Group #15762756 — subgroups

Subgroups — #15762756

Order 3328.

710 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 191 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×#11718356	C₂	1	1	yes	maximal, characteristic
2	#13115641	C₂	1	4	yes	maximal, complemented
3	C₂×#11718047	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1844	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₁₀₄	—	2	2	no	—
6	C₂×#88128	C₂²	1	2	yes	—
7	C₂×C₁₃×#106	—	2	2	no	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₁₃×#384	C₂²	1	2	yes	—
10	C₂×#88049	—	2	8	no	—
11	C₂×C₄×#1055	C₂²	1	2	yes	—
12	#11718356	—	2	2	no	—
13	#11718356	C₄	1	4	yes	complemented
14	#11718047	—	2	2	no	—
15	C₂²×D₅₂	—	2	2	no	—
16	C₂²×D₅₂	—	2	4	no	—
17	C₂²×C₈×C₁₃	D₄	1	2	yes	—
18	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
19	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
20	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
21	C₂²×#1055	D₄	1	4	yes	—
22	C₂×D₁₀₄	—	2	8	no	—
23	C₂×D₁₀₄	—	4	4	no	—
24	C₂×D₁₀₄	—	4	4	no	—
25	#88128	—	2	2	no	—
26	#88128	—	2	4	no	—
27	#88128	C₂×C₄	1	4	yes	—
28	C₁₃×#106	—	4	4	no	—
29	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₁₃	D₄	1	2	yes	—
31	#88049	—	4	16	no	—
32	C₄×#1055	—	2	2	no	—
33	C₄×#1055	—	2	2	no	—
34	#9837	—	13	1	no	maximal, Sylow
35	C₂³×D₁₃	—	4	2	no	—
36	C₂²×C₄×C₁₃	—	4	2	no	—
37	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
38	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
39	C₂×D₅₂	—	2	4	no	—
40	C₂×D₅₂	—	2	8	no	—
41	C₂×D₅₂	—	4	4	no	—
42	C₂×D₅₂	—	4	4	no	—
43	C₂×D₅₂	—	4	8	no	—
44	C₂×D₅₂	—	4	8	no	—
45	C₂×D₅₂	—	4	4	no	—
46	C₂×D₅₂	—	4	2	no	—
47	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
48	C₂×C₈×C₁₃	D₈	1	4	yes	—
49	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
50	C₁₃×C₄⋊C₄	—	4	4	no	—
51	C₁₃×C₄⋊C₄	—	4	8	no	—
52	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₁₃	#41	1	2	yes	—
55	C₂×#1055	—	2	4	no	—
56	C₂×#1055	—	2	8	no	—
57	D₁₀₄	—	4	16	no	—
58	C₂×#493	—	13	1	no	—
59	#1957	—	13	4	no	—
60	#1844	D₁₃	1	1	yes	characteristic, complemented
61	C₂×#322	—	13	1	no	—
62	C₂³×C₁₃	#127	1	1	yes	characteristic
63	C₂²×D₁₃	—	4	4	no	—
64	C₂²×D₁₃	—	8	2	no	—
65	C₂²×D₁₃	—	8	2	no	—
66	C₂²×D₁₃	—	8	4	no	—
67	C₂²×D₁₃	—	8	4	no	—
68	C₂×C₄×C₁₃	—	8	2	no	—
69	C₂×C₄×C₁₃	—	8	4	no	—
70	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
72	C₂×C₄×C₁₃	#121	1	2	yes	—
73	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
75	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
76	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	8	yes	—
77	D₅₂	—	4	8	no	—
78	D₅₂	—	4	16	no	—
79	D₅₂	—	8	8	no	—
80	D₅₂	—	8	4	no	—
81	C₈×C₁₃	#102	1	8	yes	complemented
82	#1055	—	4	4	no	—
83	C₂²×D₈	—	26	2	no	—
84	C₂×#127	—	13	2	no	—
85	#493	—	13	4	no	—
86	#493	—	26	2	no	—
87	C₂×#106	—	2	2	no	—
88	C₂×#102	—	26	8	no	—
89	C₂×C₄×C₈	—	13	2	no	—
90	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
91	#384	C₂×D₁₃	1	2	yes	—
92	#322	—	26	2	no	—
93	C₂²×C₁₃	#390	1	1	yes	characteristic
94	C₂²×C₁₃	#492	1	1	yes	characteristic
95	C₂²×C₁₃	#493	1	1	yes	characteristic
96	C₂²×C₁₃	#496	1	4	yes	—
97	C₂×D₁₃	—	8	4	no	—
98	C₂×D₁₃	—	8	8	no	—
99	C₂×D₁₃	—	16	2	no	—
100	C₂×D₁₃	—	16	2	no	—
101	C₂×D₁₃	—	16	4	no	—
102	C₄×C₁₃	—	16	2	no	—
103	C₄×C₁₃	C₂×#102	1	2	yes	—
104	C₄×C₁₃	#442	1	4	yes	—
105	C₄×C₁₃	#469	1	4	yes	—
106	C₄×C₁₃	#486	1	2	yes	—
107	C₂²×D₄	—	26	4	no	—
108	C₂²×D₄	—	26	2	no	—
109	C₂×D₈	—	26	8	no	—
110	C₂×D₈	—	52	4	no	—
111	C₂×D₈	—	52	4	no	—
112	C₂²×C₈	—	13	4	no	—
113	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
114	#127	—	13	4	no	—
115	#127	—	26	2	no	—
116	#127	—	26	4	no	—
117	C₂×C₄⋊C₄	—	2	4	no	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
120	#106	—	4	4	no	—
121	#102	—	52	16	no	—
122	C₄×C₈	—	26	2	no	—
123	C₄×C₈	—	26	2	no	—
124	C₄×C₈	C₄×D₁₃	1	2	yes	—
125	C₄×C₈	D₅₂	1	2	yes	—
126	C₂×C₁₃	#1715	1	2	yes	—
127	C₂×C₁₃	#1721	1	2	yes	—
128	C₂×C₁₃	#1957	1	1	yes	characteristic
129	C₂×C₁₃	#1972	1	2	yes	—
130	D₁₃	—	16	4	no	—
131	C₂⁴	—	52	2	no	—
132	C₂×D₄	—	26	8	no	—
133	C₂×D₄	—	26	4	no	—
134	C₂×D₄	—	52	4	no	—
135	C₂×D₄	—	52	8	no	—
136	C₂×D₄	—	52	8	no	—
137	C₂×D₄	—	52	4	no	—
138	C₂×D₄	—	52	4	no	—
139	C₂×D₄	—	52	2	no	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	D₄×D₁₃	1	2	yes	—
142	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
143	D₈	—	52	16	no	—
144	C₂×C₈	—	26	4	no	—
145	C₂×C₈	—	26	8	no	—
146	C₂×C₈	#6103	1	4	yes	—
147	C₂×C₈	#6104	1	4	yes	—
148	C₂×C₈	#6106	1	4	yes	—
149	C₄⋊C₄	—	4	8	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
152	C₄²	C₂×D₅₂	1	1	yes	characteristic
153	C₄²	#6131	1	2	yes	—
154	C₁₃	#9837	1	1	yes	characteristic, complemented, Sylow
155	C₂³	—	52	4	no	—
156	C₂³	—	104	2	no	—
157	C₂³	—	104	2	no	—
158	C₂³	—	104	4	no	—
159	C₂³	—	104	4	no	—
160	C₂³	#88194	1	1	yes	characteristic
161	D₄	—	52	16	no	—
162	D₄	—	52	8	no	—
163	D₄	—	104	4	no	—
164	D₄	—	104	8	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	8	4	no	—
167	C₂×C₄	#88147	1	2	yes	—
168	C₂×C₄	#88149	1	2	yes	—
169	C₂×C₄	#88168	1	8	yes	—
170	C₂×C₄	C₂×#6103	1	1	yes	characteristic
171	C₂×C₄	C₂×#6104	1	2	yes	—
172	C₂×C₄	C₂×#6106	1	2	yes	—
173	C₂×C₄	#88201	1	1	yes	characteristic
174	C₈	—	52	4	no	—
175	C₈	#88049	1	8	yes	complemented
176	C₂²	—	104	8	no	—
177	C₂²	—	104	4	no	—
178	C₂²	—	208	2	no	—
179	C₂²	—	208	2	no	—
180	C₂²	—	208	4	no	—
181	C₂²	#11718649	1	1	yes	characteristic
182	C₂²	#11718734	1	1	yes	characteristic
183	C₂²	#11718746	1	1	yes	characteristic
184	C₂²	#11718837	1	4	yes	—
185	C₄	—	16	2	no	—
186	C₄	#11718526	1	4	yes	—
187	C₄	#11718530	1	4	yes	—
188	C₄	C₂×#88049	1	2	yes	—
189	C₄	#11718749	1	2	yes	—
190	C₂	—	208	4	no	—
191	C₂	#13114465	1	2	yes	—
192	C₂	#13114500	1	2	yes	—
193	C₂	#13115641	1	1	yes	characteristic
194	C₂	#13115710	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718356

