Group #26225143 — subgroups

Subgroups — #26225143

Order 12032.

662 conjugacy classes of subgroups; only the 255 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 255 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	#20253912	C₂	1	1	yes	maximal, characteristic
2	#20252318	C₂	1	2	yes	maximal
3	#20244368	C₂	1	2	yes	maximal
4	#20244139	C₂	1	4	yes	maximal
5	#20243270	C₂	1	4	yes	maximal
6	#20243076	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#1849	C₂	1	1	yes	maximal, characteristic
8	#15689680	C₂²	1	2	yes	—
9	#15689532	C₄	1	8	yes	complemented
10	#15689250	—	2	4	no	—
11	#15689247	—	2	4	no	—
12	#15689161	C₂²	1	4	yes	—
13	#15689151	C₂²	1	4	yes	—
14	#15689081	C₂²	1	4	yes	—
15	C₂×#13093934	C₂²	1	2	yes	—
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
18	#15688969	C₂²	1	1	yes	characteristic
19	#15688884	C₂²	1	8	yes	—
20	C₂×#108×C₄₇	C₂²	1	4	yes	—
21	C₄₇×#423	C₂²	1	2	yes	—
22	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
24	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
25	#13094024	C₂×C₄	1	4	yes	—
26	#13094016	C₂×C₄	1	4	yes	—
27	#13094003	—	2	4	no	—
28	C₄×Dic₉₄	—	2	4	no	—
29	#13093983	C₂³	1	4	yes	—
30	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
31	#13093958	C₂×C₄	1	8	yes	—
32	#13093956	C₂×C₄	1	8	yes	—
33	Dic₄₇⋊C₈	—	2	8	no	—
34	#13093934	—	2	4	no	—
35	#13093934	—	2	4	no	—
36	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
37	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
38	C₈×Dic₄₇	—	2	4	no	—
39	C₈×Dic₄₇	Q₈	1	8	yes	complemented
40	C₄₇⋊#95	—	2	2	no	—
41	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
42	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
43	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
44	#108×C₄₇	—	2	8	no	—
45	C₄⋊C₈×C₄₇	—	2	4	no	—
46	#97×C₄₇	C₂×C₄	1	4	yes	—
47	C₂²×C₄×C₄₇	#41	1	2	yes	—
48	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
49	C₂²×Dic₄₇	—	2	2	no	—
50	C₂×Dic₉₄	—	2	4	no	—
51	M₁₆×C₄₇	—	2	8	no	—
52	C₂×C₈×C₄₇	—	2	4	no	—
53	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
54	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
55	C₂×C₈×C₄₇	#41	1	4	yes	—
56	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
57	C₄²×C₄₇	C₂×D₄	1	2	yes	—
58	C₂²⋊Dic₄₇	—	2	4	no	—
59	C₄⋊Dic₄₇	—	2	4	no	—
60	C₄⋊Dic₄₇	C₂²×C₄	1	8	yes	—
61	Dic₄₇⋊C₄	—	2	8	no	—
62	C₄×Dic₄₇	—	2	4	no	—
63	C₄×Dic₄₇	—	2	2	no	—
64	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
65	C₂×C₄₇⋊C₈	—	2	2	no	—
66	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
67	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
70	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
71	C₂×C₄×C₄₇	—	4	2	no	—
72	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
73	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
74	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
75	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
76	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
77	C₂×Dic₄₇	—	2	8	no	—
78	C₂×Dic₄₇	—	2	4	no	—
79	C₂×Dic₄₇	—	4	4	no	—
80	Dic₉₄	—	4	8	no	—
81	C₈×C₄₇	—	4	4	no	—
82	C₈×C₄₇	C₄×Q₈	1	8	yes	—
83	C₄₇⋊C₈	—	2	2	no	—
84	C₄₇⋊C₈	C₄⋊Q₈	1	2	yes	—
85	C₄₇⋊C₈	C₄⋊Q₈	1	2	yes	—
86	#21735	—	47	1	no	maximal, Sylow
87	C₂²×C₄₇	—	2	2	no	—
88	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
