Group #26104250 — subgroups

Subgroups — #26104250

Order 12032.

444 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 159 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	#20243134	C₂	1	2	yes	maximal
2	#20241211	C₂	1	4	yes	maximal
3	C₄×#125×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₄₇⋊C₈	—	2	2	no	—
5	#15688878	—	2	2	no	—
6	#15688878	C₂²	1	4	yes	—
7	C₂×#125×C₄₇	C₄	1	1	yes	characteristic
8	C₄₇×#389	C₂²	1	2	yes	—
9	C₄₇×#387	—	2	2	no	—
10	C₄₇×#382	C₄	1	1	yes	characteristic
11	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
12	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
13	C₄³×C₄₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄₇⋊C₈	—	2	4	no	—
15	C₄×C₄₇⋊C₈	—	2	8	no	—
16	#125×C₄₇	—	2	2	no	—
17	#125×C₄₇	C₈	1	4	yes	complemented
18	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
19	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
20	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
21	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₄₇	—	2	2	no	—
23	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₄₇	D₄	1	2	yes	—
26	C₂×C₄²×C₄₇	D₄	1	2	yes	—
27	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
28	#95×C₄₇	—	2	2	no	—
29	#95×C₄₇	—	2	2	no	—
30	#95×C₄₇	—	2	2	no	—
31	C₂²×C₄×C₄₇	—	2	2	no	—
32	C₂²×C₄×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×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
38	C₄⋊C₄×C₄₇	—	2	8	no	—
39	C₄⋊C₄×C₄₇	—	2	2	no	—
40	C₄⋊C₄×C₄₇	—	4	2	no	—
41	C₄⋊C₄×C₄₇	—	4	4	no	—
42	C₄⋊C₄×C₄₇	C₂×C₈	1	4	yes	—
43	C₄²×C₄₇	—	2	2	no	—
44	C₄²×C₄₇	—	2	4	no	—
45	C₄²×C₄₇	—	2	4	no	—
46	C₄²×C₄₇	—	4	1	no	—
47	C₄²×C₄₇	—	4	4	no	—
48	C₄²×C₄₇	C₂×C₈	1	2	yes	—
49	C₄²×C₄₇	M₁₆	1	2	yes	—
50	C₂×C₄₇⋊C₈	—	2	8	no	—
51	C₂×C₄₇⋊C₈	—	4	4	no	—
52	C₂×C₄₇⋊C₈	—	4	4	no	—
53	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₄₇	—	2	2	no	—
55	C₂×C₄×C₄₇	—	2	1	no	—
56	C₂×C₄×C₄₇	—	2	1	no	—
57	C₂×C₄×C₄₇	—	2	2	no	—
58	C₂×C₄×C₄₇	—	2	2	no	—
59	C₂×C₄×C₄₇	—	2	2	no	—
60	C₂×C₄×C₄₇	—	2	4	no	—
61	C₂×C₄×C₄₇	—	2	1	no	—
62	C₂×C₄×C₄₇	—	2	2	no	—
63	C₂×C₄×C₄₇	—	4	2	no	—
64	C₂×C₄×C₄₇	—	4	2	no	—
65	C₂×C₄×C₄₇	—	4	2	no	—
66	C₂×C₄×C₄₇	—	4	2	no	—
67	C₂×C₄×C₄₇	—	4	2	no	—
68	C₂×C₄×C₄₇	C₂²⋊C₈	1	4	yes	—
69	C₂×C₄×C₄₇	#101	1	2	yes	—
70	C₂×C₄×C₄₇	#104	1	4	yes	—
71	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₄₇	#135	1	4	yes	—
74	C₄₇⋊C₈	—	4	8	no	—
75	#7568	—	47	1	no	maximal, Sylow
76	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
77	C₂²×C₄₇	#410	1	1	yes	characteristic
78	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
79	C₂²×C₄₇	#416	1	2	yes	—
80	C₂²×C₄₇	C₂×#104	1	2	yes	—
81	C₄×C₄₇	—	2	1	no	—
82	C₄×C₄₇	—	2	1	no	—
83	C₄×C₄₇	—	2	4	no	—
84	C₄×C₄₇	—	2	4	no	—
85	C₄×C₄₇	—	4	1	no	—
86	C₄×C₄₇	—	4	4	no	—
87	C₄×C₄₇	—	8	2	no	—
88	C₄×C₄₇	#330	1	4	yes	—
89	C₄×#125	D₄₇	1	1	yes	characteristic
90	#1744	—	47	2	no	—
