Group #26132470 — subgroups

Subgroups — #26132470

Order 12032.

354 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 111 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×#15689541	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688952	C₂	1	2	yes	maximal, complemented
3	#20243435	C₂	1	2	yes	maximal
4	#20241492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1385	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13094030	C₂²	1	1	yes	characteristic
7	#15689541	—	2	4	no	—
8	C₂×#13093963	C₄	1	2	yes	—
9	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₆×D₄₇	C₄	1	2	yes	—
11	C₂×#13093960	C₂²	1	2	yes	—
12	#15689105	—	2	2	no	—
13	#15688952	—	2	4	no	—
14	C₄₇⋊#338	—	2	2	no	—
15	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×#108×C₄₇	C₂²	1	2	yes	—
17	C₂×#10625350	D₄	1	1	yes	characteristic
18	#13094030	—	2	2	no	—
19	#13094030	—	2	2	no	—
20	C₂×#10625329	—	2	2	no	—
21	C₂×#10625329	D₄	1	1	yes	characteristic
22	C₂×#10625324	Q₈	1	2	yes	—
23	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
24	#13093963	—	2	4	no	—
25	C₂×C₄₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄₇⋊C₁₆	D₄	1	4	yes	—
28	C₁₆×D₄₇	—	2	4	no	—
29	#13093960	—	2	2	no	—
30	#13093960	—	2	2	no	—
31	#13093942	—	4	2	no	—
32	#13093940	—	4	2	no	—
33	C₂×M₁₆×C₄₇	—	2	2	no	—
34	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
35	M₃₂×C₄₇	—	2	2	no	—
36	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
37	#108×C₄₇	—	2	2	no	—
38	#108×C₄₇	—	2	2	no	—
39	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₂²⋊D₄₇	Q₁₆	1	2	yes	—
41	#10625350	—	2	2	no	—
42	#10625350	—	2	2	no	—
43	C₂×D₁₈₈	D₈	1	1	yes	characteristic
44	C₂×C₄×D₄₇	C₄⋊C₄	1	2	yes	—
45	C₂×Dic₉₄	D₈	1	1	yes	characteristic
46	M₁₆×C₄₇	—	4	2	no	—
47	M₁₆×C₄₇	—	4	2	no	—
48	C₂×C₈×C₄₇	—	4	2	no	—
49	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
51	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₄₇	#41	1	2	yes	—
53	#10625329	—	2	2	no	—
54	#10625329	—	4	2	no	—
55	#10625329	—	4	2	no	—
56	C₂×C₄₇⋊C₈	—	4	2	no	—
57	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
58	#10625324	—	2	4	no	—
59	C₈×D₄₇	—	2	2	no	—
60	C₈×D₄₇	—	2	2	no	—
61	C₁₆×C₄₇	—	2	2	no	—
62	C₄₇⋊C₁₆	C₂²⋊C₄	1	2	yes	—
63	C₄₇⋊C₁₆	C₂²⋊C₄	1	2	yes	—
64	C₂³×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
65	C₂²×D₄₇	#107	1	2	yes	—
66	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₄₇	#121	1	2	yes	—
68	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₄₇	#135	1	2	yes	—
70	C₂²⋊D₄₇	—	2	4	no	—
71	C₂×Dic₄₇	#107	1	2	yes	—
72	D₁₈₈	—	2	2	no	—
73	C₄×D₄₇	—	2	2	no	—
74	C₄×D₄₇	—	2	2	no	—
75	Dic₉₄	—	2	2	no	—
76	C₈×C₄₇	—	8	2	no	—
77	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₈×C₄₇	#117	1	1	yes	characteristic
79	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
80	C₄₇⋊C₈	—	2	2	no	—
81	C₄₇⋊C₈	—	8	2	no	—
82	#12840	—	47	1	no	maximal, Sylow
83	C₂²×C₄₇	—	4	1	no	—
84	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
85	C₂²×C₄₇	#467	1	2	yes	—
86	C₂×D₄₇	—	2	2	no	—
87	C₂×D₄₇	—	4	2	no	—
88	C₄×C₄₇	#380	1	1	yes	characteristic
89	C₄×C₄₇	#427	1	1	yes	characteristic
90	C₄×C₄₇	#466	1	2	yes	—
91	Dic₄₇	—	2	2	no	—
92	C₂×#504	—	47	1	no	—
93	C₂×#368	—	47	2	no	—
94	#1813	—	47	2	no	—
95	#1385	—	47	1	no	—
96	#1385	D₄₇	1	1	yes	characteristic, complemented
97	C₂×C₄₇	—	2	1	no	—
98	C₂×C₄₇	—	2	2	no	—
99	C₂×C₄₇	#1808	1	1	yes	characteristic
100	D₄₇	—	4	2	no	—
101	C₂×#131	—	47	1	no	—
102	#504	—	94	4	no	—
103	C₂×M₃₂	—	47	2	no	—
104	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
105	C₂²×C₁₆	—	47	1	no	—
106	C₂²×C₁₆	—	47	2	no	—
107	C₂×#108	—	47	2	no	—
108	C₂×#108	C₂×D₄₇	1	2	yes	—
109	#413	—	94	2	no	—
110	#368	—	94	4	no	—
111	#338	—	94	2	no	—
112	C₄₇	#12840	1	1	yes	characteristic, complemented, Sylow
113	C₂×#41	—	47	1	no	—
114	#131	—	94	2	no	—
115	#131	—	94	2	no	—
116	C₂×M₁₆	—	2	2	no	—
117	C₂×M₁₆	—	47	1	no	—
118	C₂×M₁₆	—	47	2	no	—
119	C₂×M₁₆	—	94	2	no	—
120	C₂²×C₈	—	47	2	no	—
121	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
122	M₃₂	—	2	2	no	—
123	M₃₂	—	94	4	no	—
124	C₂×C₁₆	—	47	1	no	—
125	C₂×C₁₆	—	47	1	no	—
126	C₂×C₁₆	—	47	4	no	—
127	C₂×C₁₆	—	94	4	no	—
128	C₂×C₁₆	C₄×D₄₇	1	2	yes	—
129	#108	—	2	2	no	—
130	#108	—	2	2	no	—
131	#108	—	94	2	no	—
132	#108	—	94	2	no	—
133	#101	—	188	2	no	—
134	#100	—	188	2	no	—
135	#41	—	94	2	no	—
136	#41	—	94	2	no	—
137	C₂×Q₈	—	47	1	no	—
138	C₂×D₄	—	47	2	no	—
139	C₂×D₄	—	47	1	no	—
140	C₂²×C₄	—	47	2	no	—
141	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
142	M₁₆	—	4	2	no	—
143	M₁₆	—	4	2	no	—
144	M₁₆	—	94	2	no	—
145	M₁₆	—	94	4	no	—
146	M₁₆	—	188	2	no	—
147	M₁₆	—	188	2	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	47	2	no	—
150	C₂×C₈	—	94	2	no	—
151	C₂×C₈	—	94	2	no	—
152	C₂×C₈	—	188	2	no	—
153	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
154	C₂×C₈	#10625350	1	2	yes	—
155	C₂×C₈	D₄×D₄₇	1	2	yes	—
156	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
157	C₁₆	—	2	2	no	—
158	C₁₆	—	47	2	no	—
159	C₁₆	—	47	2	no	—
160	C₂³	—	47	2	no	—
161	C₂³	Q₁₆×D₄₇	1	1	yes	characteristic
162	Q₈	—	94	2	no	—
163	D₄	—	94	4	no	—
164	D₄	—	94	2	no	—
165	C₂×C₄	—	47	2	no	—
166	C₂×C₄	—	94	2	no	—
167	C₂×C₄	—	94	2	no	—
168	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
169	C₂×C₄	#13093997	1	2	yes	—
170	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
171	C₂×C₄	#13094045	1	2	yes	—
172	C₈	—	8	2	no	—
173	C₈	—	94	2	no	—
174	C₈	—	376	2	no	—
175	C₈	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
176	C₈	#13093981	1	1	yes	characteristic
177	C₈	#13094002	1	2	yes	—
178	C₂²	—	4	1	no	—
179	C₂²	—	94	2	no	—
180	C₂²	—	188	2	no	—
181	C₂²	#107×D₄₇	1	1	yes	characteristic
182	C₂²	#15689363	1	2	yes	—
183	C₄	—	94	2	no	—
184	C₄	#15688982	1	1	yes	characteristic
185	C₄	#15689158	1	1	yes	characteristic
186	C₄	#15689358	1	2	yes	—
187	C₂	—	2	1	no	—
188	C₂	—	2	2	no	—
189	C₂	—	188	2	no	—
190	C₂	#20243416	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15689541

