Group #24996227 — subgroups

Subgroups — #24996227

Order 11008.

410 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 139 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	#19135585	C₂	1	2	yes	maximal
2	#19133354	C₂	1	1	yes	maximal, characteristic
3	#19133280	C₂	1	2	yes	maximal
4	C₈×#10621877	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₃×#500	C₂	1	1	yes	maximal, characteristic
6	#14582952	C₂²	1	1	yes	characteristic
7	#14582923	C₂²	1	2	yes	—
8	C₄×#10621877	C₂²	1	1	yes	characteristic
9	#14582668	—	2	4	no	—
10	#14582652	—	2	2	no	—
11	#14582645	—	2	4	no	—
12	#14582621	—	2	1	no	—
13	C₈×C₄₃⋊C₈	—	2	1	no	—
14	C₂×C₄⋊Q₈×C₄₃	C₂²	1	2	yes	—
15	C₄₃×#500	—	2	1	no	—
16	C₄₃×#500	—	2	1	no	—
17	C₄₃×#500	—	2	2	no	—
18	C₂×#107×C₄₃	C₄	1	4	yes	—
19	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
20	C₂×#10621877	—	2	2	no	—
21	C₂×#10621877	D₄	1	2	yes	—
22	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
24	C₄⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
25	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
26	#13063236	—	2	1	no	—
27	C₄×C₄₃⋊C₈	—	2	1	no	—
28	C₄×C₄₃⋊C₈	—	2	1	no	—
29	C₂²×Q₈×C₄₃	—	2	2	no	—
30	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
31	C₄⋊Q₈×C₄₃	—	2	2	no	—
32	C₄⋊Q₈×C₄₃	—	2	2	no	—
33	C₄⋊Q₈×C₄₃	—	2	2	no	—
34	C₄⋊Q₈×C₄₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
38	#107×C₄₃	—	2	4	no	—
39	#107×C₄₃	—	2	4	no	—
40	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
42	C₄×C₈×C₄₃	D₄	1	2	yes	—
43	C₂×Q₈×C₄₃	—	4	2	no	—
44	C₂×Q₈×C₄₃	—	4	2	no	—
45	C₂×Q₈×C₄₃	—	8	4	no	—
46	C₂²×C₄×C₄₃	—	2	4	no	—
47	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
50	C₂×C₈×C₄₃	—	2	4	no	—
51	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
52	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
53	C₄⋊C₄×C₄₃	—	2	4	no	—
54	C₄⋊C₄×C₄₃	—	2	4	no	—
55	C₄⋊C₄×C₄₃	—	4	2	no	—
56	C₄⋊C₄×C₄₃	—	4	2	no	—
57	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₄₃	#41	1	1	yes	characteristic
61	#10621877	—	2	2	no	—
62	#10621877	—	4	4	no	—
63	#10621877	Q₁₆	1	4	yes	—
64	C₂×C₄₃⋊C₈	—	2	2	no	—
65	C₂×C₄₃⋊C₈	—	2	2	no	—
66	C₂×C₄₃⋊C₈	—	4	2	no	—
67	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
68	Q₈×C₄₃	—	8	4	no	—
69	Q₈×C₄₃	—	8	4	no	—
70	C₂×C₄×C₄₃	—	2	2	no	—
71	C₂×C₄×C₄₃	—	2	2	no	—
72	C₂×C₄×C₄₃	—	4	4	no	—
73	C₂×C₄×C₄₃	—	8	4	no	—
74	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
77	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
78	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
79	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
80	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
81	C₂×C₄×C₄₃	#135	1	2	yes	—
82	C₈×C₄₃	—	2	2	no	—
83	C₈×C₄₃	#101	1	4	yes	—
84	C₄₃⋊C₈	—	2	4	no	—
85	C₄₃⋊C₈	—	4	2	no	—
86	C₄₃⋊C₈	—	8	2	no	—
87	#9801	—	43	1	no	maximal, Sylow
88	C₂²×C₄₃	—	2	1	no	—
89	C₂²×C₄₃	—	2	1	no	—
90	C₂²×C₄₃	#390	1	1	yes	characteristic
91	C₂²×C₄₃	C₄×Q₁₆	1	2	yes	—
92	C₄×C₄₃	—	2	1	no	—
93	C₄×C₄₃	—	2	1	no	—
94	C₄×C₄₃	—	8	4	no	—
95	C₄×C₄₃	C₂×#101	1	2	yes	—
96	C₄×C₄₃	#467	1	2	yes	—
97	C₄×C₄₃	#472	1	2	yes	—
98	C₄×C₄₃	#494	1	1	yes	characteristic
99	C₄×C₄₃	#495	1	1	yes	characteristic
