Group #26109202 — subgroups

Subgroups — #26109202

Order 12032.

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

normal subgroups only

This group has 398 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	#20243384	C₂	1	2	yes	maximal
2	#20242144	C₂	1	2	yes	maximal
3	#20241263	C₂	1	2	yes	maximal
4	C₂×C₈×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	#15689147	C₂²	1	4	yes	—
6	C₂×#13093943	—	2	2	no	—
7	#15688892	—	2	2	no	—
8	#15688874	—	2	2	no	—
9	#15688874	—	2	2	no	—
10	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
11	C₈×Q₈×C₄₇	—	2	2	no	—
12	C₈×Q₈×C₄₇	—	2	2	no	—
13	C₂×C₄⋊C₈×C₄₇	C₄	1	2	yes	—
14	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×#10625329	—	2	4	no	—
18	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
19	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
20	#13093943	—	4	4	no	—
21	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₄×Q₈×C₄₇	—	2	2	no	—
26	C₄×Q₈×C₄₇	—	2	2	no	—
27	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₄₇	Q₈	1	2	yes	—
29	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₄₇	—	2	4	no	—
32	C₄⋊C₈×C₄₇	—	2	2	no	—
33	C₄×C₈×C₄₇	—	2	2	no	—
34	C₄×C₈×C₄₇	—	2	2	no	—
35	C₄×C₈×C₄₇	D₄	1	2	yes	—
36	C₄×C₈×C₄₇	D₄	1	2	yes	—
37	C₂×Q₈×C₄₇	—	2	2	no	—
38	C₂×Q₈×C₄₇	—	2	2	no	—
39	C₂×Q₈×C₄₇	SD₁₆	1	4	yes	—
40	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
41	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₄₇	—	2	2	no	—
46	C₂×C₈×C₄₇	—	2	2	no	—
47	C₂×C₈×C₄₇	—	4	1	no	—
48	C₂×C₈×C₄₇	—	4	2	no	—
49	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
50	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₄⋊C₄×C₄₇	—	2	2	no	—
53	C₄⋊C₄×C₄₇	—	2	4	no	—
54	C₄²×C₄₇	—	2	2	no	—
55	C₄²×C₄₇	—	2	2	no	—
56	C₄²×C₄₇	C₂×D₄	1	2	yes	—
57	C₄²×C₄₇	#41	1	2	yes	—
58	#10625329	—	4	4	no	—
59	#10625329	—	4	4	no	—
60	C₂×C₄₇⋊C₈	—	4	4	no	—
61	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	Q₈×C₄₇	—	2	8	no	—
63	C₂×C₄×C₄₇	—	2	2	no	—
64	C₂×C₄×C₄₇	—	2	2	no	—
65	C₂×C₄×C₄₇	—	2	1	no	—
66	C₂×C₄×C₄₇	—	2	1	no	—
67	C₂×C₄×C₄₇	—	2	2	no	—
68	C₂×C₄×C₄₇	—	4	1	no	—
69	C₂×C₄×C₄₇	—	4	2	no	—
70	C₂×C₄×C₄₇	#106	1	4	yes	—
71	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
73	C₂×C₄×C₄₇	#121	1	2	yes	—
74	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
75	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
76	C₂×C₄×C₄₇	#135	1	2	yes	—
77	C₈×C₄₇	—	2	1	no	—
78	C₈×C₄₇	—	2	2	no	—
79	C₈×C₄₇	—	4	2	no	—
80	C₈×C₄₇	#101	1	2	yes	complemented
81	C₄₇⋊C₈	—	8	4	no	—
82	#8615	—	47	1	no	maximal, Sylow
83	C₂²×C₄₇	—	2	1	no	—
84	C₂²×C₄₇	—	2	1	no	—
85	C₂²×C₄₇	#380	1	1	yes	characteristic
86	C₂²×C₄₇	C₂×#106	1	1	yes	characteristic
87	C₂²×C₄₇	#427	1	1	yes	characteristic
88	C₄×C₄₇	—	2	1	no	—
89	C₄×C₄₇	—	2	4	no	—
90	C₄×C₄₇	—	2	2	no	—
91	C₄×C₄₇	—	4	2	no	—
92	C₄×C₄₇	C₂×#101	1	1	yes	characteristic
93	C₄×C₄₇	#413	1	1	yes	characteristic
94	C₄×C₄₇	#465	1	4	yes	—
95	C₂×C₈×Q₈	D₄₇	1	1	yes	characteristic
96	#1801	—	47	2	no	—
97	#1566	—	47	2	no	—
98	#1292	—	47	2	no	—
99	C₂×C₄₇	—	2	1	no	—
100	C₂×C₄₇	—	2	1	no	—
101	C₂×C₄₇	#1801	1	1	yes	characteristic
102	C₂×C₄₇	#1807	1	1	yes	characteristic
