Group #26132458 — subgroups

Subgroups — #26132458

Order 12032.

394 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 119 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	C₂²×C₁₆×D₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688952	C₂	1	2	yes	maximal, complemented
3	#20243423	C₂	1	2	yes	maximal
4	#20241482	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1381	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₄₇	C₂²	1	1	yes	characteristic
7	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₁₆×D₄₇	—	2	4	no	—
9	C₂×C₁₆×D₄₇	C₄	1	4	yes	—
10	C₂×#13093960	C₂²	1	2	yes	—
11	#15689086	—	2	2	no	—
12	#15688952	—	2	4	no	—
13	#15688887	—	2	2	no	—
14	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×#108×C₄₇	C₂²	1	2	yes	—
16	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
17	C₂×#10625329	—	2	2	no	—
18	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
19	C₂×C₈×D₄₇	—	2	2	no	—
20	C₂×C₈×D₄₇	—	2	2	no	—
21	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
22	C₂×C₈×D₄₇	Q₈	1	2	yes	—
23	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₄₇⋊C₁₆	D₄	1	4	yes	—
25	C₁₆×D₄₇	—	2	8	no	—
26	#13093960	—	2	2	no	—
27	#13093960	—	2	2	no	—
28	(C₂×D₄₇)⋊C₈	—	4	4	no	—
29	C₂×M₁₆×C₄₇	—	2	2	no	—
30	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
31	C₂×C₁₆×C₄₇	—	2	2	no	—
32	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
33	#108×C₄₇	—	2	2	no	—
34	#108×C₄₇	—	2	2	no	—
35	C₂³×D₄₇	Q₁₆	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₄₇	Q₁₆	1	1	yes	characteristic
38	C₂×C₄×D₄₇	—	2	2	no	—
39	C₂×C₄×D₄₇	—	2	2	no	—
40	C₂×C₄×D₄₇	C₄⋊C₄	1	2	yes	—
41	C₂×C₄×D₄₇	D₈	1	2	yes	—
42	M₁₆×C₄₇	—	4	2	no	—
43	M₁₆×C₄₇	—	4	2	no	—
44	C₂×C₈×C₄₇	—	4	2	no	—
45	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₄₇	#41	1	2	yes	—
49	#10625329	—	4	2	no	—
50	#10625329	—	4	2	no	—
51	C₂×C₄₇⋊C₈	—	2	2	no	—
52	C₂×C₄₇⋊C₈	—	4	2	no	—
53	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
54	C₈×D₄₇	—	2	2	no	—
55	C₈×D₄₇	—	2	2	no	—
56	C₈×D₄₇	—	2	4	no	—
57	C₁₆×C₄₇	—	2	2	no	—
58	C₄₇⋊C₁₆	C₂²⋊C₄	1	4	yes	—
59	C₂³×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
60	C₂²×D₄₇	—	2	2	no	—
61	C₂²×D₄₇	—	2	4	no	—
62	C₂²×D₄₇	#107	1	2	yes	—
63	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₄₇	#121	1	2	yes	—
65	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₄₇	#135	1	2	yes	—
67	C₂×Dic₄₇	—	2	2	no	—
68	C₂×Dic₄₇	#107	1	2	yes	—
69	C₄×D₄₇	—	2	2	no	—
70	C₄×D₄₇	—	2	2	no	—
71	C₄×D₄₇	—	2	4	no	—
72	C₈×C₄₇	—	8	2	no	—
73	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₈×C₄₇	#117	1	1	yes	characteristic
75	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
76	C₄₇⋊C₈	—	2	2	no	—
77	C₄₇⋊C₈	—	8	2	no	—
78	#12837	—	47	1	no	maximal, Sylow
79	C₂²×C₄₇	—	2	2	no	—
80	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
81	C₂²×C₄₇	#467	1	2	yes	—
82	C₂×D₄₇	—	2	8	no	—
83	C₂×D₄₇	—	2	2	no	—
84	C₂×D₄₇	—	2	2	no	—
85	C₂×D₄₇	#368	1	4	yes	complemented
