Group #26247192 — subgroups

Subgroups — #26247192

Order 12032.

778 conjugacy classes of subgroups; only the 259 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 259 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	C₄×#13094024	C₂	1	2	yes	maximal
2	#20244866	C₂	1	4	yes	maximal
3	#20244722	C₂	1	4	yes	maximal
4	C₄×#13093956	C₂	1	4	yes	maximal
5	C₄₇×#1744	C₂	1	1	yes	maximal, characteristic
6	C₂×#13094024	C₂²	1	2	yes	—
7	C₂×C₄×Dic₉₄	—	2	2	no	—
8	C₂×#13093956	C₂²	1	8	yes	—
9	#15689029	—	2	2	no	—
10	#15689021	—	2	2	no	—
11	#15689021	C₂²	1	4	yes	—
12	#15688993	C₂²	1	2	yes	—
13	C₄×C₄⋊Dic₄₇	—	2	2	no	—
14	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
15	#15688886	C₂²	1	4	yes	—
16	#15688886	C₂²	1	4	yes	—
17	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
18	C₄³×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#336	C₂²	1	4	yes	—
20	C₂²×Dic₉₄	—	2	2	no	—
21	#13094024	—	2	8	no	—
22	C₄×Dic₉₄	—	4	8	no	—
23	C₂×C₄⋊Dic₄₇	—	2	2	no	—
24	C₂×C₄⋊Dic₄₇	—	2	2	no	—
25	C₂×C₄⋊Dic₄₇	—	2	2	no	—
26	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
27	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
28	C₂×Dic₄₇⋊C₄	—	2	4	no	—
29	C₂×C₄×Dic₄₇	—	2	4	no	—
30	#13093956	—	2	16	no	—
31	C₄₇⋊#95	—	2	4	no	—
32	C₄₇⋊#95	—	2	4	no	—
33	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
34	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₄₇	D₄	1	2	yes	—
37	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
38	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
39	C₄×C₈×C₄₇	Q₈	1	8	yes	complemented
40	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
42	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
45	C₂²×Dic₄₇	—	2	4	no	—
46	C₂²×Dic₄₇	—	2	4	no	—
47	C₂×Dic₉₄	—	2	8	no	—
48	C₂×Dic₉₄	—	4	4	no	—
49	C₂×Dic₉₄	—	4	4	no	—
50	C₂×C₈×C₄₇	—	2	8	no	—
51	C₂×C₈×C₄₇	C₂×Q₈	1	8	yes	—
52	C₄²×C₄₇	—	2	4	no	—
53	C₄²×C₄₇	—	4	1	no	—
54	C₄²×C₄₇	—	4	1	no	—
55	C₄²×C₄₇	SD₁₆	1	8	yes	—
56	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
57	C₄⋊Dic₄₇	—	2	8	no	—
58	C₄⋊Dic₄₇	—	2	4	no	—
59	C₄⋊Dic₄₇	—	2	8	no	—
60	C₄⋊Dic₄₇	—	4	4	no	—
61	C₄⋊Dic₄₇	—	4	4	no	—
62	C₄⋊Dic₄₇	#41	1	8	yes	—
63	Dic₄₇⋊C₄	—	4	8	no	—
64	C₄×Dic₄₇	—	4	8	no	—
65	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
66	C₂×C₄×C₄₇	—	2	2	no	—
67	C₂×C₄×C₄₇	—	2	2	no	—
68	C₂×C₄×C₄₇	—	2	2	no	—
69	C₂×C₄×C₄₇	—	2	2	no	—
70	C₂×C₄×C₄₇	—	2	1	no	—
71	C₂×C₄×C₄₇	—	2	1	no	—
72	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
73	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
74	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
75	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
76	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
77	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
78	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
79	C₂×Dic₄₇	—	2	8	no	—
80	C₂×Dic₄₇	—	4	4	no	—
81	C₂×Dic₄₇	—	4	4	no	—
82	C₂×Dic₄₇	—	4	4	no	—
83	C₂×Dic₄₇	—	4	4	no	—
84	C₂×Dic₄₇	—	4	4	no	—
85	Dic₉₄	—	4	16	no	—
86	C₈×C₄₇	—	2	8	no	—
87	#23837	—	47	1	no	maximal, Sylow
88	C₂²×C₄₇	#525	1	2	yes	—
89	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
90	C₂²×C₄₇	C₂²×SD₁₆	1	2	yes	—
91	C₂²×C₄₇	#575	1	1	yes	characteristic
92	C₂²×C₄₇	#575	1	1	yes	characteristic
93	C₄×C₄₇	—	2	4	no	—
94	C₄×C₄₇	—	2	4	no	—
95	C₄×C₄₇	—	2	2	no	—
96	C₄×C₄₇	#498	1	4	yes	—