103	C₂×C₄₇	#1813	1	1	yes	characteristic
104	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
105	C₈×Q₈	—	2	2	no	—
106	C₈×Q₈	—	2	2	no	—
107	#423	—	47	4	no	—
108	C₂×C₄⋊C₈	Dic₄₇	1	2	yes	—
109	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
110	C₂×#101	—	94	2	no	—
111	C₂×C₄×C₈	Dic₄₇	1	2	yes	—
112	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
113	#343	—	94	2	no	—
114	#334	—	94	2	no	—
115	#334	—	94	2	no	—
116	C₄₇	#8615	1	1	yes	characteristic, complemented, Sylow
117	C₂²×Q₈	D₁₈₈	1	1	yes	characteristic
118	C₂×M₁₆	—	94	4	no	—
119	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
120	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
121	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
122	C₄×Q₈	—	2	2	no	—
123	C₄×Q₈	—	2	2	no	—
124	C₂×C₄⋊C₄	Dic₉₄	1	2	yes	—
125	C₂×C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
126	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
127	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
128	C₄⋊C₈	—	2	4	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	94	4	no	—
131	C₄⋊C₈	—	94	4	no	—
132	#101	—	188	4	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	2	2	no	—
135	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
136	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
137	C₂×Q₈	—	2	2	no	—
138	C₂×Q₈	—	2	2	no	—
139	C₂×Q₈	#10625325	1	4	yes	—
140	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
141	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
142	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
143	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
144	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
145	M₁₆	—	188	4	no	—
146	M₁₆	—	188	4	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	188	4	no	—
152	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
153	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
154	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄²	—	2	2	no	—
158	C₄²	—	2	2	no	—
159	C₄²	#10625350	1	2	yes	—
160	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
161	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
162	Q₈	—	2	8	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	#13093956	1	4	yes	—
171	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
172	C₂×C₄	#13093983	1	1	yes	characteristic
173	C₂×C₄	#13093998	1	2	yes	—
174	C₂×C₄	#13094003	1	2	yes	—
175	C₂×C₄	C₂×#10625325	1	2	yes	—
176	C₂×C₄	#13094046	1	2	yes	—
177	C₈	—	2	1	no	—
178	C₈	—	2	2	no	—
179	C₈	—	4	2	no	—
180	C₈	—	376	4	no	—
181	C₈	#13093943	1	2	yes	complemented
182	C₂²	—	2	1	no	—
183	C₂²	—	2	1	no	—
184	C₂²	#15688983	1	1	yes	characteristic
185	C₂²	C₂×#13093956	1	1	yes	characteristic
186	C₂²	#15689161	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	2	4	no	—
189	C₄	—	2	2	no	—
190	C₄	—	4	2	no	—
191	C₄	C₂×#13093943	1	1	yes	characteristic
192	C₄	#15689107	1	1	yes	characteristic
193	C₄	#15689352	1	4	yes	—
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	#20243384	1	1	yes	characteristic
197	C₂	#20243413	1	1	yes	characteristic
198	C₂	#20243438	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243384

