Group #26109616 — subgroups

Subgroups — #26109616

Order 12032.

530 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 139 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	#20243456	C₂	1	2	yes	maximal
2	#20241231	C₂	1	4	yes	maximal
3	C₂×C₄₇×#382	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
5	#15688878	—	2	2	no	—
6	#15688878	—	2	2	no	—
7	C₂²×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
8	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
9	C₄₇×#382	—	2	2	no	—
10	C₄₇×#382	—	2	2	no	—
11	C₂×#95×C₄₇	C₄	1	2	yes	—
12	C₂×#95×C₄₇	C₂²	1	2	yes	—
13	C₂²×C₄₇⋊C₈	—	2	4	no	—
14	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
15	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
17	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
18	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
20	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
21	C₂×C₄⋊C₄×C₄₇	—	4	4	no	—
22	C₂×C₄²×C₄₇	—	2	4	no	—
23	C₂×C₄²×C₄₇	D₄	1	2	yes	—
24	C₂×C₄²×C₄₇	D₄	1	2	yes	—
25	#95×C₄₇	—	2	4	no	—
26	#95×C₄₇	—	2	4	no	—
27	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₄₇	—	2	2	no	—
29	C₂²×C₄×C₄₇	—	2	2	no	—
30	C₂²×C₄×C₄₇	—	2	2	no	—
31	C₂²×C₄×C₄₇	—	2	2	no	—
32	C₂²×C₄×C₄₇	—	2	4	no	—
33	C₂²×C₄×C₄₇	—	2	4	no	—
34	C₂²×C₄×C₄₇	—	2	8	no	—
35	C₂²×C₄×C₄₇	—	4	2	no	—
36	C₂²×C₄×C₄₇	—	4	4	no	—
37	C₂²×C₄×C₄₇	—	4	4	no	—
38	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
42	C₂²×C₄×C₄₇	#41	1	4	yes	—
43	C₄⋊C₄×C₄₇	—	4	4	no	—
44	C₄⋊C₄×C₄₇	—	4	4	no	—
45	C₄²×C₄₇	—	2	4	no	—
46	C₄²×C₄₇	—	2	4	no	—
47	C₂×C₄₇⋊C₈	—	4	8	no	—
48	C₂×C₄₇⋊C₈	—	4	4	no	—
49	C₂³×C₄₇	—	2	2	no	—
50	C₂³×C₄₇	—	2	1	no	—
51	C₂³×C₄₇	—	2	1	no	—
52	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂³×C₄₇	#117	1	2	yes	—
54	C₂³×C₄₇	#131	1	4	yes	—
55	C₂×C₄×C₄₇	—	2	8	no	—
56	C₂×C₄×C₄₇	—	2	1	no	—
57	C₂×C₄×C₄₇	—	2	1	no	—
58	C₂×C₄×C₄₇	—	2	4	no	—
59	C₂×C₄×C₄₇	—	2	1	no	—
60	C₂×C₄×C₄₇	—	2	1	no	—
61	C₂×C₄×C₄₇	—	2	2	no	—
62	C₂×C₄×C₄₇	—	2	4	no	—
63	C₂×C₄×C₄₇	—	4	4	no	—
64	C₂×C₄×C₄₇	—	4	4	no	—
65	C₂×C₄×C₄₇	—	4	4	no	—
66	C₂×C₄×C₄₇	—	4	4	no	—
67	C₂×C₄×C₄₇	—	4	4	no	—
68	C₂×C₄×C₄₇	—	8	8	no	—
69	C₂×C₄×C₄₇	#100	1	2	yes	—
70	C₂×C₄×C₄₇	#101	1	2	yes	—
71	C₂×C₄×C₄₇	#123	1	4	yes	—
72	C₄₇⋊C₈	—	8	4	no	—
73	#8703	—	47	1	no	maximal, Sylow
74	C₂²×C₄₇	—	2	2	no	—
75	C₂²×C₄₇	—	2	2	no	—
76	C₂²×C₄₇	—	2	4	no	—
77	C₂²×C₄₇	—	2	4	no	—
78	C₂²×C₄₇	#381	1	1	yes	characteristic
79	C₂²×C₄₇	#408	1	2	yes	—
80	C₂²×C₄₇	#410	1	2	yes	—
81	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
82	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
83	C₂²×C₄₇	#433	1	4	yes	—
84	C₄×C₄₇	—	2	4	no	—
85	C₄×C₄₇	—	2	2	no	—
86	C₄×C₄₇	—	2	2	no	—
87	C₄×C₄₇	—	4	2	no	—
88	C₄×C₄₇	—	8	4	no	—