C₂

yes

maximal, characteristic

C₂×#15688952

C₂

yes

maximal, complemented

#20243435

C₂

yes

maximal

#20241492

C₂

yes

maximal, characteristic, complemented

C₄₇×#1385

C₂

yes

maximal, characteristic, complemented

C₂×#13094030

C₂²

yes

characteristic

#15689541

—

C₂×#13093963

C₄

yes

—

C₂²×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₂×C₁₆×D₄₇

C₄

yes

—

C₂×#13093960

C₂²

yes

—

#15689105

—

#15688952

—

C₄₇⋊#338

—

C₂×M₃₂×C₄₇

C₂²

yes

characteristic

C₂×#108×C₄₇

C₂²

yes

—

C₂×#10625350

D₄

yes

—

—

—

D₄

yes

characteristic

C₂×#10625324

Q₈

yes

—

C₂×C₈×D₄₇

C₂×C₄

yes

—

#13093963

—

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₁₆

D₄

yes

—

C₁₆×D₄₇

—

—

—

—

—

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

C₂³

yes

characteristic

M₃₂×C₄₇

—

C₂×C₁₆×C₄₇

C₂×C₄

yes

—

#108×C₄₇

—

#108×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₂²⋊D₄₇

Q₁₆

yes

—

#10625350

—

#10625350

—

C₂×D₁₈₈

D₈

yes

characteristic

C₂×C₄×D₄₇

C₄⋊C₄

yes

—

C₂×Dic₉₄

D₈

yes

characteristic

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₂×C₈×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

