Group #25016204 — subgroups

Subgroups — #25016204

Order 11008.

410 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 115 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	C₂²×Dic₃₄₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582704	C₂	1	2	yes	maximal
3	#19135456	C₂	1	2	yes	maximal
4	#19132687	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₂	C₂²	1	2	yes	—
7	C₂×Dic₃₄₄	—	2	4	no	—
8	C₂×Dic₃₄₄	—	2	2	no	—
9	C₂×#13063259	C₂²	1	2	yes	—
10	C₂×#13063250	—	2	2	no	—
11	#14582805	—	2	2	no	—
12	#14582704	—	2	4	no	—
13	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
16	C₄₃×C₄⋊C₁₆	—	2	2	no	—
17	C₂²×Dic₈₆	—	2	2	no	—
18	C₂×Dic₁₇₂	—	2	2	no	—
19	C₂×Dic₁₇₂	—	2	2	no	—
20	C₂×Dic₁₇₂	—	2	2	no	—
21	C₂×Dic₁₇₂	—	2	2	no	—
22	C₂×Dic₁₇₂	—	2	2	no	—
23	C₂×Dic₁₇₂	D₄	1	4	yes	—
24	C₂×C₄⋊Dic₄₃	—	2	2	no	—
25	Dic₃₄₄	—	2	8	no	—
26	#13063259	—	2	2	no	—
27	#13063259	—	2	2	no	—
28	#13063250	—	4	4	no	—
29	C₄₃⋊#95	—	4	2	no	—
30	C₄₃⋊#95	—	4	2	no	—
31	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₄₃	—	2	2	no	—
35	C₂×C₁₆×C₄₃	—	2	1	no	—
36	C₂×C₁₆×C₄₃	—	2	2	no	—
37	C₂×C₁₆×C₄₃	D₄	1	2	yes	—
38	C₄×C₈×C₄₃	—	2	1	no	—
39	C₄×C₈×C₄₃	—	2	1	no	—
40	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
41	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₄₃	—	4	2	no	—
44	C₂²×Dic₄₃	—	4	2	no	—
45	C₂×Dic₈₆	—	2	4	no	—
46	C₂×Dic₈₆	—	4	2	no	—
47	C₂×Dic₈₆	—	4	2	no	—
48	C₂×Dic₈₆	—	4	2	no	—
49	C₂×Dic₈₆	—	4	2	no	—
50	C₂×Dic₈₆	—	4	2	no	—
51	C₂×C₈×C₄₃	—	2	1	no	—
52	C₂×C₈×C₄₃	—	2	2	no	—
53	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
54	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
56	C₂×C₈×C₄₃	#41	1	2	yes	—
57	C₄²×C₄₃	—	2	1	no	—
58	C₄²×C₄₃	—	2	1	no	—
59	C₄⋊Dic₄₃	—	4	2	no	—
60	C₄⋊Dic₄₃	—	4	2	no	—
61	Dic₁₇₂	—	2	4	no	—
62	Dic₁₇₂	—	2	4	no	—
63	Dic₁₇₂	—	4	2	no	—
64	Dic₁₇₂	—	4	2	no	—
65	C₁₆×C₄₃	—	2	2	no	—
66	C₁₆×C₄₃	—	4	1	no	—
67	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₄₃	—	2	1	no	—
69	C₂×C₄×C₄₃	—	2	1	no	—
70	C₂×C₄×C₄₃	—	2	1	no	—
71	C₂×C₄×C₄₃	—	2	2	no	—
72	C₂×C₄×C₄₃	Q₃₂	1	2	yes	complemented
73	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
74	C₂×C₄×C₄₃	#121	1	2	yes	—
75	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
76	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×Dic₄₃	—	4	4	no	—
78	C₂×Dic₄₃	—	8	2	no	—
79	C₂×Dic₄₃	—	8	2	no	—
80	C₂×Dic₄₃	—	8	2	no	—
81	C₂×Dic₄₃	—	8	2	no	—
82	C₂×Dic₄₃	—	8	2	no	—
83	Dic₈₆	—	4	4	no	—
84	Dic₈₆	—	4	4	no	—
85	Dic₈₆	—	8	2	no	—
86	Dic₈₆	—	8	2	no	—
87	C₈×C₄₃	—	4	1	no	—
88	C₈×C₄₃	#120	1	1	yes	characteristic
89	C₈×C₄₃	#123	1	2	yes	—
90	C₈×C₄₃	#124	1	1	yes	characteristic
91	#13450	—	43	1	no	maximal, Sylow
92	C₂²×C₄₃	#459	1	2	yes	—
93	C₂²×C₄₃	#466	1	1	yes	characteristic
94	C₂²×C₄₃	C₂×Q₃₂	1	1	yes	characteristic
95	C₂²×C₄₃	#508	1	1	yes	characteristic
96	C₂²×C₄₃	#510	1	2	yes	—
97	C₄×C₄₃	—	2	2	no	—
98	C₄×C₄₃	—	4	1	no	—
99	C₄×C₄₃	#394	1	1	yes	characteristic
100	C₄×C₄₃	#463	1	2	yes	—
101	C₄×C₄₃	#469	1	1	yes	characteristic
102	Dic₄₃	—	8	4	no	—
103	Dic₄₃	—	16	2	no	—
104	C₂²×Q₃₂	—	43	1	no	—
105	C₂×C₄⋊C₁₆	D₄₃	1	1	yes	characteristic
106	C₂×#358	—	43	2	no	—
107	#1969	—	43	2	no	—
108	#1380	—	43	1	no	—
109	C₂×C₄₃	#2054	1	1	yes	characteristic
