Group #25015673 — subgroups

Subgroups — #25015673

Order 11008.

382 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 117 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×#14582702	C₂	1	2	yes	maximal
2	#19135456	C₂	1	1	yes	maximal, characteristic
3	#19132686	C₂	1	2	yes	maximal
4	#19132673	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₃×#499	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
7	C₂²×C₄₃⋊C₁₆	C₂²	1	2	yes	—
8	C₂×#13063259	C₂²	1	1	yes	characteristic
9	C₂×#13063250	—	2	1	no	—
10	#14582805	—	2	1	no	—
11	#14582702	—	2	4	no	—
12	C₂×#125×C₄₃	—	2	1	no	—
13	C₄₃×#499	—	2	4	no	—
14	C₂×#107×C₄₃	C₂²	1	2	yes	—
15	C₂×#106×C₄₃	—	2	1	no	—
16	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
17	C₂²×Dic₈₆	—	2	1	no	—
18	C₂×Dic₁₇₂	—	2	2	no	—
19	C₂×Dic₁₇₂	—	2	2	no	—
20	C₂×Dic₁₇₂	—	2	1	no	—
21	C₂×Dic₁₇₂	Q₈	1	2	yes	—
22	C₂×C₄⋊Dic₄₃	—	2	1	no	—
23	C₂×C₄₃⋊C₁₆	—	2	2	no	—
24	C₂×C₄₃⋊C₁₆	—	2	2	no	—
25	C₂×C₄₃⋊C₁₆	—	2	2	no	—
26	#13063259	—	2	2	no	—
27	#13063250	—	4	2	no	—
28	C₄₃⋊#95	—	4	2	no	—
29	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
31	#125×C₄₃	—	2	4	no	—
32	#125×C₄₃	—	4	2	no	—
33	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₄₃	—	4	2	no	—
35	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
36	#107×C₄₃	—	2	2	no	—
37	#107×C₄₃	D₄	1	4	yes	—
38	#106×C₄₃	—	2	4	no	—
39	C₄×C₈×C₄₃	—	2	2	no	—
40	C₂²×C₄×C₄₃	—	4	2	no	—
41	C₂²×C₄×C₄₃	D₈	1	2	yes	—
42	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₄₃	—	4	1	no	—
44	C₂²×Dic₄₃	—	4	1	no	—
45	C₂×Dic₈₆	—	2	2	no	—
46	C₂×Dic₈₆	—	4	1	no	—
47	C₂×Dic₈₆	—	4	1	no	—
48	C₂×Dic₈₆	—	4	1	no	—
49	C₂×Dic₈₆	—	4	1	no	—
50	C₂×Dic₈₆	—	4	1	no	—
51	C₂×C₈×C₄₃	—	2	1	no	—
52	C₂×C₈×C₄₃	—	2	1	no	—
53	C₂×C₈×C₄₃	SD₁₆	1	2	yes	—
54	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
55	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₄₃	#41	1	2	yes	—
57	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
58	C₄⋊C₄×C₄₃	—	2	4	no	—
59	C₄⋊C₄×C₄₃	—	4	8	no	—
60	C₄⋊C₄×C₄₃	—	4	2	no	—
61	C₄²×C₄₃	—	2	1	no	—
62	C₄²×C₄₃	—	2	1	no	—
63	C₄⋊Dic₄₃	—	4	1	no	—
64	C₄⋊Dic₄₃	—	4	1	no	—
65	Dic₁₇₂	—	2	4	no	—
66	Dic₁₇₂	—	4	2	no	—
67	C₄₃⋊C₁₆	—	4	2	no	—
68	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₄₃	—	2	2	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	2	no	—
74	C₂×C₄×C₄₃	—	8	2	no	—
75	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
76	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
78	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
81	C₂×Dic₄₃	—	4	2	no	—
82	C₂×Dic₄₃	—	8	1	no	—
83	C₂×Dic₄₃	—	8	1	no	—
84	C₂×Dic₄₃	—	8	1	no	—
85	C₂×Dic₄₃	—	8	2	no	—
86	Dic₈₆	—	4	2	no	—
87	Dic₈₆	—	4	2	no	—
88	Dic₈₆	—	8	1	no	—
89	Dic₈₆	—	8	1	no	—
90	C₈×C₄₃	—	2	2	no	—
91	C₈×C₄₃	C₂²⋊Q₈	1	2	yes	—
92	C₈×C₄₃	#124	1	2	yes	—
93	#13366	—	43	1	no	maximal, Sylow
94	C₂²×C₄₃	#447	1	1	yes	characteristic
95	C₂²×C₄₃	#474	1	1	yes	characteristic
96	C₂²×C₄₃	#486	1	1	yes	characteristic
97	C₂²×C₄₃	#508	1	2	yes	—
98	C₂²×C₄₃	#510	1	2	yes	—
99	C₄×C₄₃	—	4	2	no	—
100	C₄×C₄₃	—	8	4	no	—
101	C₄×C₄₃	#393	1	1	yes	characteristic
102	C₄×C₄₃	#448	1	1	yes	characteristic
103	C₄×C₄₃	#476	1	1	yes	characteristic
104	C₄×C₄₃	#489	1	1	yes	characteristic
105	Dic₄₃	—	8	2	no	—
106	Dic₄₃	—	16	1	no	—
107	C₂×#499	D₄₃	1	1	yes	characteristic
