Group #26116100 — subgroups

Subgroups — #26116100

Order 12032.

494 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 163 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×#15689528	C₂	1	1	yes	maximal, characteristic
2	#20244348	C₂	1	2	yes	maximal
3	#20244189	C₂	1	2	yes	maximal
4	C₂×#15688845	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1834	C₂	1	1	yes	maximal, characteristic
6	C₂×#13094024	C₂²	1	2	yes	—
7	#15689528	—	2	2	no	—
8	#15689528	C₄	1	4	yes	complemented
9	C₂×#13093958	—	2	2	no	—
10	C₂×#13093947	—	2	4	no	—
11	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	#15688885	—	2	4	no	—
14	#15688845	—	2	2	no	—
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#384	C₂²	1	2	yes	—
17	C₄₇×#340	—	2	2	no	—
18	C₂²×Dic₉₄	—	2	2	no	—
19	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
20	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
21	#13094024	—	2	2	no	—
22	#13094024	—	2	2	no	—
23	#13094024	—	2	2	no	—
24	#13094024	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊Dic₄₇	—	2	2	no	—
26	C₂×C₄⋊Dic₄₇	—	2	2	no	—
27	#13093958	—	2	8	no	—
28	#13093947	—	4	8	no	—
29	C₄×C₄₇⋊C₈	—	2	1	no	—
30	C₄×C₄₇⋊C₈	—	2	1	no	—
31	C₄×C₄₇⋊C₈	—	2	1	no	—
32	C₄×C₄₇⋊C₈	—	2	1	no	—
33	C₂²×C₈×C₄₇	D₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
37	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
38	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₄₇	D₄	1	2	yes	—
40	C₂²×C₄×C₄₇	—	4	2	no	—
41	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₄₇	—	4	2	no	—
45	C₂×Dic₉₄	—	2	4	no	—
46	C₂×Dic₉₄	—	4	4	no	—
47	C₂×Dic₉₄	—	4	4	no	—
48	C₂×Dic₉₄	—	4	2	no	—
49	C₂×C₈×C₄₇	—	2	4	no	—
50	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
51	C₄⋊C₄×C₄₇	—	4	4	no	—
52	C₄⋊C₄×C₄₇	—	4	4	no	—
53	C₄⋊C₄×C₄₇	—	4	4	no	—
54	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₄₇	#41	1	1	yes	characteristic
57	C₄²×C₄₇	#41	1	1	yes	characteristic
58	C₄⋊Dic₄₇	—	2	4	no	—
59	C₄⋊Dic₄₇	—	2	4	no	—
60	C₄⋊Dic₄₇	—	4	2	no	—
61	C₄⋊Dic₄₇	—	4	2	no	—
62	C₂×C₄₇⋊C₈	—	2	2	no	—
63	C₂×C₄₇⋊C₈	—	2	2	no	—
64	C₂×C₄₇⋊C₈	—	2	2	no	—
65	C₂×C₄₇⋊C₈	—	2	4	no	—
66	C₂×C₄₇⋊C₈	Q₁₆	1	4	yes	—
67	C₂³×C₄₇	#127	1	1	yes	characteristic
68	C₂×C₄×C₄₇	—	8	2	no	—
69	C₂×C₄×C₄₇	—	8	2	no	—
70	C₂×C₄×C₄₇	—	8	2	no	—
71	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
75	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
76	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
78	C₂×C₄×C₄₇	#135	1	2	yes	—
79	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
80	C₂×C₄×C₄₇	#137	1	4	yes	—
81	C₂×Dic₄₇	—	4	4	no	—
82	C₂×Dic₄₇	—	8	2	no	—
83	C₂×Dic₄₇	—	8	2	no	—
84	Dic₉₄	—	4	8	no	—
85	Dic₉₄	—	8	4	no	—
86	C₈×C₄₇	—	2	4	no	—
87	C₄₇⋊C₈	—	2	4	no	—
88	C₄₇⋊C₈	—	4	2	no	—
89	#9834	—	47	1	no	maximal, Sylow
90	C₂²×C₄₇	#390	1	1	yes	characteristic
91	C₂²×C₄₇	#494	1	1	yes	characteristic
92	C₂²×C₄₇	#495	1	1	yes	characteristic
93	C₂²×C₄₇	#496	1	2	yes	—
94	C₂²×C₄₇	#497	1	2	yes	—
95	C₄×C₄₇	—	16	2	no	—
96	C₄×C₄₇	#419	1	2	yes	—
97	C₄×C₄₇	C₄×Q₁₆	1	2	yes	—
98	C₄×C₄₇	#442	1	2	yes	—
99	C₄×C₄₇	#467	1	2	yes	—
100	C₄×C₄₇	#469	1	2	yes	—
101	C₄×C₄₇	#491	1	2	yes	—
102	Dic₄₇	—	8	4	no	—
103	C₂×#500	—	47	1	no	—
104	#1990	—	47	2	no	—
105	#1956	—	47	2	no	—
106	#1834	D₄₇	1	1	yes	characteristic
107	C₂×#322	—	47	1	no	—
