Group #25000725 — subgroups

Subgroups — #25000725

Order 11008.

402 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 99 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	#19145940	C₂	1	1	yes	maximal, characteristic
2	#19135572	C₂	1	2	yes	maximal
3	#19133283	C₂	1	2	yes	maximal
4	#19132470	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#1842	C₂	1	1	yes	maximal, characteristic
6	#14583456	C₂²	1	2	yes	—
7	#14583195	—	2	2	no	—
8	#14583140	—	2	2	no	—
9	#14583110	—	2	2	no	—
10	#14582923	C₂²	1	2	yes	—
11	#14582659	—	2	2	no	—
12	#14582659	—	2	2	no	—
13	#14582647	—	2	2	no	—
14	#14582646	—	2	2	no	—
15	C₂×C₄⋊C₈×C₄₃	—	2	2	no	—
16	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
18	C₄³×C₄₃	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₇₂	—	2	2	no	—
20	C₂×#10621877	—	2	2	no	—
21	#13063325	D₄	1	2	yes	—
22	#13063325	D₄	1	2	yes	—
23	#13063317	—	2	2	no	—
24	#13063304	—	2	2	no	—
25	#13063304	—	2	2	no	—
26	C₄×Dic₈₆	—	2	2	no	—
27	C₄×Dic₈₆	—	2	2	no	—
28	#13063284	—	2	2	no	—
29	#13063259	—	2	2	no	—
30	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
31	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
32	#13063250	—	2	4	no	—
33	C₂²×C₈×C₄₃	—	2	1	no	—
34	C₂²×C₈×C₄₃	D₄	1	2	yes	—
35	C₂×C₄²×C₄₃	—	2	1	no	—
36	C₂×C₄²×C₄₃	—	2	1	no	—
37	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₄₃	—	2	2	no	—
41	C₄⋊C₈×C₄₃	—	2	2	no	—
42	C₄⋊C₈×C₄₃	—	4	4	no	—
43	C₂²×C₄×C₄₃	—	2	1	no	—
44	C₂²×C₄×C₄₃	—	2	1	no	—
45	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
48	C₂×Dic₈₆	—	2	2	no	—
49	C₂×Dic₈₆	—	2	2	no	—
50	C₂×C₈×C₄₃	—	2	2	no	—
51	C₂×C₈×C₄₃	—	4	2	no	—
52	C₂×C₈×C₄₃	—	4	2	no	—
53	C₂×C₈×C₄₃	—	4	1	no	—
54	C₄²×C₄₃	—	2	2	no	—
55	C₄²×C₄₃	—	2	2	no	—
56	C₄²×C₄₃	—	2	1	no	—
57	C₄²×C₄₃	—	2	2	no	—
58	C₄²×C₄₃	—	4	2	no	—
59	C₄²×C₄₃	Q₁₆	1	2	yes	complemented
60	C₄²×C₄₃	C₂×D₄	1	2	yes	—
61	C₄²×C₄₃	#41	1	2	yes	—
62	C₂²⋊Dic₄₃	—	4	2	no	—
63	C₄⋊Dic₄₃	—	2	2	no	—
64	C₄⋊Dic₄₃	—	2	2	no	—
65	C₄⋊Dic₄₃	—	2	2	no	—
66	C₄⋊Dic₄₃	—	2	2	no	—
67	Dic₄₃⋊C₄	—	4	2	no	—
68	Dic₄₃⋊C₄	—	4	2	no	—
69	C₄×Dic₄₃	—	4	2	no	—
70	#10621877	—	4	2	no	—
71	#10621877	—	4	2	no	—
72	C₂×C₄₃⋊C₈	—	4	2	no	—
73	Dic₁₇₂	—	8	2	no	—
74	C₂³×C₄₃	#121	1	1	yes	characteristic
75	C₂×C₄×C₄₃	—	2	2	no	—
76	C₂×C₄×C₄₃	—	2	2	no	—
77	C₂×C₄×C₄₃	—	2	2	no	—
78	C₂×C₄×C₄₃	—	2	1	no	—
79	C₂×C₄×C₄₃	—	2	1	no	—
80	C₂×C₄×C₄₃	—	2	2	no	—
81	C₂×C₄×C₄₃	—	4	1	no	—
82	C₂×C₄×C₄₃	—	4	1	no	—
83	C₂×C₄×C₄₃	—	4	1	no	—
84	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
85	C₂×C₄×C₄₃	#121	1	2	yes	—
86	C₂×C₄×C₄₃	#123	1	2	yes	—
87	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
88	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
89	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
90	C₂×C₄×C₄₃	#137	1	2	yes	—
91	C₂×Dic₄₃	—	4	2	no	—
92	C₂×Dic₄₃	—	4	2	no	—
93	Dic₈₆	—	4	2	no	—
94	Dic₈₆	—	4	2	no	—
95	Dic₈₆	—	8	2	no	—
96	C₈×C₄₃	—	4	2	no	—
97	C₈×C₄₃	—	8	1	no	—
98	C₄₃⋊C₈	—	8	2	no	—
99	#10522	—	43	1	no	maximal, Sylow
100	C₂²×C₄₃	—	2	1	no	—
101	C₂²×C₄₃	—	2	1	no	—
102	C₂²×C₄₃	#394	1	1	yes	characteristic
103	C₂²×C₄₃	#467	1	1	yes	characteristic
104	C₂²×C₄₃	#470	1	1	yes	characteristic
105	C₄×C₄₃	—	2	4	no	—
106	C₄×C₄₃	—	2	2	no	—
107	C₄×C₄₃	—	2	1	no	—
108	C₄×C₄₃	—	4	2	no	—
109	C₄×C₄₃	#453	1	1	yes	characteristic
110	C₄×C₄₃	#456	1	1	yes	characteristic
111	C₄×C₄₃	#462	1	2	yes	—
112	C₄×C₄₃	#464	1	2	yes	—
113	Dic₄₃	—	8	2	no	—