108	C₂×#358	—	43	2	no	—
109	#1969	—	43	1	no	—
110	#1380	—	43	2	no	—
111	#1374	—	43	1	no	—
112	C₂×C₄₃	#2023	1	1	yes	characteristic
113	C₂×C₄₃	#2189	1	2	yes	—
114	C₂×C₄₃	#2229	1	2	yes	—
115	C₂×C₄₃	#2242	1	1	yes	characteristic
116	C₂×C₄₃	#2245	1	1	yes	characteristic
117	C₂²×Q₁₆	—	43	1	no	—
118	C₂×#125	—	2	1	no	—
119	C₂²×C₁₆	—	43	2	no	—
120	#499	—	2	4	no	—
121	C₂×#107	—	43	1	no	—
122	C₂×#107	C₂×D₄₃	1	2	yes	—
123	C₂×#106	—	2	1	no	—
124	C₂×#103	—	86	1	no	—
125	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
126	#391	—	86	1	no	—
127	#358	—	86	4	no	—
128	C₄₃	#13366	1	1	yes	characteristic, complemented, Sylow
129	C₂²×Q₈	—	86	1	no	—
130	C₂×Q₁₆	—	43	2	no	—
131	C₂×Q₁₆	—	86	2	no	—
132	C₂×Q₁₆	—	86	1	no	—
133	C₂×Q₁₆	—	86	2	no	—
134	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
136	#125	—	2	4	no	—
137	#125	—	4	2	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	4	2	no	—
140	C₂×C₄⋊C₄	—	86	1	no	—
141	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
142	C₂×C₁₆	—	86	2	no	—
143	C₂×C₁₆	—	86	2	no	—
144	C₂×C₁₆	—	86	2	no	—
145	#107	—	2	2	no	—
146	#107	—	86	2	no	—
147	#107	C₂²⋊D₄₃	1	4	yes	—
148	#106	—	2	4	no	—
149	#103	—	172	2	no	—
150	C₄×C₈	—	2	2	no	—
151	#95	—	172	2	no	—
152	C₂×Q₈	—	86	2	no	—
153	C₂×Q₈	—	172	1	no	—
154	C₂×Q₈	—	172	1	no	—
155	C₂×Q₈	—	172	1	no	—
156	C₂×Q₈	—	172	1	no	—
157	C₂×Q₈	—	172	1	no	—
158	C₂²×C₄	—	4	2	no	—
159	C₂²×C₄	—	172	1	no	—
160	C₂²×C₄	—	172	1	no	—
161	C₂²×C₄	#10621882	1	2	yes	—
162	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
163	Q₁₆	—	86	4	no	—
164	Q₁₆	—	172	2	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	#10621884	1	2	yes	—
168	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
169	C₂×C₈	#10621902	1	2	yes	—
170	C₂×C₈	#10621902	1	1	yes	characteristic
171	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	4	8	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	172	1	no	—
176	C₄⋊C₄	—	172	1	no	—
177	C₄²	—	2	1	no	—
178	C₄²	—	2	1	no	—
179	C₁₆	—	172	2	no	—
180	C₂³	C₂×#10621882	1	1	yes	characteristic
181	Q₈	—	344	1	no	—
182	Q₈	—	344	1	no	—
183	Q₈	—	172	2	no	—
184	Q₈	—	172	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	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	2	no	—
194	C₂×C₄	—	344	1	no	—
195	C₂×C₄	—	172	2	no	—
196	C₂×C₄	#13063294	1	1	yes	characteristic
197	C₂×C₄	#13063301	1	1	yes	characteristic
198	C₂×C₄	#13063314	1	1	yes	characteristic
199	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
200	C₂×C₄	#13063351	1	1	yes	characteristic
201	C₂×C₄	#13063356	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	#13063301	1	2	yes	—
204	C₈	#13063314	1	2	yes	—
205	C₂²	#14583033	1	1	yes	characteristic
206	C₂²	#14583176	1	1	yes	characteristic
207	C₂²	#14583247	1	1	yes	characteristic
208	C₂²	#14583329	1	2	yes	—
209	C₂²	#14583339	1	2	yes	—
210	C₄	—	4	2	no	—
211	C₄	—	8	4	no	—
212	C₄	—	344	2	no	—
213	C₄	—	688	1	no	—
214	C₄	#14582810	1	1	yes	characteristic
215	C₄	#14583038	1	1	yes	characteristic
216	C₄	#14583186	1	1	yes	characteristic
217	C₄	#14583261	1	1	yes	characteristic
218	C₂	#19135728	1	1	yes	characteristic
219	C₂	#19136535	1	2	yes	—
220	C₂	#19136756	1	2	yes	—
221	C₂	#19136829	1	1	yes	characteristic
222	C₂	#19136842	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14582702

