Group #26121051 — subgroups

Subgroups — #26121051

Order 12032.

292 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 95 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	#20243542	C₂	1	1	yes	maximal, characteristic
2	#20242045	C₂	1	2	yes	maximal
3	#20241283	C₂	1	2	yes	maximal
4	#20241217	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1995	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
8	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
10	#15688872	—	2	2	no	—
11	#15688869	—	2	2	no	—
12	C₄₇×#533	C₂²	1	1	yes	characteristic
13	C₄₇×#423	C₂²	1	1	yes	characteristic
14	C₄₇×#376	C₂²	1	1	yes	characteristic
15	C₄₇×#339	—	2	1	no	—
16	C₄₇×#339	—	2	1	no	—
17	C₂²×C₄₇⋊C₈	—	2	2	no	—
18	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
19	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
20	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
21	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
22	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
23	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
24	C₄×C₄₇⋊C₈	—	2	2	no	—
25	C₂×M₁₆×C₄₇	—	2	1	no	—
26	C₄⋊Q₈×C₄₇	D₄	1	2	yes	—
27	#125×C₄₇	—	2	1	no	—
28	C₂²⋊Q₈×C₄₇	—	2	2	no	—
29	C₄×Q₈×C₄₇	—	2	2	no	—
30	#117×C₄₇	—	2	1	no	—
31	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₄₇	—	2	2	no	—
35	#95×C₄₇	—	2	2	no	—
36	C₂×Q₈×C₄₇	—	2	2	no	—
37	C₂²×C₄×C₄₇	—	2	1	no	—
38	C₂²×C₄×C₄₇	—	2	1	no	—
39	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	#41	1	2	yes	—
41	M₁₆×C₄₇	—	4	1	no	—
42	M₁₆×C₄₇	—	4	1	no	—
43	C₂×C₈×C₄₇	—	4	1	no	—
44	C₄⋊C₄×C₄₇	—	2	2	no	—
45	C₄⋊C₄×C₄₇	—	2	2	no	—
46	C₄⋊C₄×C₄₇	—	4	2	no	—
47	C₂²⋊C₄×C₄₇	—	4	1	no	—
48	C₄²×C₄₇	—	2	1	no	—
49	C₄²×C₄₇	—	2	1	no	—
50	C₄²×C₄₇	—	4	1	no	—
51	C₄²×C₄₇	SD₁₆	1	2	yes	—
52	C₄²×C₄₇	Q₁₆	1	2	yes	—
53	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₄₇	#41	1	1	yes	characteristic
55	C₄²×C₄₇	#41	1	2	yes	—
56	C₂×C₄₇⋊C₈	—	4	2	no	—
57	C₂×C₄₇⋊C₈	—	4	2	no	—
58	C₂×C₄₇⋊C₈	—	4	2	no	—
59	C₂×C₄₇⋊C₈	—	4	2	no	—
60	C₂³×C₄₇	#123	1	1	yes	characteristic
61	Q₈×C₄₇	—	8	2	no	—
62	C₂×C₄×C₄₇	—	2	2	no	—
63	C₂×C₄×C₄₇	—	2	2	no	—
64	C₂×C₄×C₄₇	—	4	2	no	—
65	C₂×C₄×C₄₇	—	4	2	no	—
66	C₂×C₄×C₄₇	—	4	2	no	—
67	C₂×C₄×C₄₇	—	4	2	no	—
68	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
69	C₂×C₄×C₄₇	#123	1	2	yes	—
70	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
71	C₂×C₄×C₄₇	#126	1	2	yes	—
72	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
74	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
76	C₈×C₄₇	—	8	1	no	—
77	C₄₇⋊C₈	—	8	2	no	—
78	C₄₇⋊C₈	—	8	2	no	—
79	#10628	—	47	1	no	maximal, Sylow
80	C₂²×C₄₇	—	2	2	no	—
81	C₂²×C₄₇	#397	1	1	yes	characteristic
82	C₂²×C₄₇	#483	1	1	yes	characteristic
83	C₂²×C₄₇	#484	1	1	yes	characteristic
84	C₄×C₄₇	—	2	2	no	—
85	C₄×C₄₇	—	2	2	no	—
86	C₄×C₄₇	—	4	2	no	—
87	C₄×C₄₇	—	4	2	no	—
88	C₄×C₄₇	—	8	2	no	—
89	C₄×C₄₇	#460	1	1	yes	characteristic