100	C₂×#500	D₄₃	1	1	yes	characteristic
101	#1997	—	43	2	no	—
102	#1567	—	43	1	no	—
103	#1549	—	43	2	no	—
104	C₈×M₁₆	—	43	1	no	—
105	C₂×C₄₃	—	2	2	no	—
106	C₂×C₄₃	#1966	1	1	yes	characteristic
107	C₂×C₄₃	#1969	1	1	yes	characteristic
108	C₂×C₄₃	#1978	1	1	yes	characteristic
109	C₂×C₄⋊Q₈	C₂×D₄₃	1	2	yes	—
110	#500	—	2	1	no	—
111	#500	—	2	1	no	—
112	#500	—	2	2	no	—
113	#431	—	43	1	no	—
114	C₂×#107	Dic₄₃	1	4	yes	—
115	#423	—	43	2	no	—
116	C₄×M₁₆	—	43	1	no	—
117	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
118	#343	—	86	4	no	—
119	#335	—	86	2	no	—
120	#332	—	86	4	no	—
121	#322	—	86	1	no	—
122	C₈²	—	86	1	no	—
123	C₄₃	#9801	1	1	yes	characteristic, complemented, Sylow
124	C₂²×Q₈	—	2	2	no	—
125	C₂×M₁₆	—	43	2	no	—
126	C₂×M₁₆	—	86	2	no	—
127	C₂²×C₈	C₂×Dic₄₃	1	2	yes	—
128	C₄⋊Q₈	—	2	2	no	—
129	C₄⋊Q₈	—	2	2	no	—
130	C₄⋊Q₈	—	2	2	no	—
131	C₄⋊Q₈	—	2	2	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	C₂×Dic₄₃	1	4	yes	—
134	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
135	#107	—	2	4	no	—
136	#107	—	2	4	no	—
137	C₄⋊C₈	—	43	2	no	—
138	C₄⋊C₈	—	86	2	no	—
139	C₄⋊C₈	—	86	2	no	—
140	C₂²⋊C₈	—	86	1	no	—
141	#97	—	86	1	no	—
142	C₄×C₈	—	86	1	no	—
143	C₄×C₈	—	86	1	no	—
144	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
145	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
146	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
147	C₂×Q₈	—	4	2	no	—
148	C₂×Q₈	—	4	2	no	—
149	C₂×Q₈	—	8	4	no	—
150	C₂²×C₄	—	2	4	no	—
151	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
152	C₂²×C₄	#10621900	1	1	yes	characteristic
153	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
154	M₁₆	—	43	4	no	—
155	M₁₆	—	86	2	no	—
156	M₁₆	—	172	4	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	86	2	no	—
159	C₂×C₈	—	86	2	no	—
160	C₂×C₈	—	172	2	no	—
161	C₂×C₈	C₂²⋊Dic₄₃	1	2	yes	—
162	C₂×C₈	C₂²⋊Dic₄₃	1	2	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	D₄×D₄₃	1	1	yes	characteristic
168	C₄²	#10621900	1	1	yes	characteristic
169	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
170	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
171	C₂³	D₄×Dic₄₃	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⋊Dic₄₃	1	1	yes	characteristic
179	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
180	C₂×C₄	#13063296	1	1	yes	characteristic
181	C₂×C₄	#13063296	1	1	yes	characteristic
182	C₂×C₄	#13063312	1	1	yes	characteristic
183	C₂×C₄	#13063321	1	1	yes	characteristic
184	C₂×C₄	Q₁₆×D₄₃	1	2	yes	—
185	C₂×C₄	#13063348	1	2	yes	—
186	C₈	—	2	2	no	—
187	C₈	—	86	4	no	—
188	C₈	—	344	2	no	—
189	C₈	—	172	2	no	—
190	C₈	#13063244	1	4	yes	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	#14582802	1	1	yes	characteristic
194	C₂²	Q₁₆×Dic₄₃	1	2	yes	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	8	4	no	—
198	C₄	C₂×#13063244	1	2	yes	—
199	C₄	#14583137	1	2	yes	—
200	C₄	#14583166	1	2	yes	—
201	C₄	#14583277	1	1	yes	characteristic
202	C₄	#14583284	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	#19135440	1	1	yes	characteristic
205	C₂	#19135457	1	1	yes	characteristic
206	C₂	#19135506	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19135585

