Group #26132617 — subgroups

Subgroups — #26132617

Order 12032.

366 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 119 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	#20251999	C₂	1	1	yes	maximal, characteristic
2	#20243434	C₂	1	1	yes	maximal, characteristic
3	#20241503	C₂	1	2	yes	maximal, complemented
4	#20241484	C₂	1	2	yes	maximal, complemented
5	C₄₇×#2113	C₂	1	1	yes	maximal, characteristic
6	#15689890	—	2	1	no	—
7	#15689172	—	2	1	no	—
8	#15689172	C₄	1	4	yes	complemented
9	C₂×#13093960	—	2	2	no	—
10	C₂×#13093960	C₂²	1	2	yes	—
11	#15689161	C₂²	1	2	yes	—
12	#15689097	—	2	1	no	—
13	#15688880	—	2	1	no	—
14	C₂×#131×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×C₂²⋊C₁₆	—	2	2	no	—
18	#13094031	—	4	4	no	—
19	C₂×#10625329	—	2	4	no	—
20	C₂×#10625329	—	2	2	no	—
21	#13093983	—	2	2	no	—
22	#13093960	—	2	4	no	—
23	#13093960	—	2	2	no	—
24	#13093960	C₂×C₄	1	4	yes	—
25	#13093958	—	2	2	no	—
26	#13093956	—	2	2	no	—
27	#13093939	—	4	2	no	—
28	C₂×#41×C₄₇	D₄	1	1	yes	characteristic
29	#131×C₄₇	—	2	2	no	—
30	#131×C₄₇	—	2	2	no	—
31	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
32	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₄₇	D₄	1	2	yes	—
34	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
35	M₃₂×C₄₇	—	2	2	no	—
36	C₂×C₁₆×C₄₇	—	2	1	no	—
37	C₂×C₁₆×C₄₇	—	2	1	no	—
38	C₂×C₁₆×C₄₇	—	2	2	no	—
39	#41×C₄₇	—	2	2	no	—
40	#41×C₄₇	—	2	2	no	—
41	C₂×Q₈×C₄₇	Q₁₆	1	1	yes	characteristic
42	C₂×D₄×C₄₇	SD₁₆	1	2	yes	—
43	C₂×D₄×C₄₇	Q₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
46	M₁₆×C₄₇	—	2	2	no	—
47	M₁₆×C₄₇	—	2	4	no	—
48	C₂×C₈×C₄₇	—	2	2	no	—
49	C₂×C₈×C₄₇	—	2	2	no	—
50	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₄₇	#41	1	2	yes	—
54	C₂×C₈×C₄₇	#41	1	2	yes	—
55	C₂²⋊Dic₄₇	—	4	2	no	—
56	C₄⋊Dic₄₇	—	4	2	no	—
57	C₄×Dic₄₇	—	4	2	no	—
58	#10625329	—	2	4	no	—
59	#10625329	—	4	4	no	—
60	#10625329	—	4	2	no	—
61	C₂×C₄₇⋊C₈	—	4	4	no	—
62	C₂×C₄₇⋊C₈	—	4	2	no	—
63	C₁₆×C₄₇	—	4	1	no	—
64	C₁₆×C₄₇	—	4	1	no	—
65	C₂³×C₄₇	#103	1	2	yes	—
66	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
67	Q₈×C₄₇	—	2	2	no	—
68	D₄×C₄₇	—	2	4	no	—
69	D₄×C₄₇	—	2	2	no	—
70	C₂×C₄×C₄₇	—	2	2	no	—
71	C₂×C₄×C₄₇	—	2	2	no	—
72	C₂×C₄×C₄₇	#103	1	2	yes	—
73	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₄₇	#123	1	2	yes	—
75	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×Dic₄₇	—	8	2	no	—
78	C₈×C₄₇	—	2	2	no	—
79	C₈×C₄₇	#117	1	2	yes	—
80	C₈×C₄₇	#123	1	1	yes	characteristic
81	C₈×C₄₇	#123	1	1	yes	characteristic
82	C₄₇⋊C₈	—	4	4	no	—
83	#12877	—	47	1	no	maximal, Sylow
84	C₂²×C₄₇	—	2	2	no	—
85	C₂²×C₄₇	—	4	1	no	—
86	C₂²×C₄₇	—	4	2	no	—
87	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
88	C₂²×C₄₇	#481	1	2	yes	—
89	C₄×C₄₇	—	2	2	no	—
90	C₄×C₄₇	#381	1	1	yes	characteristic
91	C₄×C₄₇	#418	1	1	yes	characteristic
92	C₄×C₄₇	#484	1	2	yes	—
93	Dic₄₇	—	16	2	no	—
94	#2916	—	47	1	no	—
95	#2113	D₄₇	1	1	yes	characteristic
96	#1812	—	47	1	no	—
