Group #26134800 — subgroups

Subgroups — #26134800

Order 12032.

386 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 155 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	C₂×C₈×Dic₉₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688899	C₂	1	1	yes	maximal, characteristic
3	#20241465	C₂	1	4	yes	maximal
4	C₂×C₄₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₉₄	C₄	1	1	yes	characteristic
6	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
7	C₈×Dic₉₄	—	2	2	no	—
8	C₈×Dic₉₄	—	2	2	no	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₄	1	1	yes	characteristic
10	C₂×Dic₄₇⋊C₈	C₄	1	2	yes	—
11	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
12	#15688899	—	2	2	no	—
13	C₂²×C₁₆×C₄₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×C₄⋊C₁₆	—	2	2	no	—
16	C₂²×Dic₉₄	C₈	1	1	yes	characteristic
17	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
18	C₄×Dic₉₄	—	2	2	no	—
19	C₄×Dic₉₄	—	2	2	no	—
20	C₂×C₄⋊Dic₄₇	C₈	1	1	yes	characteristic
21	C₂×Dic₄₇⋊C₄	C₈	1	2	yes	—
22	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
23	C₂×C₄₇⋊C₁₆	—	2	2	no	—
24	C₂×C₄₇⋊C₁₆	—	2	2	no	—
25	C₂×C₄₇⋊C₁₆	Q₈	1	4	yes	complemented
26	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
27	Dic₄₇⋊C₈	—	2	4	no	—
28	C₈×Dic₄₇	—	2	2	no	—
29	C₈×Dic₄₇	—	2	2	no	—
30	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₄₇	—	2	2	no	—
34	C₂×C₁₆×C₄₇	—	2	4	no	—
35	C₄×C₈×C₄₇	D₄	1	2	yes	—
36	C₄×C₈×C₄₇	D₄	1	2	yes	—
37	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
40	C₂²×Dic₄₇	C₂×C₈	1	2	yes	—
41	C₂²×Dic₄₇	C₂×C₈	1	2	yes	—
42	C₂×Dic₉₄	—	2	2	no	—
43	C₂×Dic₉₄	—	2	4	no	—
44	C₂×C₈×C₄₇	—	2	2	no	—
45	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
46	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
49	C₂×C₈×C₄₇	#41	1	2	yes	—
50	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
51	C₄²×C₄₇	M₁₆	1	2	yes	—
52	C₄⋊Dic₄₇	—	2	2	no	—
53	Dic₄₇⋊C₄	—	2	4	no	—
54	C₄×Dic₄₇	—	2	2	no	—
55	C₄×Dic₄₇	—	2	2	no	—
56	C₂×C₄₇⋊C₈	—	2	2	no	—
57	C₂×C₄₇⋊C₈	—	2	4	no	—
58	C₁₆×C₄₇	—	4	2	no	—
59	C₄₇⋊C₁₆	—	2	4	no	—
60	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₄₇	—	2	2	no	—
62	C₂×C₄×C₄₇	—	2	2	no	—
63	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
64	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
65	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
67	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
68	C₂×C₄×C₄₇	#131	1	2	yes	—
69	C₂×Dic₄₇	—	2	2	no	—
70	C₂×Dic₄₇	—	2	2	no	—
71	C₂×Dic₄₇	—	2	2	no	—
72	C₂×Dic₄₇	—	2	4	no	—
73	C₂×Dic₄₇	M₃₂	1	4	yes	—
74	Dic₉₄	—	2	8	no	—
75	C₈×C₄₇	—	2	2	no	—
76	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
77	C₈×C₄₇	#124	1	1	yes	characteristic
78	C₈×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
79	C₄₇⋊C₈	—	4	2	no	—
80	#13264	—	47	1	no	maximal, Sylow
81	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
82	C₂²×C₄₇	C₈×Q₈	1	2	yes	—
83	C₂²×C₄₇	C₂×M₃₂	1	2	yes	—
84	C₂²×C₄₇	#504	1	2	yes	—
85	C₄×C₄₇	—	2	2	no	—
86	C₄×C₄₇	—	2	2	no	—
87	C₄×C₄₇	#391	1	1	yes	characteristic
88	C₄×C₄₇	#408	1	1	yes	characteristic
89	C₄×C₄₇	#446	1	2	yes	—
90	Dic₄₇	—	2	4	no	—