108	C₂×C₄₇	#1719	1	2	yes	—
109	C₂×C₄₇	#1723	1	2	yes	—
110	C₂×C₄₇	#1960	1	1	yes	characteristic
111	C₂×C₄₇	#1969	1	1	yes	characteristic
112	C₂×C₄₇	#1972	1	1	yes	characteristic
113	C₂×C₄⋊Q₈	—	47	2	no	—
114	#500	—	47	4	no	—
115	#500	—	94	2	no	—
116	C₂×#107	—	94	2	no	—
117	C₂×#103	—	94	4	no	—
118	C₂×C₄×C₈	—	47	1	no	—
119	C₂×C₄×C₈	—	47	1	no	—
120	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
121	#384	C₂×D₄₇	1	2	yes	—
122	#340	—	2	2	no	—
123	#340	—	94	4	no	—
124	#322	—	94	2	no	—
125	C₄₇	#9834	1	1	yes	characteristic, complemented, Sylow
126	C₂²×Q₈	—	94	2	no	—
127	C₂²×C₈	—	47	2	no	—
128	C₂²×C₈	—	47	2	no	—
129	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
130	C₄⋊Q₈	—	47	4	no	—
131	C₄⋊Q₈	—	94	2	no	—
132	C₄⋊Q₈	—	94	2	no	—
133	C₄⋊Q₈	—	94	2	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	94	2	no	—
138	C₂×C₄⋊C₄	—	94	2	no	—
139	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
140	#107	—	94	8	no	—
141	#103	—	188	8	no	—
142	C₄×C₈	—	94	1	no	—
143	C₄×C₈	—	94	1	no	—
144	C₄×C₈	—	94	1	no	—
145	C₄×C₈	—	94	1	no	—
146	C₄×C₈	C₄×D₄₇	1	2	yes	—
147	C₄×C₈	D₁₈₈	1	2	yes	—
148	C₂×Q₈	—	94	4	no	—
149	C₂×Q₈	—	188	4	no	—
150	C₂×Q₈	—	188	4	no	—
151	C₂×Q₈	—	188	2	no	—
152	C₂²×C₄	—	4	2	no	—
153	C₂²×C₄	—	188	2	no	—
154	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
155	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
156	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	47	4	no	—
159	C₂×C₈	—	94	2	no	—
160	C₂×C₈	—	94	2	no	—
161	C₂×C₈	—	94	2	no	—
162	C₂×C₈	—	94	4	no	—
163	C₂×C₈	(C₂×D₄₇)⋊C₄	1	4	yes	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄⋊C₄	—	4	4	no	—
166	C₄⋊C₄	—	4	4	no	—
167	C₄⋊C₄	—	94	4	no	—
168	C₄⋊C₄	—	94	4	no	—
169	C₄⋊C₄	—	188	2	no	—
170	C₄⋊C₄	—	188	2	no	—
171	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
172	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
173	C₄²	#10625354	1	1	yes	characteristic
174	C₄²	#10625354	1	1	yes	characteristic
175	C₂³	#13094020	1	1	yes	characteristic
176	Q₈	—	376	4	no	—
177	Q₈	—	188	8	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	—	8	2	no	—
181	C₂×C₄	—	376	2	no	—
182	C₂×C₄	—	376	2	no	—
183	C₂×C₄	—	188	4	no	—
184	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
185	C₂×C₄	#13093986	1	1	yes	characteristic
186	C₂×C₄	#13093986	1	1	yes	characteristic
187	C₂×C₄	#13093999	1	1	yes	characteristic
188	C₂×C₄	#13093999	1	1	yes	characteristic
189	C₂×C₄	#13094013	1	1	yes	characteristic
190	C₂×C₄	Q₁₆×D₄₇	1	2	yes	—
191	C₂×C₄	#13094043	1	2	yes	—
192	C₂×C₄	#13094051	1	4	yes	—
193	C₂×C₄	#13094054	1	4	yes	—
194	C₈	—	2	4	no	—
195	C₈	—	94	4	no	—
196	C₈	—	188	2	no	—
197	C₂²	#15689024	1	1	yes	characteristic
198	C₂²	#15689501	1	1	yes	characteristic
199	C₂²	#15689506	1	1	yes	characteristic
200	C₂²	#15689509	1	2	yes	—
201	C₂²	#15689518	1	2	yes	—
202	C₄	—	16	2	no	—
203	C₄	—	376	4	no	—
204	C₄	#15689131	1	2	yes	—
205	C₄	#15689217	1	2	yes	—
206	C₄	#15689234	1	2	yes	—
207	C₄	#15689365	1	2	yes	—
208	C₄	#15689377	1	2	yes	—
209	C₄	#15689493	1	2	yes	—
210	C₂	#20243040	1	2	yes	—
211	C₂	#20243068	1	2	yes	—
212	C₂	#20244207	1	1	yes	characteristic
213	C₂	#20244247	1	1	yes	characteristic
214	C₂	#20244262	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15689528

