Group #26368856 — subgroups

Subgroups — #26368856

Order 12032.

862 conjugacy classes of subgroups; only the 299 normal subgroups are shown.

normal subgroups only

This group has 862 conjugacy classes of subgroups — too many to draw. Only its 299 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×#15689526	C₂	1	2	yes	maximal
2	C₂×#15689519	C₂	1	2	yes	maximal
3	#20251766	C₂	1	1	yes	maximal, characteristic
4	#20243550	C₂	1	8	yes	maximal
5	C₄²×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₄₇×#2862	C₂	1	1	yes	maximal, characteristic
7	C₂×#13094024	C₂²	1	2	yes	—
8	#15689526	—	2	8	no	—
9	#15689519	—	2	8	no	—
10	C₂×#13093957	C₂²	1	8	yes	—
11	C₂×#13093955	C₂²	1	8	yes	—
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
15	#15689029	—	2	4	no	—
16	#15688989	C₂²	1	4	yes	—
17	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
18	C₄₇×#391	—	2	4	no	—
19	C₄₇×#382	C₂²	1	4	yes	—
20	C₄³×C₄₇	C₂²	1	1	yes	characteristic
21	C₂²×Dic₉₄	—	2	2	no	—
22	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
23	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
24	#13094024	—	2	4	no	—
25	#13094024	—	2	4	no	—
26	C₂×C₄⋊Dic₄₇	—	2	2	no	—
27	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
28	#13093957	—	2	16	no	—
29	#13093955	—	2	16	no	—
30	C₄×C₄₇⋊C₈	D₄	1	4	yes	—
31	C₄×C₄₇⋊C₈	D₄	1	4	yes	—
32	C₄×C₄₇⋊C₈	Q₈	1	16	yes	complemented
33	C₄₇⋊#95	—	2	8	no	—
34	C₂²×Q₈×C₄₇	—	2	2	no	—
35	C₄⋊Q₈×C₄₇	—	2	4	no	—
36	C₄⋊Q₈×C₄₇	—	2	4	no	—
37	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
39	C₂×C₄²×C₄₇	D₄	1	4	yes	—
40	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
42	#95×C₄₇	—	2	8	no	—
43	C₂×Q₈×C₄₇	—	4	4	no	—
44	C₂×Q₈×C₄₇	—	4	8	no	—
45	C₂²×C₄×C₄₇	—	2	4	no	—
46	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
47	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
49	C₂²×Dic₄₇	—	2	4	no	—
50	C₂×Dic₉₄	—	4	4	no	—
51	C₂×Dic₉₄	—	4	8	no	—
52	C₄⋊C₄×C₄₇	—	2	4	no	—
53	C₄⋊C₄×C₄₇	—	2	4	no	—
54	C₄⋊C₄×C₄₇	—	4	4	no	—
55	C₄²×C₄₇	—	2	8	no	—
56	C₄²×C₄₇	C₂×D₄	1	2	yes	—
57	C₄²×C₄₇	C₂×D₄	1	2	yes	—
58	C₄²×C₄₇	C₂×Q₈	1	8	yes	—
59	C₄⋊Dic₄₇	—	2	4	no	—
60	C₄⋊Dic₄₇	—	2	4	no	—
61	C₄⋊Dic₄₇	—	4	4	no	—
62	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
63	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
64	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	—
65	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	—
66	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
67	Q₈×C₄₇	—	8	8	no	—
68	C₂×C₄×C₄₇	—	2	8	no	—
69	C₂×C₄×C₄₇	—	4	4	no	—
70	C₂×C₄×C₄₇	—	4	8	no	—
71	C₂×C₄×C₄₇	#127	1	4	yes	—
72	C₂×C₄×C₄₇	C₄⋊Q₈	1	8	yes	—
73	C₂×C₄×C₄₇	#135	1	8	yes	—
74	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
75	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
77	C₂×Dic₄₇	—	4	4	no	—
78	C₂×Dic₄₇	—	4	8	no	—
79	Dic₉₄	—	8	8	no	—
80	C₄₇⋊C₈	C₄⋊Q₈	1	8	yes	—
81	C₄₇⋊C₈	C₄⋊Q₈	1	8	yes	—
82	#33313	—	47	1	no	maximal, Sylow
83	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
84	C₂²×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
85	C₂²×C₄₇	C₂×#135	1	4	yes	—
86	C₄×C₄₇	—	2	8	no	—
87	C₄×C₄₇	—	8	4	no	—
