Group #24996459 — subgroups

Subgroups — #24996459

Order 11008.

510 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 199 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	#19135557	C₂	1	2	yes	maximal
2	#19135398	C₂	1	2	yes	maximal
3	#19134755	C₂	1	1	yes	maximal, characteristic
4	C₂×#14582620	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₃×#498	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063259	—	2	2	no	—
7	C₂×#13063249	—	2	4	no	—
8	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
9	#14582769	C₂²	1	1	yes	characteristic
10	#14582769	C₂²	1	1	yes	characteristic
11	#14582661	—	2	4	no	—
12	#14582620	—	2	2	no	—
13	C₂×C₄⋊Q₈×C₄₃	C₂²	1	2	yes	—
14	C₄₃×#498	—	2	2	no	—
15	C₄₃×#498	C₄	1	4	yes	complemented
16	C₂×#106×C₄₃	—	2	2	no	—
17	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄₃⋊C₈	D₄	1	4	yes	—
19	C₂×C₄⋊Dic₄₃	—	2	2	no	—
20	C₂×C₄⋊Dic₄₃	—	2	2	no	—
21	C₂×C₄⋊Dic₄₃	—	2	1	no	—
22	C₂×C₄⋊Dic₄₃	—	2	1	no	—
23	#13063259	—	4	4	no	—
24	#13063249	—	4	8	no	—
25	C₄×C₄₃⋊C₈	—	2	2	no	—
26	C₄×C₄₃⋊C₈	—	2	2	no	—
27	C₂²×Q₈×C₄₃	—	2	2	no	—
28	C₂²×C₈×C₄₃	D₄	1	2	yes	—
29	C₄⋊Q₈×C₄₃	—	2	2	no	—
30	C₄⋊Q₈×C₄₃	—	2	2	no	—
31	C₄⋊Q₈×C₄₃	—	2	2	no	—
32	C₄⋊Q₈×C₄₃	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
35	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
36	#106×C₄₃	—	2	8	no	—
37	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₄₃	D₄	1	2	yes	—
39	C₂×Q₈×C₄₃	—	2	4	no	—
40	C₂×Q₈×C₄₃	—	4	2	no	—
41	C₂×Q₈×C₄₃	—	4	4	no	—
42	C₂×Q₈×C₄₃	—	4	4	no	—
43	C₂²×C₄×C₄₃	—	4	2	no	—
44	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×Dic₄₃	—	4	1	no	—
47	C₂²×Dic₄₃	—	4	1	no	—
48	C₂×C₈×C₄₃	C₂²⋊C₄	1	4	yes	—
49	C₂×C₈×C₄₃	SD₁₆	1	4	yes	—
50	C₂×C₈×C₄₃	Q₁₆	1	4	yes	—
51	C₄⋊C₄×C₄₃	—	2	4	no	—
52	C₄⋊C₄×C₄₃	—	2	4	no	—
53	C₄⋊C₄×C₄₃	—	4	2	no	—
54	C₄⋊C₄×C₄₃	—	4	2	no	—
55	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₄₃	#41	1	2	yes	—
58	C₄⋊Dic₄₃	—	4	2	no	—
59	C₄⋊Dic₄₃	—	4	2	no	—
60	C₄⋊Dic₄₃	—	4	4	no	—
61	C₄⋊Dic₄₃	—	4	4	no	—
62	C₂×C₄₃⋊C₈	—	2	4	no	—
63	C₂×C₄₃⋊C₈	—	2	4	no	—
64	C₂×C₄₃⋊C₈	—	2	4	no	—
65	C₂³×C₄₃	#127	1	1	yes	characteristic
66	Q₈×C₄₃	—	4	8	no	—
67	Q₈×C₄₃	—	8	4	no	—
68	C₂×C₄×C₄₃	—	4	4	no	—
69	C₂×C₄×C₄₃	—	8	2	no	—
70	C₂×C₄×C₄₃	—	8	2	no	—
71	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
73	C₂×C₄×C₄₃	#121	1	2	yes	—
74	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
75	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
76	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
77	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
78	C₂×C₄×C₄₃	#137	1	4	yes	—
79	C₂×Dic₄₃	—	8	1	no	—
80	C₂×Dic₄₃	—	8	1	no	—
81	C₂×Dic₄₃	—	8	2	no	—
82	C₂×Dic₄₃	—	8	2	no	—
83	C₈×C₄₃	#103	1	8	yes	complemented
84	C₄₃⋊C₈	—	4	4	no	—
85	#9841	—	43	1	no	maximal, Sylow
86	C₂²×C₄₃	#390	1	1	yes	characteristic
87	C₂²×C₄₃	#492	1	1	yes	characteristic
88	C₂²×C₄₃	#494	1	1	yes	characteristic
89	C₂²×C₄₃	#496	1	2	yes	—
90	C₂²×C₄₃	#497	1	2	yes	—
91	C₄×C₄₃	—	8	4	no	—
92	C₄×C₄₃	C₂×#103	1	2	yes	—
93	C₄×C₄₃	#440	1	4	yes	—
94	C₄×C₄₃	#468	1	4	yes	—
95	C₄×C₄₃	#487	1	2	yes	—
96	Dic₄₃	—	16	1	no	—
97	Dic₄₃	—	16	1	no	—
98	C₂×#498	D₄₃	1	1	yes	characteristic
99	#1990	—	43	2	no	—
100	#1956	—	43	2	no	—
