74	C₂×Dic₄₃	—	4	2	no	—
75	C₂×Dic₄₃	#108	1	4	yes	—
76	Dic₈₆	—	2	8	no	—
77	C₈×C₄₃	—	4	2	no	—
78	C₈×C₄₃	—	8	2	no	—
79	C₄₃⋊C₈	—	2	1	no	—
80	C₄₃⋊C₈	—	2	2	no	—
81	C₄₃⋊C₈	—	8	2	no	—
82	C₄₃⋊C₈	#101	1	2	yes	—
83	#8623	—	43	1	no	maximal, Sylow
84	C₂²×C₄₃	—	2	1	no	—
85	C₂²×C₄₃	—	2	1	no	—
86	C₂²×C₄₃	#380	1	1	yes	characteristic
87	C₂²×C₄₃	C₂×#108	1	2	yes	—
88	C₄×C₄₃	—	2	1	no	—
89	C₄×C₄₃	—	2	2	no	—
90	C₄×C₄₃	—	2	2	no	—
91	C₄×C₄₃	C₂×#101	1	1	yes	characteristic
92	C₄×C₄₃	#413	1	1	yes	characteristic
93	Dic₄₃	—	2	4	no	—
94	Dic₄₃	—	4	2	no	—
95	C₂×C₈×Q₈	—	43	1	no	—
96	#1568	—	43	2	no	—
97	#1467	—	43	2	no	—
98	#1290	—	43	1	no	—
99	#1290	D₄₃	1	1	yes	characteristic
100	C₂×C₄₃	—	2	2	no	—
101	C₂×C₄₃	#1801	1	1	yes	characteristic
102	C₂×C₄₃	#1810	1	1	yes	characteristic
103	C₂×C₄₃	#1812	1	1	yes	characteristic
104	C₂×C₄×Q₈	—	43	1	no	—
105	C₈×Q₈	—	86	2	no	—
106	C₈×Q₈	—	86	2	no	—
107	#432	—	43	2	no	—
108	#432	C₂×D₄₃	1	2	yes	—
109	C₂×C₄⋊C₈	—	43	2	no	—
110	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
111	C₂×C₄×C₈	—	43	1	no	—
112	C₂×C₄×C₈	—	43	2	no	—
113	#335	—	86	2	no	—
114	#334	—	86	2	no	—
115	#324	—	86	4	no	—
116	C₄₃	#8623	1	1	yes	characteristic, complemented, Sylow
117	C₂²×Q₈	—	43	1	no	—
118	C₂²×C₈	—	43	1	no	—
119	C₂²×C₈	—	43	1	no	—
120	C₂²×C₈	C₄×D₄₃	1	2	yes	—
121	C₄×Q₈	—	86	2	no	—
122	C₄×Q₈	—	86	2	no	—
123	C₂×C₄⋊C₄	—	43	2	no	—
124	C₂×C₄⋊C₄	—	43	1	no	—
125	C₂×C₄²	—	43	2	no	—
126	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
127	C₄⋊C₈	—	2	2	no	—
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	—	86	4	no	—
130	C₄⋊C₈	—	86	2	no	—
131	C₂²⋊C₈	—	2	2	no	—
132	C₂²⋊C₈	—	86	2	no	—
133	#97	—	2	2	no	—
134	#97	—	86	2	no	—
135	C₄×C₈	—	43	2	no	—
136	C₄×C₈	—	43	2	no	—
137	C₄×C₈	—	86	2	no	—
138	C₄×C₈	—	86	2	no	—
139	C₂×Q₈	—	86	4	no	—
140	C₂×Q₈	—	86	2	no	—
141	C₂²×C₄	—	43	2	no	—
142	C₂²×C₄	—	43	2	no	—
143	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
144	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
145	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	4	2	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	43	1	no	—
150	C₂×C₈	—	43	1	no	—
151	C₂×C₈	—	43	2	no	—
152	C₂×C₈	—	86	2	no	—
153	C₂×C₈	—	172	1	no	—
154	C₂×C₈	—	172	2	no	—
155	C₄⋊C₄	—	86	4	no	—
156	C₄⋊C₄	—	86	2	no	—
157	C₄²	—	86	2	no	—
158	C₄²	—	86	2	no	—
159	C₄²	#10621898	1	2	yes	—
160	C₄²	D₄×D₄₃	1	2	yes	—
161	C₂³	C₄⋊C₄×D₄₃	1	1	yes	characteristic
162	Q₈	—	86	8	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	4	1	no	—
167	C₂×C₄	—	43	4	no	—
168	C₂×C₄	—	86	2	no	—
169	C₂×C₄	—	86	2	no	—
170	C₂×C₄	—	86	2	no	—
171	C₂×C₄	—	172	2	no	—
172	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
173	C₂×C₄	#13063282	1	1	yes	characteristic
174	C₂×C₄	#13063298	1	2	yes	—
175	C₂×C₄	#13063303	1	2	yes	—
176	C₈	—	4	2	no	—
177	C₈	—	8	2	no	—
178	C₈	—	43	2	no	—
179	C₈	—	86	1	no	—
180	C₈	—	86	2	no	—
181	C₈	—	344	2	no	—
182	C₂²	—	2	1	no	—
183	C₂²	—	2	1	no	—
184	C₂²	#14582758	1	1	yes	characteristic
185	C₂²	#108×D₄₃	1	2	yes	—
186	C₄	—	2	1	no	—
187	C₄	—	2	2	no	—
188	C₄	—	2	2	no	—
189	C₄	—	86	4	no	—
190	C₄	—	172	2	no	—
191	C₄	#101×D₄₃	1	1	yes	characteristic
192	C₄	#14582880	1	1	yes	characteristic
193	C₂	—	2	2	no	—
194	C₂	#19134590	1	1	yes	characteristic
195	C₂	#19134632	1	1	yes	characteristic
196	C₂	#19134640	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14583023