C₂

yes

maximal, characteristic

#20244348

C₂

yes

maximal

#20244189

C₂

yes

maximal

C₂×#15688845

C₂

yes

maximal, characteristic

C₄₇×#1834

C₂

yes

maximal, characteristic

C₂×#13094024

C₂²

yes

—

#15689528

—

#15689528

C₄

yes

complemented

C₂×#13093958

—

C₂×#13093947

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₄₇⋊C₈

C₂²

yes

characteristic

#15688885

—

#15688845

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#384

C₂²

yes

—

C₄₇×#340

—

C₂²×Dic₉₄

—

C₂²×C₄₇⋊C₈

D₄

yes

—

C₂²×C₄₇⋊C₈

D₄

yes

—

—

—

—

C₂×C₄

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

#13093958

—

#13093947

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

C₄²×C₄₇

yes

C₄²×C₄₇

yes

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

Q₁₆

yes

—

C₂³×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

C₂×Q₁₆

yes

—

C₂×C₄×C₄₇

#135

yes

—

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₄₇

#137

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#497

yes

—

C₄×C₄₇

—

C₄×C₄₇

#419

yes

—

C₄×C₄₇

C₄×Q₁₆

yes

—

C₄×C₄₇

#442

yes

—

C₄×C₄₇

#467

yes

—

100

C₄×C₄₇

#469

yes

—

101

C₄×C₄₇

#491

yes

—

102

Dic₄₇

—

103

C₂×#500

—

104

#1990

—

105

#1956

—

106

#1834

D₄₇

yes

characteristic

107

C₂×#322

—

108

C₂×C₄₇

#1719

yes

—

109

C₂×C₄₇

#1723

yes

—

110

C₂×C₄₇

#1960

yes

characteristic

111

C₂×C₄₇

#1969

yes

characteristic

112

C₂×C₄₇

#1972

yes

characteristic

113

C₂×C₄⋊Q₈

—

114

#500

—

115

#500

—

116

C₂×#107

—

117

C₂×#103

—

118

C₂×C₄×C₈

—

119

C₂×C₄×C₈

—

120

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

121

#384

C₂×D₄₇

yes

—

122

#340

—

123

#340

—

124

#322

—

125

C₄₇

#9834

yes

characteristic, complemented, Sylow

126

C₂²×Q₈

—

127

C₂²×C₈

—

128

C₂²×C₈

—

129

C₂²×C₈

C₂²⋊D₄₇

yes

—

130

C₄⋊Q₈

—

131

C₄⋊Q₈

—

132

C₄⋊Q₈

—

133

C₄⋊Q₈

—

134

C₂×C₄⋊C₄

—

135

C₂×C₄⋊C₄

—

136

C₂×C₄⋊C₄

—

137

C₂×C₄⋊C₄

—

138

C₂×C₄⋊C₄

—

139

C₂×C₄²

C₂²×D₄₇

yes

characteristic

140

#107

—

141

#103

—

188

—

142

C₄×C₈

—

143

C₄×C₈

—

144

C₄×C₈

—

145

C₄×C₈

—

146

C₄×C₈

C₄×D₄₇

yes

—

147

C₄×C₈

D₁₈₈

yes

—

148

C₂×Q₈

—

149

C₂×Q₈

—

188

—

150

C₂×Q₈

—

188

—

151

C₂×Q₈

—

188

—

152

C₂²×C₄

—

153

C₂²×C₄

—

188

—

154

C₂²×C₄

D₄×D₄₇

yes

characteristic