90	C₄×C₄₇	#462	1	1	yes	characteristic
91	C₄×C₄₇	#487	1	1	yes	characteristic
92	C₄×C₄₇	#489	1	1	yes	characteristic
93	#1995	D₄₇	1	1	yes	characteristic
94	#1843	—	47	1	no	—
95	#1542	—	47	2	no	—
96	#1301	—	47	2	no	—
97	#1274	—	47	1	no	—
98	C₂×C₄₇	—	2	1	no	—
99	C₂×C₄₇	—	2	1	no	—
100	C₂×C₄₇	#2072	1	1	yes	characteristic
101	C₂×C₄₇	#2080	1	2	yes	—
102	#533	C₂×D₄₇	1	1	yes	characteristic
103	#431	—	47	2	no	—
104	#423	C₂×D₄₇	1	1	yes	characteristic
105	C₂×C₄⋊C₈	—	47	2	no	—
106	C₂×C₄⋊C₈	—	94	1	no	—
107	C₂×C₄⋊C₈	—	94	1	no	—
108	#376	C₂×D₄₇	1	1	yes	characteristic
109	#339	—	2	1	no	—
110	#339	—	2	1	no	—
111	#333	—	94	2	no	—
112	#332	—	94	2	no	—
113	C₄₇	#10628	1	1	yes	characteristic, complemented, Sylow
114	C₂×M₁₆	—	2	1	no	—
115	C₂²×C₈	—	94	2	no	—
116	C₄⋊Q₈	D₁₈₈	1	2	yes	—
117	#125	—	2	1	no	—
118	C₂²⋊Q₈	—	2	2	no	—
119	C₄×Q₈	—	2	2	no	—
120	#117	—	2	1	no	—
121	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
122	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
123	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	94	2	no	—
126	C₄⋊C₈	—	94	2	no	—
127	C₄⋊C₈	—	94	2	no	—
128	C₄⋊C₈	—	188	2	no	—
129	C₄⋊C₈	—	188	2	no	—
130	C₂²⋊C₈	—	94	2	no	—
131	C₄×C₈	—	94	2	no	—
132	#95	—	2	2	no	—
133	C₂×Q₈	—	2	2	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	#10625352	1	2	yes	—
137	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
138	M₁₆	—	4	1	no	—
139	M₁₆	—	4	1	no	—
140	C₂×C₈	—	4	1	no	—
141	C₂×C₈	—	188	2	no	—
142	C₂×C₈	—	188	2	no	—
143	C₂×C₈	—	188	2	no	—
144	C₂×C₈	—	188	2	no	—
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₂²⋊C₄	—	4	1	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	2	1	no	—
151	C₄²	—	4	1	no	—
152	C₄²	#10625336	1	2	yes	—
153	C₄²	#10625337	1	2	yes	—
154	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
155	C₄²	#10625350	1	1	yes	characteristic
156	C₄²	#10625352	1	2	yes	—
157	C₂³	#13094005	1	1	yes	characteristic
158	Q₈	—	8	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	#13093998	1	1	yes	characteristic
166	C₂×C₄	#13094008	1	2	yes	—
167	C₂×C₄	#13094011	1	1	yes	characteristic
168	C₂×C₄	#13094017	1	2	yes	—
169	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
170	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
171	C₂×C₄	#13094049	1	1	yes	characteristic
172	C₂×C₄	#13094054	1	1	yes	characteristic
173	C₈	—	8	1	no	—
174	C₈	—	376	2	no	—
175	C₈	—	376	2	no	—
176	C₂²	—	2	2	no	—
177	C₂²	#15689048	1	1	yes	characteristic
178	C₂²	#15689451	1	1	yes	characteristic
179	C₂²	#15689456	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₄	—	8	2	no	—
185	C₄	#15689323	1	1	yes	characteristic
186	C₄	#15689333	1	1	yes	characteristic
187	C₄	#15689473	1	1	yes	characteristic
188	C₄	#15689482	1	1	yes	characteristic
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	#20244796	1	1	yes	characteristic
192	C₂	#20244838	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243542