91	Dic₄₇	—	4	2	no	—
92	C₂×C₈×Q₈	—	47	1	no	—
93	C₂×C₄⋊C₁₆	D₄₇	1	1	yes	characteristic
94	C₂×#346	—	47	1	no	—
95	#1374	—	47	4	no	—
96	C₂×C₄₇	#1984	1	1	yes	characteristic
97	C₂×C₄₇	#2112	1	1	yes	characteristic
98	C₂×C₄₇	#2113	1	1	yes	characteristic
99	C₂×C₄₇	#2183	1	4	yes	—
100	C₂×C₄×Q₈	—	47	1	no	—
101	C₂²×C₁₆	—	47	2	no	—
102	C₂²×C₁₆	C₂×D₄₇	1	2	yes	—
103	C₈×Q₈	—	94	2	no	—
104	C₈×Q₈	—	94	2	no	—
105	C₂×C₄⋊C₈	—	47	2	no	—
106	C₂×C₄⋊C₈	—	47	1	no	—
107	C₂×C₄×C₈	—	47	2	no	—
108	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
109	C₄⋊C₁₆	—	2	2	no	—
110	#346	—	94	2	no	—
111	C₄₇	#13264	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₈	—	47	1	no	—
113	C₂²×C₈	—	47	2	no	—
114	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
116	C₄×Q₈	—	94	2	no	—
117	C₄×Q₈	—	94	2	no	—
118	C₂×C₄⋊C₄	—	47	2	no	—
119	C₂×C₄⋊C₄	—	47	1	no	—
120	C₂×C₄²	—	47	2	no	—
121	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
122	C₂×C₁₆	—	2	4	no	—
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	—	47	4	no	—
125	C₂×C₁₆	—	94	2	no	—
126	C₂×C₁₆	—	94	2	no	—
127	C₄⋊C₈	—	94	4	no	—
128	C₄⋊C₈	—	94	2	no	—
129	C₄×C₈	—	94	2	no	—
130	C₄×C₈	—	94	2	no	—
131	C₄×C₈	D₁₈₈	1	2	yes	—
132	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
133	C₂×Q₈	—	94	4	no	—
134	C₂×Q₈	—	94	2	no	—
135	C₂²×C₄	—	47	2	no	—
136	C₂²×C₄	—	47	2	no	—
137	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
138	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	94	2	no	—
142	C₂×C₈	—	94	4	no	—
143	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
144	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
145	C₂×C₈	#10625352	1	2	yes	—
146	C₂×C₈	Q₈×D₄₇	1	2	yes	—
147	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
148	C₄⋊C₄	—	94	4	no	—
149	C₄⋊C₄	—	94	2	no	—
150	C₄²	—	94	2	no	—
151	C₄²	—	94	2	no	—
152	C₄²	#10625324	1	2	yes	—
153	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
154	C₁₆	—	4	2	no	—
155	C₁₆	—	94	4	no	—
156	C₂³	C₂×C₈×D₄₇	1	1	yes	characteristic
157	Q₈	—	94	8	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	47	4	no	—
161	C₂×C₄	—	94	2	no	—
162	C₂×C₄	—	94	2	no	—
163	C₂×C₄	—	94	4	no	—
164	C₂×C₄	—	94	2	no	—
165	C₂×C₄	(C₂×D₄₇)⋊C₈	1	2	yes	—
166	C₂×C₄	(C₂×D₄₇)⋊C₈	1	2	yes	—
167	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
168	C₂×C₄	#13093989	1	2	yes	—
169	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
170	C₂×C₄	#13094032	1	2	yes	—
171	C₈	—	2	2	no	—
172	C₈	—	188	2	no	—
173	C₈	#13094004	1	2	yes	—
174	C₈	#13094011	1	1	yes	characteristic
175	C₈	#13094022	1	1	yes	characteristic
176	C₂²	C₂×(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
177	C₂²	#15689247	1	2	yes	—
178	C₂²	M₃₂×D₄₇	1	2	yes	—
179	C₂²	#15689539	1	2	yes	—
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	94	4	no	—
183	C₄	—	188	2	no	—
184	C₄	#15689028	1	1	yes	characteristic
185	C₄	#15689091	1	1	yes	characteristic
186	C₄	#15689250	1	2	yes	—
187	C₂	#20244323	1	1	yes	characteristic
188	C₂	#20244998	1	1	yes	characteristic
189	C₂	#20245001	1	1	yes	characteristic
190	C₂	#20245295	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₉₄