97	#1389	—	47	2	no	—
98	#1382	—	47	2	no	—
99	C₂×C₄₇	—	2	2	no	—
100	C₂×C₄₇	—	2	1	no	—
101	C₂×C₄₇	—	4	2	no	—
102	C₂×C₄₇	#1824	1	1	yes	characteristic
103	#568	—	94	1	no	—
104	C₂×#131	C₂×D₄₇	1	1	yes	characteristic
105	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
106	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
107	#430	—	47	4	no	—
108	#430	—	94	1	no	—
109	C₂×#108	—	47	2	no	—
110	C₂×#108	—	94	2	no	—
111	#427	—	47	2	no	—
112	#410	—	94	1	no	—
113	C₂²⋊C₁₆	—	2	2	no	—
114	H(3,ℤ/4)	—	94	1	no	—
115	C₄₇	#12877	1	1	yes	characteristic, complemented, Sylow
116	C₂×#41	D₁₈₈	1	1	yes	characteristic
117	#131	—	2	2	no	—
118	#131	—	2	2	no	—
119	#131	—	188	4	no	—
120	C₂×M₁₆	—	94	4	no	—
121	C₂×M₁₆	—	94	2	no	—
122	C₂×M₁₆	C₄×D₄₇	1	2	yes	—
123	C₂×M₁₆	D₁₈₈	1	1	yes	characteristic
124	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
125	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
126	#117	—	94	2	no	—
127	M₃₂	—	2	2	no	—
128	C₂×C₁₆	—	2	2	no	—
129	C₂×C₁₆	—	2	1	no	—
130	C₂×C₁₆	—	2	1	no	—
131	#108	—	47	4	no	—
132	#108	—	94	4	no	—
133	#108	—	94	2	no	—
134	#107	—	94	2	no	—
135	#106	—	94	2	no	—
136	#99	—	188	2	no	—
137	#41	—	2	2	no	—
138	#41	—	2	2	no	—
139	C₂×Q₈	Dic₁₈₈	1	1	yes	characteristic
140	C₂×D₄	#10625325	1	2	yes	—
141	C₂×D₄	Dic₁₈₈	1	1	yes	characteristic
142	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
143	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
144	M₁₆	—	2	2	no	—
145	M₁₆	—	2	4	no	—
146	M₁₆	—	94	4	no	—
147	M₁₆	—	188	4	no	—
148	M₁₆	—	188	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	188	4	no	—
152	C₂×C₈	—	188	2	no	—
153	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
154	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
155	C₂×C₈	#10625352	1	2	yes	—
156	C₂×C₈	#10625352	1	2	yes	—
157	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
158	C₄⋊C₄	—	188	2	no	—
159	C₂²⋊C₄	—	188	2	no	—
160	C₄²	—	188	2	no	—
161	C₁₆	—	4	1	no	—
162	C₁₆	—	4	1	no	—
163	C₂³	#13093949	1	2	yes	—
164	C₂³	C₂×#10625325	1	1	yes	characteristic
165	Q₈	—	2	2	no	—
166	D₄	—	2	4	no	—
167	D₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	376	2	no	—
171	C₂×C₄	#13093949	1	2	yes	—
172	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
173	C₂×C₄	#13094008	1	2	yes	—
174	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
175	C₂×C₄	#13094052	1	2	yes	—
176	C₈	—	2	2	no	—
177	C₈	—	188	4	no	—
178	C₈	#13093980	1	2	yes	—
179	C₈	#13094005	1	1	yes	characteristic
180	C₈	#13094005	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	—	4	1	no	—
183	C₂²	—	4	2	no	—
184	C₂²	C₂×#13093949	1	1	yes	characteristic
185	C₂²	#15689442	1	2	yes	—
186	C₄	—	2	2	no	—
187	C₄	—	752	2	no	—
188	C₄	#15688984	1	1	yes	characteristic
189	C₄	#15689129	1	1	yes	characteristic
190	C₄	#15689459	1	2	yes	—
191	C₂	—	2	2	no	—
192	C₂	—	2	1	no	—
193	C₂	—	4	2	no	—
194	C₂	#20243479	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20251999

