Group #26135039 — subgroups

Subgroups — #26135039

Order 12032.

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

normal subgroups only

This group has 516 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	C₂×#15689395	C₂	1	2	yes	maximal
2	C₂×#15688939	C₂	1	2	yes	maximal, complemented
3	#20245001	C₂	1	1	yes	maximal, characteristic
4	#20241482	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×Q₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13094055	—	2	2	no	—
7	C₂×#13094030	C₂²	1	1	yes	characteristic
8	C₂×#13093964	C₂²	1	1	yes	characteristic
9	#15689395	—	2	4	no	—
10	#15689395	—	2	4	no	—
11	C₂×#13093960	C₂²	1	2	yes	—
12	C₂×#13093943	—	2	2	no	—
13	#15688939	—	2	4	no	—
14	(C₂×D₄₇)⋊C₁₆	—	2	2	no	—
15	C₂²×Q₁₆×C₄₇	C₂²	1	2	yes	—
16	C₂×Q₃₂×C₄₇	—	2	4	no	—
17	C₂×Q₃₂×C₄₇	—	2	2	no	—
18	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×#10625350	D₄	1	1	yes	characteristic
20	#13094055	—	2	8	no	—
21	#13094055	—	4	4	no	—
22	C₂×#10625337	—	4	2	no	—
23	C₂×#10625336	—	4	2	no	—
24	#13094030	—	2	2	no	—
25	#13094030	—	2	2	no	—
26	C₂×#10625329	—	2	2	no	—
27	C₂×#10625329	D₄	1	1	yes	characteristic
28	C₂×#10625324	—	2	1	no	—
29	C₂×C₈×D₄₇	—	2	1	no	—
30	#13093964	—	2	2	no	—
31	C₂×C₄₇⋊C₁₆	—	2	1	no	—
32	#13093960	—	2	2	no	—
33	#13093960	D₄	1	4	yes	—
34	#13093943	—	2	8	no	—
35	C₂²×Q₈×C₄₇	—	2	2	no	—
36	C₂×Q₁₆×C₄₇	—	2	2	no	—
37	C₂×Q₁₆×C₄₇	—	2	2	no	—
38	C₂×Q₁₆×C₄₇	—	2	2	no	—
39	C₂×Q₁₆×C₄₇	—	2	2	no	—
40	C₂×Q₁₆×C₄₇	—	2	2	no	—
41	C₂×Q₁₆×C₄₇	D₄	1	4	yes	—
42	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
43	Q₃₂×C₄₇	—	2	8	no	—
44	C₂×C₁₆×C₄₇	—	2	2	no	—
45	C₂×C₁₆×C₄₇	D₄	1	2	yes	—
46	C₂×Q₈×C₄₇	—	2	4	no	—
47	C₂×Q₈×C₄₇	—	4	2	no	—
48	C₂×Q₈×C₄₇	—	4	2	no	—
49	C₂×Q₈×C₄₇	—	4	2	no	—
50	C₂×Q₈×C₄₇	—	4	2	no	—
51	C₂×Q₈×C₄₇	—	4	2	no	—
52	C₂²×C₄×C₄₇	—	4	2	no	—
53	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₄₇	—	2	1	no	—
55	#10625350	—	2	2	no	—
56	#10625350	—	2	2	no	—
57	C₂×D₁₈₈	D₈	1	1	yes	characteristic
58	C₂×C₄×D₄₇	—	2	1	no	—
59	C₂×Dic₉₄	D₈	1	1	yes	characteristic
60	Q₁₆×C₄₇	—	2	4	no	—
61	Q₁₆×C₄₇	—	2	4	no	—
62	Q₁₆×C₄₇	—	4	2	no	—
63	Q₁₆×C₄₇	—	4	2	no	—
64	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
68	#10625337	—	4	8	no	—
69	#10625336	—	4	8	no	—