101	#1845	—	43	1	no	—
102	C₂×#322	—	43	1	no	—
103	C₂×C₄₃	#1715	1	1	yes	characteristic
104	C₂×C₄₃	#1716	1	1	yes	characteristic
105	C₂×C₄₃	#1723	1	2	yes	—
106	C₂×C₄₃	#1956	1	1	yes	characteristic
107	C₂×C₄₃	#1973	1	2	yes	—
108	C₂×C₄⋊Q₈	C₂×D₄₃	1	2	yes	—
109	#498	—	2	2	no	—
110	#498	Dic₄₃	1	4	yes	complemented
111	C₂×#107	—	86	2	no	—
112	C₂×#106	—	2	2	no	—
113	C₂×#103	—	86	4	no	—
114	C₂×C₄×C₈	—	43	2	no	—
115	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
116	#384	—	43	1	no	—
117	#384	—	43	1	no	—
118	#340	—	86	4	no	—
119	#322	—	86	2	no	—
120	C₄₃	#9841	1	1	yes	characteristic, complemented, Sylow
121	C₂²×Q₈	—	2	2	no	—
122	C₂²×C₈	—	43	4	no	—
123	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
124	C₄⋊Q₈	—	2	2	no	—
125	C₄⋊Q₈	—	2	2	no	—
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	C₂×Dic₄₃	1	4	yes	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	86	2	no	—
131	C₂×C₄⋊C₄	—	86	2	no	—
132	C₂×C₄⋊C₄	—	86	1	no	—
133	C₂×C₄⋊C₄	—	86	1	no	—
134	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
135	#107	—	172	4	no	—
136	#106	—	2	8	no	—
137	#103	—	172	8	no	—
138	C₄×C₈	—	86	2	no	—
139	C₄×C₈	—	86	2	no	—
140	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
141	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
142	C₂×Q₈	—	2	4	no	—
143	C₂×Q₈	—	4	2	no	—
144	C₂×Q₈	—	4	4	no	—
145	C₂×Q₈	—	4	4	no	—
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	—	172	1	no	—
148	C₂²×C₄	—	172	1	no	—
149	C₂²×C₄	D₄×D₄₃	1	2	yes	—
150	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
151	C₂×C₈	—	86	4	no	—
152	C₂×C₈	—	86	4	no	—
153	C₂×C₈	—	86	4	no	—
154	C₂×C₈	#10621884	1	4	yes	—
155	C₂×C₈	#10621885	1	4	yes	—
156	C₂×C₈	C₂²⋊Dic₄₃	1	4	yes	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	172	2	no	—
162	C₄⋊C₄	—	172	2	no	—
163	C₄⋊C₄	—	172	4	no	—
164	C₄⋊C₄	—	172	4	no	—
165	C₄²	#10621900	1	2	yes	—
166	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
167	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
168	C₂³	#13063321	1	1	yes	characteristic
169	Q₈	—	4	8	no	—
170	Q₈	—	8	4	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	344	1	no	—
175	C₂×C₄	—	344	1	no	—
176	C₂×C₄	—	344	2	no	—
177	C₂×C₄	—	344	2	no	—
178	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
179	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
180	C₂×C₄	#13063296	1	2	yes	—
181	C₂×C₄	#13063312	1	1	yes	characteristic
182	C₂×C₄	C₂×#10621884	1	2	yes	—
183	C₂×C₄	C₂×#10621885	1	2	yes	—
184	C₂×C₄	#13063352	1	4	yes	—
185	C₂×C₄	#13063354	1	4	yes	—
186	C₈	—	172	4	no	—
187	C₈	#13063249	1	8	yes	complemented
188	C₂²	#14582802	1	1	yes	characteristic
189	C₂²	#14583273	1	1	yes	characteristic
190	C₂²	#14583278	1	1	yes	characteristic
191	C₂²	#14583285	1	2	yes	—
192	C₂²	#14583290	1	2	yes	—
193	C₄	—	8	4	no	—
194	C₄	—	688	1	no	—
195	C₄	—	688	1	no	—
196	C₄	C₂×#13063249	1	2	yes	—
197	C₄	#14583000	1	4	yes	—
198	C₄	#14583142	1	4	yes	—
199	C₄	#14583249	1	2	yes	—
200	C₂	#19134224	1	1	yes	characteristic
201	C₂	#19134229	1	1	yes	characteristic
202	C₂	#19134269	1	2	yes	—
203	C₂	#19135398	1	1	yes	characteristic
204	C₂	#19135478	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19135557