91	#1271	—	47	4	no	—
92	C₂×C₄₇	#1461	1	1	yes	characteristic
93	C₂×C₄₇	#1476	1	2	yes	—
94	C₂×C₄₇	#1500	1	2	yes	—
95	C₂×C₄₇	C₂×#330	1	1	yes	characteristic
96	C₂×C₄₇	#1544	1	1	yes	characteristic
97	C₂×#125	Dic₄₇	1	1	yes	characteristic
98	C₂×C₄×C₈	—	94	2	no	—
99	#389	C₂×D₄₇	1	2	yes	—
100	#387	—	2	2	no	—
101	#382	Dic₄₇	1	1	yes	characteristic
102	C₄×C₄⋊C₄	—	2	2	no	—
103	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
104	C₄³	C₂×D₄₇	1	1	yes	characteristic
105	#336	—	47	4	no	—
106	#336	—	94	2	no	—
107	C₄₇	#7568	1	1	yes	characteristic, complemented, Sylow
108	C₂²×C₈	—	94	4	no	—
109	#125	—	2	2	no	—
110	#125	C₄₇⋊C₈	1	4	yes	complemented
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄⋊C₄	—	2	2	no	—
113	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
114	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
115	C₂×C₄²	—	2	2	no	—
116	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
117	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
118	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
119	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
120	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
121	C₄×C₈	—	94	8	no	—
122	#95	—	2	2	no	—
123	#95	—	2	2	no	—
124	#95	—	2	2	no	—
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
128	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
129	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
130	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
131	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
132	C₂×C₈	—	94	8	no	—
133	C₂×C₈	—	188	4	no	—
134	C₂×C₈	—	188	4	no	—
135	C₄⋊C₄	—	2	8	no	—
136	C₄⋊C₄	—	2	2	no	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄⋊C₄	—	4	4	no	—
139	C₄⋊C₄	C₂×C₄₇⋊C₈	1	4	yes	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	4	no	—
142	C₄²	—	2	4	no	—
143	C₄²	—	4	1	no	—
144	C₄²	—	4	4	no	—
145	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
146	C₄²	#10625329	1	2	yes	—
147	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
163	C₂×C₄	#13093943	1	2	yes	—
164	C₂×C₄	#13093951	1	4	yes	—
165	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
166	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
167	C₂×C₄	#13094044	1	4	yes	—
168	C₈	—	188	8	no	—
169	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
170	C₂²	#15689097	1	1	yes	characteristic
171	C₂²	C₂×#13093943	1	1	yes	characteristic
172	C₂²	#15689119	1	2	yes	—
173	C₂²	C₂×#13093951	1	2	yes	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	—	2	4	no	—
177	C₄	—	2	4	no	—
178	C₄	—	4	4	no	—
179	C₄	—	4	1	no	—
180	C₄	—	8	2	no	—
181	C₄	#15688865	1	4	yes	—
182	C₂	#20241712	1	1	yes	characteristic
183	C₂	#20241770	1	2	yes	—
184	C₂	#20241874	1	2	yes	—
185	C₂	C₂×#15688865	1	1	yes	characteristic
186	C₂	#20242053	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243134