C₂

yes

maximal

#19135456

C₂

yes

maximal, characteristic

#19132686

C₂

yes

maximal

#19132673

C₂

yes

maximal, characteristic

C₂×C₄₃×#499

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₂

C₂²

yes

characteristic

C₂²×C₄₃⋊C₁₆

C₂²

yes

—

C₂×#13063259

C₂²

yes

—

—

—

C₂×#125×C₄₃

—

C₄₃×#499

—

C₂×#107×C₄₃

C₂²

yes

—

C₂×#106×C₄₃

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂²×Dic₈₆

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

Q₈

yes

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₁₆

—

C₂×C₄₃⋊C₁₆

—

C₂×C₄₃⋊C₁₆

—

#13063259

—

#13063250

—

C₄₃⋊#95

—

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂²×C₈×C₄₃

C₂³

yes

characteristic

#125×C₄₃

—

#125×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄²×C₄₃

D₄

yes

characteristic

#107×C₄₃

—

#107×C₄₃

D₄

yes

—

#106×C₄₃

—

C₄×C₈×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

D₈

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

SD₁₆

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₂×C₈×C₄₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₃

#41

yes

—

C₂×C₈×C₄₃

#41

yes

characteristic

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₁₇₂

—

Dic₁₇₂

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

C₂×D₈

yes

characteristic

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

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₄₃

#124

yes

characteristic

C₂×C₄×C₄₃

C₂×SD₁₆

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

C₂²⋊Q₈

yes

—

C₈×C₄₃

#124

yes

—

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

#510

yes

—

C₄×C₄₃

—

100

C₄×C₄₃

—

101

C₄×C₄₃

#393

yes

characteristic

102

C₄×C₄₃

#448

yes

characteristic

103

C₄×C₄₃

#476

yes

characteristic

104

C₄×C₄₃

#489

yes

characteristic

105

Dic₄₃

—

106

Dic₄₃

—

107

C₂×#499

D₄₃

yes

characteristic

108

C₂×#358

—

109

#1969

—

110

#1380

—

111

#1374

—

112

C₂×C₄₃

#2023

yes

characteristic

113

C₂×C₄₃

#2189

yes

—

114

C₂×C₄₃

#2229

yes

—

115

C₂×C₄₃

#2242

yes

characteristic

116

C₂×C₄₃

#2245

yes

characteristic

117

C₂²×Q₁₆

—

118

C₂×#125

—

119

C₂²×C₁₆

—

120

#499

—

121

C₂×#107

—

122

C₂×#107

C₂×D₄₃

yes

—

123

C₂×#106

—

124

C₂×#103

—

125

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

126

#391

—

127

#358

—

128

C₄₃

#13366

yes

characteristic, complemented, Sylow

129

C₂²×Q₈

—

130

C₂×Q₁₆

—

131

C₂×Q₁₆

—

132

C₂×Q₁₆

—

133

C₂×Q₁₆

—

134

C₂²×C₈

C₂²⋊D₄₃

yes

characteristic

135

C₂²×C₈

C₂²×D₄₃

yes

characteristic

136

#125

—

137

#125

—

138

C₂×C₄⋊C₄

—

139

C₂×C₄⋊C₄

—

140

C₂×C₄⋊C₄

—

141

C₂×C₄²

C₂²⋊D₄₃

yes

characteristic

142

C₂×C₁₆

—

143

C₂×C₁₆

—

144

C₂×C₁₆

—

145

#107

—

146

#107

—

147

#107

C₂²⋊D₄₃

