Group #26128278 — subgroups

Subgroups — #26128278

Order 12032.

206 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 73 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	C₄×#13093970	C₂	1	1	yes	maximal, characteristic
2	#20242285	C₂	1	1	yes	maximal, characteristic
3	#20242284	C₂	1	1	yes	maximal, characteristic
4	#20241461	C₂	1	1	yes	maximal, characteristic
5	#20241353	C₂	1	1	yes	maximal, characteristic
6	#20241304	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#1332	C₂	1	1	yes	maximal, characteristic
8	C₂×#13093970	C₄	1	1	yes	characteristic
9	#15689254	—	2	1	no	—
10	#15689250	—	2	1	no	—
11	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
12	#15688948	C₄	1	1	yes	characteristic
13	#15688927	C₄	1	1	yes	characteristic
14	#15688926	C₄	1	1	yes	characteristic
15	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	#15688876	C₂²	1	1	yes	characteristic
17	Q₈⋊(C₄₇⋊C₈)	—	2	1	no	—
18	#15688853	—	2	1	no	—
19	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
21	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
22	C₄₇×#335	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₈₈	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₉₄	—	2	1	no	—
25	#13093970	—	2	2	no	—
26	C₂×C₄₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
28	#13093958	C₂×C₄	1	1	yes	characteristic
29	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
30	C₄⋊(C₄₇⋊C₈)	—	4	1	no	—
31	Dic₄₇⋊C₈	—	4	1	no	—
32	#13093949	—	2	1	no	—
33	#13093935	—	4	1	no	—
34	#13093934	—	4	1	no	—
35	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
36	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
37	C₂×Q₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₄×Q₈×C₄₇	—	2	1	no	—
39	C₂×C₁₆×C₄₇	—	2	1	no	—
40	#107×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₄⋊C₈×C₄₇	—	2	1	no	—
42	#103×C₄₇	—	2	1	no	—
43	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₄₇	—	2	1	no	—
45	C₂×Dic₉₄	—	2	1	no	—
46	Q₁₆×C₄₇	—	2	1	no	—
47	Q₁₆×C₄₇	—	2	1	no	—
48	C₂×C₈×C₄₇	—	4	1	no	—
49	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
51	C₄⋊C₄×C₄₇	—	2	1	no	—
52	C₄⋊C₄×C₄₇	—	4	1	no	—
53	C₄²×C₄₇	—	4	1	no	—
54	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₄⋊Dic₄₇	—	2	1	no	—
56	Dic₄₇⋊C₄	—	4	1	no	—
57	C₄×Dic₄₇	—	4	1	no	—
58	C₂×C₄₇⋊C₈	—	2	1	no	—
59	C₂×C₄₇⋊C₈	—	4	1	no	—
60	C₂×C₄₇⋊C₈	D₈	1	1	yes	characteristic
61	C₂×C₄₇⋊C₈	D₈	1	1	yes	characteristic
62	Dic₁₈₈	—	2	1	no	—
63	Dic₁₈₈	—	2	1	no	—
64	C₁₆×C₄₇	—	2	2	no	—
65	C₄₇⋊C₁₆	—	2	1	no	—
66	C₄₇⋊C₁₆	M₁₆	1	2	yes	—
67	Q₈×C₄₇	—	4	1	no	—
68	Q₈×C₄₇	—	4	1	no	—
69	C₂×C₄×C₄₇	—	4	1	no	—
70	C₂×C₄×C₄₇	—	4	1	no	—
71	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
74	C₂×Dic₄₇	—	4	1	no	—
75	C₂×Dic₄₇	—	4	1	no	—
76	Dic₉₄	—	4	1	no	—
77	Dic₉₄	—	4	1	no	—
78	C₈×C₄₇	—	2	1	no	—
79	C₈×C₄₇	—	8	1	no	—
80	C₈×C₄₇	C₂×M₁₆	1	1	yes	characteristic
81	C₈×C₄₇	#131	1	1	yes	characteristic
82	C₄₇⋊C₈	—	2	1	no	—
83	C₄₇⋊C₈	—	2	1	no	—
84	C₄₇⋊C₈	—	4	1	no	—
85	C₄₇⋊C₈	—	8	1	no	—
86	#12085	—	47	1	no	maximal, Sylow
87	C₂²×C₄₇	C₄×D₈	1	1	yes	characteristic
88	C₄×C₄₇	—	2	1	no	—
89	C₄×C₄₇	—	8	1	no	—