C₂

yes

maximal

#19133354

C₂

yes

maximal, characteristic

#19133280

C₂

yes

maximal

C₈×#10621877

C₂

yes

maximal, characteristic

C₂×C₄₃×#500

C₂

yes

maximal, characteristic

#14582952

C₂²

yes

characteristic

#14582923

C₂²

yes

—

C₄×#10621877

C₂²

yes

—

—

—

—

C₈×C₄₃⋊C₈

—

C₂×C₄⋊Q₈×C₄₃

C₂²

yes

—

C₄₃×#500

—

C₄₃×#500

—

C₄₃×#500

—

C₂×#107×C₄₃

C₄

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂×#10621877

—

C₂×#10621877

D₄

yes

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

D₄

yes

—

C₂²⋊(C₄₃⋊C₈)

—

#13063236

—

C₄×C₄₃⋊C₈

—

C₄×C₄₃⋊C₈

—

C₂²×Q₈×C₄₃

—

C₂²×C₈×C₄₃

C₂×C₄

yes

—

C₄⋊Q₈×C₄₃

—

C₄⋊Q₈×C₄₃

—

C₄⋊Q₈×C₄₃

—

C₄⋊Q₈×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

#107×C₄₃

—

#107×C₄₃

—

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

characteristic

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

C₄²×C₄₃

yes

—

—

Q₁₆

yes

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₄×D₄

yes

characteristic

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₄×D₄

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

C₂×Q₁₆

yes

—

C₂×C₄×C₄₃

#135

yes

—

C₈×C₄₃

—

C₈×C₄₃

#101

yes

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#9801

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#390

yes

characteristic

C₂²×C₄₃

C₄×Q₁₆

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×#101

yes

—

C₄×C₄₃

#467

yes

—

C₄×C₄₃

#472

yes

—

C₄×C₄₃

yes

C₄×C₄₃

yes

100

D₄₃

yes

characteristic

101

#1997

—

102

#1567

—

103

#1549

—

104

C₈×M₁₆

—

105

C₂×C₄₃

—

106

C₂×C₄₃

#1966

yes

characteristic

107

C₂×C₄₃

#1969

yes

characteristic

108

C₂×C₄₃

#1978

yes

characteristic

109

C₂×C₄⋊Q₈

C₂×D₄₃

yes

—

110

#500

—

111

#500

—

112

#500

—

113

#431

—

114

C₂×#107

Dic₄₃

yes

—

115

#423

—

116

C₄×M₁₆

—

117

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

118

#343

—

119

#335

—

120

#332

—

121

#322

—

122

C₈²

—

123

C₄₃

#9801

yes

characteristic, complemented, Sylow

124

C₂²×Q₈

—

125

C₂×M₁₆

—

126

C₂×M₁₆

—

127

C₂²×C₈

C₂×Dic₄₃

yes

—

128

C₄⋊Q₈

—

129

C₄⋊Q₈

—

130

C₄⋊Q₈

—

131

C₄⋊Q₈

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

C₂×Dic₄₃

yes

—

134

C₂×C₄²

C₂²×D₄₃

yes

characteristic

135

#107

—

136

#107

—

137

C₄⋊C₈

—

138

C₄⋊C₈

—

139

C₄⋊C₈

—

140

C₂²⋊C₈

—

141

#97

—

142

C₄×C₈

—

143

C₄×C₈

—

144

C₄×C₈

C₂²⋊D₄₃

yes

characteristic

145

