Group #26136170 — subgroups

Subgroups — #26136170

Order 12032.

292 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 83 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	#20245183	C₂	1	1	yes	maximal, characteristic, complemented
2	#20244998	C₂	1	1	yes	maximal, characteristic
3	#20243423	C₂	1	2	yes	maximal
4	#20241501	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1385	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₄₇	C₂²	1	1	yes	characteristic
7	C₂×#13093964	C₂²	1	1	yes	characteristic
8	C₂×#13093960	C₂²	1	2	yes	—
9	#15689086	—	2	2	no	—
10	#15688953	—	2	2	no	—
11	(C₂×D₄₇)⋊C₁₆	—	2	2	no	—
12	#15688887	—	2	2	no	—
13	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×#108×C₄₇	C₂²	1	2	yes	—
15	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
16	C₂×#10625329	—	2	2	no	—
17	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
18	C₂×C₈×D₄₇	—	2	1	no	—
19	C₂×C₈×D₄₇	—	2	2	no	—
20	C₂×C₈×D₄₇	—	2	2	no	—
21	C₂×C₈×D₄₇	—	2	1	no	—
22	#13093964	—	2	2	no	—
23	C₂×C₄₇⋊C₁₆	—	2	1	no	—
24	#13093960	—	2	2	no	—
25	#13093960	—	2	2	no	—
26	(C₂×D₄₇)⋊C₈	—	4	4	no	—
27	C₂×M₁₆×C₄₇	—	2	2	no	—
28	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
29	M₃₂×C₄₇	—	2	2	no	—
30	C₂×C₁₆×C₄₇	D₄	1	2	yes	—
31	#108×C₄₇	—	2	2	no	—
32	#108×C₄₇	—	2	2	no	—
33	C₂³×D₄₇	Q₁₆	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×Dic₄₇	Q₁₆	1	1	yes	characteristic
36	C₂×C₄×D₄₇	—	2	1	no	—
37	C₂×C₄×D₄₇	—	2	2	no	—
38	C₂×C₄×D₄₇	—	2	1	no	—
39	C₂×C₄×D₄₇	—	2	2	no	—
40	M₁₆×C₄₇	—	4	2	no	—
41	M₁₆×C₄₇	—	4	2	no	—
42	C₂×C₈×C₄₇	—	4	2	no	—
43	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₄₇	#41	1	2	yes	—
45	C₂×C₈×C₄₇	#41	1	2	yes	—
46	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
47	#10625329	—	4	2	no	—
48	#10625329	—	4	2	no	—
49	C₂×C₄₇⋊C₈	—	2	1	no	—
50	C₂×C₄₇⋊C₈	—	2	2	no	—
51	C₂×C₄₇⋊C₈	—	4	2	no	—
52	C₈×D₄₇	—	4	1	no	—
53	C₈×D₄₇	—	4	1	no	—
54	C₈×D₄₇	—	4	2	no	—
55	C₁₆×C₄₇	—	2	2	no	—
56	C₄₇⋊C₁₆	—	4	1	no	—
57	C₂³×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
58	C₂²×D₄₇	—	2	1	no	—
59	C₂²×D₄₇	—	2	2	no	—
60	C₂²×D₄₇	—	4	2	no	—
61	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
62	C₂×C₄×C₄₇	#123	1	2	yes	—
63	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
64	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
65	C₂×Dic₄₇	—	2	1	no	—
66	C₂×Dic₄₇	—	2	2	no	—
67	C₄×D₄₇	—	4	1	no	—
68	C₄×D₄₇	—	4	1	no	—
69	C₄×D₄₇	—	4	2	no	—
70	C₈×C₄₇	—	8	2	no	—
71	C₈×C₄₇	#123	1	1	yes	characteristic
72	C₈×C₄₇	#124	1	1	yes	characteristic
73	C₈×C₄₇	#126	1	2	yes	—
74	C₄₇⋊C₈	—	4	1	no	—
75	C₄₇⋊C₈	—	8	2	no	—
76	#13498	—	47	1	no	maximal, Sylow
77	C₂²×C₄₇	—	4	1	no	—
78	C₂²×C₄₇	#467	1	1	yes	characteristic
79	C₂²×C₄₇	#484	1	2	yes	—
80	C₂×D₄₇	—	2	2	no	—
81	C₂×D₄₇	—	2	4	no	—
82	C₂×D₄₇	—	4	1	no	—
83	C₂×D₄₇	—	4	2	no	—
84	C₂×D₄₇	—	4	1	no	—