C₂

yes

maximal, characteristic

#19133356

C₂

yes

maximal

#19132938

C₂

yes

maximal

#19132465

C₂

yes

maximal, characteristic

C₄₃×#1290

C₂

yes

maximal, characteristic

C₂×C₄×Dic₈₆

C₂²

yes

—

—

C₂²

yes

—

C₂×Dic₄₃⋊C₈

C₄

yes

—

C₂×C₄×C₄₃⋊C₈

C₂²

yes

characteristic

C₂×C₈×Dic₄₃

C₄

yes

—

#14582651

—

#14582649

—

#14582624

—

C₄₃×#432

C₂²

yes

—

C₂×C₄⋊C₈×C₄₃

C₂²

yes

characteristic

C₂²×Dic₈₆

Q₈

yes

characteristic

C₂²×C₄₃⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₄₃⋊C₈

C₂×C₄

yes

characteristic

C₄×Dic₈₆

—

C₄×Dic₈₆

—

C₂×C₄⋊Dic₄₃

Q₈

yes

characteristic

C₂×Dic₄₃⋊C₄

D₄

yes

—

C₂×C₄×Dic₄₃

C₂×C₄

yes

—

C₄⋊(C₄₃⋊C₈)

—

Dic₄₃⋊C₈

—

C₂²⋊(C₄₃⋊C₈)