97	C₄×C₄₇	#533	1	4	yes	—
98	Dic₄₇	—	4	8	no	—
99	Dic₄₇	—	8	4	no	—
100	C₄×C₄⋊Q₈	—	47	2	no	—
101	#2084	—	47	4	no	—
102	#2058	—	47	4	no	—
103	C₄×#106	—	47	4	no	—
104	#1744	D₄₇	1	1	yes	characteristic
105	C₂×C₄₇	#2399	1	1	yes	characteristic
106	C₂×C₄₇	#3101	1	2	yes	—
107	C₂×C₄₇	#3103	1	2	yes	—
108	C₂×C₄₇	C₂×#498	1	1	yes	characteristic
109	C₂×C₄₇	#3162	1	1	yes	characteristic
110	C₂×C₄⋊Q₈	—	47	2	no	—
111	C₂×C₄×Q₈	—	94	2	no	—
112	C₂×#106	—	47	8	no	—
113	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
114	#391	—	94	2	no	—
115	#389	—	47	4	no	—
116	#389	—	94	2	no	—
117	#384	—	47	2	no	—
118	C₄×C₄⋊C₄	—	47	4	no	—
119	C₄×C₄⋊C₄	—	94	2	no	—
120	C₄³	C₂×D₄₇	1	1	yes	characteristic
121	#340	—	47	4	no	—
122	#340	—	47	4	no	—
123	#336	C₂×D₄₇	1	4	yes	—
124	C₄₇	#23837	1	1	yes	characteristic, complemented, Sylow
125	C₂²×Q₈	—	94	2	no	—
126	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
127	C₄⋊Q₈	—	94	8	no	—
128	C₄×Q₈	—	188	8	no	—
129	C₂×C₄⋊C₄	—	47	4	no	—
130	C₂×C₄⋊C₄	—	47	4	no	—
131	C₂×C₄⋊C₄	—	94	2	no	—
132	C₂×C₄⋊C₄	—	94	4	no	—
133	C₂×C₄⋊C₄	—	94	2	no	—
134	C₂×C₄⋊C₄	—	94	2	no	—
135	C₂×C₄²	—	94	4	no	—
136	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
137	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
138	C₂×C₄²	D₁₈₈	1	2	yes	—
139	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
140	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
141	#106	—	94	16	no	—
142	C₄×C₈	Dic₉₄	1	8	yes	complemented
143	#95	—	94	4	no	—
144	#95	—	94	4	no	—
145	C₂×Q₈	—	94	8	no	—
146	C₂×Q₈	—	188	4	no	—
147	C₂×Q₈	—	188	4	no	—
148	C₂²×C₄	—	94	4	no	—
149	C₂²×C₄	—	94	4	no	—
150	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
151	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
152	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
153	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
154	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
155	C₂×C₈	—	2	8	no	—
156	C₂×C₈	C₂×Dic₉₄	1	8	yes	—
157	C₄⋊C₄	—	47	8	no	—
158	C₄⋊C₄	—	94	8	no	—
159	C₄⋊C₄	—	94	4	no	—
160	C₄⋊C₄	—	94	8	no	—
161	C₄⋊C₄	—	188	4	no	—
162	C₄⋊C₄	—	188	8	no	—
163	C₄⋊C₄	—	188	4	no	—
164	C₄²	—	2	4	no	—
165	C₄²	—	4	1	no	—
166	C₄²	—	4	1	no	—
167	C₄²	—	188	8	no	—
168	C₄²	#10625325	1	8	yes	—
169	C₄²	C₂×Dic₉₄	1	4	yes	—
170	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
171	Q₈	—	188	16	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	94	8	no	—
179	C₂×C₄	—	188	4	no	—
180	C₂×C₄	—	188	4	no	—
181	C₂×C₄	—	188	4	no	—
182	C₂×C₄	—	188	4	no	—
183	C₂×C₄	—	188	4	no	—
184	C₂×C₄	#13094024	1	2	yes	—
185	C₂×C₄	#13094024	1	2	yes	—
186	C₂×C₄	C₂×#10625325	1	4	yes	—
187	C₂×C₄	C₂×#10625325	1	4	yes	—
188	C₂×C₄	C₂×#10625325	1	4	yes	—
189	C₂×C₄	C₂²×Dic₉₄	1	2	yes	—
190	C₂×C₄	#41×D₄₇	1	4	yes	—
191	C₈	—	2	8	no	—
192	C₂²	#15689635	1	2	yes	—
193	C₂²	C₂×#13094024	1	1	yes	characteristic
194	C₂²	C₂²×#10625325	1	2	yes	—
195	C₂²	#15689923	1	1	yes	characteristic
196	C₂²	#15689923	1	1	yes	characteristic
197	C₄	—	2	4	no	—
198	C₄	—	2	2	no	—
199	C₄	—	2	4	no	—
200	C₄	—	376	4	no	—
201	C₄	—	188	8	no	—
202	C₄	#15689521	1	4	yes	—
203	C₄	#15689676	1	4	yes	—
204	C₂	#20246681	1	1	yes	characteristic
205	C₂	#20253397	1	2	yes	—
206	C₂	#20253420	1	2	yes	—
207	C₂	C₂×#15689521	1	1	yes	characteristic
208	C₂	#20253912	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13094024