70	#10625329	—	2	2	no	—
71	#10625329	—	2	4	no	—
72	#10625329	—	4	2	no	—
73	C₂×C₄₇⋊C₈	—	2	1	no	—
74	C₂×C₄₇⋊C₈	—	4	2	no	—
75	#10625324	—	4	2	no	—
76	C₈×D₄₇	—	4	1	no	—
77	C₈×D₄₇	—	4	1	no	—
78	C₁₆×C₄₇	—	2	2	no	—
79	C₄₇⋊C₁₆	—	4	1	no	—
80	C₂³×C₄₇	#135	1	1	yes	characteristic
81	C₂²×D₄₇	—	2	1	no	—
82	Q₈×C₄₇	—	4	4	no	—
83	Q₈×C₄₇	—	4	4	no	—
84	Q₈×C₄₇	—	8	2	no	—
85	Q₈×C₄₇	—	8	2	no	—
86	C₂×C₄×C₄₇	—	4	4	no	—
87	C₂×C₄×C₄₇	—	8	2	no	—
88	C₂×C₄×C₄₇	—	8	2	no	—
89	C₂×C₄×C₄₇	#120	1	2	yes	—
90	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
93	C₂²⋊D₄₇	—	4	2	no	—
94	C₂×Dic₄₇	—	2	1	no	—
95	D₁₈₈	—	2	2	no	—
96	C₄×D₄₇	—	4	1	no	—
97	C₄×D₄₇	—	4	1	no	—
98	Dic₉₄	—	2	2	no	—
99	C₈×C₄₇	#120	1	1	yes	characteristic
100	C₈×C₄₇	#121	1	2	yes	—
101	C₈×C₄₇	#127	1	1	yes	characteristic
102	C₄₇⋊C₈	—	4	1	no	—
103	C₄₇⋊C₈	—	4	4	no	—
104	#13315	—	47	1	no	maximal, Sylow
105	C₂²×C₄₇	—	2	2	no	—
106	C₂²×C₄₇	#449	1	2	yes	—
107	C₂²×C₄₇	#466	1	1	yes	characteristic
108	C₂×D₄₇	—	4	2	no	—
109	C₂×D₄₇	—	4	1	no	—
110	C₄×C₄₇	—	8	4	no	—
111	C₄×C₄₇	#392	1	1	yes	characteristic
112	C₄×C₄₇	#447	1	2	yes	—
113	C₄×C₄₇	#469	1	1	yes	characteristic
114	Dic₄₇	—	4	1	no	—
115	C₂²×Q₃₂	D₄₇	1	1	yes	characteristic, complemented
116	C₂×#473	—	47	2	no	—
117	C₂×#362	—	47	2	no	—
118	#2113	—	47	1	no	—
119	#1381	—	47	1	no	—
120	C₂×C₄₇	—	2	2	no	—
121	C₂×C₄₇	#1999	1	1	yes	characteristic
122	C₂×C₄₇	#2222	1	2	yes	—
123	D₄₇	—	8	1	no	—
124	C₂×#137	—	94	2	no	—
125	C₂²×Q₁₆	C₂×D₄₇	1	2	yes	—
126	C₂×#131	—	47	1	no	—
127	C₂×Q₃₂	—	2	4	no	—
128	C₂×Q₃₂	—	2	2	no	—
129	C₂×M₃₂	—	47	1	no	—
130	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
131	#473	—	94	4	no	—
132	#473	—	94	4	no	—
133	C₂×#108	—	47	2	no	—
134	C₂×#101	—	94	2	no	—
135	#362	—	94	4	no	—
136	C₂²⋊C₁₆	—	94	2	no	—
137	C₄₇	#13315	1	1	yes	characteristic, complemented, Sylow
138	C₂×#41	—	47	1	no	—
139	C₂²×Q₈	—	2	2	no	—
140	#137	—	94	8	no	—
141	#137	—	188	4	no	—
142	C₂×Q₁₆	—	2	2	no	—
143	C₂×Q₁₆	—	2	2	no	—
144	C₂×Q₁₆	—	2	2	no	—
145	C₂×Q₁₆	—	2	2	no	—
146	C₂×Q₁₆	—	2	2	no	—
147	C₂×Q₁₆	—	188	2	no	—
148	C₂×Q₁₆	C₂²⋊D₄₇	1	4	yes	—
149	C₂×SD₁₆	—	188	2	no	—
150	#131	—	94	2	no	—
151	#131	—	94	2	no	—
152	C₂×M₁₆	—	47	1	no	—
153	C₂×M₁₆	—	94	1	no	—
154	C₂×M₁₆	—	94	2	no	—
155	C₂²×C₈	—	94	1	no	—