89	C₂²×C₄₇	#533	1	2	yes	—
90	C₄×C₄₇	—	2	4	no	—
91	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
92	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
93	C₄×C₄₇	#532	1	1	yes	characteristic
94	Dic₄₇	—	4	4	no	—
95	Dic₄₇	—	4	8	no	—
96	#3162	—	47	1	no	—
97	#2960	—	47	2	no	—
98	#1995	—	47	2	no	—
99	#1946	—	47	4	no	—
100	#1849	D₄₇	1	1	yes	characteristic
101	#1772	—	47	4	no	—
102	#1726	—	47	1	no	—
103	C₂×C₄₇	—	2	2	no	—
104	C₂×C₄₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
105	C₂×C₄₇	#2955	1	2	yes	—
106	#533	—	47	2	no	—
107	#501	—	47	8	no	—
108	#446	—	94	4	no	—
109	C₈×Q₈	—	94	4	no	—
110	C₂×#108	C₂×D₄₇	1	4	yes	—
111	#427	—	47	4	no	—
112	#424	—	47	4	no	—
113	#423	C₂×D₄₇	1	2	yes	—
114	#405	—	47	4	no	—
115	C₂×#97	—	47	2	no	—
116	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
117	C₂×C₄×C₈	—	47	2	no	—
118	C₂×C₄×C₈	—	47	1	no	—
119	#376	—	47	1	no	—
120	#339	—	47	8	no	—
121	C₄₇	#21735	1	1	yes	characteristic, complemented, Sylow
122	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
123	C₂²×C₈	—	47	1	no	—
124	C₂²×C₈	—	47	1	no	—
125	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
126	C₄⋊Q₈	—	47	4	no	—
127	#125	—	47	4	no	—
128	C₂²⋊Q₈	—	94	4	no	—
129	C₄×Q₈	—	94	4	no	—
130	#117	—	47	4	no	—
131	C₂×C₄²	—	47	2	no	—
132	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
133	#108	—	2	8	no	—
134	#107	—	47	8	no	—
135	#106	—	47	8	no	—
136	C₄⋊C₈	—	2	4	no	—
137	C₄⋊C₈	—	94	8	no	—
138	#97	—	94	4	no	—
139	#97	—	94	4	no	—
140	#97	C₄×D₄₇	1	4	yes	—
141	C₄×C₈	—	47	8	no	—
142	C₄×C₈	—	47	2	no	—
143	C₄×C₈	—	47	2	no	—
144	C₄×C₈	—	94	4	no	—
145	#95	—	94	2	no	—
146	C₂×Q₈	—	94	4	no	—
147	C₂²×C₄	—	94	2	no	—
148	C₂²×C₄	#10625354	1	2	yes	—
149	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
150	M₁₆	—	2	8	no	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	47	1	no	—
153	C₂×C₈	—	47	1	no	—
154	C₂×C₈	—	47	4	no	—
155	C₂×C₈	—	47	2	no	—
156	C₂×C₈	—	94	2	no	—
157	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
158	C₂×C₈	#10625352	1	4	yes	—
159	C₂×C₈	Q₈×D₄₇	1	4	yes	—
160	C₄⋊C₄	—	47	8	no	—
161	C₄⋊C₄	—	94	8	no	—
162	C₄⋊C₄	—	94	4	no	—
163	C₂²⋊C₄	—	94	4	no	—
164	C₄²	—	47	4	no	—
165	C₄²	—	94	4	no	—
166	C₄²	—	94	2	no	—
167	C₄²	C₂×C₄×D₄₇	1	2	yes	—
168	C₄²	D₄×D₄₇	1	2	yes	—
169	C₂³	C₂×#10625354	1	1	yes	characteristic
170	Q₈	—	188	8	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	94	8	no	—
173	C₂×C₄	—	94	4	no	—
174	C₂×C₄	—	188	4	no	—
175	C₂×C₄	#13093986	1	4	yes	—
176	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
177	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
178	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
179	C₂×C₄	C₂×#10625352	1	2	yes	—
180	C₈	—	4	4	no	—
181	C₈	—	47	2	no	—
182	C₈	—	47	2	no	—
183	C₈	—	94	2	no	—
184	C₈	#13093989	1	8	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	C₂×#13093986	1	1	yes	characteristic
187	C₂²	#15689678	1	2	yes	—
188	C₄	—	2	4	no	—
189	C₄	—	188	4	no	—
190	C₄	—	188	8	no	—
191	C₄	C₂×#13093989	1	2	yes	—
192	C₄	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
193	C₄	#15689671	1	1	yes	characteristic
194	C₂	—	2	2	no	—
195	C₂	#20245955	1	1	yes	characteristic
196	C₂	#20252290	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20253912