88	C₄×C₄₇	#495	1	8	yes	—
89	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
90	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
91	Dic₄₇	—	8	4	no	—
92	C₂×#500	—	47	2	no	—
93	C₂×#498	—	47	2	no	—
94	#2862	—	47	1	no	—
95	#2862	D₄₇	1	1	yes	characteristic
96	#1846	—	47	8	no	—
97	C₄²×C₈	—	47	1	no	—
98	C₂×C₄₇	#2868	1	1	yes	characteristic
99	C₂×C₄₇	C₂×#495	1	2	yes	—
100	C₂×C₄₇	#3160	1	4	yes	—
101	C₂×C₄⋊Q₈	—	47	2	no	—
102	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
103	#500	—	94	8	no	—
104	#498	—	94	8	no	—
105	C₂×#107	—	47	8	no	—
106	C₂×#106	—	47	8	no	—
107	C₂×C₄×C₈	—	47	4	no	—
108	C₂×C₄×C₈	—	47	1	no	—
109	C₂×C₄×C₈	—	47	1	no	—
110	#391	—	2	4	no	—
111	#391	—	94	4	no	—
112	#382	—	47	4	no	—
113	#382	C₂×D₄₇	1	4	yes	—
114	C₄³	C₂×D₄₇	1	1	yes	characteristic
115	C₄₇	#33313	1	1	yes	characteristic, complemented, Sylow
116	C₂²×Q₈	—	2	2	no	—
117	C₂²×Q₈	—	94	2	no	—
118	C₂²×C₈	—	47	2	no	—
119	C₂²×C₈	—	47	2	no	—
120	C₄⋊Q₈	—	2	4	no	—
121	C₄⋊Q₈	—	2	4	no	—
122	C₄⋊Q₈	—	94	4	no	—
123	C₄⋊Q₈	—	94	4	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄⋊C₄	—	47	4	no	—
126	C₂×C₄⋊C₄	—	94	2	no	—
127	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
128	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
129	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
130	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
131	#107	—	94	16	no	—
132	#106	—	94	16	no	—
133	C₄×C₈	—	47	4	no	—
134	C₄×C₈	—	47	4	no	—
135	C₄×C₈	—	47	16	no	—
136	#95	—	2	8	no	—
137	#95	—	94	8	no	—
138	C₂×Q₈	—	4	4	no	—
139	C₂×Q₈	—	4	8	no	—
140	C₂×Q₈	—	188	4	no	—
141	C₂×Q₈	—	188	8	no	—
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	—	94	4	no	—
144	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
145	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
146	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
147	C₂×C₈	—	47	4	no	—
148	C₂×C₈	—	47	4	no	—
149	C₂×C₈	—	47	8	no	—
150	C₂×C₈	—	47	8	no	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	4	4	no	—
154	C₄⋊C₄	—	94	4	no	—
155	C₄⋊C₄	—	94	4	no	—
156	C₄⋊C₄	—	188	4	no	—
157	C₄²	—	2	8	no	—
158	C₄²	D₄×D₄₇	1	2	yes	—
159	C₄²	D₄×D₄₇	1	2	yes	—
160	C₄²	Q₈×D₄₇	1	8	yes	—
161	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
162	Q₈	—	8	8	no	—
163	Q₈	—	376	8	no	—
164	C₂×C₄	—	2	8	no	—
165	C₂×C₄	—	4	8	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	188	8	no	—
168	C₂×C₄	—	188	4	no	—
169	C₂×C₄	#13094020	1	4	yes	—
170	C₂×C₄	#13094022	1	8	yes	—
171	C₂×C₄	#13094044	1	8	yes	—
172	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
173	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
174	C₂×C₄	C₂×Q₈×D₄₇	1	4	yes	—
175	C₈	—	47	8	no	—
176	C₈	—	47	8	no	—
177	C₂²	C₂×#13094020	1	1	yes	characteristic
178	C₂²	C₂×#13094022	1	2	yes	—
179	C₂²	C₂×#13094044	1	4	yes	—
180	C₄	—	2	8	no	—
181	C₄	—	8	4	no	—
182	C₄	—	376	4	no	—
183	C₄	#15689505	1	8	yes	—
184	C₄	C₄⋊Q₈×D₄₇	1	2	yes	—
185	C₄	C₄⋊Q₈×D₄₇	1	2	yes	—
186	C₂	#20251784	1	1	yes	characteristic
187	C₂	C₂×#15689505	1	2	yes	—
188	C₂	#20253898	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15689526