C₂

yes

maximal, characteristic

#20242045

C₂

yes

maximal

#20241283

C₂

yes

maximal

#20241217

C₂

yes

maximal, characteristic

C₄₇×#1995

C₂

yes

maximal, characteristic

#15689176

C₂²

yes

—

C₂×C₄⋊(C₄₇⋊C₈)

—

C₂×C₄⋊(C₄₇⋊C₈)

—

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

#15688872

—

#15688869

—

C₄₇×#533

C₂²

yes

characteristic

C₄₇×#423

C₂²

yes

characteristic

C₄₇×#376

C₂²

yes

characteristic

C₄₇×#339

—

C₄₇×#339

—

C₂²×C₄₇⋊C₈

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₂×M₁₆×C₄₇

—

C₄⋊Q₈×C₄₇

D₄

yes

—

#125×C₄₇

—

C₂²⋊Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

#117×C₄₇

—

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

#95×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

SD₁₆

yes

—

C₄²×C₄₇

Q₁₆

yes

—

C₄²×C₄₇

C₂×D₄

yes

C₄²×C₄₇

yes

C₄²×C₄₇

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#123

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

#121

yes

characteristic

C₂×C₄×C₄₇

#123

yes

—

C₂×C₄×C₄₇

#124

yes

characteristic

C₂×C₄×C₄₇

#126

yes

—

C₂×C₄×C₄₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₁₆

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#10628

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

D₄₇

yes

—

—

—

—

C₂×C₄₇

—

C₂×C₄₇

—

100

C₂×C₄₇

#2072

yes

characteristic

101

C₂×C₄₇

#2080

yes

—

102

#533

C₂×D₄₇

yes

characteristic

103

#431

—

104

#423

C₂×D₄₇

yes

characteristic

105

C₂×C₄⋊C₈

—

106

C₂×C₄⋊C₈

—

107

C₂×C₄⋊C₈

—

108

#376

C₂×D₄₇

yes

characteristic

109

#339

—

110

#339

—

111

#333

—

112

#332

—

113

C₄₇

#10628

yes

characteristic, complemented, Sylow

114

C₂×M₁₆

—

115

C₂²×C₈

—

116

C₄⋊Q₈

D₁₈₈

yes

—

117

#125

—

118

C₂²⋊Q₈

—

119

C₄×Q₈

—

120

#117

—

121

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

122

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

123

C₂×C₄²

C₂²×D₄₇

yes

characteristic

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₄⋊C₈

—

128

C₄⋊C₈

—

188

—

129

C₄⋊C₈

—

188

—

130

C₂²⋊C₈

—

131

C₄×C₈

—

132

#95

—

133

C₂×Q₈

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

#10625352

yes

—

137

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

138

M₁₆

—

139

M₁₆

—

140

C₂×C₈

—

141

C₂×C₈

—

188

—

142

C₂×C₈

—

188

—

143

C₂×C₈

—

188

—

144

C₂×C₈

—

188

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₂²⋊C₄

—

149

C₄²

—

150

C₄²

—

151

C₄²

—

152

C₄²

#10625336

yes

—

153

C₄²

#10625337

yes

—

154

C₄²

C₂×D₁₈₈

yes

characteristic

155

C₄²

#10625350

yes

characteristic

156

C₄²

#10625352

yes

—

157

C₂³

#13094005

yes

characteristic

158

Q₈

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

#13093998

yes

characteristic

166

C₂×C₄

#13094008

yes

—

167

C₂×C₄

#13094011

yes

characteristic

168

C₂×C₄

#13094017

yes

—

169

C₂×C₄

C₂×#10625336

yes

characteristic

170

C₂×C₄

C₂×#10625337

yes

characteristic

171

C₂×C₄

#13094049

yes

characteristic

172

C₂×C₄

#13094054

yes

characteristic

173

C₈

—

174

C₈

—

376

—

175

C₈

—

376

—

176

C₂²

—

177

C₂²

#15689048

yes

characteristic

178

C₂²

#15689451

yes

characteristic

179

C₂²

#15689456

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

#15689323

yes

characteristic

186

C₄

#15689333

yes

characteristic

187

C₄

#15689473

yes

characteristic

188

C₄

#15689482

yes

characteristic

189

C₂

—

190

C₂

—

191

C₂

#20244796

yes

characteristic

192

C₂

#20244838

yes

—

	subgroup	G/H
center	C₂×C₄	#13094011
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1995	C₂
socle	C₂²×C₄₇	#397
prime radical	C₂³×C₄₇	#123
hypercenter	#1995	D₄₇
hypocenter	C₄₇	#10628
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1995

subgroup

G/H

center

C₂×C₄

#13094011

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1995

C₂

C₂²×C₄₇

C₂³×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1995

N	H ≅ G/N	type
113. C₄₇	#10628	semidirect

H ≅ G/N

type

113. C₄₇

#10628

semidirect

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

subgroup	index
#20243542	2
#20242045	2
#20241283	2
#20241217	2
C₄₇×#1995	2
#10628	47

subgroup

index

C₄₇×#1995

Subgroups — #26121051

Conjugacy classes of subgroups

Complements

Maximal subgroups