110	C₂×C₄₃	#2209	1	2	yes	—
111	C₂×C₄₃	#2211	1	2	yes	—
112	C₂×C₄₃	#2216	1	1	yes	characteristic
113	C₂×C₄₃	#2219	1	1	yes	characteristic
114	C₂²×Q₁₆	—	43	2	no	—
115	C₂×Q₃₂	—	86	4	no	—
116	C₂×Q₃₂	—	86	2	no	—
117	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
118	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
119	C₂×#107	—	43	2	no	—
120	C₂×#103	—	86	2	no	—
121	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
122	#391	—	86	2	no	—
123	C₄⋊C₁₆	—	2	2	no	—
124	#358	—	86	4	no	—
125	C₄₃	#13450	1	1	yes	characteristic, complemented, Sylow
126	C₂²×Q₈	—	86	2	no	—
127	C₂×Q₁₆	—	43	4	no	—
128	C₂×Q₁₆	—	86	2	no	—
129	C₂×Q₁₆	—	86	2	no	—
130	C₂×Q₁₆	—	86	2	no	—
131	C₂×Q₁₆	—	86	2	no	—
132	C₂×Q₁₆	—	86	2	no	—
133	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
134	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
135	C₂×C₄⋊C₄	—	86	2	no	—
136	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
137	Q₃₂	—	86	8	no	—
138	C₂×C₁₆	—	2	2	no	—
139	C₂×C₁₆	—	2	2	no	—
140	C₂×C₁₆	—	2	1	no	—
141	C₂×C₁₆	C₂²⋊D₄₃	1	2	yes	—
142	#107	—	86	2	no	—
143	#107	—	86	2	no	—
144	#103	—	172	4	no	—
145	C₄×C₈	—	2	1	no	—
146	C₄×C₈	—	2	1	no	—
147	#95	—	172	2	no	—
148	#95	—	172	2	no	—
149	C₂×Q₈	—	86	4	no	—
150	C₂×Q₈	—	172	2	no	—
151	C₂×Q₈	—	172	2	no	—
152	C₂×Q₈	—	172	2	no	—
153	C₂×Q₈	—	172	2	no	—
154	C₂×Q₈	—	172	2	no	—
155	C₂²×C₄	—	172	2	no	—
156	C₂²×C₄	—	172	2	no	—
157	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
158	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
159	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
160	Q₁₆	—	86	4	no	—
161	Q₁₆	—	86	4	no	—
162	Q₁₆	—	172	2	no	—
163	Q₁₆	—	172	2	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	#10621898	1	1	yes	characteristic
167	C₂×C₈	D₄×D₄₃	1	2	yes	—
168	C₂×C₈	#10621902	1	2	yes	—
169	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
170	C₄⋊C₄	—	172	2	no	—
171	C₄⋊C₄	—	172	2	no	—
172	C₄²	—	2	1	no	—
173	C₄²	—	2	1	no	—
174	C₁₆	—	2	2	no	—
175	C₁₆	—	4	1	no	—
176	C₂³	C₂×D₃₄₄	1	1	yes	characteristic
177	Q₈	—	344	2	no	—
178	Q₈	—	344	2	no	—
179	Q₈	—	172	4	no	—
180	Q₈	—	172	4	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	344	2	no	—
186	C₂×C₄	—	344	2	no	—
187	C₂×C₄	—	344	2	no	—
188	C₂×C₄	—	344	2	no	—
189	C₂×C₄	—	344	2	no	—
190	C₂×C₄	—	172	4	no	—
191	C₂×C₄	Dic₃₄₄	1	2	yes	complemented
192	C₂×C₄	#13063299	1	1	yes	characteristic
193	C₂×C₄	#13063300	1	2	yes	—
194	C₂×C₄	#13063347	1	1	yes	characteristic
195	C₂×C₄	#13063353	1	2	yes	—
196	C₈	—	4	1	no	—
197	C₈	#13063293	1	1	yes	characteristic
198	C₈	#13063310	1	2	yes	—
199	C₈	#13063314	1	1	yes	characteristic
200	C₂²	#14583095	1	2	yes	—
201	C₂²	#14583135	1	1	yes	characteristic
202	C₂²	C₂×Dic₃₄₄	1	1	yes	characteristic
203	C₂²	#14583331	1	1	yes	characteristic
204	C₂²	#14583342	1	2	yes	—
205	C₄	—	2	2	no	—
206	C₄	—	4	1	no	—
207	C₄	—	344	4	no	—
208	C₄	—	688	2	no	—
209	C₄	#14582816	1	1	yes	characteristic
210	C₄	#14583115	1	2	yes	—
211	C₄	#14583152	1	1	yes	characteristic
212	C₂	#19135906	1	1	yes	characteristic
213	C₂	#19136646	1	2	yes	—
214	C₂	#19136658	1	2	yes	—
215	C₂	#19136686	1	1	yes	characteristic
216	C₂	#19136703	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₃₄₄