85	C₄×C₄₇	#397	1	1	yes	characteristic
86	C₄×C₄₇	#469	1	1	yes	characteristic
87	C₄×C₄₇	#482	1	2	yes	—
88	Dic₄₇	—	4	1	no	—
89	#2161	—	47	1	no	—
90	#2112	—	47	1	no	—
91	#1810	—	47	2	no	—
92	#1388	—	47	2	no	—
93	#1385	D₄₇	1	1	yes	characteristic, complemented
94	C₂×C₄₇	—	2	1	no	—
95	C₂×C₄₇	—	2	2	no	—
96	C₂×C₄₇	#2076	1	1	yes	characteristic
97	D₄₇	—	4	2	no	—
98	C₂³×C₈	—	47	1	no	—
99	C₂×M₃₂	—	47	1	no	—
100	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
101	C₂×#108	—	47	2	no	—
102	C₂×#108	C₂×D₄₇	1	2	yes	—
103	#407	—	94	2	no	—
104	#368	—	94	2	no	—
105	C₂²⋊C₁₆	—	94	2	no	—
106	#341	—	94	2	no	—
107	C₄₇	#13498	1	1	yes	characteristic, complemented, Sylow
108	C₂³×C₄	—	47	1	no	—
109	C₂×M₁₆	—	2	2	no	—
110	C₂×M₁₆	—	94	2	no	—
111	C₂²×C₈	—	47	1	no	—
112	C₂²×C₈	—	94	1	no	—
113	C₂²×C₈	—	94	2	no	—
114	C₂²×C₈	—	94	2	no	—
115	C₂²×C₈	—	94	1	no	—
116	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
117	M₃₂	—	2	2	no	—
118	M₃₂	—	94	2	no	—
119	C₂×C₁₆	—	94	1	no	—
120	C₂×C₁₆	D₁₈₈	1	2	yes	—
121	#108	—	2	2	no	—
122	#108	—	2	2	no	—
123	#108	—	94	2	no	—
124	#108	—	94	2	no	—
125	C₂²⋊C₈	—	188	4	no	—
126	C₂⁴	—	47	1	no	—
127	C₂²×C₄	—	47	1	no	—
128	C₂²×C₄	—	94	1	no	—
129	C₂²×C₄	—	94	2	no	—
130	C₂²×C₄	—	94	1	no	—
131	C₂²×C₄	—	94	2	no	—
132	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
133	M₁₆	—	4	2	no	—
134	M₁₆	—	4	2	no	—
135	M₁₆	—	188	2	no	—
136	M₁₆	—	188	2	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	94	1	no	—
139	C₂×C₈	—	94	2	no	—
140	C₂×C₈	—	188	2	no	—
141	C₂×C₈	—	188	1	no	—
142	C₂×C₈	—	188	1	no	—
143	C₂×C₈	—	188	2	no	—
144	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
145	C₂×C₈	#10625350	1	2	yes	—
146	C₂×C₈	#10625352	1	2	yes	—
147	C₂×C₈	#10625354	1	1	yes	characteristic
148	C₁₆	—	2	2	no	—
149	C₁₆	—	188	1	no	—
150	C₂³	—	94	1	no	—
151	C₂³	—	94	2	no	—
152	C₂³	—	188	2	no	—
153	C₂³	Q₁₆×D₄₇	1	1	yes	characteristic
154	C₂×C₄	—	94	1	no	—
155	C₂×C₄	—	94	2	no	—
156	C₂×C₄	—	188	1	no	—
157	C₂×C₄	—	188	1	no	—
158	C₂×C₄	—	188	2	no	—
159	C₂×C₄	#13093999	1	1	yes	characteristic
160	C₂×C₄	#13094006	1	2	yes	—
161	C₂×C₄	#13094043	1	1	yes	characteristic
162	C₂×C₄	#13094051	1	2	yes	—
163	C₈	—	8	2	no	—
164	C₈	—	376	2	no	—
165	C₈	—	188	1	no	—
166	C₈	#13094008	1	1	yes	characteristic
167	C₈	#13094012	1	1	yes	characteristic
168	C₈	#13094019	1	2	yes	—
169	C₂²	—	4	1	no	—
170	C₂²	—	94	2	no	—
171	C₂²	—	94	4	no	—
172	C₂²	—	188	1	no	—
173	C₂²	—	188	2	no	—
174	C₂²	—	188	1	no	—
175	C₂²	#15689365	1	1	yes	characteristic
176	C₂²	#15689457	1	2	yes	—
177	C₄	—	188	1	no	—
178	C₄	#15689050	1	1	yes	characteristic
179	C₄	#15689377	1	1	yes	characteristic
180	C₄	#15689445	1	2	yes	—
181	C₂	—	2	1	no	—
182	C₂	—	2	2	no	—
183	C₂	—	188	2	no	—
184	C₂	#20244818	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245183