C₂

yes

maximal

#19135398

C₂

yes

maximal

#19134755

C₂

yes

maximal, characteristic

C₂×#14582620

C₂

yes

maximal, characteristic

C₂×C₄₃×#498

C₂

yes

maximal, characteristic

C₂×#13063259

—

C₂×#13063249

—

C₂×C₄×C₄₃⋊C₈

C₂²

yes

—

#14582769

C₂²

yes

characteristic

#14582769

C₂²

yes

characteristic

#14582661

—

#14582620

—

C₂×C₄⋊Q₈×C₄₃

C₂²

yes

—

C₄₃×#498

—

C₄₃×#498

C₄

yes

complemented

C₂×#106×C₄₃

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂²×C₄₃⋊C₈

D₄

yes

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

#13063259

—

#13063249

—

C₄×C₄₃⋊C₈

—

C₄×C₄₃⋊C₈

—

C₂²×Q₈×C₄₃

—

C₂²×C₈×C₄₃

D₄

yes

—

C₄⋊Q₈×C₄₃

—

C₄⋊Q₈×C₄₃

—

C₄⋊Q₈×C₄₃

—

C₄⋊Q₈×C₄₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

#106×C₄₃

—

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

D₄

yes

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

—

C₂×C₈×C₄₃

SD₁₆

yes

—

C₂×C₈×C₄₃

Q₁₆

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂²×C₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