C₂

yes

maximal

#20244866

C₂

yes

maximal

#20244722

C₂

yes

maximal

C₄×#13093956

C₂

yes

maximal

C₄₇×#1744

C₂

yes

maximal, characteristic

C₂×#13094024

C₂²

yes

—

C₂×C₄×Dic₉₄

—

C₂×#13093956

C₂²

yes

—

#15689029

—

#15689021

—

#15689021

C₂²

yes

—

#15688993

C₂²

yes

—

C₄×C₄⋊Dic₄₇

—

C₄×C₄⋊Dic₄₇

C₂²

yes

—

#15688886

C₂²

yes

—

#15688886

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

—

C₄³×C₄₇

C₂²

yes

characteristic

C₄₇×#336

C₂²

yes

—

C₂²×Dic₉₄

—

#13094024

—

C₄×Dic₉₄

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

C₂³

yes

—

C₂×C₄⋊Dic₄₇

C₂³

yes

—

C₂×Dic₄₇⋊C₄

—

C₂×C₄×Dic₄₇

—

#13093956

—

C₄₇⋊#95

—

C₄₇⋊#95

—

C₂²×C₈×C₄₇

C₂³

yes

—

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

Q₈

yes

complemented

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂⁴

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×Q₈

yes

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

SD₁₆

yes

—

C₄²×C₄₇

C₂×Q₈

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

#41

yes

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂³×C₄₇

C₂²×D₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄⋊Q₈

yes

—

C₂×C₄×C₄₇

C₄⋊Q₈

yes

—

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

C₂²×Q₈

yes

—

C₂×C₄×C₄₇

C₂×#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

C₈×C₄₇

—

#23837

—

maximal, Sylow

C₂²×C₄₇

#525

yes

—

C₂²×C₄₇

C₂×C₄⋊Q₈

yes

characteristic

C₂²×C₄₇

C₂²×SD₁₆

yes

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#498

yes

—

C₄×C₄₇

#533

yes

—

Dic₄₇

—

Dic₄₇

—

100

C₄×C₄⋊Q₈

—

101

#2084

—

102

#2058

—

103

C₄×#106

—

104

#1744

D₄₇

yes

characteristic

105

C₂×C₄₇

#2399

yes

characteristic

106

C₂×C₄₇

#3101

yes

—

107

C₂×C₄₇

#3103

yes

—

108

C₂×C₄₇

C₂×#498

yes

characteristic

109

C₂×C₄₇

#3162

yes

characteristic

110

C₂×C₄⋊Q₈

—

111

C₂×C₄×Q₈

—

112

C₂×#106

—

113

C₂×C₄×C₈

C₂×D₄₇

yes

—

114

#391

—

115

#389

—

116

#389

—

117

#384

—

118

C₄×C₄⋊C₄

—

119

C₄×C₄⋊C₄

—

120

C₄³

C₂×D₄₇

yes

characteristic

121

#340

—

122

#340

—

123

#336

C₂×D₄₇

yes

—

124

C₄₇

#23837

yes

characteristic, complemented, Sylow

125

C₂²×Q₈

—

126

C₂²×C₈

C₂²×D₄₇

yes

—

127

C₄⋊Q₈

—

128

C₄×Q₈

—

188

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

134

C₂×C₄⋊C₄

—

135

C₂×C₄²

—

136

C₂×C₄²

D₁₈₈

yes

characteristic

137

C₂×C₄²

D₁₈₈

yes

characteristic

138

C₂×C₄²