90	C₄×C₄₇	—	8	1	no	—
91	C₄×C₄₇	#436	1	1	yes	characteristic
92	C₄×C₄₇	#443	1	1	yes	characteristic
93	C₄×C₄₇	#510	1	1	yes	characteristic
94	C₄×C₄₇	#510	1	1	yes	characteristic
95	Dic₄₇	—	8	1	no	—
96	Dic₄₇	—	8	1	no	—
97	C₄×Q₃₂	—	47	1	no	—
98	#1591	—	47	1	no	—
99	#1591	—	47	1	no	—
100	#1372	—	47	1	no	—
101	#1332	—	47	1	no	—
102	#1332	D₄₇	1	1	yes	characteristic
103	#1311	—	47	1	no	—
104	C₂×C₄₇	#1589	1	1	yes	characteristic
105	C₂×C₄₇	#2176	1	1	yes	characteristic
106	C₂×C₄₇	#2179	1	1	yes	characteristic
107	C₂×Q₃₂	—	47	1	no	—
108	#446	—	94	1	no	—
109	#446	—	94	1	no	—
110	C₄×Q₁₆	—	47	1	no	—
111	C₄×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
112	#366	—	47	1	no	—
113	#358	—	47	1	no	—
114	#358	—	47	1	no	—
115	C₄×C₁₆	—	47	1	no	—
116	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
117	#335	—	47	1	no	—
118	#335	C₂×D₄₇	1	1	yes	characteristic
119	Q₈⋊C₈	—	94	1	no	—
120	Q₈⋊C₈	—	94	1	no	—
121	C₈²	—	47	1	no	—
122	C₄₇	#12085	1	1	yes	characteristic, complemented, Sylow
123	C₂×Q₁₆	—	47	1	no	—
124	C₂×Q₁₆	C₄×D₄₇	1	1	yes	characteristic
125	C₄×Q₈	—	2	1	no	—
126	C₄×Q₈	—	94	1	no	—
127	Q₃₂	—	94	2	no	—
128	C₂×C₁₆	—	2	1	no	—
129	C₂×C₁₆	—	47	1	no	—
130	C₂×C₁₆	—	47	1	no	—
131	#107	—	47	1	no	—
132	#107	C₄×D₄₇	1	1	yes	characteristic
133	C₄⋊C₈	—	2	1	no	—
134	C₄⋊C₈	—	94	1	no	—
135	C₄⋊C₈	—	188	1	no	—
136	C₄⋊C₈	—	188	1	no	—
137	#103	—	2	1	no	—
138	#103	—	94	1	no	—
139	#97	—	188	1	no	—
140	#97	—	188	1	no	—
141	C₄×C₈	—	47	1	no	—
142	C₄×C₈	—	47	1	no	—
143	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
144	C₂×Q₈	—	2	1	no	—
145	C₂×Q₈	—	94	1	no	—
146	Q₁₆	—	2	1	no	—
147	Q₁₆	—	2	1	no	—
148	Q₁₆	—	94	1	no	—
149	Q₁₆	—	94	1	no	—
150	C₂×C₈	—	4	1	no	—
151	C₂×C₈	—	47	1	no	—
152	C₂×C₈	—	47	1	no	—
153	C₂×C₈	—	94	1	no	—
154	C₂×C₈	—	188	1	no	—
155	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
156	C₂×C₈	#10625352	1	1	yes	characteristic
157	C₄⋊C₄	—	2	1	no	—
158	C₄⋊C₄	—	4	1	no	—
159	C₄⋊C₄	—	94	1	no	—
160	C₄⋊C₄	—	188	1	no	—
161	C₄²	—	4	1	no	—
162	C₄²	—	188	1	no	—
163	C₄²	D₄×D₄₇	1	1	yes	characteristic
164	C₁₆	—	2	2	no	—
165	C₁₆	—	47	2	no	—
166	C₁₆	—	94	1	no	—
167	Q₈	—	4	1	no	—
168	Q₈	—	4	1	no	—
169	Q₈	—	188	1	no	—
170	Q₈	—	188	1	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	188	1	no	—
174	C₂×C₄	—	188	1	no	—
175	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
176	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
177	C₂×C₄	#13094045	1	1	yes	characteristic
178	C₈	—	2	1	no	—
179	C₈	—	8	1	no	—
180	C₈	—	94	1	no	—
181	C₈	—	94	1	no	—
182	C₈	—	376	1	no	—
183	C₈	—	188	1	no	—
184	C₈	M₁₆×D₄₇	1	1	yes	characteristic
185	C₈	#13094030	1	1	yes	characteristic
186	C₂²	#15689203	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	8	1	no	—
189	C₄	—	8	1	no	—
190	C₄	—	376	1	no	—
191	C₄	—	376	1	no	—
192	C₄	#15689198	1	1	yes	characteristic
193	C₄	#15689237	1	1	yes	characteristic
194	C₄	#15689562	1	1	yes	characteristic
195	C₄	#15689565	1	1	yes	characteristic
196	C₂	#20242270	1	1	yes	characteristic
197	C₂	#20245256	1	1	yes	characteristic
198	C₂	#20245271	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13093970

