Group #26121053 — subgroups

Subgroups — #26121053

Order 12032.

292 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 93 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	#20244369	C₂	1	1	yes	maximal, characteristic
2	#20243542	C₂	1	1	yes	maximal, characteristic
3	#20242043	C₂	1	2	yes	maximal
4	#20241284	C₂	1	2	yes	maximal
5	C₄₇×#1274	C₂	1	1	yes	maximal, characteristic
6	#15689680	C₂²	1	1	yes	characteristic
7	#15689147	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
9	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	#15688883	—	2	1	no	—
12	#15688883	—	2	1	no	—
13	#15688871	—	2	2	no	—
14	#15688870	—	2	2	no	—
15	C₄₇×#431	C₂²	1	2	yes	—
16	C₄₇×#376	C₂²	1	1	yes	characteristic
17	C₂×#10625329	—	2	1	no	—
18	C₂²×C₄₇⋊C₈	—	2	2	no	—
19	#13094024	D₄	1	2	yes	—
20	#13094016	—	2	1	no	—
21	#13094003	—	2	2	no	—
22	C₄×Dic₉₄	—	2	2	no	—
23	#13093983	—	2	1	no	—
24	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
26	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
27	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
28	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
29	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₄₇	—	2	2	no	—
33	C₂²⋊C₈×C₄₇	—	2	2	no	—
34	C₄×C₈×C₄₇	—	2	2	no	—
35	#95×C₄₇	—	2	2	no	—
36	C₂²×C₄×C₄₇	—	2	1	no	—
37	C₂²×C₄×C₄₇	—	2	1	no	—
38	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	#41	1	2	yes	—
40	C₂×Dic₉₄	—	2	2	no	—
41	C₂×C₈×C₄₇	—	4	2	no	—
42	C₄²×C₄₇	—	2	1	no	—
43	C₄²×C₄₇	—	2	1	no	—
44	C₄²×C₄₇	SD₁₆	1	2	yes	—
45	C₄²×C₄₇	Q₁₆	1	2	yes	—
46	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₄₇	#41	1	1	yes	characteristic
48	C₄²×C₄₇	#41	1	2	yes	—
49	C₂²⋊Dic₄₇	—	4	1	no	—
50	C₄⋊Dic₄₇	—	2	2	no	—
51	C₄⋊Dic₄₇	—	2	2	no	—
52	Dic₄₇⋊C₄	—	4	2	no	—
53	C₄×Dic₄₇	—	4	1	no	—
54	#10625329	—	4	1	no	—
55	#10625329	—	4	1	no	—
56	C₂×C₄₇⋊C₈	—	4	2	no	—
57	C₂×C₄₇⋊C₈	—	4	1	no	—
58	C₂×C₄₇⋊C₈	—	4	2	no	—
59	C₂×C₄₇⋊C₈	—	4	2	no	—
60	C₂³×C₄₇	#123	1	1	yes	characteristic
61	C₂×C₄×C₄₇	—	2	2	no	—
62	C₂×C₄×C₄₇	—	2	2	no	—
63	C₂×C₄×C₄₇	—	4	2	no	—
64	C₂×C₄×C₄₇	—	4	2	no	—
65	C₂×C₄×C₄₇	—	4	2	no	—
66	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
67	C₂×C₄×C₄₇	#123	1	2	yes	—
68	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
69	C₂×C₄×C₄₇	#126	1	2	yes	—
70	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
74	C₂×Dic₄₇	—	4	2	no	—
75	Dic₉₄	—	8	2	no	—
76	C₈×C₄₇	—	8	2	no	—
77	C₄₇⋊C₈	—	8	2	no	—
78	C₄₇⋊C₈	—	8	1	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	#460	1	1	yes	characteristic
89	C₄×C₄₇	#462	1	1	yes	characteristic
90	C₄×C₄₇	#487	1	1	yes	characteristic
91	C₄×C₄₇	#489	1	1	yes	characteristic
92	Dic₄₇	—	8	2	no	—
93	#1995	—	47	1	no	—
94	#1843	—	47	1	no	—
95	#1542	—	47	2	no	—
96	#1301	—	47	2	no	—
97	#1274	D₄₇	1	1	yes	characteristic
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	—	47	1	no	—
103	#431	C₂×D₄₇	1	2	yes	—
104	#423	—	47	1	no	—
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	—	94	1	no	—
110	#339	—	94	1	no	—
111	#333	—	94	2	no	—
112	#332	—	94	2	no	—
113	C₄₇	#10628	1	1	yes	characteristic, complemented, Sylow
114	C₂×M₁₆	—	94	1	no	—
115	C₂²×C₈	—	94	2	no	—
116	C₄⋊Q₈	—	47	2	no	—
117	#125	—	94	1	no	—
118	C₂²⋊Q₈	—	94	2	no	—
119	C₄×Q₈	—	94	2	no	—
120	#117	—	94	1	no	—
121	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
122	C₂×C₄²	D₁₈₈	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	—	2	2	no	—
131	C₄×C₈	—	2	2	no	—
132	#95	—	2	2	no	—
133	C₂×Q₈	—	94	2	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
137	C₂²×C₄	#10625352	1	2	yes	—
138	M₁₆	—	188	1	no	—
139	M₁₆	—	188	1	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	188	2	no	—
142	C₂×C₈	—	188	1	no	—
143	C₂×C₈	—	188	2	no	—
144	C₂×C₈	—	188	2	no	—
145	C₄⋊C₄	—	94	2	no	—
146	C₄⋊C₄	—	94	2	no	—
147	C₄⋊C₄	—	188	2	no	—
148	C₂²⋊C₄	—	188	1	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	2	1	no	—
151	C₄²	—	188	1	no	—
152	C₄²	#10625325	1	2	yes	—
153	C₄²	Dic₁₈₈	1	2	yes	—
154	C₄²	#10625350	1	2	yes	—
155	C₄²	D₄×D₄₇	1	1	yes	characteristic
156	C₄²	#10625354	1	1	yes	characteristic
157	C₂³	#13094008	1	1	yes	characteristic
158	Q₈	—	376	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	—	188	2	no	—
165	C₂×C₄	#13093999	1	1	yes	characteristic
166	C₂×C₄	#13094006	1	2	yes	—
167	C₂×C₄	#13094012	1	1	yes	characteristic
168	C₂×C₄	#13094019	1	2	yes	—
169	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
170	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
171	C₂×C₄	#13094052	1	1	yes	characteristic
172	C₂×C₄	#13094057	1	1	yes	characteristic
173	C₈	—	8	2	no	—
174	C₈	—	376	1	no	—
175	C₈	—	376	2	no	—
176	C₂²	—	2	2	no	—
177	C₂²	#15689050	1	1	yes	characteristic
178	C₂²	#15689454	1	1	yes	characteristic
179	C₂²	#15689459	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	—	376	2	no	—
185	C₄	#15689324	1	1	yes	characteristic
186	C₄	#15689334	1	1	yes	characteristic
187	C₄	#15689474	1	1	yes	characteristic
188	C₄	#15689483	1	1	yes	characteristic
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	#20244798	1	1	yes	characteristic
192	C₂	#20244840	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20244369