C₂

yes

maximal

C₂×#15689519

C₂

yes

maximal

#20251766

C₂

yes

maximal, characteristic

#20243550

C₂

yes

maximal

C₄²×C₄₇⋊C₈

C₂

yes

maximal, characteristic

C₄₇×#2862

C₂

yes

maximal, characteristic

C₂×#13094024

C₂²

yes

—

#15689526

—

#15689519

—

C₂×#13093957

C₂²

yes

—

C₂×#13093955

C₂²

yes

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

#15689029

—

#15688989

C₂²

yes

—

C₂×C₄⋊Q₈×C₄₇

C₂²

yes

—

C₄₇×#391

—

C₄₇×#382

C₂²

yes

—

C₄³×C₄₇

C₂²

yes

characteristic

C₂²×Dic₉₄

—

C₂²×C₄₇⋊C₈

C₂³

yes

—

C₂²×C₄₇⋊C₈

C₂³

yes

—

#13094024

—

#13094024

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

C₂³

yes

—

#13093957

—

#13093955

—

C₄×C₄₇⋊C₈

D₄

yes

—

C₄×C₄₇⋊C₈

D₄

yes

—

C₄×C₄₇⋊C₈

Q₈

yes

complemented

C₄₇⋊#95

—

C₂²×Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂³

yes

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₂×C₄²×C₄₇

C₂³

yes

—

#95×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂⁴

yes

characteristic

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×Q₈

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

C₂×D₄

yes

—

C₂×C₄₇⋊C₈

C₂×D₄

yes

—

C₂×C₄₇⋊C₈

C₂×Q₈

yes

—

C₂×C₄₇⋊C₈

C₂×Q₈

yes

—

C₂³×C₄₇

C₂²×D₄

yes

characteristic

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#127

yes

—

C₂×C₄×C₄₇

C₄⋊Q₈

yes

—

C₂×C₄×C₄₇

#135

yes

—

C₂×C₄×C₄₇

C₂²×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×Q₈

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

C₄₇⋊C₈

C₄⋊Q₈

yes

—

C₄₇⋊C₈

C₄⋊Q₈

yes

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

C₂×C₄⋊Q₈

yes

—

C₂²×C₄₇

C₂×#135

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#495

yes

—

C₄×C₄₇

C₂×C₄⋊Q₈

yes

—

C₄×C₄₇

C₂×C₄⋊Q₈

yes

—

Dic₄₇

—

—

—

—

D₄₇

yes

characteristic

#1846

—

C₄²×C₈

—

C₂×C₄₇

#2868

yes

characteristic

C₂×C₄₇

C₂×#495

yes

—

100

C₂×C₄₇

#3160

yes

—

101

C₂×C₄⋊Q₈

—

102

C₂×C₄⋊Q₈

C₂×D₄₇

yes

—

103

#500

—

104

#498

—

105

C₂×#107

—

106

C₂×#106

—

107

C₂×C₄×C₈

—

108

C₂×C₄×C₈

—

109

C₂×C₄×C₈

—

110

#391

—

111

#391

—

112

#382

—

113

#382

C₂×D₄₇

yes

—

114

C₄³

C₂×D₄₇

yes

characteristic

115

C₄₇

#33313

yes

characteristic, complemented, Sylow

116

C₂²×Q₈

—

117

C₂²×Q₈

—

118

C₂²×C₈

—

119

C₂²×C₈

—