C₂

yes

maximal, characteristic

#20242285

C₂

yes

maximal, characteristic

#20242284

C₂

yes

maximal, characteristic

#20241461

C₂

yes

maximal, characteristic

#20241353

C₂

yes

maximal, characteristic

#20241304

C₂

yes

maximal, characteristic

C₄₇×#1332

C₂

yes

maximal, characteristic

C₂×#13093970

C₄

yes

characteristic

#15689254

—

#15689250

—

C₄×Dic₁₈₈

C₂²

yes

characteristic

#15688948

C₄

yes

characteristic

#15688927

C₄

yes

characteristic

#15688926

C₄

yes

characteristic

C₄×C₄₇⋊C₁₆

C₂²

yes

characteristic

#15688876

C₂²

yes

characteristic

Q₈⋊(C₄₇⋊C₈)

—

#15688853

—

C₈×C₄₇⋊C₈

C₂²

yes

characteristic

C₄×Q₁₆×C₄₇

C₂²

yes

characteristic

C₄×C₁₆×C₄₇

C₂²

yes

characteristic

C₄₇×#335

C₂²

yes

characteristic

C₂×Dic₁₈₈

C₂×C₄

yes

characteristic

C₄×Dic₉₄

—

#13093970

—

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

characteristic

#13093958

C₂×C₄

yes

characteristic

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

Dic₄₇⋊C₈

—

#13093949

—

#13093935

—

#13093934

—

C₄×C₄₇⋊C₈

D₄

yes

characteristic

C₄×C₄₇⋊C₈

D₄

yes

characteristic

C₂×Q₁₆×C₄₇

C₂×C₄

yes

characteristic

C₄×Q₈×C₄₇

—

C₂×C₁₆×C₄₇

—

#107×C₄₇

C₂×C₄

yes

characteristic

C₄⋊C₈×C₄₇

—

#103×C₄₇

—

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Dic₉₄

—

Q₁₆×C₄₇

—

Q₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

#41

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₈

yes

characteristic

C₂×C₄₇⋊C₈

D₈

yes

characteristic

Dic₁₈₈

—

Dic₁₈₈

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₄₇⋊C₁₆

M₁₆

yes

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

#135

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₂×M₁₆

yes

characteristic

C₈×C₄₇

#131

yes

characteristic

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#12085

—

maximal, Sylow

C₂²×C₄₇

C₄×D₈

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

C₄×Q₃₂

—

#1591

—

#1591

—

100

#1372

—

101

#1332

—

102

#1332

D₄₇

yes

characteristic

103

#1311

—

104

C₂×C₄₇

#1589

yes

characteristic

105

C₂×C₄₇

#2176

yes

characteristic

106

C₂×C₄₇

#2179

yes

characteristic

107

C₂×Q₃₂

—

108

#446

—

109

#446

—

110

C₄×Q₁₆

—

111

C₄×Q₁₆

C₂×D₄₇

yes

characteristic

112

#366

—

113

#358

—

114

#358

—

115

C₄×C₁₆

—

116

C₄×C₁₆

C₂×D₄₇

yes

characteristic

117

#335

—

118

#335

C₂×D₄₇

yes

characteristic

119

Q₈⋊C₈

—

120

Q₈⋊C₈

—

121

C₈²

—

122

C₄₇

#12085

yes

characteristic, complemented, Sylow

123

C₂×Q₁₆

—

124

C₂×Q₁₆

C₄×D₄₇

yes