yes

—

148

#106

—

149

#103

—

172

—

150

C₄×C₈

—

151

#95

—

172

—

152

C₂×Q₈

—

153

C₂×Q₈

—

172

—

154

C₂×Q₈

—

172

—

155

C₂×Q₈

—

172

—

156

C₂×Q₈

—

172

—

157

C₂×Q₈

—

172

—

158

C₂²×C₄

—

159

C₂²×C₄

—

172

—

160

C₂²×C₄

—

172

—

161

C₂²×C₄

#10621882

yes

—

162

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

163

Q₁₆

—

164

Q₁₆

—

172

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

#10621884

yes

—

168

C₂×C₈

Q₈×D₄₃

yes

characteristic

169

C₂×C₈

#10621902

yes

—

170

C₂×C₈

#10621902

yes

characteristic

171

C₂×C₈

C₂×C₂²⋊D₄₃

yes

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

172

—

176

C₄⋊C₄

—

172

—

177

C₄²

—

178

C₄²

—

179

C₁₆

—

172

—

180

C₂³

C₂×#10621882

yes

characteristic

181

Q₈

—

344

—

182

Q₈

—

344

—

183

Q₈

—

172

—

184

Q₈

—

172

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

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

#13063294

yes

characteristic

197

C₂×C₄

#13063301

yes

characteristic

198

C₂×C₄

#13063314

yes

characteristic

199

C₂×C₄

C₂×#10621884

yes

characteristic

200

C₂×C₄

#13063351

yes

characteristic

201

C₂×C₄

#13063356

yes

characteristic

202

C₈

—

203

C₈

#13063301

yes

—

204

C₈

#13063314

yes

—

205

C₂²

#14583033

yes

characteristic

206

C₂²

#14583176

yes

characteristic

207

C₂²

#14583247

yes

characteristic

208

C₂²

#14583329

yes

—

209

C₂²

#14583339

yes

—

210

C₄

—

211

C₄

—

212

C₄

—

344

—

213

C₄

—

688

—

214

C₄

#14582810

yes

characteristic

215

C₄

#14583038

yes

characteristic

216

C₄

#14583186

yes

characteristic

217

C₄

#14583261

yes

characteristic

218

C₂

#19135728

yes

characteristic

219

C₂

#19136535

yes

—

220

C₂

#19136756

yes

—

221

C₂

#19136829

yes

characteristic

222

C₂

#19136842

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#10621882
commutator, cosocle	C₂²×C₈×C₄₃	C₂³
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂×C₄₃×#499	C₂
socle, prime radical	C₂³×C₄₃	C₂×D₈
hypercenter	C₂×#499	D₄₃
hypocenter	C₄₃	#13366
LCS	G ▷ C₂²×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

subgroup

G/H

center

C₂³

C₂×#10621882

commutator, cosocle

C₂²×C₈×C₄₃

C₂³

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting

C₂×C₄₃×#499

C₂

socle, prime radical

C₂³×C₄₃

C₂×D₈

D₄₃

C₄₃

G ▷ C₂²×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

N	H ≅ G/N	type
128. C₄₃	#13366	semidirect

H ≅ G/N

type

128. C₄₃

#13366

semidirect

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

subgroup	index
C₂×#14582702	2
#19135456	2
#19132686	2
#19132673	2
C₂×C₄₃×#499	2
#13366	43

subgroup

index

C₂×C₄₃×#499

Subgroups — #25015673

Conjugacy classes of subgroups

Complements

Maximal subgroups