89	C₂×#382	D₄₇	1	1	yes	characteristic
90	#1817	—	47	2	no	—
91	#1280	—	47	4	no	—
92	C₂×C₄₇	—	2	4	no	—
93	C₂×C₄₇	#1469	1	2	yes	—
94	C₂×C₄₇	#1470	1	2	yes	—
95	C₂×C₄₇	#1818	1	1	yes	characteristic
96	C₂×C₄₇	#1820	1	1	yes	characteristic
97	C₂×C₄₇	#1821	1	1	yes	characteristic
98	C₂²×C₄⋊C₄	Dic₄₇	1	2	yes	—
99	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
100	C₂×C₂²⋊C₈	—	47	4	no	—
101	#382	—	2	2	no	—
102	#382	—	2	2	no	—
103	C₂×#95	Dic₄₇	1	2	yes	—
104	C₂×#95	C₂×D₄₇	1	2	yes	—
105	#336	—	94	2	no	—
106	#336	—	94	2	no	—
107	C₄₇	#8703	1	1	yes	characteristic, complemented, Sylow
108	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
109	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
110	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
111	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
112	C₂²×C₈	—	94	4	no	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	4	4	no	—
116	C₂×C₄²	—	2	4	no	—
117	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
118	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
119	C₂²⋊C₈	—	94	8	no	—
120	#95	—	2	4	no	—
121	#95	—	2	4	no	—
122	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	—	2	4	no	—
128	C₂²×C₄	—	2	4	no	—
129	C₂²×C₄	—	2	8	no	—
130	C₂²×C₄	—	4	2	no	—
131	C₂²×C₄	—	4	4	no	—
132	C₂²×C₄	—	4	4	no	—
133	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
134	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
135	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
136	C₂²×C₄	#10625350	1	4	yes	—
137	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
138	C₂×C₈	—	188	8	no	—
139	C₂×C₈	—	188	4	no	—
140	C₄⋊C₄	—	4	4	no	—
141	C₄⋊C₄	—	4	4	no	—
142	C₄²	—	2	4	no	—
143	C₄²	—	2	4	no	—
144	C₂³	—	2	2	no	—
145	C₂³	—	2	1	no	—
146	C₂³	—	2	1	no	—
147	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
148	C₂³	#13093983	1	2	yes	—
149	C₂³	#13094031	1	4	yes	—
150	C₂×C₄	—	2	8	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	8	8	no	—
164	C₂×C₄	#13093941	1	2	yes	—
165	C₂×C₄	#13093943	1	2	yes	—
166	C₂×C₄	#13094007	1	4	yes	—
167	C₈	—	376	4	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	4	no	—
171	C₂²	—	2	4	no	—
172	C₂²	#15688986	1	1	yes	characteristic
173	C₂²	#15689090	1	2	yes	—
174	C₂²	#15689097	1	2	yes	—
175	C₂²	C₂×#13093941	1	1	yes	characteristic
176	C₂²	C₂×#13093943	1	1	yes	characteristic
177	C₂²	#15689184	1	4	yes	—
178	C₄	—	2	4	no	—
179	C₄	—	2	2	no	—
180	C₄	—	2	2	no	—
181	C₄	—	4	2	no	—
182	C₄	—	8	4	no	—
183	C₂	—	2	4	no	—
184	C₂	#20241740	1	2	yes	—
185	C₂	#20241744	1	2	yes	—
186	C₂	#20243459	1	1	yes	characteristic
187	C₂	#20243465	1	1	yes	characteristic
188	C₂	#20243468	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243456