characteristic

125

C₄×Q₈

—

126

C₄×Q₈

—

127

Q₃₂

—

128

C₂×C₁₆

—

129

C₂×C₁₆

—

130

C₂×C₁₆

—

131

#107

—

132

#107

C₄×D₄₇

yes

characteristic

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

C₄⋊C₈

—

188

—

136

C₄⋊C₈

—

188

—

137

#103

—

138

#103

—

139

#97

—

188

—

140

#97

—

188

—

141

C₄×C₈

—

142

C₄×C₈

—

143

C₄×C₈

C₂²×D₄₇

yes

characteristic

144

C₂×Q₈

—

145

C₂×Q₈

—

146

Q₁₆

—

147

Q₁₆

—

148

Q₁₆

—

149

Q₁₆

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

188

—

155

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

156

C₂×C₈

#10625352

yes

characteristic

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

188

—

161

C₄²

—

162

C₄²

—

188

—

163

C₄²

D₄×D₄₇

yes

characteristic

164

C₁₆

—

165

C₁₆

—

166

C₁₆

—

167

Q₈

—

168

Q₈

—

169

Q₈

—

188

—

170

Q₈

—

188

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

188

—

174

C₂×C₄

—

188

—

175

C₂×C₄

(C₂²⋊D₄₇)⋊C₄

yes

characteristic

176

C₂×C₄

D₈×D₄₇

yes

characteristic

177

C₂×C₄

#13094045

yes

characteristic

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

182

C₈

—

376

—

183

C₈

—

188

—

184

C₈

M₁₆×D₄₇

yes

characteristic

185

C₈

#13094030

yes

characteristic

186

C₂²

#15689203

yes

characteristic

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

376

—

191

C₄

—

376

—

192

C₄

#15689198

yes

characteristic

193

C₄

#15689237

yes

characteristic

194

C₄

#15689562

yes

characteristic

195

C₄

#15689565

yes

characteristic

196

C₂

#20242270

yes

characteristic

197

C₂

#20245256

yes

characteristic

198

C₂

#20245271

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₄₇
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#1332	C₂
socle, prime radical	C₂²×C₄₇	C₄×D₈
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	#1332	D₄₇
hypocenter	C₄₇	#12085
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1332

subgroup

G/H

center

C₂×C₄

D₈×D₄₇

commutator

C₂×C₈×C₄₇

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#1332

C₂

socle, prime radical

C₂²×C₄₇

C₄×D₈

cosocle

C₄×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1332

N	H ≅ G/N	type
122. C₄₇	#12085	semidirect

H ≅ G/N

type

122. C₄₇

#12085

semidirect

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

subgroup	index
C₄×#13093970	2
#20242285	2
#20242284	2
#20241461	2
#20241353	2
#20241304	2
C₄₇×#1332	2
#12085	47

subgroup

index

C₄₇×#1332

Subgroups — #26128278

Conjugacy classes of subgroups

Complements

Maximal subgroups