C₂

yes

maximal, characteristic

#20252318

C₂

yes

maximal

#20244368

C₂

yes

maximal

#20244139

C₂

yes

maximal

#20243270

C₂

yes

maximal

#20243076

C₂

yes

maximal, characteristic

C₄₇×#1849

C₂

yes

maximal, characteristic

#15689680

C₂²

yes

—

#15689532

C₄

yes

—

—

C₂²

yes

—

#15689151

C₂²

yes

—

#15689081

C₂²

yes

—

C₂×#13093934

C₂²

yes

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×C₈×Dic₄₇

C₂²

yes

—

#15688969

C₂²

yes

characteristic

#15688884

C₂²

yes

—

C₂×#108×C₄₇

C₂²

yes

—

C₄₇×#423

C₂²

yes

—

C₂×#97×C₄₇

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂³

yes

characteristic

C₂²×C₄₇⋊C₈

C₂³

yes

characteristic

#13094024

C₂×C₄

yes

—

#13094016

C₂×C₄

yes

—

#13094003

—

C₄×Dic₉₄

—

#13093983

C₂³

yes

—

C₂×C₄×Dic₄₇

C₂³

yes

—

#13093958

C₂×C₄

yes

—

#13093956

C₂×C₄

yes

—

Dic₄₇⋊C₈

—

#13093934

—

#13093934

—

C₄×C₄₇⋊C₈

D₄

yes

—

C₄×C₄₇⋊C₈

D₄

yes

—

C₈×Dic₄₇

—

C₈×Dic₄₇

Q₈

yes

complemented

C₄₇⋊#95

—

C₂×M₁₆×C₄₇

C₂³

yes

—

C₂²×C₈×C₄₇

C₂³

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#108×C₄₇

—

C₄⋊C₈×C₄₇

—

#97×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

#41

yes

—

C₂²×C₄×C₄₇

C₂⁴

yes

characteristic

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²×C₄

yes

—

C₂×C₈×C₄₇

C₂×Q₈

yes

—

C₂×C₈×C₄₇

#41

yes

—

C₄²×C₄₇

C₂²×C₄

yes

—

C₄²×C₄₇

C₂×D₄

yes

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂²×C₄

yes

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

C₂×Q₈

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂×D₄

yes

—

C₂×C₄₇⋊C₈

C₂×D₄

yes

characteristic

C₂×C₄₇⋊C₈

C₂×D₄

yes

characteristic

C₂×C₄₇⋊C₈

C₂×Q₈

yes

—

C₂³×C₄₇

C₂×#41

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

C₂³×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×Q₈

yes

—

C₂×C₄×C₄₇

C₂×#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

C₄×Q₈

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

C₄⋊Q₈

yes

—

C₄₇⋊C₈

C₄⋊Q₈

yes

—

#21735

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄×D₄

yes

characteristic

C₂²×C₄₇

#533

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₄×Q₈

yes

—

C₄×C₄₇

C₂×C₄⋊Q₈

yes

characteristic

C₄×C₄₇

#532

yes

characteristic

Dic₄₇

—

Dic₄₇

—

—

—

—

—

100

#1849

D₄₇

yes

characteristic

101

#1772

—

102

#1726

—

103

C₂×C₄₇

—

104

C₂×C₄₇

C₄×C₄⋊Q₈

yes

characteristic

105

C₂×C₄₇

#2955

yes

—

106

#533

—

107

#501

—

108

#446

—

109

C₈×Q₈

—

110

C₂×#108

C₂×D₄₇

yes

—

111

#427

—

112

#424

—

113

#423

C₂×D₄₇

yes

—

114

#405

—

115

C₂×#97

—

116

C₂×#97

C₂×D₄₇

yes

characteristic

117

C₂×C₄×C₈