C₂

yes

maximal

#20241211

C₂

yes

maximal

C₄×#125×C₄₇

C₂

yes

maximal, characteristic

C₂×C₄×C₄₇⋊C₈

—

#15688878

—

#15688878

C₂²

yes

—

C₂×#125×C₄₇

C₄

yes

characteristic

C₄₇×#389

C₂²

yes

—

C₄₇×#387

—

C₄₇×#382

C₄

yes

characteristic

C₄×C₄⋊C₄×C₄₇

—

C₄×C₄⋊C₄×C₄₇

C₄

yes

—

C₄³×C₄₇

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

#125×C₄₇

—

#125×C₄₇

C₈

yes

complemented

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#95×C₄₇

—

#95×C₄₇

—

#95×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

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

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂×C₈

yes

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×C₈

yes

—

C₄²×C₄₇

M₁₆

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

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

C₂²⋊C₈

yes

—

C₂×C₄×C₄₇

#101

yes

—

C₂×C₄×C₄₇

#104

yes

—

C₂×C₄×C₄₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₄₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₄₇

#135

yes

—

C₄₇⋊C₈

—

#7568

—

maximal, Sylow

C₂²×C₄₇

C₂×C₂²⋊C₈

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

C₂×#104

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#330

yes

—

C₄×#125

D₄₇

yes

characteristic

#1744

—

#1271

—

C₂×C₄₇

#1461

yes

characteristic

C₂×C₄₇

#1476

yes

—

C₂×C₄₇

#1500

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

Dic₄₇

yes

characteristic

C₂×C₄×C₈

—

#389

C₂×D₄₇

yes

—

100

#387

—

101

#382

Dic₄₇

yes

characteristic

102

C₄×C₄⋊C₄

—

103

C₄×C₄⋊C₄

Dic₄₇

yes

—

104

C₄³

C₂×D₄₇

yes

characteristic

105

#336

—

106

#336

—

107

C₄₇

#7568

yes

characteristic, complemented, Sylow

108

C₂²×C₈

—

109

#125

—

110

#125

C₄₇⋊C₈

yes

complemented

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

C₂×Dic₄₇

yes

—

114

C₂×C₄⋊C₄

C₂×Dic₄₇

yes

—

115

C₂×C₄²

—

116

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

117

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

118

C₂×C₄²

C₂²⋊D₄₇

yes

—

119

C₂×C₄²

C₂²⋊D₄₇

yes

—

120

C₂×C₄²

C₂²×D₄₇

yes

characteristic

121

C₄×C₈

—

122

#95

—

123

#95

—

124

#95

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

C₂²⋊Dic₄₇

yes

characteristic

128

C₂²×C₄

C₂²⋊Dic₄₇

yes

—

129

C₂²×C₄

C₂²⋊Dic₄₇

yes

characteristic

130

C₂²×C₄

C₂²×Dic₄₇

yes

characteristic

131

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

132

C₂×C₈

—

133

C₂×C₈

—

188

—

134

C₂×C₈

—

188

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

C₂×C₄₇⋊C₈

yes

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

—

145

C₄²

C₂×C₄₇⋊C₈

yes

—

146

C₄²

#10625329

yes

—

147

C₂³

C₂×C₂²⋊Dic₄₇

yes

characteristic

148

C₂×C₄

—

149

C₂×C₄

—

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

C₂²⋊(C₄₇⋊C₈)

yes

—

163

C₂×C₄

#13093943

yes

—

164

C₂×C₄

#13093951

yes

—

165

C₂×C₄

C₂²×C₄₇⋊C₈

yes

characteristic

166

C₂×C₄

C₂×#10625329

yes

characteristic

167

C₂×C₄

#13094044

yes

—

168

C₈

—

188

—

169

C₂²

C₂×C₂²⋊(C₄₇⋊C₈)

yes

characteristic

170

C₂²

#15689097

yes

characteristic

171

C₂²

C₂×#13093943

yes

characteristic

172

C₂²

#15689119

yes

—

173

C₂²

C₂×#13093951

yes

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

#15688865

yes

—

182

C₂

#20241712

yes

characteristic

183

C₂

#20241770

yes

—

184

C₂

#20241874

yes

—

185

C₂

C₂×#15688865

yes

characteristic

186

C₂

#20242053

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×#125×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×#125	D₄₇
hypocenter	C₄₇	#7568
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂²×C₄

C₂²⋊Dic₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₈

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄×#125×C₄₇

C₂

socle, prime radical

C₂³×C₄₇

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

N	H ≅ G/N	type
17. #125×C₄₇	C₈	semidirect
107. C₄₇	#7568	semidirect
110. #125	C₄₇⋊C₈	semidirect

H ≅ G/N

type

17. #125×C₄₇

C₈

semidirect

107. C₄₇

#7568

semidirect

110. #125

C₄₇⋊C₈

semidirect

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

subgroup	index
#20243134	2
#20241211	2
C₄×#125×C₄₇	2
#7568	47

subgroup

index

#20243134

#20241211

C₄×#125×C₄₇

#7568

Subgroups — #26104250

Conjugacy classes of subgroups

Complements

Maximal subgroups