C₂

yes

maximal, characteristic

#20243542

C₂

yes

maximal, characteristic

#20242043

C₂

yes

maximal

#20241284

C₂

yes

maximal

C₄₇×#1274

C₂

yes

maximal, characteristic

#15689680

C₂²

yes

characteristic

#15689147

C₂²

yes

characteristic

C₂×C₄⋊(C₄₇⋊C₈)

—

C₂×C₄⋊(C₄₇⋊C₈)

—

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

—

—

—

—

C₄₇×#431

C₂²

yes

—

C₄₇×#376

C₂²

yes

characteristic

C₂×#10625329

—

C₂²×C₄₇⋊C₈

—

#13094024

D₄

yes

—

#13094016

—

#13094003

—

C₄×Dic₉₄

—

#13093983

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

C₄×C₈×C₄₇

—

#95×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

SD₁₆

yes

—

C₄²×C₄₇

Q₁₆

yes

—

C₄²×C₄₇

C₂×D₄

yes

C₄²×C₄₇

yes

C₄²×C₄₇

yes

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#123

yes

characteristic

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

C₂×Dic₄₇

—

Dic₉₄

—

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

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

—

—

—

—

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

—

100

C₂×C₄₇

#2072

yes

characteristic

101

C₂×C₄₇

#2080

yes

—

102

#533

—

103

#431

C₂×D₄₇

yes

—

104

#423

—

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

—

117

#125

—

118

C₂²⋊Q₈

—

119

C₄×Q₈

—

120

#117

—

121

C₂×C₄²

D₁₈₈

yes

characteristic

122

C₂×C₄²

D₁₈₈

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

C₂×D₁₈₈

yes

characteristic

137

C₂²×C₄

#10625352

yes

—

138

M₁₆

—

188

—

139

M₁₆

—

188

—

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

—

188

—

148

C₂²⋊C₄

—

188

—

149

C₄²

—

150

C₄²

—

151

C₄²

—

188

—

152

C₄²

#10625325

yes

—

153

C₄²

Dic₁₈₈

yes

—

154

C₄²

#10625350

yes

—

155

C₄²

D₄×D₄₇

yes

characteristic

156

C₄²

#10625354

yes

characteristic

157

C₂³

#13094008

yes

characteristic

158

Q₈

—

376

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

188

—

165

C₂×C₄

#13093999

yes

characteristic

166

C₂×C₄

#13094006

yes

—

167

C₂×C₄

#13094012

yes

characteristic

168

C₂×C₄

#13094019

yes

—

169

C₂×C₄

C₂×#10625325

yes

characteristic

170

C₂×C₄

C₂×Dic₁₈₈

yes

characteristic

171

C₂×C₄

#13094052

yes

characteristic

172

C₂×C₄

#13094057

yes

characteristic

173

C₈

—

174

C₈

—

376

—

175

C₈

—

376

—

176

C₂²

—

177

C₂²

#15689050

yes

characteristic

178

C₂²

#15689454

yes

characteristic

179

C₂²

#15689459

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

376

—

185

C₄

#15689324

yes

characteristic

186

C₄

#15689334

yes

characteristic

187

C₄

#15689474

yes

characteristic

188

C₄

#15689483

yes

characteristic

189

C₂

—

190

C₂

—

191

C₂

#20244798

yes

characteristic

192

C₂

#20244840

yes

—

	subgroup	G/H
center	C₂×C₄	#13094012
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1274	C₂
socle	C₂²×C₄₇	#397
prime radical	C₂³×C₄₇	#123
hypercenter	#1274	D₄₇
hypocenter	C₄₇	#10628
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

subgroup

G/H

center

C₂×C₄

#13094012

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1274

C₂

C₂²×C₄₇

C₂³×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

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
#20244369	2
#20243542	2
#20242043	2
#20241284	2
C₄₇×#1274	2
#10628	47

subgroup

index

C₄₇×#1274

Subgroups — #26121053

Conjugacy classes of subgroups

Complements

Maximal subgroups