155

C₂²×C₄

D₄×D₄₇

yes

characteristic

156

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

188

—

170

C₄⋊C₄

—

188

—

171

C₄²

C₂×C₄×D₄₇

yes

characteristic

172

C₄²

C₂×D₁₈₈

yes

characteristic

173

C₄²

#10625354

yes

characteristic

174

C₄²

#10625354

yes

characteristic

175

C₂³

#13094020

yes

characteristic

176

Q₈

—

376

—

177

Q₈

—

188

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

376

—

182

C₂×C₄

—

376

—

183

C₂×C₄

—

188

—

184

C₂×C₄

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

185

C₂×C₄

#13093986

yes

characteristic

186

C₂×C₄

#13093986

yes

characteristic

187

C₂×C₄

#13093999

yes

characteristic

188

C₂×C₄

#13093999

yes

characteristic

189

C₂×C₄

#13094013

yes

characteristic

190

C₂×C₄

Q₁₆×D₄₇

yes

—

191

C₂×C₄

#13094043

yes

—

192

C₂×C₄

#13094051

yes

—

193

C₂×C₄

#13094054

yes

—

194

C₈

—

195

C₈

—

196

C₈

—

188

—

197

C₂²

#15689024

yes

characteristic

198

C₂²

#15689501

yes

characteristic

199

C₂²

#15689506

yes

characteristic

200

C₂²

#15689509

yes

—

201

C₂²

#15689518

yes

—

202

C₄

—

203

C₄

—

376

—

204

C₄

#15689131

yes

—

205

C₄

#15689217

yes

—

206

C₄

#15689234

yes

—

207

C₄

#15689365

yes

—

208

C₄

#15689377

yes

—

209

C₄

#15689493

yes

—

210

C₂

#20243040

yes

—

211

C₂

#20243068

yes

—

212

C₂

#20244207

yes

characteristic

213

C₂

#20244247

yes

characteristic

214

C₂

#20244262

yes

characteristic

	subgroup	G/H
center	C₂³	#13094020
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1834	C₂
socle, prime radical	C₂³×C₄₇	#127
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1834	D₄₇
hypocenter	C₄₇	#9834
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

subgroup

G/H

center

C₂³

#13094020

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1834

C₂

socle, prime radical

C₂³×C₄₇

#127

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

N	H ≅ G/N	type
8. #15689528	C₄	semidirect
125. C₄₇	#9834	semidirect

H ≅ G/N

type

8. #15689528

C₄

semidirect

125. C₄₇

#9834

semidirect

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

subgroup	index
C₂×#15689528	2
#20244348	2
#20244189	2
C₂×#15688845	2
C₄₇×#1834	2
#9834	47

subgroup

index

C₄₇×#1834

Subgroups — #26116100

Conjugacy classes of subgroups

Complements

Maximal subgroups