—

118

C₂×C₄×C₈

—

119

#376

—

120

#339

—

121

C₄₇

#21735

yes

characteristic, complemented, Sylow

122

C₂×M₁₆

C₂²×D₄₇

yes

—

123

C₂²×C₈

—

124

C₂²×C₈

—

125

C₂²×C₈

C₂²×D₄₇

yes

—

126

C₄⋊Q₈

—

127

#125

—

128

C₂²⋊Q₈

—

129

C₄×Q₈

—

130

#117

—

131

C₂×C₄²

—

132

C₂×C₄²

C₂²×D₄₇

yes

characteristic

133

#108

—

134

#107

—

135

#106

—

136

C₄⋊C₈

—

137

C₄⋊C₈

—

138

#97

—

139

#97

—

140

#97

C₄×D₄₇

yes

—

141

C₄×C₈

—

142

C₄×C₈

—

143

C₄×C₈

—

144

C₄×C₈

—

145

#95

—

146

C₂×Q₈

—

147

C₂²×C₄

—

148

C₂²×C₄

#10625354

yes

—

149

C₂²×C₄

C₂³×D₄₇

yes

characteristic

150

M₁₆

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

C₂×C₄×D₄₇

yes

—

158

C₂×C₈

#10625352

yes

—

159

C₂×C₈

Q₈×D₄₇

yes

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₂²⋊C₄

—

164

C₄²

—

165

C₄²

—

166

C₄²

—

167

C₄²

C₂×C₄×D₄₇

yes

—

168

C₄²

D₄×D₄₇

yes

—

169

C₂³

C₂×#10625354

yes

characteristic

170

Q₈

—

188

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

188

—

175

C₂×C₄

#13093986

yes

—

176

C₂×C₄

C₂²×C₄×D₄₇

yes

characteristic

177

C₂×C₄

C₂×D₄×D₄₇

yes

characteristic

178

C₂×C₄

C₂×Q₈×D₄₇

yes

—

179

C₂×C₄

C₂×#10625352

yes

—

180

C₈

—

181

C₈

—

182

C₈

—

183

C₈

—

184

C₈

#13093989

yes

—

185

C₂²

—

186

C₂²

C₂×#13093986

yes

characteristic

187

C₂²

#15689678

yes

—

188

C₄

—

189

C₄

—

188

—

190

C₄

—

188

—

191

C₄

C₂×#13093989

yes

—

192

C₄

C₄⋊Q₈×D₄₇

yes

characteristic

193

C₄

#15689671

yes

characteristic

194

C₂

—

195

C₂

#20245955

yes

characteristic

196

C₂

#20252290

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1849	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂³×C₄₇	C₂×#41
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1849	D₄₇
hypocenter	C₄₇	#21735
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1849

subgroup

G/H

center

C₂×C₄

C₂×D₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂³×C₄

Frattini

C₂²×C₄

C₂³×D₄₇

Fitting

C₄₇×#1849

C₂

socle

C₂²×C₄₇

C₂×C₄×D₄

prime radical

C₂³×C₄₇

C₂×#41

cosocle

C₂²×C₄×C₄₇

C₂⁴

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1849

N	H ≅ G/N	type
9. #15689532	C₄	semidirect
39. C₈×Dic₄₇	Q₈	semidirect
121. C₄₇	#21735	semidirect

H ≅ G/N

type

9. #15689532

C₄

semidirect

39. C₈×Dic₄₇

Q₈

semidirect

121. C₄₇

#21735

semidirect

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

subgroup	index
#20253912	2
#20252318	2
#20244368	2
#20244139	2
#20243270	2
#20243076	2
C₄₇×#1849	2
#21735	47

subgroup

index

C₄₇×#1849

Subgroups — #26225143

Conjugacy classes of subgroups

Complements

Maximal subgroups