C₂

yes

maximal

#20242144

C₂

yes

maximal

#20241263

C₂

yes

maximal

C₂×C₈×Q₈×C₄₇

C₂

yes

maximal, characteristic

#15689147

C₂²

yes

—

—

—

—

—

C₂×C₄×Q₈×C₄₇

C₂²

yes

characteristic

C₈×Q₈×C₄₇

—

C₈×Q₈×C₄₇

—

C₂×C₄⋊C₈×C₄₇

C₄

yes

—

C₂×C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₄₇

C₄

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×#10625329

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

#13093943

—

C₂²×Q₈×C₄₇

D₄

yes

characteristic

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

C₂×C₄⋊C₄×C₄₇

D₄

yes

characteristic

C₂×C₄⋊C₄×C₄₇

Q₈

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

SD₁₆

yes

—

C₂²×C₄×C₄₇

C₄⋊C₄

yes

—

C₂²×C₄×C₄₇

C₄⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₄₇

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

#41

yes

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×C₄⋊C₄

yes

characteristic

Q₈×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

#106

yes

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

#135

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

#101

yes

complemented

C₄₇⋊C₈

—

#8615

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₂×C₈×Q₈

D₄₇

yes

—

—

—

C₂×C₄₇

—

100

C₂×C₄₇

—

101

C₂×C₄₇

#1801

yes

characteristic

102

C₂×C₄₇

#1807

yes

characteristic

103

C₂×C₄₇

#1813

yes

characteristic

104

C₂×C₄×Q₈

C₂×D₄₇

yes

characteristic

105

C₈×Q₈

—

106

C₈×Q₈

—

107

#423

—

108

C₂×C₄⋊C₈

Dic₄₇

yes

—

109

C₂×C₄⋊C₈

C₂×D₄₇

yes

characteristic

110

C₂×#101

—

111

C₂×C₄×C₈

Dic₄₇

yes

—

112

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

113

#343

—

114

#334

—

115

#334

—

116

C₄₇

#8615

yes

characteristic, complemented, Sylow

117

C₂²×Q₈

D₁₈₈

yes

characteristic

118

C₂×M₁₆

—

119

C₂²×C₈

C₂×Dic₄₇

yes

characteristic

120

C₂²×C₈

C₂×Dic₄₇

yes

—

121

C₂²×C₈

C₂×Dic₄₇

yes

characteristic

122

C₄×Q₈

—

123

C₄×Q₈

—

124

C₂×C₄⋊C₄

Dic₉₄

yes

—

125

C₂×C₄⋊C₄

D₁₈₈

yes

characteristic

126

C₂×C₄²

C₂×Dic₄₇

yes

—

127

C₂×C₄²

C₂²×D₄₇

yes

characteristic

128

C₄⋊C₈

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

131

C₄⋊C₈

—

132

#101

—

188

—

133

C₄×C₈

—

134

C₄×C₈

—

135

C₄×C₈

C₂²⋊D₄₇

yes

—

136

C₄×C₈

C₂²⋊D₄₇

yes

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

#10625325

yes

—

140

C₂²×C₄

C₄⋊Dic₄₇

yes

—

141

C₂²×C₄

C₄⋊Dic₄₇

yes

—

142

C₂²×C₄

C₂×Dic₉₄

yes

characteristic

143

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

144

C₂²×C₄

C₂²×Dic₄₇

yes

characteristic

145

M₁₆

—

188

—

146

M₁₆

—

188

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

188

—

152

C₂×C₈

C₂²⋊Dic₄₇

yes

characteristic

153

C₂×C₈

C₂²⋊Dic₄₇

yes

—

154

C₂×C₈

C₂²⋊Dic₄₇

yes

characteristic

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄²

—

158

C₄²

—

159

C₄²

#10625350

yes

—

160

C₄²

C₂×C₂²⋊D₄₇

yes

—

161

C₂³

C₂×C₄⋊Dic₄₇

yes

characteristic

162

Q₈

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

#13093956

yes

—

171

C₂×C₄

C₂×C₂²⋊Dic₄₇

yes

characteristic

172

C₂×C₄

#13093983

yes

characteristic

173

C₂×C₄

#13093998

yes

—

174

C₂×C₄

#13094003

yes

—

175

C₂×C₄

C₂×#10625325

yes

—

176

C₂×C₄

#13094046

yes

—

177

C₈

—

178

C₈

—

179

C₈

—

180

C₈

—

376

—

181

C₈

#13093943

yes

complemented

182

C₂²

—

183

C₂²

—

184

C₂²

#15688983

yes

characteristic

185

C₂²

C₂×#13093956

yes

characteristic

186

C₂²

#15689161

yes

characteristic

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

C₂×#13093943

yes

characteristic

192

C₄

#15689107

yes

characteristic

193

C₄

#15689352

yes

—

194

C₂

—

195

C₂

—

196

C₂

#20243384

yes

characteristic

197

C₂

#20243413

yes

characteristic

198

C₂

#20243438

yes

characteristic

	subgroup	G/H
center	C₂²	#15688983
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₈×Q₈×C₄₇	C₂
socle	C₂²×C₄₇	#380
prime radical	C₂³×C₄₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×C₈×Q₈	D₄₇
hypocenter	C₄₇	#8615
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈

subgroup

G/H

center

C₂²

#15688983

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₂×C₈×Q₈×C₄₇

C₂

socle

C₂²×C₄₇

#380

prime radical

C₂³×C₄₇

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₄₇

C₂³

hypercenter

C₂×C₈×Q₈

D₄₇

hypocenter

C₄₇

#8615

LCS

G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈

N	H ≅ G/N	type
80. C₈×C₄₇	#101	semidirect
116. C₄₇	#8615	semidirect
181. C₈	#13093943	semidirect

H ≅ G/N

type

80. C₈×C₄₇

#101

semidirect

116. C₄₇

#8615

semidirect

181. C₈

#13093943

semidirect

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

subgroup	index
#20243384	2
#20242144	2
#20241263	2
C₂×C₈×Q₈×C₄₇	2
#8615	47

subgroup

index

C₂×C₈×Q₈×C₄₇

Subgroups — #26109202

Conjugacy classes of subgroups

Complements

Maximal subgroups