#41

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

#127

yes

characteristic

Q₈×C₄₃

—

Q₈×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

#121

yes

—

C₂×C₄×C₄₃

#124

yes

characteristic

C₂×C₄×C₄₃

C₂×SD₁₆

yes

—

C₂×C₄×C₄₃

C₂×Q₁₆

yes

—

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₄₃

#137

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₈×C₄₃

#103

yes

complemented

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

#497

yes

—

C₄×C₄₃

—

C₄×C₄₃

C₂×#103

yes

—

C₄×C₄₃

#440

yes

—

C₄×C₄₃

#468

yes

—

C₄×C₄₃

#487

yes

—

Dic₄₃

—

Dic₄₃

—

C₂×#498

D₄₃

yes

characteristic

#1990

—

100

#1956

—

101

#1845

—

102

C₂×#322

—

103

C₂×C₄₃

#1715

yes

characteristic

104

C₂×C₄₃

#1716

yes

characteristic

105

C₂×C₄₃

#1723

yes

—

106

C₂×C₄₃

#1956

yes

characteristic

107

C₂×C₄₃

#1973

yes

—

108

C₂×C₄⋊Q₈

C₂×D₄₃

yes

—

109

#498

—

110

#498

Dic₄₃

yes

complemented

111

C₂×#107

—

112

C₂×#106

—

113

C₂×#103

—

114

C₂×C₄×C₈

—

115

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

116

#384

—

117

#384

—

118

#340

—

119

#322

—

120

C₄₃

#9841

yes

characteristic, complemented, Sylow

121

C₂²×Q₈

—

122

C₂²×C₈

—

123

C₂²×C₈

C₂²⋊D₄₃

yes

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

C₂×Dic₄₃

yes

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

134

C₂×C₄²

C₂²×D₄₃

yes

characteristic

135

#107

—

172

—

136

#106

—

137

#103

—

172

—

138

C₄×C₈

—

139

C₄×C₈

—

140

C₄×C₈

C₂×Dic₄₃

yes

—

141

C₄×C₈

C₂²⋊D₄₃

yes

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂²×C₄

—

147

C₂²×C₄

—

172

—

148

C₂²×C₄

—

172

—

149

C₂²×C₄

D₄×D₄₃

yes

—

150

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

#10621884

yes

—

155

C₂×C₈

#10621885

yes

—

156

C₂×C₈

C₂²⋊Dic₄₃

yes

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

172

—

162

C₄⋊C₄

—

172

—

163

C₄⋊C₄

—

172

—

164

C₄⋊C₄

—

172

—

165

C₄²

#10621900

yes

—

166

C₄²

C₂²×Dic₄₃

yes

characteristic

167

C₄²

C₂×C₂²⋊D₄₃

yes

characteristic

168

C₂³

#13063321

yes

characteristic

169

Q₈

—

170

Q₈

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

344

—

175

C₂×C₄

—

344

—

176

C₂×C₄

—

344

—

177

C₂×C₄

—

344

—

178

C₂×C₄

C₂×C₂²⋊Dic₄₃

yes

characteristic

179

C₂×C₄

D₄×Dic₄₃

yes

—

180

C₂×C₄

#13063296

yes

—

181

C₂×C₄

#13063312

yes

characteristic

182

C₂×C₄

C₂×#10621884

yes

—

183

C₂×C₄

C₂×#10621885

yes

—

184

C₂×C₄

#13063352

yes

—

185

C₂×C₄

#13063354

yes

—

186

C₈

—

172

—

187

C₈

#13063249

yes

complemented

188

C₂²

#14582802

yes

characteristic

189

C₂²

#14583273

yes

characteristic

190

C₂²

#14583278

yes

characteristic

191

C₂²

#14583285

yes

—

192

C₂²

#14583290

yes

—

193

C₄

—

194

C₄

—

688

—

195

C₄

—

688

—

196

C₄

C₂×#13063249

yes

—

197

C₄

#14583000

yes

—

198

C₄

#14583142

yes

—

199

C₄

#14583249

yes

—

200

C₂

#19134224

yes

characteristic

201

C₂

#19134229

yes

characteristic

202

C₂

#19134269

yes

—

203

C₂

#19135398

yes

characteristic

204

C₂

#19135478

yes

—

	subgroup	G/H
center	C₂³	#13063321
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₂×C₄₃×#498	C₂
socle, prime radical	C₂³×C₄₃	#127
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₂×#498	D₄₃
hypocenter	C₄₃	#9841
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

subgroup

G/H

center

C₂³

#13063321

commutator

C₄²×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₂×C₄₃×#498

C₂

socle, prime radical

C₂³×C₄₃

#127

cosocle

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

N	H ≅ G/N	type
15. C₄₃×#498	C₄	semidirect
83. C₈×C₄₃	#103	semidirect
110. #498	Dic₄₃	semidirect
120. C₄₃	#9841	semidirect
187. C₈	#13063249	semidirect

H ≅ G/N

type

15. C₄₃×#498

C₄

semidirect

83. C₈×C₄₃

#103

semidirect

110. #498

Dic₄₃

semidirect

120. C₄₃

#9841

semidirect

187. C₈

#13063249

semidirect

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

subgroup	index
#19135557	2
#19135398	2
#19134755	2
C₂×#14582620	2
C₂×C₄₃×#498	2
#9841	43

subgroup

index

C₂×C₄₃×#498

Subgroups — #24996459

Conjugacy classes of subgroups

Complements

Maximal subgroups