Group #24999437 — subgroups

Subgroups — #24999437

Order 11008.

554 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 153 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×#14583249	C₂	1	2	yes	maximal
2	#19135557	C₂	1	2	yes	maximal
3	#19135397	C₂	1	1	yes	maximal, characteristic
4	C₄₃⋊#1273	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₃×#498	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063325	C₂²	1	1	yes	characteristic
7	#14583249	—	2	4	no	—
8	#14583249	—	2	4	no	—
9	C₂×#13063250	—	2	2	no	—
10	C₂×#13063249	—	2	4	no	—
11	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
12	#14582769	C₂²	1	1	yes	characteristic
13	#14582661	—	2	4	no	—
14	C₂×C₄⋊Q₈×C₄₃	C₂²	1	2	yes	—
15	C₄₃×#498	—	2	4	no	—
16	C₂×#106×C₄₃	—	2	2	no	—
17	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
18	C₂²×Dic₈₆	—	2	1	no	—
19	C₂²×C₄₃⋊C₈	D₄	1	4	yes	—
20	#13063325	—	2	2	no	—
21	#13063325	—	2	1	no	—
22	#13063325	D₄	1	2	yes	—
23	C₂×C₄⋊Dic₄₃	—	2	2	no	—
24	C₂×C₄⋊Dic₄₃	—	2	1	no	—
25	C₂×C₄⋊Dic₄₃	—	2	2	no	—
26	#13063250	—	4	4	no	—
27	#13063249	—	2	16	no	—
28	C₄×C₄₃⋊C₈	—	2	2	no	—
29	C₄×C₄₃⋊C₈	—	2	2	no	—
30	C₂²×Q₈×C₄₃	—	2	2	no	—
31	C₂²×C₈×C₄₃	D₄	1	2	yes	—
32	C₄⋊Q₈×C₄₃	—	2	2	no	—
33	C₄⋊Q₈×C₄₃	—	2	2	no	—
34	C₄⋊Q₈×C₄₃	—	2	2	no	—
35	C₄⋊Q₈×C₄₃	D₄	1	4	yes	—
36	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
38	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
39	#106×C₄₃	—	2	8	no	—
40	C₄×C₈×C₄₃	—	2	2	no	—
41	C₂×Q₈×C₄₃	—	2	4	no	—
42	C₂×Q₈×C₄₃	—	4	2	no	—
43	C₂×Q₈×C₄₃	—	4	4	no	—
44	C₂×Q₈×C₄₃	—	4	4	no	—
45	C₂²×C₄×C₄₃	—	4	2	no	—
46	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×Dic₄₃	—	4	1	no	—
49	C₂²×Dic₄₃	—	4	1	no	—
50	C₂×Dic₈₆	—	2	2	no	—
51	C₂×Dic₈₆	—	4	2	no	—
52	C₂×Dic₈₆	—	4	2	no	—
53	C₂×Dic₈₆	—	4	1	no	—
54	C₂×C₈×C₄₃	—	2	2	no	—
55	C₂×C₈×C₄₃	—	2	2	no	—
56	C₂×C₈×C₄₃	SD₁₆	1	4	yes	—
57	C₄⋊C₄×C₄₃	—	2	4	no	—
58	C₄⋊C₄×C₄₃	—	2	4	no	—
59	C₄⋊C₄×C₄₃	—	4	2	no	—
60	C₄⋊C₄×C₄₃	—	4	2	no	—
61	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₄₃	C₂×D₄	1	2	yes	—
63	C₄²×C₄₃	#41	1	1	yes	characteristic
64	C₄⋊Dic₄₃	—	2	4	no	—
65	C₄⋊Dic₄₃	—	4	2	no	—
66	C₄⋊Dic₄₃	—	4	2	no	—
67	C₄⋊Dic₄₃	—	4	4	no	—
68	C₂×C₄₃⋊C₈	—	2	4	no	—
69	C₂×C₄₃⋊C₈	—	2	4	no	—