#41

yes

—

#10625329

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

#10625324

—

C₈×D₄₇

—

C₈×D₄₇

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

C₂²⋊C₄

yes

—

C₄₇⋊C₁₆

C₂²⋊C₄

yes

—

C₂³×C₄₇

C₂×Q₁₆

yes

characteristic

C₂²×D₄₇

#107

yes

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#121

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

#135

yes

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

#107

yes

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₄₇

#117

yes

characteristic

C₈×C₄₇

C₂²⋊Q₈

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#12840

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

C₂×#107

yes

characteristic

C₂²×C₄₇

#467

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

—

—

—

—

D₄₇

yes

characteristic, complemented

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#1808

yes

characteristic

100

D₄₇

—

101

C₂×#131

—

102

#504

—

103

C₂×M₃₂

—

104

C₂×M₃₂

C₂×D₄₇

yes

characteristic

105

C₂²×C₁₆

—

106

C₂²×C₁₆

—

107

C₂×#108

—

108

C₂×#108

C₂×D₄₇

yes

—

109

#413

—

110

#368

—

111

#338

—

112

C₄₇

#12840

yes

characteristic, complemented, Sylow

113

C₂×#41

—

114

#131

—

115

#131

—

116

C₂×M₁₆

—

117

C₂×M₁₆

—

118

C₂×M₁₆

—

119

C₂×M₁₆

—

120

C₂²×C₈

—

121

C₂²×C₈

C₂²×D₄₇

yes

characteristic

122

M₃₂

—

123

M₃₂

—

124

C₂×C₁₆

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

127

C₂×C₁₆

—

128

C₂×C₁₆

C₄×D₄₇

yes

—

129

#108

—

130

#108

—

131

#108

—

132

#108

—

133

#101

—

188

—

134

#100

—

188

—

135

#41

—

136

#41

—

137

C₂×Q₈

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂²×C₄

—

141

C₂²×C₄

D₄×D₄₇

yes

characteristic

142

M₁₆

—

143

M₁₆

—

144

M₁₆

—

145

M₁₆

—

146

M₁₆

—

188

—

147

M₁₆

—

188

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

188

—

153

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

154

C₂×C₈

#10625350

yes

—

155

C₂×C₈

D₄×D₄₇

yes

—

156

C₂×C₈

Q₈×D₄₇

yes

characteristic

157

C₁₆

—

158

C₁₆

—

159

C₁₆

—

160

C₂³

—

161

C₂³

Q₁₆×D₄₇

yes

characteristic

162

Q₈

—

163

D₄

—

164

D₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

C₄⋊C₄×D₄₇

yes

characteristic

169

C₂×C₄

#13093997

yes

—

170

C₂×C₄

D₈×D₄₇

yes

characteristic

171

C₂×C₄

#13094045

yes

—

172

C₈

—

173

C₈

—

174

C₈

—

376

—

175

C₈

C₂²⋊C₄×D₄₇

yes

characteristic

176

C₈

#13093981

yes

characteristic

177

C₈

#13094002

yes

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

188

—

181

C₂²

#107×D₄₇

yes

characteristic

182

C₂²

#15689363

yes

—

183

C₄

—

184

C₄

#15688982

yes

characteristic

185

C₄

#15689158

yes

characteristic

186

C₄

#15689358

yes

—

187

C₂

—

188

C₂

—

189

C₂

—

188

—

190

C₂

#20243416

yes

characteristic

	subgroup	G/H
center	C₄	#15689158
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1385	C₂
socle	C₂×C₄₇	#1808
prime radical	C₂×#10625350	D₄
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1385	D₄₇
hypocenter	C₄₇	#12840
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

subgroup

G/H

center

C₄

#15689158

commutator

C₂×C₈×C₄₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#1385

C₂

C₂×C₄₇

D₄

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

N	H ≅ G/N	type
2. C₂×#15688952	C₂	semidirect
4. #20241492	C₂	semidirect
5. C₄₇×#1385	C₂	semidirect
96. #1385	D₄₇	semidirect
112. C₄₇	#12840	semidirect

H ≅ G/N

type

2. C₂×#15688952

C₂

semidirect

4. #20241492

C₂

semidirect

5. C₄₇×#1385

C₂

semidirect

96. #1385

D₄₇

semidirect

112. C₄₇

#12840

semidirect

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

subgroup	index
C₂×#15689541	2
C₂×#15688952	2
#20243435	2
#20241492	2
C₄₇×#1385	2
#12840	47

subgroup

index

C₄₇×#1385

Subgroups — #26132470

Conjugacy classes of subgroups

Complements

Maximal subgroups