120

C₄⋊Q₈

—

121

C₄⋊Q₈

—

122

C₄⋊Q₈

—

123

C₄⋊Q₈

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

C₂²×D₄₇

yes

—

128

C₂×C₄²

C₂²⋊D₄₇

yes

—

129

C₂×C₄²

C₂²×D₄₇

yes

characteristic

130

C₂×C₄²

C₂²×D₄₇

yes

—

131

#107

—

132

#106

—

133

C₄×C₈

—

134

C₄×C₈

—

135

C₄×C₈

—

136

#95

—

137

#95

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

188

—

141

C₂×Q₈

—

188

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

145

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

146

C₂²×C₄

C₂³×D₄₇

yes

characteristic

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

188

—

157

C₄²

—

158

C₄²

D₄×D₄₇

yes

—

159

C₄²

D₄×D₄₇

yes

—

160

C₄²

Q₈×D₄₇

yes

—

161

C₂³

C₂²×C₂²⋊D₄₇

yes

characteristic

162

Q₈

—

163

Q₈

—

376

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

188

—

168

C₂×C₄

—

188

—

169

C₂×C₄

#13094020

yes

—

170

C₂×C₄

#13094022

yes

—

171

C₂×C₄

#13094044

yes

—

172

C₂×C₄

C₂×D₄×D₄₇

yes

characteristic

173

C₂×C₄

C₂×D₄×D₄₇

yes

characteristic

174

C₂×C₄

C₂×Q₈×D₄₇

yes

—

175

C₈

—

176

C₈

—

177

C₂²

C₂×#13094020

yes

characteristic

178

C₂²

C₂×#13094022

yes

—

179

C₂²

C₂×#13094044

yes

—

180

C₄

—

181

C₄

—

182

C₄

—

376

—

183

C₄

#15689505

yes

—

184

C₄

C₄⋊Q₈×D₄₇

yes

—

185

C₄

C₄⋊Q₈×D₄₇

yes

—

186

C₂

#20251784

yes

characteristic

187

C₂

C₂×#15689505

yes

—

188

C₂

#20253898

yes

—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#2862	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	#2862	D₄₇
hypocenter	C₄₇	#33313
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2862

subgroup

G/H

center

C₂³

C₂²×C₂²⋊D₄₇

commutator, cosocle

C₂²×C₄×C₄₇

C₂⁴

Frattini

C₂²×C₄

C₂³×D₄₇

Fitting

C₄₇×#2862

C₂

socle, prime radical

C₂³×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #2862

N	H ≅ G/N	type
32. C₄×C₄₇⋊C₈	Q₈	semidirect
115. C₄₇	#33313	semidirect

H ≅ G/N

type

32. C₄×C₄₇⋊C₈

Q₈

semidirect

115. C₄₇

#33313

semidirect

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

subgroup	index
C₂×#15689526	2
C₂×#15689519	2
#20251766	2
#20243550	2
C₄²×C₄₇⋊C₈	2
C₄₇×#2862	2
#33313	47

subgroup

index

C₄²×C₄₇⋊C₈

C₄₇×#2862

#33313

Subgroups — #26368856

Conjugacy classes of subgroups

Complements

Maximal subgroups