86	C₄×C₄₇	#380	1	1	yes	characteristic
87	C₄×C₄₇	#427	1	1	yes	characteristic
88	C₄×C₄₇	#466	1	2	yes	—
89	Dic₄₇	—	2	2	no	—
90	C₂³×C₁₆	—	47	1	no	—
91	C₂×#368	—	47	2	no	—
92	#1810	—	47	2	no	—
93	#1381	—	47	1	no	—
94	#1381	D₄₇	1	1	yes	characteristic, complemented
95	C₂×C₄₇	—	2	2	no	—
96	C₂×C₄₇	#1808	1	1	yes	characteristic
97	C₂×C₄₇	C₂×#368	1	2	yes	—
98	D₄₇	—	2	4	no	—
99	C₂³×C₈	—	47	1	no	—
100	C₂²×C₁₆	—	47	1	no	—
101	C₂²×C₁₆	—	47	4	no	—
102	C₂²×C₁₆	—	94	4	no	—
103	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
104	C₂×#108	—	47	2	no	—
105	C₂×#108	C₂×D₄₇	1	2	yes	—
106	#407	—	94	2	no	—
107	#368	—	94	4	no	—
108	#341	—	94	2	no	—
109	C₄₇	#12837	1	1	yes	characteristic, complemented, Sylow
110	C₂³×C₄	—	47	1	no	—
111	C₂×M₁₆	—	2	2	no	—
112	C₂×M₁₆	—	94	2	no	—
113	C₂²×C₈	—	47	1	no	—
114	C₂²×C₈	—	47	2	no	—
115	C₂²×C₈	—	47	2	no	—
116	C₂²×C₈	—	94	2	no	—
117	C₂²×C₈	—	94	2	no	—
118	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
119	C₂×C₁₆	—	2	2	no	—
120	C₂×C₁₆	—	47	2	no	—
121	C₂×C₁₆	—	47	4	no	—
122	C₂×C₁₆	—	94	8	no	—
123	C₂×C₁₆	C₄×D₄₇	1	2	yes	—
124	#108	—	2	2	no	—
125	#108	—	2	2	no	—
126	#108	—	94	2	no	—
127	#108	—	94	2	no	—
128	C₂²⋊C₈	—	188	4	no	—
129	C₂⁴	—	47	1	no	—
130	C₂²×C₄	—	47	1	no	—
131	C₂²×C₄	—	47	2	no	—
132	C₂²×C₄	—	47	2	no	—
133	C₂²×C₄	—	94	2	no	—
134	C₂²×C₄	—	94	2	no	—
135	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
136	M₁₆	—	4	2	no	—
137	M₁₆	—	4	2	no	—
138	M₁₆	—	188	2	no	—
139	M₁₆	—	188	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	47	2	no	—
142	C₂×C₈	—	94	2	no	—
143	C₂×C₈	—	94	2	no	—
144	C₂×C₈	—	94	2	no	—
145	C₂×C₈	—	94	4	no	—
146	C₂×C₈	—	188	2	no	—
147	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
148	C₂×C₈	#10625350	1	2	yes	—
149	C₂×C₈	D₄×D₄₇	1	2	yes	—
150	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
151	C₁₆	—	2	2	no	—
152	C₁₆	—	47	4	no	—
153	C₂³	—	47	2	no	—
154	C₂³	—	94	4	no	—
155	C₂³	—	94	2	no	—
156	C₂³	Q₁₆×D₄₇	1	1	yes	characteristic
157	C₂×C₄	—	47	2	no	—
158	C₂×C₄	—	94	2	no	—
159	C₂×C₄	—	94	2	no	—
160	C₂×C₄	—	94	4	no	—
161	C₂×C₄	—	94	2	no	—
162	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
163	C₂×C₄	#13093997	1	2	yes	—
164	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
165	C₂×C₄	#13094045	1	2	yes	—
166	C₈	—	8	2	no	—
167	C₈	—	94	2	no	—
168	C₈	—	376	2	no	—
169	C₈	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
170	C₈	#13093981	1	1	yes	characteristic
171	C₈	#13094002	1	2	yes	—
172	C₂²	—	2	2	no	—
173	C₂²	—	47	4	no	—
174	C₂²	—	94	8	no	—
175	C₂²	—	94	2	no	—
176	C₂²	—	94	2	no	—
177	C₂²	#107×D₄₇	1	1	yes	characteristic
178	C₂²	#15689363	1	2	yes	—
179	C₄	—	94	2	no	—
180	C₄	#15688982	1	1	yes	characteristic
181	C₄	#15689158	1	1	yes	characteristic
182	C₄	#15689358	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	—	94	4	no	—
185	C₂	#20243416	1	1	yes	characteristic
186	C₂	#368×D₄₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₁₆×D₄₇