156	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
157	Q₃₂	—	2	8	no	—
158	M₃₂	—	94	2	no	—
159	C₂×C₁₆	—	2	2	no	—
160	C₂×C₁₆	—	94	1	no	—
161	C₂×C₁₆	C₂²⋊D₄₇	1	2	yes	—
162	#108	—	47	4	no	—
163	#108	—	94	2	no	—
164	#101	—	94	8	no	—
165	#41	—	94	2	no	—
166	#41	—	94	2	no	—
167	C₂×Q₈	—	2	4	no	—
168	C₂×Q₈	—	4	2	no	—
169	C₂×Q₈	—	4	2	no	—
170	C₂×Q₈	—	4	2	no	—
171	C₂×Q₈	—	4	2	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂×Q₈	—	47	1	no	—
174	C₂×D₄	—	47	1	no	—
175	C₂×D₄	—	94	1	no	—
176	C₂²×C₄	—	4	2	no	—
177	C₂²×C₄	—	94	1	no	—
178	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
179	Q₁₆	—	2	4	no	—
180	Q₁₆	—	2	4	no	—
181	Q₁₆	—	4	2	no	—
182	Q₁₆	—	4	2	no	—
183	Q₁₆	—	188	8	no	—
184	SD₁₆	—	188	8	no	—
185	M₁₆	—	94	2	no	—
186	M₁₆	—	94	4	no	—
187	M₁₆	—	188	2	no	—
188	M₁₆	—	188	2	no	—
189	C₂×C₈	—	94	1	no	—
190	C₂×C₈	—	188	2	no	—
191	C₂×C₈	—	188	1	no	—
192	C₂×C₈	—	188	1	no	—
193	C₂×C₈	D₄×D₄₇	1	2	yes	—
194	C₂×C₈	#10625352	1	1	yes	characteristic
195	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
196	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
197	C₁₆	—	2	2	no	—
198	C₁₆	—	188	1	no	—
199	C₂³	—	94	1	no	—
200	C₂³	#13094043	1	1	yes	characteristic
201	Q₈	—	4	4	no	—
202	Q₈	—	4	4	no	—
203	Q₈	—	8	2	no	—
204	Q₈	—	8	2	no	—
205	Q₈	—	94	2	no	—
206	D₄	—	94	2	no	—
207	D₄	—	188	2	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	8	2	no	—
210	C₂×C₄	—	8	2	no	—
211	C₂×C₄	—	94	1	no	—
212	C₂×C₄	—	188	1	no	—
213	C₂×C₄	—	188	1	no	—
214	C₂×C₄	#13093992	1	2	yes	—
215	C₂×C₄	#13093995	1	1	yes	characteristic
216	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
217	C₂×C₄	#13094048	1	2	yes	—
218	C₈	—	188	4	no	—
219	C₈	—	188	1	no	—
220	C₈	#13093993	1	1	yes	characteristic
221	C₈	#13093996	1	2	yes	—
222	C₈	#13094020	1	1	yes	characteristic
223	C₂²	—	2	2	no	—
224	C₂²	—	188	2	no	—
225	C₂²	—	188	1	no	—
226	C₂²	#15689265	1	2	yes	—
227	C₂²	#15689356	1	1	yes	characteristic
228	C₄	—	8	4	no	—
229	C₄	—	188	1	no	—
230	C₄	#15689031	1	1	yes	characteristic
231	C₄	#15689255	1	2	yes	—
232	C₄	#15689373	1	1	yes	characteristic
233	C₂	—	2	2	no	—
234	C₂	—	376	1	no	—
235	C₂	#20244383	1	1	yes	characteristic
236	C₂	#20245510	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15689395