D₁₈₈

yes

—

139

C₂×C₄²

C₂²×D₄₇

yes

—

140

C₂×C₄²

C₂²×D₄₇

yes

characteristic

141

#106

—

142

C₄×C₈

Dic₉₄

yes

complemented

143

#95

—

144

#95

—

145

C₂×Q₈

—

146

C₂×Q₈

—

188

—

147

C₂×Q₈

—

188

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

151

C₂²×C₄

C₂×D₁₈₈

yes

—

152

C₂²×C₄

C₂×D₁₈₈

yes

—

153

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

154

C₂²×C₄

C₂³×D₄₇

yes

characteristic

155

C₂×C₈

—

156

C₂×C₈

C₂×Dic₉₄

yes

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

188

—

162

C₄⋊C₄

—

188

—

163

C₄⋊C₄

—

188

—

164

C₄²

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

188

—

168

C₄²

#10625325

yes

—

169

C₄²

C₂×Dic₉₄

yes

—

170

C₂³

C₂²×D₁₈₈

yes

characteristic

171

Q₈

—

188

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

188

—

180

C₂×C₄

—

188

—

181

C₂×C₄

—

188

—

182

C₂×C₄

—

188

—

183

C₂×C₄

—

188

—

184

C₂×C₄

#13094024

yes

—

185

C₂×C₄

#13094024

yes

—

186

C₂×C₄

C₂×#10625325

yes

—

187

C₂×C₄

C₂×#10625325

yes

—

188

C₂×C₄

C₂×#10625325

yes

—

189

C₂×C₄

C₂²×Dic₉₄

yes

—

190

C₂×C₄

#41×D₄₇

yes

—

191

C₈

—

192

C₂²

#15689635

yes

—

193

C₂²

C₂×#13094024

yes

characteristic

194

C₂²

C₂²×#10625325

yes

—

195

C₂²

#15689923

yes

characteristic

196

C₂²

#15689923

yes

characteristic

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

376

—

201

C₄

—

188

—

202

C₄

#15689521

yes

—

203

C₄

#15689676

yes

—

204

C₂

#20246681

yes

characteristic

205

C₂

#20253397

yes

—

206

C₂

#20253420

yes

—

207

C₂

C₂×#15689521

yes

characteristic

208

C₂

#20253912

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1744	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	#1744	D₄₇
hypocenter	C₄₇	#23837
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₁₈₈

commutator, cosocle

C₂²×C₄×C₄₇

C₂⁴

Frattini

C₂²×C₄

C₂³×D₄₇

Fitting

C₄₇×#1744

C₂

socle, prime radical

C₂³×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
39. C₄×C₈×C₄₇	Q₈	semidirect
124. C₄₇	#23837	semidirect
142. C₄×C₈	Dic₉₄	semidirect

H ≅ G/N

type

39. C₄×C₈×C₄₇

Q₈

semidirect

124. C₄₇

#23837

semidirect

142. C₄×C₈

Dic₉₄

semidirect

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

subgroup	index
C₄×#13094024	2
#20244866	2
#20244722	2
C₄×#13093956	2
C₄₇×#1744	2
#23837	47

subgroup

index

C₄₇×#1744

Subgroups — #26247192

Conjugacy classes of subgroups

Complements

Maximal subgroups