Group #26106709 — subgroups

Subgroups — #26106709

Order 12032.

414 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 179 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	#20243133	C₂	1	2	yes	maximal
2	C₄×#13093934	C₂	1	1	yes	maximal, characteristic
3	C₄₇⋊#1273	C₂	1	2	yes	maximal
4	C₄₇⋊#1273	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093934	C₄	1	2	yes	—
7	C₂×#13093934	C₄	1	2	yes	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
10	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
11	#15688877	C₄	1	4	yes	—
12	#15688877	C₄	1	4	yes	—
13	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
15	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
16	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
17	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
21	#13093934	—	2	4	no	—
22	#13093934	—	2	4	no	—
23	C₄×C₄₇⋊C₈	—	2	1	no	—
24	C₄×C₄₇⋊C₈	—	2	1	no	—
25	C₄×C₄₇⋊C₈	—	2	2	no	—
26	C₄×C₄₇⋊C₈	—	2	2	no	—
27	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
30	C₄×C₈×C₄₇	—	2	2	no	—
31	C₄×C₈×C₄₇	—	2	2	no	—
32	C₄×C₈×C₄₇	—	2	2	no	—
33	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
35	C₂²×Dic₄₇	C₄²	1	1	yes	characteristic
36	C₂²×Dic₄₇	C₄²	1	1	yes	characteristic
37	C₂²×Dic₄₇	C₄²	1	2	yes	—
38	C₂×C₈×C₄₇	—	2	2	no	—
39	C₂×C₈×C₄₇	—	2	2	no	—
40	C₂×C₈×C₄₇	—	2	4	no	—
41	C₂×C₈×C₄₇	—	2	4	no	—
42	C₂×C₈×C₄₇	—	2	4	no	—
43	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
44	C₄²×C₄₇	#41	1	2	yes	—
45	C₄²×C₄₇	#41	1	1	yes	characteristic
46	C₄²×C₄₇	#41	1	1	yes	characteristic
47	C₄×Dic₄₇	—	2	2	no	—
48	C₄×Dic₄₇	—	2	2	no	—
49	C₄×Dic₄₇	—	2	4	no	—
50	C₄×Dic₄₇	—	2	4	no	—
51	C₂×C₄₇⋊C₈	—	2	2	no	—
52	C₂×C₄₇⋊C₈	—	2	2	no	—
53	C₂×C₄₇⋊C₈	—	2	4	no	—
54	C₂×C₄₇⋊C₈	—	2	4	no	—
55	C₂×C₄₇⋊C₈	—	2	4	no	—
56	C₂×C₄₇⋊C₈	M₁₆	1	4	yes	—
57	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
58	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
59	C₂×C₄×C₄₇	#117	1	2	yes	—
60	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
61	C₂×C₄×C₄₇	#117	1	2	yes	—
62	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
63	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
64	C₂×C₄×C₄₇	#131	1	4	yes	—
65	C₂×C₄×C₄₇	#131	1	4	yes	—
66	C₂×Dic₄₇	—	2	1	no	—
67	C₂×Dic₄₇	—	2	1	no	—
68	C₂×Dic₄₇	—	2	2	no	—
69	C₂×Dic₄₇	—	2	2	no	—
70	C₂×Dic₄₇	—	2	2	no	—
71	C₂×Dic₄₇	—	2	2	no	—
72	C₂×Dic₄₇	—	2	2	no	—
73	C₈×C₄₇	—	2	4	no	—
74	C₈×C₄₇	—	4	4	no	—
75	C₄₇⋊C₈	—	2	4	no	—
76	C₄₇⋊C₈	—	4	4	no	—
77	#8134	—	47	1	no	maximal, Sylow
78	C₂²×C₄₇	#376	1	1	yes	characteristic
79	C₂²×C₄₇	C₄×M₁₆	1	1	yes	characteristic
80	C₂²×C₄₇	C₄×M₁₆	1	1	yes	characteristic
81	C₂²×C₄₇	#405	1	2	yes	—
82	C₂²×C₄₇	#405	1	2	yes	—
83	C₄×C₄₇	#432	1	2	yes	—
84	C₄×C₄₇	#432	1	2	yes	—
85	C₄×C₄₇	#433	1	4	yes	—
86	C₄×C₄₇	#433	1	4	yes	—