C₂

yes

maximal, characteristic

C₂×#14582704

C₂

yes

maximal

#19135456

C₂

yes

maximal

#19132687

C₂

yes

maximal, characteristic

C₂×C₄₃×C₄⋊C₁₆

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₂

C₂²

yes

—

C₂×Dic₃₄₄

—

C₂×Dic₃₄₄

—

C₂×#13063259

C₂²

yes

—

C₂×#13063250

—

#14582805

—

#14582704

—

C₂²×C₁₆×C₄₃

C₂²

yes

characteristic

C₂²×C₁₆×C₄₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×C₄⋊C₁₆

—

C₂²×Dic₈₆

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

D₄

yes

—

C₂×C₄⋊Dic₄₃

—

Dic₃₄₄

—

#13063259

—

#13063259

—

#13063250

—

C₄₃⋊#95

—

C₄₃⋊#95

—

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂²×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₄²×C₄₃

D₄

yes

characteristic

C₂×C₁₆×C₄₃

—

C₂×C₁₆×C₄₃

—

C₂×C₁₆×C₄₃

—

C₂×C₁₆×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

—

C₄×C₈×C₄₃

—

C₂²×C₄×C₄₃

D₈

yes

characteristic

C₂²×C₄×C₄₃

D₈

yes

characteristic

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

C₂×D₄

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

C₂×C₈×C₄₃

yes

C₂×C₈×C₄₃

yes

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₁₇₂

—

Dic₁₇₂

—

Dic₁₇₂

—

Dic₁₇₂

—

C₁₆×C₄₃

—

C₁₆×C₄₃

—

C₂³×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

Q₃₂

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

—

C₂×C₄×C₄₃

#135

yes

characteristic

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

#120

yes

characteristic

C₈×C₄₃

#123

yes

—

C₈×C₄₃

yes

—

C₂²×C₄₃

yes

—

C₂²×C₄₃

#466

yes

characteristic

C₂²×C₄₃

C₂×Q₃₂

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#394

yes

characteristic

100

C₄×C₄₃

#463

yes

—

101

C₄×C₄₃

#469

yes

characteristic

102

Dic₄₃

—

103

Dic₄₃

—

104

C₂²×Q₃₂

—

105

C₂×C₄⋊C₁₆

D₄₃

yes

characteristic

106

C₂×#358

—

107

#1969

—

108

#1380

—

109

C₂×C₄₃

#2054

yes

characteristic

110

C₂×C₄₃

#2209

yes

—

111

C₂×C₄₃

#2211

yes

—

112

C₂×C₄₃

#2216

yes

characteristic

113

C₂×C₄₃

#2219

yes

characteristic

114

C₂²×Q₁₆

—

115

C₂×Q₃₂

—

116

C₂×Q₃₂

—

117

C₂²×C₁₆

C₂×D₄₃

yes

characteristic

118

C₂²×C₁₆

C₂×D₄₃

yes

characteristic

119

C₂×#107

—

120

C₂×#103

—

121

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

122

#391

—

123

C₄⋊C₁₆

—

124

#358

—

125

C₄₃

#13450

yes

characteristic, complemented, Sylow

126

C₂²×Q₈

—

127

C₂×Q₁₆

—

128

C₂×Q₁₆

—

129

C₂×Q₁₆

—

130

C₂×Q₁₆

—

131

C₂×Q₁₆

—

132

C₂×Q₁₆

—

133

C₂²×C₈

D₁₇₂

yes

characteristic

134

C₂²×C₈

C₂²×D₄₃

yes

characteristic

135

C₂×C₄⋊C₄

—

136

C₂×C₄²

D₁₇₂

yes

characteristic

137

Q₃₂

—

138

C₂×C₁₆

—

139

C₂×C₁₆

—

140

C₂×C₁₆

—

141

C₂×C₁₆

C₂²⋊D₄₃

yes

—

142

#107

—

143

#107

—

144

#103

—

172

—

145

C₄×C₈

—

146

C₄×C₈

—

147

#95

—

172

—

148

#95

—

172

—

149

C₂×Q₈

—

150

C₂×Q₈

—

172

—

151

C₂×Q₈

—

172

—

152

C₂×Q₈