—

#13063236

—

C₄×C₄₃⋊C₈

D₄

yes

—

C₄×C₄₃⋊C₈

D₄

yes

—

C₈×Dic₄₃

—

C₈×Dic₄₃

—

C₂²×C₈×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

—

C₂²⋊C₈×C₄₃

—

#97×C₄₃

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×Q₈

yes

characteristic

C₂²×Dic₄₃

C₄⋊C₄

yes

—

C₂²×Dic₄₃

C₄⋊C₄

yes

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

#41

yes

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×C₄₃⋊C₈

C₂²⋊C₄

yes

—

C₂×C₄₃⋊C₈

C₂²⋊C₄

yes

characteristic

C₂³×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×C₂²⋊C₄

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

—

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

#108

yes

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

#101

yes

—

#8623

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#380

yes

characteristic

C₂²×C₄₃

C₂×#108

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

C₂×C₈×Q₈

—

—

—

—

D₄₃

yes

characteristic

100

C₂×C₄₃

—

101

C₂×C₄₃

#1801

yes

characteristic

102

C₂×C₄₃

#1810

yes

characteristic

103

C₂×C₄₃

#1812

yes

characteristic

104

C₂×C₄×Q₈

—

105

C₈×Q₈

—

106

C₈×Q₈

—

107

#432

—

108

#432

C₂×D₄₃

yes

—

109

C₂×C₄⋊C₈

—

110

C₂×C₄⋊C₈

C₂×D₄₃

yes

characteristic

111

C₂×C₄×C₈

—

112

C₂×C₄×C₈

—

113

#335

—

114

#334

—

115

#324

—

116

C₄₃

#8623

yes

characteristic, complemented, Sylow

117

C₂²×Q₈

—

118

C₂²×C₈

—

119

C₂²×C₈

—

120

C₂²×C₈

C₄×D₄₃

yes

—

121

C₄×Q₈

—

122

C₄×Q₈

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄²

—

126

C₂×C₄²

C₂²×D₄₃

yes

characteristic

127

C₄⋊C₈

—

128

C₄⋊C₈

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

131

C₂²⋊C₈

—

132

C₂²⋊C₈

—

133

#97

—

134

#97

—

135

C₄×C₈

—

136

C₄×C₈

—

137

C₄×C₈

—

138

C₄×C₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

C₂×C₄×D₄₃

yes

characteristic

144

C₂²×C₄

D₄×D₄₃

yes

characteristic

145

C₂²×C₄

Q₈×D₄₃

yes

characteristic

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

172

—

154

C₂×C₈

—

172

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄²

—

158

C₄²

—

159

C₄²

#10621898

yes

—

160

C₄²

D₄×D₄₃

yes

—

161

C₂³

C₄⋊C₄×D₄₃

yes

characteristic

162

Q₈

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

—

172

C₂×C₄

C₂²⋊C₄×D₄₃

yes

characteristic

173

C₂×C₄

#13063282

yes

characteristic

174

C₂×C₄

#13063298

yes

—

175

C₂×C₄

#13063303

yes

—

176

C₈

—

177

C₈

—

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

344

—

182

C₂²

—

183

C₂²

—

184

C₂²

#14582758

yes

characteristic

185

C₂²

#108×D₄₃

yes

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

172

—

191

C₄

#101×D₄₃

yes

characteristic

192

C₄

#14582880

yes

characteristic

193

C₂

—

194

C₂

#19134590

yes

characteristic

195

C₂

#19134632

yes

characteristic

196

C₂

#19134640

yes

characteristic

	subgroup	G/H
center	C₂²	#14582758
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1290	C₂
socle	C₂²×C₄₃	#380
prime radical	C₂³×C₄₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1290	D₄₃
hypocenter	C₄₃	#8623
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1290

subgroup

G/H

center

C₂²

#14582758

commutator

C₂²×C₄×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1290

C₂

socle

C₂²×C₄₃

#380

prime radical

C₂³×C₄₃

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1290

N	H ≅ G/N	type
116. C₄₃	#8623	semidirect

H ≅ G/N

type

116. C₄₃

#8623

semidirect

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

subgroup	index
C₂×#14583023	2
#19133356	2
#19132938	2
#19132465	2
C₄₃×#1290	2
#8623	43

subgroup

index

C₄₃×#1290

Subgroups — #24989554

Conjugacy classes of subgroups

Complements

Maximal subgroups