114	Dic₄₃	—	8	2	no	—
115	#3249	—	43	1	no	—
116	#1994	—	43	2	no	—
117	#1842	D₄₃	1	1	yes	characteristic
118	#1550	—	43	2	no	—
119	#1292	—	43	1	no	—
120	C₂×C₄₃	—	2	1	no	—
121	C₂×C₄₃	—	2	1	no	—
122	C₂×C₄₃	#2039	1	1	yes	characteristic
123	C₂×C₄₃	#2050	1	1	yes	characteristic
124	C₂×C₄₃	#2056	1	1	yes	characteristic
125	#533	—	43	2	no	—
126	#477	—	86	2	no	—
127	#467	—	86	2	no	—
128	#462	—	86	2	no	—
129	#423	—	43	2	no	—
130	C₂×C₄⋊C₈	—	2	2	no	—
131	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
132	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
133	C₄³	C₂×D₄₃	1	1	yes	characteristic
134	#339	—	86	2	no	—
135	#339	—	86	2	no	—
136	#333	—	86	2	no	—
137	#332	—	86	2	no	—
138	C₄₃	#10522	1	1	yes	characteristic, complemented, Sylow
139	C₂×Q₁₆	—	86	2	no	—
140	C₂×M₁₆	—	86	2	no	—
141	C₂²×C₈	—	2	1	no	—
142	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
143	C₄⋊Q₈	—	43	2	no	—
144	C₄⋊Q₈	—	43	2	no	—
145	#125	—	86	2	no	—
146	C₂²⋊Q₈	—	86	2	no	—
147	C₂²⋊Q₈	—	86	2	no	—
148	C₄×Q₈	—	86	2	no	—
149	C₄×Q₈	—	86	2	no	—
150	#117	—	86	2	no	—
151	C₂×C₄²	—	2	1	no	—
152	C₂×C₄²	—	2	1	no	—
153	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
154	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
155	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
156	#107	—	86	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	4	4	no	—
160	C₄⋊C₈	—	86	2	no	—
161	C₄⋊C₈	—	86	2	no	—
162	#103	—	86	4	no	—
163	C₂×Q₈	—	86	2	no	—
164	C₂×Q₈	—	86	2	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
168	C₂²×C₄	#10621898	1	1	yes	characteristic
169	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
170	Q₁₆	—	344	2	no	—
171	M₁₆	—	172	2	no	—
172	M₁₆	—	172	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	—	4	2	no	—
176	C₂×C₈	—	4	1	no	—
177	C₂×C₈	—	172	2	no	—
178	C₄⋊C₄	—	86	2	no	—
179	C₄⋊C₄	—	86	2	no	—
180	C₄⋊C₄	—	86	2	no	—
181	C₄⋊C₄	—	86	2	no	—
182	C₄⋊C₄	—	172	2	no	—
183	C₄⋊C₄	—	172	2	no	—
184	C₂²⋊C₄	—	172	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	2	2	no	—
187	C₄²	—	2	1	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	4	2	no	—
190	C₄²	—	172	2	no	—
191	C₄²	Dic₁₇₂	1	2	yes	complemented
192	C₄²	D₄×D₄₃	1	2	yes	—
193	C₄²	#10621902	1	2	yes	—
194	C₂³	#13063299	1	1	yes	characteristic
195	Q₈	—	344	2	no	—
196	Q₈	—	172	2	no	—
197	Q₈	—	172	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	172	2	no	—
208	C₂×C₄	—	172	2	no	—
209	C₂×C₄	#13063293	1	1	yes	characteristic
210	C₂×C₄	#13063300	1	2	yes	—
211	C₂×C₄	#13063310	1	2	yes	—
212	C₂×C₄	#13063314	1	1	yes	characteristic
213	C₂×C₄	C₂×Dic₁₇₂	1	1	yes	characteristic
214	C₂×C₄	#13063347	1	1	yes	characteristic
215	C₂×C₄	#13063358	1	2	yes	—
216	C₈	—	4	2	no	—
217	C₈	—	8	1	no	—
218	C₈	—	344	2	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	#14582816	1	1	yes	characteristic
222	C₂²	#14583140	1	1	yes	characteristic
223	C₂²	#14583157	1	1	yes	characteristic
224	C₄	—	2	4	no	—
225	C₄	—	2	1	no	—
226	C₄	—	2	2	no	—
227	C₄	—	4	2	no	—
228	C₄	—	344	2	no	—
229	C₄	—	344	2	no	—
230	C₄	#14583066	1	1	yes	characteristic
231	C₄	#14583081	1	1	yes	characteristic
232	C₄	#14583110	1	2	yes	—
233	C₄	#14583122	1	2	yes	—
234	C₂	—	2	1	no	—
235	C₂	—	2	1	no	—
236	C₂	#19135819	1	1	yes	characteristic
237	C₂	#19135882	1	1	yes	characteristic
238	C₂	#19135918	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19145940