70	C₂×C₄₃⋊C₈	—	2	4	no	—
71	C₂³×C₄₃	#127	1	1	yes	characteristic
72	Q₈×C₄₃	—	4	8	no	—
73	Q₈×C₄₃	—	8	4	no	—
74	C₂×C₄×C₄₃	—	4	4	no	—
75	C₂×C₄×C₄₃	—	8	2	no	—
76	C₂×C₄×C₄₃	—	8	2	no	—
77	C₂×C₄×C₄₃	#120	1	2	yes	—
78	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
79	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
80	C₂×C₄×C₄₃	#121	1	2	yes	—
81	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
82	C₂×C₄×C₄₃	#135	1	4	yes	—
83	C₂×C₄×C₄₃	#137	1	4	yes	—
84	C₂×C₄×C₄₃	#137	1	2	yes	—
85	C₂×Dic₄₃	—	4	2	no	—
86	C₂×Dic₄₃	—	8	1	no	—
87	C₂×Dic₄₃	—	8	1	no	—
88	C₂×Dic₄₃	—	8	1	no	—
89	C₂×Dic₄₃	—	8	2	no	—
90	Dic₈₆	—	4	4	no	—
91	Dic₈₆	—	8	2	no	—
92	C₈×C₄₃	—	2	4	no	—
93	C₄₃⋊C₈	—	4	4	no	—
94	#10317	—	43	1	no	maximal, Sylow
95	C₂²×C₄₃	#392	1	1	yes	characteristic
96	C₂²×C₄₃	#492	1	1	yes	characteristic
97	C₂²×C₄₃	#495	1	2	yes	—
98	C₂²×C₄₃	#497	1	2	yes	—
99	C₂²×C₄₃	#497	1	1	yes	characteristic
100	C₄×C₄₃	—	8	4	no	—
101	C₄×C₄₃	#448	1	2	yes	—
102	C₄×C₄₃	#452	1	4	yes	—
103	C₄×C₄₃	#461	1	2	yes	—
104	C₄×C₄₃	#464	1	4	yes	—
105	Dic₄₃	—	8	2	no	—
106	Dic₄₃	—	16	1	no	—
107	C₂×#498	D₄₃	1	1	yes	characteristic
108	C₂×#487	—	43	2	no	—
109	#1990	—	43	2	no	—
110	#1956	—	43	1	no	—
111	#1273	—	43	1	no	—
112	C₂×C₄₃	#1708	1	1	yes	characteristic
113	C₂×C₄₃	#1716	1	1	yes	characteristic
114	C₂×C₄₃	#1718	1	2	yes	—
115	C₂×C₄₃	#2013	1	1	yes	characteristic
116	C₂×C₄₃	#2015	1	2	yes	—
117	C₂×C₄⋊Q₈	—	43	1	no	—
118	C₂×C₄⋊Q₈	C₂×D₄₃	1	2	yes	—
119	#498	—	2	4	no	—
120	#487	—	86	4	no	—
121	#487	—	86	4	no	—
122	C₂×#106	—	2	2	no	—
123	C₂×#103	—	86	4	no	—
124	C₂×#103	—	86	2	no	—
125	C₂×C₄×C₈	—	43	2	no	—
126	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
127	#384	—	43	1	no	—
128	#340	—	86	4	no	—
129	C₄₃	#10317	1	1	yes	characteristic, complemented, Sylow
130	C₂²×Q₈	—	2	2	no	—
131	C₂²×Q₈	—	86	1	no	—
132	C₂²×C₈	—	43	4	no	—
133	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
134	C₄⋊Q₈	—	2	2	no	—
135	C₄⋊Q₈	—	2	2	no	—
136	C₄⋊Q₈	—	2	2	no	—
137	C₄⋊Q₈	—	43	2	no	—
138	C₄⋊Q₈	—	86	1	no	—
139	C₄⋊Q₈	—	86	2	no	—
140	C₄⋊Q₈	C₂²⋊D₄₃	1	4	yes	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	—	2	2	no	—
143	C₂×C₄⋊C₄	—	86	2	no	—
144	C₂×C₄⋊C₄	—	86	2	no	—