C₂

yes

maximal, characteristic, complemented

#20244998

C₂

yes

maximal, characteristic

#20243423

C₂

yes

maximal

#20241501

C₂

yes

maximal, complemented

C₄₇×#1385

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₄₇

C₂²

yes

characteristic

C₂×#13093964

C₂²

yes

characteristic

C₂×#13093960

C₂²

yes

—

#15689086

—

#15688953

—

(C₂×D₄₇)⋊C₁₆

—

#15688887

—

C₂×M₃₂×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×C₈×D₄₇

—

#13093964

—

C₂×C₄₇⋊C₁₆

—

#13093960

—

#13093960

—

(C₂×D₄₇)⋊C₈

—

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

C₂³

yes

characteristic

M₃₂×C₄₇

—

C₂×C₁₆×C₄₇

D₄

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

—

C₂×C₄×D₄₇

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

#41

yes

—

C₂×C₈×C₄₇

#41

yes

—

C₂×C₈×C₄₇

yes

—

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₈×D₄₇

—

C₈×D₄₇

—

C₈×D₄₇

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

C₂×Q₁₆

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

#121

yes

characteristic

C₂×C₄×C₄₇

#123

yes

—

C₂×C₄×C₄₇

#135

yes

characteristic

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

—

C₈×C₄₇

yes

C₈×C₄₇

yes

C₈×C₄₇

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#13498

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#467

yes

characteristic

C₂²×C₄₇

#484

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

—

—

—

—

D₄₇

yes

characteristic, complemented

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#2076

yes

characteristic

D₄₇

—

C₂³×C₈

—

C₂×M₃₂

—

100

C₂×M₃₂

C₂×D₄₇

yes

characteristic

101

C₂×#108

—

102

C₂×#108

C₂×D₄₇

yes

—

103

#407

—

104

#368

—

105

C₂²⋊C₁₆

—

106

#341

—

107

C₄₇

#13498

yes

characteristic, complemented, Sylow

108

C₂³×C₄

—

109

C₂×M₁₆

—

110

C₂×M₁₆

—

111

C₂²×C₈

—

112

C₂²×C₈

—

113

C₂²×C₈

—

114

C₂²×C₈

—

115

C₂²×C₈

—

116

C₂²×C₈

C₂²×D₄₇

yes

characteristic

117

M₃₂

—

118

M₃₂

—

119

C₂×C₁₆

—

120

C₂×C₁₆

D₁₈₈

yes

—

121

#108

—

122

#108

—

123

#108

—

124

#108

—

125

C₂²⋊C₈

—

188

—

126

C₂⁴

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

D₄×D₄₇

yes

characteristic

133

M₁₆

—

134

M₁₆

—

135

M₁₆

—

188

—

136

M₁₆

—

188

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

188

—

141

C₂×C₈

—

188

—

142

C₂×C₈

—

188

—

143

C₂×C₈

—

188

—

144

C₂×C₈

C₂×D₁₈₈

yes

characteristic

145

C₂×C₈

#10625350

yes

—

146

C₂×C₈

#10625352

yes

—

147

C₂×C₈

#10625354

yes

characteristic

148

C₁₆

—

149

C₁₆

—

188

—

150

C₂³

—

151

C₂³

—

152

C₂³

—

188

—

153

C₂³

Q₁₆×D₄₇

yes

characteristic

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

188

—

157

C₂×C₄

—

188

—

158

C₂×C₄

—

188

—

159

C₂×C₄

#13093999

yes

characteristic

160

C₂×C₄

#13094006

yes

—

161

C₂×C₄

#13094043

yes

characteristic

162

C₂×C₄

#13094051

yes

—

163

C₈

—

164

C₈

—

376

—

165

C₈

—

188

—

166

C₈

#13094008

yes

characteristic

167

C₈

#13094012

yes

characteristic

168

C₈

#13094019

yes

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

188

—

173

C₂²

—

188

—

174

C₂²

—

188

—

175

C₂²

#15689365

yes

characteristic

176

C₂²

#15689457

yes

—

177

C₄

—

188

—

178

C₄

#15689050

yes

characteristic

179

C₄

#15689377

yes

characteristic

180

C₄

#15689445

yes

—

181

C₂

—

182

C₂

—

183

C₂

—

188

—

184

C₂

#20244818

yes

characteristic

	subgroup	G/H
center	C₄	#15689377
commutator, cosocle	C₂²×C₈×C₄₇	C₂³
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1385	C₂
socle	C₂×C₄₇	#2076
prime radical	C₂³×D₄₇	Q₁₆
hypercenter	#1385	D₄₇
hypocenter	C₄₇	#13498
LCS	G ▷ C₂²×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

subgroup

G/H

center

C₄

#15689377

commutator, cosocle

C₂²×C₈×C₄₇

C₂³

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#1385

C₂

socle

C₂×C₄₇

#2076

prime radical

C₂³×D₄₇

Q₁₆

D₄₇

C₄₇

G ▷ C₂²×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

N	H ≅ G/N	type
1. #20245183	C₂	semidirect
4. #20241501	C₂	semidirect
5. C₄₇×#1385	C₂	semidirect
93. #1385	D₄₇	semidirect
107. C₄₇	#13498	semidirect

H ≅ G/N

type

1. #20245183

C₂

semidirect

4. #20241501

C₂

semidirect

5. C₄₇×#1385

C₂

semidirect

93. #1385

D₄₇

semidirect

107. C₄₇

#13498

semidirect

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

subgroup	index
#20245183	2
#20244998	2
#20243423	2
#20241501	2
C₄₇×#1385	2
#13498	47

subgroup

index

C₄₇×#1385

Subgroups — #26136170

Conjugacy classes of subgroups

Complements

Maximal subgroups