87	Dic₄₇	—	4	1	no	—
88	Dic₄₇	—	4	2	no	—
89	Dic₄₇	—	4	1	no	—
90	#1744	—	47	2	no	—
91	C₄×#97	—	47	1	no	—
92	#1273	—	47	2	no	—
93	#1273	—	47	1	no	—
94	#1273	D₄₇	1	1	yes	characteristic
95	C₂×C₄₇	#1453	1	1	yes	characteristic
96	C₂×C₄₇	#1453	1	2	yes	—
97	C₂×C₄₇	#1453	1	1	yes	characteristic
98	C₂×C₄₇	#1745	1	1	yes	characteristic
99	C₂×C₄₇	#1746	1	2	yes	—
100	C₂×#97	—	47	2	no	—
101	C₂×#97	—	47	2	no	—
102	C₂×C₄×C₈	—	47	1	no	—
103	C₂×C₄×C₈	—	47	2	no	—
104	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
106	C₄³	—	47	1	no	—
107	#336	—	47	4	no	—
108	#336	—	47	4	no	—
109	C₄₇	#8134	1	1	yes	characteristic, complemented, Sylow
110	C₂²×C₈	—	47	2	no	—
111	C₂²×C₈	—	47	4	no	—
112	C₂²×C₈	C₄×D₄₇	1	4	yes	—
113	C₂²×C₈	C₄×D₄₇	1	2	yes	—
114	C₂×C₄²	—	47	2	no	—
115	C₂×C₄²	—	47	2	no	—
116	C₂×C₄²	—	47	1	no	—
117	C₂×C₄²	—	47	1	no	—
118	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
119	#97	—	94	4	no	—
120	#97	—	94	4	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	—	2	2	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	94	1	no	—
125	C₄×C₈	—	94	2	no	—
126	C₄×C₈	—	94	1	no	—
127	C₄×C₈	—	94	2	no	—
128	C₂²×C₄	—	47	1	no	—
129	C₂²×C₄	—	47	1	no	—
130	C₂²×C₄	—	47	2	no	—
131	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	47	4	no	—
139	C₂×C₈	—	94	2	no	—
140	C₂×C₈	—	94	2	no	—
141	C₂×C₈	—	94	4	no	—
142	C₂×C₈	—	94	4	no	—
143	C₂×C₈	—	94	4	no	—
144	C₂×C₈	#10625324	1	4	yes	—
145	C₄²	—	94	2	no	—
146	C₄²	—	94	2	no	—
147	C₄²	—	94	4	no	—
148	C₄²	—	94	4	no	—
149	C₄²	#10625352	1	2	yes	—
150	C₄²	#10625352	1	1	yes	characteristic
151	C₄²	#10625354	1	1	yes	characteristic
152	C₂³	C₄²×D₄₇	1	1	yes	characteristic
153	C₂×C₄	—	94	1	no	—
154	C₂×C₄	—	94	1	no	—
155	C₂×C₄	—	94	2	no	—
156	C₂×C₄	—	94	2	no	—
157	C₂×C₄	—	94	2	no	—
158	C₂×C₄	—	94	2	no	—
159	C₂×C₄	—	94	2	no	—
160	C₂×C₄	#13093980	1	1	yes	characteristic
161	C₂×C₄	#13093980	1	2	yes	—
162	C₂×C₄	#13093982	1	1	yes	characteristic
163	C₂×C₄	#13093982	1	2	yes	—
164	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
165	C₂×C₄	C₂×#10625324	1	2	yes	—
166	C₂×C₄	#13094030	1	4	yes	—
167	C₂×C₄	#13094032	1	4	yes	—
168	C₈	—	2	4	no	—
169	C₈	—	4	4	no	—
170	C₈	—	94	4	no	—
171	C₈	—	188	4	no	—
172	C₂²	#15688968	1	1	yes	characteristic
173	C₂²	C₄×#10625324	1	1	yes	characteristic
174	C₂²	#15689075	1	1	yes	characteristic
175	C₂²	#15689077	1	2	yes	—
176	C₂²	#15689080	1	2	yes	—
177	C₄	—	188	1	no	—
178	C₄	—	188	2	no	—
179	C₄	—	188	1	no	—
180	C₄	#15689177	1	2	yes	—
181	C₄	#15689179	1	2	yes	—
182	C₄	#15689181	1	4	yes	—
183	C₄	#15689183	1	4	yes	—
184	C₂	#20241682	1	1	yes	characteristic
185	C₂	#20241684	1	2	yes	—
186	C₂	#20241686	1	1	yes	characteristic
187	C₂	#20243135	1	1	yes	characteristic
188	C₂	#20243138	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243133