—

172

—

153

C₂×Q₈

—

172

—

154

C₂×Q₈

—

172

—

155

C₂²×C₄

—

172

—

156

C₂²×C₄

—

172

—

157

C₂²×C₄

D₃₄₄

yes

characteristic

158

C₂²×C₄

D₃₄₄

yes

characteristic

159

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

160

Q₁₆

—

161

Q₁₆

—

162

Q₁₆

—

172

—

163

Q₁₆

—

172

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

#10621898

yes

characteristic

167

C₂×C₈

D₄×D₄₃

yes

—

168

C₂×C₈

#10621902

yes

—

169

C₂×C₈

C₂×C₂²⋊D₄₃

yes

characteristic

170

C₄⋊C₄

—

172

—

171

C₄⋊C₄

—

172

—

172

C₄²

—

173

C₄²

—

174

C₁₆

—

175

C₁₆

—

176

C₂³

C₂×D₃₄₄

yes

characteristic

177

Q₈

—

344

—

178

Q₈

—

344

—

179

Q₈

—

172

—

180

Q₈

—

172

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

344

—

186

C₂×C₄

—

344

—

187

C₂×C₄

—

344

—

188

C₂×C₄

—

344

—

189

C₂×C₄

—

344

—

190

C₂×C₄

—

172

—

191

C₂×C₄

Dic₃₄₄

yes

complemented

192

C₂×C₄

#13063299

yes

characteristic

193

C₂×C₄

#13063300

yes

—

194

C₂×C₄

#13063347

yes

characteristic

195

C₂×C₄

#13063353

yes

—

196

C₈

—

197

C₈

#13063293

yes

characteristic

198

C₈

#13063310

yes

—

199

C₈

#13063314

yes

characteristic

200

C₂²

#14583095

yes

—

201

C₂²

#14583135

yes

characteristic

202

C₂²

C₂×Dic₃₄₄

yes

characteristic

203

C₂²

#14583331

yes

characteristic

204

C₂²

#14583342

yes

—

205

C₄

—

206

C₄

—

207

C₄

—

344

—

208

C₄

—

688

—

209

C₄

#14582816

yes

characteristic

210

C₄

#14583115

yes

—

211

C₄

#14583152

yes

characteristic

212

C₂

#19135906

yes

characteristic

213

C₂

#19136646

yes

—

214

C₂

#19136658

yes

—

215

C₂

#19136686

yes

characteristic

216

C₂

#19136703

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₄₄
commutator, cosocle	C₂²×C₈×C₄₃	C₂³
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂×C₄₃×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₄₃	C₂×D₈
hypercenter	C₂×C₄⋊C₁₆	D₄₃
hypocenter	C₄₃	#13450
LCS	G ▷ C₂²×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

subgroup

G/H

center

C₂³

C₂×D₃₄₄

commutator, cosocle

C₂²×C₈×C₄₃

C₂³

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting

C₂×C₄₃×C₄⋊C₁₆

C₂

socle, prime radical

C₂³×C₄₃

C₂×D₈

hypercenter

C₂×C₄⋊C₁₆

D₄₃

hypocenter

C₄₃

#13450

LCS

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

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

N	H ≅ G/N	type
72. C₂×C₄×C₄₃	Q₃₂	semidirect
125. C₄₃	#13450	semidirect
191. C₂×C₄	Dic₃₄₄	semidirect

H ≅ G/N

type

72. C₂×C₄×C₄₃

Q₃₂

semidirect

125. C₄₃

#13450

semidirect

191. C₂×C₄

Dic₃₄₄

semidirect

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

subgroup	index
C₂²×Dic₃₄₄	2
C₂×#14582704	2
#19135456	2
#19132687	2
C₂×C₄₃×C₄⋊C₁₆	2
#13450	43

subgroup

index

C₂²×Dic₃₄₄

C₂×C₄₃×C₄⋊C₁₆

Subgroups — #25016204

Conjugacy classes of subgroups

Complements

Maximal subgroups