C₂

yes

maximal, characteristic

#19135572

C₂

yes

maximal

#19133283

C₂

yes

maximal

#19132470

C₂

yes

maximal, characteristic

C₄₃×#1842

C₂

yes

maximal, characteristic

#14583456

C₂²

yes

—

—

—

—

C₂²

yes

—

—

—

—

—

C₂×C₄⋊C₈×C₄₃

—

C₂×C₄⋊C₈×C₄₃

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₄₃

C₂²

yes

characteristic

C₄³×C₄₃

C₂²

yes

characteristic

C₂×Dic₁₇₂

—

C₂×#10621877

—

#13063325

D₄

yes

—

#13063325

D₄

yes

—

#13063317

—

#13063304

—

#13063304

—

C₄×Dic₈₆

—

C₄×Dic₈₆

—

#13063284

—

#13063259

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

#13063250

—

C₂²×C₈×C₄₃

—

C₂²×C₈×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

—

C₂×C₄²×C₄₃

—

C₂×C₄²×C₄₃

D₄

yes

characteristic

C₂×C₄²×C₄₃

D₄

yes

characteristic

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

characteristic

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

Q₁₆

yes

complemented

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

#41

yes

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

#10621877

—

#10621877

—

C₂×C₄₃⋊C₈

—

Dic₁₇₂

—

C₂³×C₄₃

#121

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

#120

yes

characteristic

C₂×C₄×C₄₃

#121

yes

—

C₂×C₄×C₄₃

#123

yes

—

C₂×C₄×C₄₃

#124

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₁₆

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

#10522

—

maximal, Sylow

100

C₂²×C₄₃

—

101

C₂²×C₄₃

—

102

C₂²×C₄₃

#394

yes

characteristic

103

C₂²×C₄₃

#467

yes

characteristic

104

C₂²×C₄₃

#470

yes

characteristic

105

C₄×C₄₃

—

106

C₄×C₄₃

—

107

C₄×C₄₃

—

108

C₄×C₄₃

—

109

C₄×C₄₃

#453

yes

characteristic

110

C₄×C₄₃

#456

yes

characteristic

111

C₄×C₄₃

#462

yes

—

112

C₄×C₄₃

#464

yes

—

113

Dic₄₃

—

114

Dic₄₃

—

115

#3249

—

116

#1994

—

117

#1842

D₄₃

yes

characteristic

118

#1550

—

119

#1292

—

120

C₂×C₄₃

—

121

C₂×C₄₃

—

122

C₂×C₄₃

#2039

yes

characteristic

123

C₂×C₄₃

#2050

yes

characteristic

124

C₂×C₄₃

#2056

yes

characteristic

125

#533

—

126

#477

—

127

#467

—

128

#462

—

129

#423

—

130

C₂×C₄⋊C₈

—

131

C₂×C₄⋊C₈

C₂×D₄₃

yes

characteristic

132

C₂×C₄⋊C₈

C₂×D₄₃

yes

characteristic

133

C₄³

C₂×D₄₃

yes

characteristic

134

#339

—

135

#339

—

136

#333

—

137

#332

—

138

C₄₃

#10522

yes

characteristic, complemented, Sylow

139

C₂×Q₁₆

—

140

C₂×M₁₆

—

141

C₂²×C₈

—

142

C₂²×C₈

C₂²⋊D₄₃

yes

—

143

C₄⋊Q₈

—

144

C₄⋊Q₈

—

145

#125

—

146

C₂²⋊Q₈

—

147

C₂²⋊Q₈

—

148

C₄×Q₈

—

149

C₄×Q₈

—

150

#117

—

151

C₂×C₄²

—

152

C₂×C₄²

—

153

C₂×C₄²

D₁₇₂

yes

characteristic

154

C₂×C₄²

D₁₇₂

yes

characteristic

155

C₂×C₄²

C₂²×D₄₃