C₂

yes

maximal

#20241231

C₂

yes

maximal

C₂×C₄₇×#382

C₂

yes

maximal, characteristic

C₂×C₂²⋊(C₄₇⋊C₈)

C₂²

yes

—

#15688878

—

#15688878

—

C₂²×C₄⋊C₄×C₄₇

C₄

yes

—

C₂²×C₄²×C₄₇

C₂²

yes

characteristic

C₄₇×#382

—

C₄₇×#382

—

C₂×#95×C₄₇

C₄

yes

—

C₂×#95×C₄₇

C₂²

yes

—

C₂²×C₄₇⋊C₈

—

C₂²⋊(C₄₇⋊C₈)

—

C₂³×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂³×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂³×C₄×C₄₇

C₂×C₄

yes

—

C₂³×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

—

#95×C₄₇

—

#95×C₄₇

—

C₂⁴×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

#41

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂³×C₄₇

#117

yes

—

C₂³×C₄₇

#131

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#100

yes

—

C₂×C₄×C₄₇

#101

yes

—

C₂×C₄×C₄₇

#123

yes

—

C₄₇⋊C₈

—

#8703

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#381

yes

characteristic

C₂²×C₄₇

#408

yes

—

C₂²×C₄₇

#410

yes

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₂×#382

D₄₇

yes

characteristic

#1817

—

#1280

—

C₂×C₄₇

—

C₂×C₄₇

#1469

yes

—

C₂×C₄₇

#1470

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂²×C₄⋊C₄

Dic₄₇

yes

—

C₂²×C₄²

C₂×D₄₇

yes

characteristic

100

C₂×C₂²⋊C₈

—

101

#382

—

102

#382

—

103

C₂×#95

Dic₄₇

yes

—

104

C₂×#95

C₂×D₄₇

yes

—

105

#336

—

106

#336

—

107

C₄₇

#8703

yes

characteristic, complemented, Sylow

108

C₂³×C₄

C₂×Dic₄₇

yes

characteristic

109

C₂³×C₄

C₂×Dic₄₇

yes

characteristic

110

C₂³×C₄

C₂×Dic₄₇

yes

—

111

C₂³×C₄

C₂²×D₄₇

yes

characteristic

112

C₂²×C₈

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄²

—

117

C₂×C₄²

C₂²⋊D₄₇

yes

—

118

C₂×C₄²

C₂²⋊D₄₇

yes

—

119

C₂²⋊C₈

—

120

#95

—

121

#95

—

122

C₂⁴

C₂²×Dic₄₇

yes

characteristic

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

C₂²⋊Dic₄₇

yes

—

134

C₂²×C₄

C₂²⋊Dic₄₇

yes

characteristic

135

C₂²×C₄

C₂²⋊Dic₄₇

yes

characteristic

136

C₂²×C₄

#10625350

yes

—

137

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

138

C₂×C₈

—

188

—

139

C₂×C₈

—

188

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄²

—

143

C₄²

—

144

C₂³

—

145

C₂³

—

146

C₂³

—

147

C₂³

C₂×C₂²⋊Dic₄₇

yes

characteristic

148

C₂³

#13093983

yes

—

149

C₂³

#13094031

yes

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

#13093941

yes

—

165

C₂×C₄

#13093943

yes

—

166

C₂×C₄

#13094007

yes

—

167

C₈

—

376

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

#15688986

yes

characteristic

173

C₂²

#15689090

yes

—

174

C₂²

#15689097

yes

—

175

C₂²

C₂×#13093941

yes

characteristic

176

C₂²

C₂×#13093943

yes

characteristic

177

C₂²

#15689184

yes

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₂

—

184

C₂

#20241740

yes

—

185

C₂

#20241744

yes

—

186

C₂

#20243459

yes

characteristic

187

C₂

#20243465

yes

characteristic

188

C₂

#20243468

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator, prime radical	C₂⁴×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#382	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#382	D₄₇
hypocenter	C₄₇	#8703
LCS	G ▷ C₂⁴×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#382

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₄₇

commutator, prime radical

C₂⁴×C₄₇

C₂²×C₄

Frattini

C₂³×C₄

C₂²×D₄₇

Fitting

C₂×C₄₇×#382

C₂

socle

C₂³×C₄₇

C₂×C₂²⋊C₄

cosocle

C₂³×C₄×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂⁴×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#382

N	H ≅ G/N	type
107. C₄₇	#8703	semidirect

H ≅ G/N

type

107. C₄₇

#8703

semidirect

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

subgroup	index
#20243456	2
#20241231	2
C₂×C₄₇×#382	2
#8703	47

subgroup

index

#20243456

#20241231

C₂×C₄₇×#382

#8703

Subgroups — #26109616

Conjugacy classes of subgroups

Complements

Maximal subgroups