C₂

yes

maximal

C₂×#15688939

C₂

yes

maximal, complemented

#20245001

C₂

yes

maximal, characteristic

#20241482

C₂

yes

maximal, characteristic, complemented

C₂²×Q₃₂×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂×#13094055

—

C₂×#13094030

C₂²

yes

characteristic

C₂×#13093964

C₂²

yes

—

—

C₂²

yes

—

C₂×#13093943

—

#15688939

—

(C₂×D₄₇)⋊C₁₆

—

C₂²×Q₁₆×C₄₇

C₂²

yes

—

C₂×Q₃₂×C₄₇

—

C₂×Q₃₂×C₄₇

—

C₂²×C₁₆×C₄₇

C₂²

yes

characteristic

C₂×#10625350

D₄

yes

—

—

—

—

—

—

—

D₄

yes

characteristic

C₂×#10625324

—

C₂×C₈×D₄₇

—

#13093964

—

C₂×C₄₇⋊C₁₆

—

#13093960

—

#13093960

D₄

yes

—

#13093943

—

C₂²×Q₈×C₄₇

—

C₂×Q₁₆×C₄₇

—

C₂×Q₁₆×C₄₇

—

C₂×Q₁₆×C₄₇

—

C₂×Q₁₆×C₄₇

—

C₂×Q₁₆×C₄₇

—

C₂×Q₁₆×C₄₇

D₄

yes

—

C₂²×C₈×C₄₇

C₂³

yes

characteristic

Q₃₂×C₄₇

—

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

D₄

yes

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₂²⋊D₄₇

—

#10625350

—

#10625350

—

C₂×D₁₈₈

D₈

yes

characteristic

C₂×C₄×D₄₇

—

C₂×Dic₉₄

D₈

yes

characteristic

Q₁₆×C₄₇

—

Q₁₆×C₄₇

—

Q₁₆×C₄₇

—

Q₁₆×C₄₇

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₂×C₈×C₄₇

yes

—

—

—

—

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

#10625324

—

C₈×D₄₇

—

C₈×D₄₇

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

#135

yes

characteristic

C₂²×D₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#120

yes

—

C₂×C₄×C₄₇

#121

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

C₈×C₄₇

#120

yes

characteristic

100

C₈×C₄₇

#121

yes

—

101

C₈×C₄₇

#127

yes

characteristic

102

C₄₇⋊C₈

—

103

C₄₇⋊C₈

—

104

#13315

—

maximal, Sylow

105

C₂²×C₄₇

—

106

C₂²×C₄₇

#449

yes

—

107

C₂²×C₄₇

#466

yes

characteristic

108

C₂×D₄₇

—

109

C₂×D₄₇

—

110

C₄×C₄₇

—

111

C₄×C₄₇

#392

yes

characteristic

112

C₄×C₄₇

#447

yes

—

113

C₄×C₄₇

#469

yes

characteristic

114

Dic₄₇

—

115

C₂²×Q₃₂

D₄₇

yes

characteristic, complemented

116

C₂×#473

—

117

C₂×#362

—

118

#2113

—

119

#1381

—

120

C₂×C₄₇

—

121

C₂×C₄₇

#1999

yes

characteristic

122

C₂×C₄₇

#2222

yes

—

123

D₄₇

—

124

C₂×#137

—

125

C₂²×Q₁₆

C₂×D₄₇

yes

—

126

C₂×#131

—

127

C₂×Q₃₂

—

128

C₂×Q₃₂

—

129

C₂×M₃₂

—

130

C₂²×C₁₆

C₂×D₄₇

yes

characteristic

131

#473

—

132

#473

—

133

C₂×#108

—

134

C₂×#101

—

135

#362

—

136

C₂²⋊C₁₆

—

137

C₄₇

#13315

yes

characteristic, complemented, Sylow

138

C₂×#41

—

139

C₂²×Q₈

—

140

#137

—

141

#137

—

188

—

142

C₂×Q₁₆

—

143

C₂×Q₁₆

—

144

C₂×Q₁₆

—

145

C₂×Q₁₆

—

146

C₂×Q₁₆

—

147

C₂×Q₁₆

—

188

—

148

C₂×Q₁₆

C₂²⋊D₄₇

yes

—

149

C₂×SD₁₆

—

188

—

150

#131

—

151

#131

—

152

C₂×M₁₆

—

153

C₂×M₁₆

—

154

C₂×M₁₆

—

155

C₂²×C₈

—

156

C₂²×C₈

C₂²×D₄₇

yes

characteristic

157

Q₃₂

—

158

M₃₂

—

159

C₂×C₁₆

—