C₂

yes

maximal, characteristic

#20243434

C₂

yes

maximal, characteristic

#20241503

C₂

yes

maximal, complemented

#20241484

C₂

yes

maximal, complemented

C₄₇×#2113

C₂

yes

maximal, characteristic

#15689890

—

#15689172

—

#15689172

C₄

yes

complemented

C₂×#13093960

—

C₂×#13093960

C₂²

yes

—

#15689161

C₂²

yes

—

#15689097

—

#15688880

—

C₂×#131×C₄₇

C₂²

yes

characteristic

C₂×M₃₂×C₄₇

C₂²

yes

characteristic

C₂²×C₁₆×C₄₇

C₂²

yes

characteristic

C₄₇×C₂²⋊C₁₆

—

—

—

—

—

—

—

C₂×C₄

yes

—

#13093958

—

#13093956

—

#13093939

—

C₂×#41×C₄₇

D₄

yes

characteristic

#131×C₄₇

—

#131×C₄₇

—

C₂×M₁₆×C₄₇

C₂×C₄

yes

—

C₂×M₁₆×C₄₇

D₄

yes

characteristic

C₂²×C₈×C₄₇

D₄

yes

—

C₂²×C₈×C₄₇

C₂³

yes

characteristic

M₃₂×C₄₇

—

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

—

#41×C₄₇

—

#41×C₄₇

—

C₂×Q₈×C₄₇

Q₁₆

yes

characteristic

C₂×D₄×C₄₇

SD₁₆

yes

—

C₂×D₄×C₄₇

Q₁₆

yes

characteristic

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

#41

yes

—

C₂×C₈×C₄₇

#41

yes

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

#10625329

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₁₆×C₄₇

—

C₁₆×C₄₇

—

C₂³×C₄₇

#103

yes

—

C₂³×C₄₇

C₂×SD₁₆

yes

characteristic

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#103

yes

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#123

yes

—

C₂×C₄×C₄₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₈×C₄₇

#117

yes

—

C₈×C₄₇

yes

C₈×C₄₇

yes

C₄₇⋊C₈

—

#12877

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×#103

yes

characteristic

C₂²×C₄₇

#481

yes

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

#2916

—

#2113

D₄₇

yes

—

—

—

C₂×C₄₇

—

100

C₂×C₄₇

—

101

C₂×C₄₇

—

102

C₂×C₄₇

#1824

yes

characteristic

103

#568

—

104

C₂×#131

C₂×D₄₇

yes

characteristic

105

C₂×M₃₂

C₂×D₄₇

yes

characteristic

106

C₂²×C₁₆

C₂×D₄₇

yes

characteristic

107

#430

—

108

#430

—

109

C₂×#108

—

110

C₂×#108

—

111

#427

—

112

#410

—

113

C₂²⋊C₁₆

—

114

H(3,ℤ/4)