C₄×C₈

C₂²⋊D₄₃

yes

characteristic

146

C₄×C₈

C₂²⋊D₄₃

yes

—

147

C₂×Q₈

—

148

C₂×Q₈

—

149

C₂×Q₈

—

150

C₂²×C₄

—

151

C₂²×C₄

D₄×D₄₃

yes

characteristic

152

C₂²×C₄

#10621900

yes

characteristic

153

C₂²×C₄

C₂²×Dic₄₃

yes

characteristic

154

M₁₆

—

155

M₁₆

—

156

M₁₆

—

172

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

172

—

161

C₂×C₈

C₂²⋊Dic₄₃

yes

—

162

C₂×C₈

C₂²⋊Dic₄₃

yes

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄²

D₄×D₄₃

yes

characteristic

168

C₄²

#10621900

yes

characteristic

169

C₄²

C₂×C₂²⋊D₄₃

yes

characteristic

170

C₄²

C₂×C₂²⋊D₄₃

yes

characteristic

171

C₂³

D₄×Dic₄₃

yes

characteristic

172

Q₈

—

173

Q₈

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

C₂×C₂²⋊Dic₄₃

yes

characteristic

179

C₂×C₄

D₄×Dic₄₃

yes

characteristic

180

C₂×C₄

#13063296

yes

characteristic

181

C₂×C₄

#13063296

yes

characteristic

182

C₂×C₄

#13063312

yes

characteristic

183

C₂×C₄

#13063321

yes

characteristic

184

C₂×C₄

Q₁₆×D₄₃

yes

—

185

C₂×C₄

#13063348

yes

—

186

C₈

—

187

C₈

—

188

C₈

—

344

—

189

C₈

—

172

—

190

C₈

#13063244

yes

—

191

C₂²

—

192

C₂²

—

193

C₂²

#14582802

yes

characteristic

194

C₂²

Q₁₆×Dic₄₃

yes

—

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

C₂×#13063244

yes

—

199

C₄

#14583137

yes

—

200

C₄

#14583166

yes

—

201

C₄

#14583277

yes

characteristic

202

C₄

#14583284

yes

characteristic

203

C₂

—

204

C₂

#19135440

yes

characteristic

205

C₂

#19135457

yes

characteristic

206

C₂

#19135506

yes

characteristic

	subgroup	G/H
center	C₂²	#14582802
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₂×C₄₃×#500	C₂
socle	C₂²×C₄₃	#390
prime radical	C₂³×C₄₃	C₄×D₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₂×#500	D₄₃
hypocenter	C₄₃	#9801
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

subgroup

G/H

center

C₂²

#14582802

commutator

C₂²×C₄×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₂×C₄₃×#500

C₂

socle

C₂²×C₄₃

#390

prime radical

C₂³×C₄₃

C₄×D₄

cosocle

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

N	H ≅ G/N	type
123. C₄₃	#9801	semidirect

H ≅ G/N

type

123. C₄₃

#9801

semidirect

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

subgroup	index
#19135585	2
#19133354	2
#19133280	2
C₈×#10621877	2
C₂×C₄₃×#500	2
#9801	43

subgroup

index

C₂×C₄₃×#500

Subgroups — #24996227

Conjugacy classes of subgroups

Complements

Maximal subgroups