160

C₂×C₁₆

—

161

C₂×C₁₆

C₂²⋊D₄₇

yes

—

162

#108

—

163

#108

—

164

#101

—

165

#41

—

166

#41

—

167

C₂×Q₈

—

168

C₂×Q₈

—

169

C₂×Q₈

—

170

C₂×Q₈

—

171

C₂×Q₈

—

172

C₂×Q₈

—

173

C₂×Q₈

—

174

C₂×D₄

—

175

C₂×D₄

—

176

C₂²×C₄

—

177

C₂²×C₄

—

178

C₂²×C₄

D₄×D₄₇

yes

characteristic

179

Q₁₆

—

180

Q₁₆

—

181

Q₁₆

—

182

Q₁₆

—

183

Q₁₆

—

188

—

184

SD₁₆

—

188

—

185

M₁₆

—

186

M₁₆

—

187

M₁₆

—

188

—

188

M₁₆

—

188

—

189

C₂×C₈

—

190

C₂×C₈

—

188

—

191

C₂×C₈

—

188

—

192

C₂×C₈

—

188

—

193

C₂×C₈

D₄×D₄₇

yes

—

194

C₂×C₈

#10625352

yes

characteristic

195

C₂×C₈

C₂×C₂²⋊D₄₇

yes

—

196

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

197

C₁₆

—

198

C₁₆

—

188

—

199

C₂³

—

200

C₂³

#13094043

yes

characteristic

201

Q₈

—

202

Q₈

—

203

Q₈

—

204

Q₈

—

205

Q₈

—

206

D₄

—

207

D₄

—

188

—

208

C₂×C₄

—

209

C₂×C₄

—

210

C₂×C₄

—

211

C₂×C₄

—

212

C₂×C₄

—

188

—

213

C₂×C₄

—

188

—

214

C₂×C₄

#13093992

yes

—

215

C₂×C₄

#13093995

yes

characteristic

216

C₂×C₄

D₈×D₄₇

yes

characteristic

217

C₂×C₄

#13094048

yes

—

218

C₈

—

188

—

219

C₈

—

188

—

220

C₈

#13093993

yes

characteristic

221

C₈

#13093996

yes

—

222

C₈

#13094020

yes

characteristic

223

C₂²

—

224

C₂²

—

188

—

225

C₂²

—

188

—

226

C₂²

#15689265

yes

—

227

C₂²

#15689356

yes

characteristic

228

C₄

—

229

C₄

—

188

—

230

C₄

#15689031

yes

characteristic

231

C₄

#15689255

yes

—

232

C₄

#15689373

yes

characteristic

233

C₂

—

234

C₂

—

376

—

235

C₂

#20244383

yes

characteristic

236

C₂

#20245510

yes

—

	subgroup	G/H
center	C₂²	#15689356
commutator, cosocle	C₂²×C₈×C₄₇	C₂³
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂²×Q₃₂×C₄₇	C₂
socle	C₂²×C₄₇	#466
prime radical	C₂×#10625350	D₄
hypercenter	C₂²×Q₃₂	D₄₇
hypocenter	C₄₇	#13315
LCS	G ▷ C₂²×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

subgroup

G/H

center

C₂²

#15689356

commutator, cosocle

C₂²×C₈×C₄₇

C₂³

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₂²×Q₃₂×C₄₇

C₂

C₂²×C₄₇

D₄

C₂²×Q₃₂

D₄₇

hypocenter

C₄₇

#13315

LCS

G ▷ C₂²×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

N	H ≅ G/N	type
2. C₂×#15688939	C₂	semidirect
4. #20241482	C₂	semidirect
5. C₂²×Q₃₂×C₄₇	C₂	semidirect
115. C₂²×Q₃₂	D₄₇	semidirect
137. C₄₇	#13315	semidirect

H ≅ G/N

type

2. C₂×#15688939

C₂

semidirect

4. #20241482

C₂

semidirect

5. C₂²×Q₃₂×C₄₇

C₂

semidirect

115. C₂²×Q₃₂

D₄₇

semidirect

137. C₄₇

#13315

semidirect

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

subgroup	index
C₂×#15689395	2
C₂×#15688939	2
#20245001	2
#20241482	2
C₂²×Q₃₂×C₄₇	2
#13315	47

subgroup

index

C₂²×Q₃₂×C₄₇

Subgroups — #26135039

Conjugacy classes of subgroups

Complements

Maximal subgroups