C₂

yes

maximal

C₄×#13093934

C₂

yes

maximal, characteristic

C₄₇⋊#1273

C₂

yes

maximal

C₄₇⋊#1273

C₂

yes

maximal, characteristic

C₄₇×#1273

C₂

yes

maximal, characteristic

C₂×#13093934

C₄

yes

—

C₂×#13093934

C₄

yes

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

C₄²×Dic₄₇

C₂²

yes

characteristic

#15688877

C₄

yes

—

#15688877

C₄

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₄₇

C₂²

yes

—

C₂²×C₄₇⋊C₈

C₂×C₄

yes

—

C₂²×C₄₇⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₄₇

C₂×C₄

yes

characteristic

C₂×C₄×Dic₄₇

C₂×C₄

yes

characteristic

C₂×C₄×Dic₄₇

C₂×C₄

yes

—

C₂×C₄×Dic₄₇

C₂×C₄

yes

—

#13093934

—

#13093934

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂²×C₄

yes

—

C₂²×Dic₄₇

C₄²

yes

characteristic

C₂²×Dic₄₇

C₄²

yes

characteristic

C₂²×Dic₄₇

C₄²

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

M₁₆

yes

—

C₄²×C₄₇

#41

yes

—

C₄²×C₄₇

yes

C₄²×C₄₇

yes

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

M₁₆

yes

—

C₂³×C₄₇

C₂×C₄²

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×C₄×C₄₇

#117

yes

characteristic

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

C₄×M₁₆

yes

characteristic

C₂²×C₄₇

C₄×M₁₆

yes

characteristic

C₂²×C₄₇

#405

yes

—

C₂²×C₄₇

#405

yes

—

C₄×C₄₇

#432

yes

—

C₄×C₄₇

#432

yes

—

C₄×C₄₇

#433

yes

—

C₄×C₄₇

#433

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

#1744

—

C₄×#97

—

#1273

—

#1273

—

#1273

D₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

100

C₂×#97

—

101

C₂×#97

—

102

C₂×C₄×C₈

—

103

C₂×C₄×C₈

—

104

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

105

C₂×C₄×C₈

C₂×D₄₇

yes

—

106

C₄³

—

107

#336

—

108

#336

—

109

C₄₇

#8134

yes

characteristic, complemented, Sylow

110

C₂²×C₈

—

111

C₂²×C₈

—

112

C₂²×C₈

C₄×D₄₇

yes

—

113

C₂²×C₈

C₄×D₄₇

yes

—

114

C₂×C₄²

—

115

C₂×C₄²

—

116

C₂×C₄²

—

117

C₂×C₄²

—

118

C₂×C₄²

C₂²×D₄₇

yes

characteristic

119

#97

—

120

#97

—

121

C₄×C₈

—

122

C₄×C₈

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

132

C₂²×C₄

C₂×C₄×D₄₇

yes

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

#10625324

yes

—

145

C₄²

—

146

C₄²

—

147

C₄²

—

148

C₄²

—

149

C₄²

#10625352

yes

—

150

C₄²

#10625352

yes

characteristic

151

C₄²

#10625354

yes

characteristic

152

C₂³

C₄²×D₄₇

yes

characteristic

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

#13093980

yes

characteristic

161

C₂×C₄

#13093980

yes

—

162

C₂×C₄

#13093982

yes

characteristic

163

C₂×C₄

#13093982

yes

—

164

C₂×C₄

M₁₆×D₄₇

yes

—

165

C₂×C₄

C₂×#10625324

yes

—

166

C₂×C₄

#13094030

yes

—

167

C₂×C₄

#13094032

yes

—

168

C₈

—

169

C₈

—

170

C₈

—

171

C₈

—

188

—

172

C₂²

#15688968

yes

characteristic

173

C₂²

C₄×#10625324

yes

characteristic

174

C₂²

#15689075

yes

characteristic

175

C₂²

#15689077

yes

—

176

C₂²

#15689080

yes

—

177

C₄

—

188

—

178

C₄

—

188

—

179

C₄

—

188

—

180

C₄

#15689177

yes

—

181

C₄

#15689179

yes

—

182

C₄

#15689181

yes

—

183

C₄

#15689183

yes

—

184

C₂

#20241682

yes

characteristic

185

C₂

#20241684

yes

—

186

C₂

#20241686

yes

characteristic

187

C₂

#20243135

yes

characteristic

188

C₂

#20243138

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₄₇
commutator, socle, prime radical	C₂³×C₄₇	C₂×C₄²
Fitting	C₄₇×#1273	C₂
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1273	D₄₇
hypocenter	C₄₇	#8134
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₄₇

commutator, socle, prime radical

C₂³×C₄₇

C₂×C₄²

Fitting

C₄₇×#1273

C₂

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
109. C₄₇	#8134	semidirect

H ≅ G/N

type

109. C₄₇

#8134

semidirect

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

subgroup	index
#20243133	2
C₄×#13093934	2
C₄₇⋊#1273	2
C₄₇×#1273	2
#8134	47

subgroup

index

#20243133

C₄×#13093934

C₄₇⋊#1273

C₄₇×#1273

#8134

Subgroups — #26106709

Conjugacy classes of subgroups

Complements

Maximal subgroups