—

115

C₄₇

#12877

yes

characteristic, complemented, Sylow

116

C₂×#41

D₁₈₈

yes

characteristic

117

#131

—

118

#131

—

119

#131

—

188

—

120

C₂×M₁₆

—

121

C₂×M₁₆

—

122

C₂×M₁₆

C₄×D₄₇

yes

—

123

C₂×M₁₆

D₁₈₈

yes

characteristic

124

C₂²×C₈

C₂²⋊D₄₇

yes

—

125

C₂²×C₈

C₂²×D₄₇

yes

characteristic

126

#117

—

127

M₃₂

—

128

C₂×C₁₆

—

129

C₂×C₁₆

—

130

C₂×C₁₆

—

131

#108

—

132

#108

—

133

#108

—

134

#107

—

135

#106

—

136

#99

—

188

—

137

#41

—

138

#41

—

139

C₂×Q₈

Dic₁₈₈

yes

characteristic

140

C₂×D₄

#10625325

yes

—

141

C₂×D₄

Dic₁₈₈

yes

characteristic

142

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

—

143

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

144

M₁₆

—

145

M₁₆

—

146

M₁₆

—

147

M₁₆

—

188

—

148

M₁₆

—

188

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

188

—

152

C₂×C₈

—

188

—

153

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

—

154

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

155

C₂×C₈

#10625352

yes

—

156

C₂×C₈

#10625352

yes

—

157

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

158

C₄⋊C₄

—

188

—

159

C₂²⋊C₄

—

188

—

160

C₄²

—

188

—

161

C₁₆

—

162

C₁₆

—

163

C₂³

#13093949

yes

—

164

C₂³

C₂×#10625325

yes

characteristic

165

Q₈

—

166

D₄

—

167

D₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

376

—

171

C₂×C₄

#13093949

yes

—

172

C₂×C₄

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

173

C₂×C₄

#13094008

yes

—

174

C₂×C₄

C₂×Dic₁₈₈

yes

characteristic

175

C₂×C₄

#13094052

yes

—

176

C₈

—

177

C₈

—

188

—

178

C₈

#13093980

yes

—

179

C₈

#13094005

yes

characteristic

180

C₈

#13094005

yes

characteristic

181

C₂²

—

182

C₂²

—

183

C₂²

—

184

C₂²

C₂×#13093949

yes

characteristic

185

C₂²

#15689442

yes

—

186

C₄

—

187

C₄

—

752

—

188

C₄

#15688984

yes

characteristic

189

C₄

#15689129

yes

characteristic

190

C₄

#15689459

yes

—

191

C₂

—

192

C₂

—

193

C₂

—

194

C₂

#20243479

yes

characteristic

	subgroup	G/H
center	C₄	#15689129
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#2113	C₂
socle	C₂×C₄₇	#1824
prime radical	C₂×#41×C₄₇	D₄
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#2113	D₄₇
hypocenter	C₄₇	#12877
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2113

subgroup

G/H

center

C₄

#15689129

commutator

C₂×C₈×C₄₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#2113

C₂

socle

C₂×C₄₇

#1824

prime radical

C₂×#41×C₄₇

D₄

cosocle

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2113

N	H ≅ G/N	type
3. #20241503	C₂	semidirect
4. #20241484	C₂	semidirect
8. #15689172	C₄	semidirect
115. C₄₇	#12877	semidirect

H ≅ G/N

type

3. #20241503

C₂

semidirect

4. #20241484

C₂

semidirect

8. #15689172

C₄

semidirect

115. C₄₇

#12877

semidirect

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

subgroup	index
#20251999	2
#20243434	2
#20241503	2
#20241484	2
C₄₇×#2113	2
#12877	47

subgroup

index

C₄₇×#2113

Subgroups — #26132617

Conjugacy classes of subgroups

Complements

Maximal subgroups