145	C₂×C₄⋊C₄	—	86	1	no	—
146	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
147	#106	—	2	8	no	—
148	#103	—	86	16	no	—
149	#103	—	172	4	no	—
150	C₄×C₈	—	2	2	no	—
151	C₄×C₈	—	86	2	no	—
152	C₄×C₈	—	86	2	no	—
153	C₂×Q₈	—	2	4	no	—
154	C₂×Q₈	—	4	2	no	—
155	C₂×Q₈	—	4	4	no	—
156	C₂×Q₈	—	4	4	no	—
157	C₂×Q₈	—	86	2	no	—
158	C₂×Q₈	—	172	2	no	—
159	C₂×Q₈	—	172	2	no	—
160	C₂×Q₈	—	172	1	no	—
161	C₂²×C₄	—	4	2	no	—
162	C₂²×C₄	—	172	1	no	—
163	C₂²×C₄	—	172	1	no	—
164	C₂²×C₄	D₄×D₄₃	1	2	yes	—
165	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	86	4	no	—
169	C₂×C₈	—	86	4	no	—
170	C₂×C₈	—	86	4	no	—
171	C₂×C₈	#10621884	1	4	yes	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	2	4	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	86	4	no	—
177	C₄⋊C₄	—	172	2	no	—
178	C₄⋊C₄	—	172	4	no	—
179	C₄⋊C₄	—	172	2	no	—
180	C₄²	D₄×D₄₃	1	1	yes	characteristic
181	C₄²	#10621900	1	1	yes	characteristic
182	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
183	C₂³	#13063321	1	1	yes	characteristic
184	Q₈	—	4	8	no	—
185	Q₈	—	8	4	no	—
186	Q₈	—	344	2	no	—
187	Q₈	—	172	4	no	—
188	C₂×C₄	—	4	4	no	—
189	C₂×C₄	—	8	2	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	344	1	no	—
192	C₂×C₄	—	344	1	no	—
193	C₂×C₄	—	344	1	no	—
194	C₂×C₄	—	344	2	no	—
195	C₂×C₄	—	172	2	no	—
196	C₂×C₄	#13063293	1	2	yes	—
197	C₂×C₄	#13063294	1	1	yes	characteristic
198	C₂×C₄	#13063296	1	1	yes	characteristic
199	C₂×C₄	#13063297	1	2	yes	—
200	C₂×C₄	C₂×#10621884	1	2	yes	—
201	C₂×C₄	#13063346	1	4	yes	—
202	C₂×C₄	#13063354	1	4	yes	—
203	C₂×C₄	#13063356	1	2	yes	—
204	C₈	—	2	4	no	—
205	C₈	—	172	4	no	—
206	C₂²	#14582807	1	1	yes	characteristic
207	C₂²	#14583273	1	1	yes	characteristic
208	C₂²	#14583280	1	2	yes	—
209	C₂²	#14583290	1	2	yes	—
210	C₂²	#14583291	1	1	yes	characteristic
211	C₄	—	8	4	no	—
212	C₄	—	344	2	no	—
213	C₄	—	688	1	no	—
214	C₄	#14583038	1	2	yes	—
215	C₄	#14583058	1	4	yes	—
216	C₄	#14583101	1	2	yes	—
217	C₄	#14583119	1	4	yes	—
218	C₂	#19134197	1	1	yes	characteristic
219	C₂	#19134229	1	1	yes	characteristic
220	C₂	#19134239	1	2	yes	—
221	C₂	#19135678	1	1	yes	characteristic
222	C₂	#19135687	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14583249