C₂

yes

maximal, characteristic

C₂×#15688952

C₂

yes

maximal, complemented

#20243423

C₂

yes

maximal

#20241482

C₂

yes

maximal, characteristic, complemented

C₄₇×#1381

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₄₇

C₂²

yes

characteristic

C₂²×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₂×C₁₆×D₄₇

—

C₂×C₁₆×D₄₇

C₄

yes

—

C₂×#13093960

C₂²

yes

—

#15689086

—

#15688952

—

#15688887

—

C₂²×C₁₆×C₄₇

C₂²

yes

characteristic

C₂×#108×C₄₇

C₂²

yes

—

C₂²×C₄×D₄₇

D₄

yes

characteristic

C₂×#10625329

—

C₂²×C₄₇⋊C₈

D₄

yes

characteristic

C₂×C₈×D₄₇

—

C₂×C₈×D₄₇

—

C₂×C₈×D₄₇

C₂×C₄

yes

—

C₂×C₈×D₄₇

Q₈

yes

—

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

—

C₂×C₄₇⋊C₁₆

D₄

yes

—

C₁₆×D₄₇

—

#13093960

—

#13093960

—

(C₂×D₄₇)⋊C₈

—

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

C₂×C₄

yes

—

#108×C₄₇

—

#108×C₄₇

—

C₂³×D₄₇

Q₁₆

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×Dic₄₇

Q₁₆

yes

characteristic

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

C₄⋊C₄

yes

—

C₂×C₄×D₄₇

D₈

yes

—

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

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

C₈×D₄₇

—

C₈×D₄₇

—

C₈×D₄₇

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

C₂²⋊C₄

yes

—

C₂³×C₄₇

C₂×Q₁₆

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

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

—

C₂×Dic₄₇

—

C₂×Dic₄₇

#107

yes

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₄₇

#117

yes

characteristic

C₈×C₄₇

C₂²⋊Q₈

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#12837

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

C₂×#107

yes

characteristic

C₂²×C₄₇

#467

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

C₂³×C₁₆

—

—

—

—

D₄₇

yes

characteristic, complemented

C₂×C₄₇

—

C₂×C₄₇

#1808

yes

characteristic

C₂×C₄₇

C₂×#368

yes

—

D₄₇

—

C₂³×C₈

—

100

C₂²×C₁₆

—

101

C₂²×C₁₆

—

102

C₂²×C₁₆

—

103

C₂²×C₁₆

C₂×D₄₇

yes

characteristic

104

C₂×#108

—

105

C₂×#108

C₂×D₄₇

yes

—

106

#407

—

107

#368

—

108

#341

—

109

C₄₇

#12837

yes

characteristic, complemented, Sylow

110

C₂³×C₄

—

111

C₂×M₁₆

—

112

C₂×M₁₆

—

113

C₂²×C₈

—

114

C₂²×C₈

—

115

C₂²×C₈

—

116

C₂²×C₈

—

117

C₂²×C₈

—

118

C₂²×C₈

C₂²×D₄₇

yes

characteristic

119

C₂×C₁₆

—

120

C₂×C₁₆

—

121

C₂×C₁₆

—

122

C₂×C₁₆

—

123

C₂×C₁₆

C₄×D₄₇

yes

—

124

#108

—

125

#108

—

126

#108

—

127

#108

—

128

C₂²⋊C₈

—

188

—

129

C₂⁴

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

D₄×D₄₇

yes

characteristic

136

M₁₆

—

137

M₁₆

—

138

M₁₆

—

188

—

139

M₁₆

—

188

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

188

—

147

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

148

C₂×C₈

#10625350

yes

—

149

C₂×C₈

D₄×D₄₇

yes

—

150

C₂×C₈

Q₈×D₄₇

yes

characteristic

151

C₁₆

—

152

C₁₆

—

153

C₂³

—

154

C₂³

—

155

C₂³

—

156

C₂³

Q₁₆×D₄₇

yes

characteristic

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

C₄⋊C₄×D₄₇

yes

characteristic

163

C₂×C₄

#13093997

yes

—

164

C₂×C₄

D₈×D₄₇

yes

characteristic

165

C₂×C₄

#13094045

yes

—

166

C₈

—

167

C₈

—

168

C₈

—

376

—

169

C₈

C₂²⋊C₄×D₄₇

yes

characteristic

170

C₈

#13093981

yes

characteristic

171

C₈

#13094002

yes

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

—

176

C₂²

—

177

C₂²

#107×D₄₇

yes

characteristic

178

C₂²

#15689363

yes

—

179

C₄

—

180

C₄

#15688982

yes

characteristic

181

C₄

#15689158

yes

characteristic

182

C₄

#15689358

yes

—

183

C₂

—

184

C₂

—

185

C₂

#20243416

yes

characteristic

186

C₂

#368×D₄₇

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₄₇
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1381	C₂
socle	C₂²×C₄₇	C₂×#107
prime radical	C₂³×D₄₇	Q₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1381	D₄₇
hypocenter	C₄₇	#12837
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂×C₄

D₈×D₄₇

commutator

C₂×C₈×C₄₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#1381

C₂

socle

C₂²×C₄₇

C₂×#107

prime radical

C₂³×D₄₇

Q₁₆

cosocle

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
2. C₂×#15688952	C₂	semidirect
4. #20241482	C₂	semidirect
5. C₄₇×#1381	C₂	semidirect
85. C₂×D₄₇	#368	semidirect
94. #1381	D₄₇	semidirect
109. C₄₇	#12837	semidirect

H ≅ G/N

type

2. C₂×#15688952

C₂

semidirect

4. #20241482

C₂

semidirect

5. C₄₇×#1381

C₂

semidirect

85. C₂×D₄₇

#368

semidirect

94. #1381

D₄₇

semidirect

109. C₄₇

#12837

semidirect

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

subgroup	index
C₂²×C₁₆×D₄₇	2
C₂×#15688952	2
#20243423	2
#20241482	2
C₄₇×#1381	2
#12837	47

subgroup

index

C₂²×C₁₆×D₄₇

C₄₇×#1381

Subgroups — #26132458

Conjugacy classes of subgroups

Complements

Maximal subgroups