C₂

yes

maximal, characteristic

C₂×#15688899

C₂

yes

maximal, characteristic

#20241465

C₂

yes

maximal

C₂×C₄₇×C₄⋊C₁₆

C₂

yes

maximal, characteristic

C₂×C₄×Dic₉₄

C₄

yes

characteristic

C₂²×C₄₇⋊C₁₆

C₂²

yes

—

C₈×Dic₉₄

—

C₈×Dic₉₄

—

C₂×C₄⋊(C₄₇⋊C₈)

C₄

yes

characteristic

C₂×Dic₄₇⋊C₈

C₄

yes

—

C₂×C₈×Dic₄₇

C₂²

yes

—

#15688899

—

C₂²×C₁₆×C₄₇

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×C₄⋊C₁₆

—

C₂²×Dic₉₄

C₈

yes

characteristic

C₂²×C₄₇⋊C₈

C₂×C₄

yes

—

C₄×Dic₉₄

—

C₄×Dic₉₄

—

C₂×C₄⋊Dic₄₇

C₈

yes

characteristic

C₂×Dic₄₇⋊C₄

C₈

yes

—

C₂×C₄×Dic₄₇

C₂×C₄

yes

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

Q₈

yes

complemented

C₄⋊(C₄₇⋊C₈)

—

Dic₄₇⋊C₈

—

C₈×Dic₄₇

—

C₈×Dic₄₇

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

—

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂×C₈

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₈

yes

characteristic

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×Dic₄₇

C₂×C₈

yes

—

C₂²×Dic₄₇

C₂×C₈

yes

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×Q₈

yes

—

C₂×C₈×C₄₇

#41

yes

—

C₄²×C₄₇

C₂²⋊C₄

yes

—

C₄²×C₄₇

M₁₆

yes

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₈

yes

—

C₂×C₄×C₄₇

C₂²⋊C₈

yes

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×Q₈

yes

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₄₇

#131

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

M₃₂

yes

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

C₂²⋊Q₈

yes

—

C₈×C₄₇

#124

yes

characteristic

C₈×C₄₇

C₄⋊Q₈

yes

characteristic

C₄₇⋊C₈

—

#13264

—

maximal, Sylow

C₂²×C₄₇

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₄₇

C₈×Q₈

yes

—

C₂²×C₄₇

C₂×M₃₂

yes

—

C₂²×C₄₇

#504

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

Dic₄₇

—

C₂×C₈×Q₈

—

C₂×C₄⋊C₁₆

D₄₇

yes

characteristic

C₂×#346

—

#1374

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

100

C₂×C₄×Q₈

—

101

C₂²×C₁₆

—

102

C₂²×C₁₆

C₂×D₄₇

yes

—

103

C₈×Q₈

—

104

C₈×Q₈

—

105

C₂×C₄⋊C₈

—

106

C₂×C₄⋊C₈

—

107

C₂×C₄×C₈

—

108

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

109

C₄⋊C₁₆

—

110

#346

—

111

C₄₇

#13264

yes

characteristic, complemented, Sylow

112

C₂²×Q₈

—

113

C₂²×C₈

—

114

C₂²×C₈

C₄×D₄₇

yes

characteristic

115

C₂²×C₈

C₂²×D₄₇

yes

characteristic

116

C₄×Q₈

—

117

C₄×Q₈

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄²

—

121

C₂×C₄²

C₄×D₄₇

yes

characteristic

122

C₂×C₁₆

—

123

C₂×C₁₆

—

124

C₂×C₁₆

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

127

C₄⋊C₈

—

128

C₄⋊C₈

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

D₁₈₈

yes

—

132

C₄×C₈

C₂²⋊D₄₇

yes

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

C₈×D₄₇

yes

characteristic

138

C₂²×C₄

C₈×D₄₇

yes

characteristic

139

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

—

144

C₂×C₈

C₂×D₁₈₈

yes

characteristic

145

C₂×C₈

#10625352

yes

—

146

C₂×C₈

Q₈×D₄₇

yes

—

147

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄²

—

151

C₄²

—

152

C₄²

#10625324

yes

—

153

C₄²

(C₂×D₄₇)⋊C₄

yes

—

154

C₁₆

—

155

C₁₆

—

156

C₂³

C₂×C₈×D₄₇

yes

characteristic

157

Q₈

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

(C₂×D₄₇)⋊C₈

yes

—

166

C₂×C₄

(C₂×D₄₇)⋊C₈

yes

—

167

C₂×C₄

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

168

C₂×C₄

#13093989

yes

—

169

C₂×C₄

C₂×#10625324

yes

characteristic

170

C₂×C₄

#13094032

yes

—

171

C₈

—

172

C₈

—

188

—

173

C₈

#13094004

yes

—

174

C₈

#13094011

yes

characteristic

175

C₈

#13094022

yes

characteristic

176

C₂²

C₂×(C₂×D₄₇)⋊C₈

yes

characteristic

177

C₂²

#15689247

yes

—

178

C₂²

M₃₂×D₄₇

yes

—

179

C₂²

#15689539

yes

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

188

—

184

C₄

#15689028

yes

characteristic

185

C₄

#15689091

yes

characteristic

186

C₄

#15689250

yes

—

187

C₂

#20244323

yes

characteristic

188

C₂

#20244998

yes

characteristic

189

C₂

#20245001

yes

characteristic

190

C₂

#20245295

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₄₇
commutator, socle, prime radical	C₂³×C₄₇	C₂²×C₈
Fitting	C₂×C₄₇×C₄⋊C₁₆	C₂
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₄₇
hypocenter	C₄₇	#13264
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₄₇

commutator, socle, prime radical

C₂³×C₄₇

C₂²×C₈

Fitting

C₂×C₄₇×C₄⋊C₁₆

C₂

cosocle

C₂²×C₈×C₄₇

C₂³

hypercenter

C₂×C₄⋊C₁₆

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

N	H ≅ G/N	type
25. C₂×C₄₇⋊C₁₆	Q₈	semidirect
111. C₄₇	#13264	semidirect

H ≅ G/N

type

25. C₂×C₄₇⋊C₁₆

Q₈

semidirect

111. C₄₇

#13264

semidirect

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

subgroup	index
C₂×C₈×Dic₉₄	2
C₂×#15688899	2
#20241465	2
C₂×C₄₇×C₄⋊C₁₆	2
#13264	47

subgroup

index

C₂×C₈×Dic₉₄

C₂×#15688899

#20241465

C₂×C₄₇×C₄⋊C₁₆

#13264

Subgroups — #26134800

Conjugacy classes of subgroups

Complements

Maximal subgroups