C₂

yes

maximal

#19135557

C₂

yes

maximal

#19135397

C₂

yes

maximal, characteristic

C₄₃⋊#1273

C₂

yes

maximal, characteristic

C₂×C₄₃×#498

C₂

yes

maximal, characteristic

C₂×#13063325

C₂²

yes

—

—

—

—

C₂×C₄×C₄₃⋊C₈

C₂²

yes

—

#14582769

C₂²

yes

characteristic

#14582661

—

C₂×C₄⋊Q₈×C₄₃

C₂²

yes

—

C₄₃×#498

—

C₂×#106×C₄₃

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂²×Dic₈₆

—

C₂²×C₄₃⋊C₈

D₄

yes

—

#13063325

—

#13063325

—

#13063325

D₄

yes

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

#13063250

—

#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

D₄

yes

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

#106×C₄₃

—

C₄×C₈×C₄₃

—

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×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

SD₁₆

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

#41

yes

characteristic

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

#120

yes

—

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

—

C₂×C₄×C₄₃

C₂×SD₁₆

yes

—

C₂×C₄×C₄₃

#135

yes

—

C₂×C₄×C₄₃

#137

yes

—

C₂×C₄×C₄₃

#137

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

#497

yes

—

C₂²×C₄₃

#497

yes

characteristic

100

C₄×C₄₃

—

101

C₄×C₄₃

#448

yes

—

102

C₄×C₄₃

#452

yes

—

103

C₄×C₄₃

#461

yes

—

104

C₄×C₄₃

#464

yes

—

105

Dic₄₃

—

106

Dic₄₃

—

107

C₂×#498

D₄₃

yes

characteristic

108

C₂×#487

—

109

#1990

—

110

#1956

—

111

#1273

—

112

C₂×C₄₃

#1708

yes

characteristic

113

C₂×C₄₃

#1716

yes

characteristic

114

C₂×C₄₃

#1718

yes

—

115

C₂×C₄₃

#2013

yes

characteristic

116

C₂×C₄₃

#2015

yes

—

117

C₂×C₄⋊Q₈

—

118

C₂×C₄⋊Q₈

C₂×D₄₃

yes

—

119

#498

—

120

#487

—

121

#487

—

122

C₂×#106

—

123

C₂×#103

—

124

C₂×#103

—

125

C₂×C₄×C₈

—

126

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

127

#384

—

128

#340

—

129

C₄₃

#10317

yes

characteristic, complemented, Sylow

130

C₂²×Q₈

—

131

C₂²×Q₈

—

132

C₂²×C₈

—

133

C₂²×C₈

C₂²⋊D₄₃

yes

—

134

C₄⋊Q₈

—

135

C₄⋊Q₈

—

136

C₄⋊Q₈

—

137

C₄⋊Q₈

—

138

C₄⋊Q₈

—

139

C₄⋊Q₈

—

140

C₄⋊Q₈

C₂²⋊D₄₃

yes

—

141

C₂×C₄⋊C₄

—

142

C₂×C₄⋊C₄

—

143

C₂×C₄⋊C₄

—

144

C₂×C₄⋊C₄

—

145

C₂×C₄⋊C₄

—

146

C₂×C₄²

C₂²×D₄₃

yes

characteristic

147

#106

—

148

#103

—

149

#103

—

172

—

150

C₄×C₈

—

151

C₄×C₈

—

152

C₄×C₈

—

153

C₂×Q₈

—

154

C₂×Q₈

—

155

C₂×Q₈

—

156

C₂×Q₈

—

157

C₂×Q₈

—

158

C₂×Q₈

—

172

—

159

C₂×Q₈

—

172

—

160

C₂×Q₈

—

172

—

161

C₂²×C₄

—

162

C₂²×C₄

—

172

—

163

C₂²×C₄

—

172

—

164

C₂²×C₄

D₄×D₄₃

yes

—

165

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

#10621884

yes

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

176

C₄⋊C₄

—

177

C₄⋊C₄

—

172

—

178

C₄⋊C₄

—

172

—

179

C₄⋊C₄

—

172

—

180

C₄²

D₄×D₄₃

yes

characteristic

181

C₄²

#10621900

yes

characteristic

182

C₄²

C₂×C₂²⋊D₄₃

yes

—

183

C₂³

#13063321

yes

characteristic

184

Q₈

—

185

Q₈

—

186

Q₈

—

344

—

187

Q₈

—

172

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

344

—

192

C₂×C₄

—

344

—

193

C₂×C₄

—

344

—

194

C₂×C₄

—

344

—

195

C₂×C₄

—

172

—

196

C₂×C₄

#13063293

yes

—

197

C₂×C₄

#13063294

yes

characteristic

198

C₂×C₄

#13063296

yes

characteristic

199

C₂×C₄

#13063297

yes

—

200

C₂×C₄

C₂×#10621884

yes

—

201

C₂×C₄

#13063346

yes

—

202

C₂×C₄

#13063354

yes

—

203

C₂×C₄

#13063356

yes

—

204

C₈

—

205

C₈

—

172

—

206

C₂²

#14582807

yes

characteristic

207

C₂²

#14583273

yes

characteristic

208

C₂²

#14583280

yes

—

209

C₂²

#14583290

yes

—

210

C₂²

#14583291

yes

characteristic

211

C₄

—

212

C₄

—

344

—

213

C₄

—

688

—

214

C₄

#14583038

yes

—

215

C₄

#14583058

yes

—

216

C₄

#14583101

yes

—

217

C₄

#14583119

yes

—

218

C₂

#19134197

yes

characteristic

219

C₂

#19134229

yes

characteristic

220

C₂

#19134239

yes

—

221

C₂

#19135678

yes

characteristic

222

C₂

#19135687

yes

—

	subgroup	G/H
center	C₂³	#13063321
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₂×C₄₃×#498	C₂
socle, prime radical	C₂³×C₄₃	#127
hypercenter	C₂×#498	D₄₃
hypocenter	C₄₃	#10317
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

subgroup

G/H

center

C₂³

#13063321

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₂×C₄₃×#498

C₂

C₂³×C₄₃

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

N	H ≅ G/N	type
129. C₄₃	#10317	semidirect

H ≅ G/N

type

129. C₄₃

#10317

semidirect

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

subgroup	index
C₂×#14583249	2
#19135557	2
#19135397	2
C₄₃⋊#1273	2
C₂×C₄₃×#498	2
#10317	43

subgroup

index

C₂×#14583249

#19135557

#19135397

C₄₃⋊#1273

C₂×C₄₃×#498

#10317

Subgroups — #24999437

Conjugacy classes of subgroups

Complements

Maximal subgroups