yes

characteristic

156

#107

—

157

C₄⋊C₈

—

158

C₄⋊C₈

—

159

C₄⋊C₈

—

160

C₄⋊C₈

—

161

C₄⋊C₈

—

162

#103

—

163

C₂×Q₈

—

164

C₂×Q₈

—

165

C₂²×C₄

—

166

C₂²×C₄

—

167

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

168

C₂²×C₄

#10621898

yes

characteristic

169

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

170

Q₁₆

—

344

—

171

M₁₆

—

172

—

172

M₁₆

—

172

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

176

C₂×C₈

—

177

C₂×C₈

—

172

—

178

C₄⋊C₄

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₄⋊C₄

—

172

—

183

C₄⋊C₄

—

172

—

184

C₂²⋊C₄

—

172

—

185

C₄²

—

186

C₄²

—

187

C₄²

—

188

C₄²

—

189

C₄²

—

190

C₄²

—

172

—

191

C₄²

Dic₁₇₂

yes

complemented

192

C₄²

D₄×D₄₃

yes

—

193

C₄²

#10621902

yes

—

194

C₂³

#13063299

yes

characteristic

195

Q₈

—

344

—

196

Q₈

—

172

—

197

Q₈

—

172

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

—

172

—

208

C₂×C₄

—

172

—

209

C₂×C₄

#13063293

yes

characteristic

210

C₂×C₄

#13063300

yes

—

211

C₂×C₄

#13063310

yes

—

212

C₂×C₄

#13063314

yes

characteristic

213

C₂×C₄

C₂×Dic₁₇₂

yes

characteristic

214

C₂×C₄

#13063347

yes

characteristic

215

C₂×C₄

#13063358

yes

—

216

C₈

—

217

C₈

—

218

C₈

—

344

—

219

C₂²

—

220

C₂²

—

221

C₂²

#14582816

yes

characteristic

222

C₂²

#14583140

yes

characteristic

223

C₂²

#14583157

yes

characteristic

224

C₄

—

225

C₄

—

226

C₄

—

227

C₄

—

228

C₄

—

344

—

229

C₄

—

344

—

230

C₄

#14583066

yes

characteristic

231

C₄

#14583081

yes

characteristic

232

C₄

#14583110

yes

—

233

C₄

#14583122

yes

—

234

C₂

—

235

C₂

—

236

C₂

#19135819

yes

characteristic

237

C₂

#19135882

yes

characteristic

238

C₂

#19135918

yes

characteristic

	subgroup	G/H
center	C₂²	#14582816
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1842	C₂
socle	C₂²×C₄₃	#394
prime radical	C₂³×C₄₃	#121
hypercenter	#1842	D₄₃
hypocenter	C₄₃	#10522
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1842

subgroup

G/H

center

C₂²

#14582816

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1842

C₂

C₂²×C₄₃

C₂³×C₄₃

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1842

N	H ≅ G/N	type
59. C₄²×C₄₃	Q₁₆	semidirect
138. C₄₃	#10522	semidirect
191. C₄²	Dic₁₇₂	semidirect

H ≅ G/N

type

59. C₄²×C₄₃

Q₁₆

semidirect

138. C₄₃

#10522

semidirect

191. C₄²

Dic₁₇₂

semidirect

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

subgroup	index
#19145940	2
#19135572	2
#19133283	2
#19132470	2
C₄₃×#1842	2
#10522	43

subgroup

index

C₄₃×#1842

Subgroups — #25000725

Conjugacy classes of subgroups

Complements

Maximal subgroups