C₂

yes

maximal, characteristic

#13115641

C₂

yes

maximal, complemented

C₂×#11718047

C₂

yes

maximal, characteristic, complemented

C₁₃×#1844

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₀₄

—

C₂×#88128

C₂²

yes

—

C₂×C₁₃×#106

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₁₃×#384

C₂²

yes

—

C₂×#88049

—

C₂×C₄×#1055

C₂²

yes

—

#11718356

—

#11718356

C₄

yes

complemented

#11718047

—

C₂²×D₅₂

—

C₂²×D₅₂

—

C₂²×C₈×C₁₃

D₄

yes

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂²×#1055

D₄

yes

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

#88128

—

#88128

—

#88128

C₂×C₄

yes

—

C₁₃×#106

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

D₄

yes

—

—

—

—

—

C₂³×D₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×C₈×C₁₃

C₂²⋊C₄

yes

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

SD₁₆

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

#41

yes

—

C₂×#1055

—

C₂×#1055

—

D₁₀₄

—

C₂×#493

—

#1957

—

#1844

D₁₃

yes

characteristic, complemented

C₂×#322

—

C₂³×C₁₃

#127

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₄×D₄

yes

—

C₂×C₄×C₁₃

#121

yes

—

C₂×C₄×C₁₃

#124

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

—

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

D₅₂

—

D₅₂

—

D₅₂

—

D₅₂

—

C₈×C₁₃

#102

yes

complemented

#1055

—

C₂²×D₈

—

—

—

—

—

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

#384

C₂×D₁₃

yes

—

#322

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

100

C₂×D₁₃

—

101

C₂×D₁₃

—

102

C₄×C₁₃

—

103

C₄×C₁₃

C₂×#102

yes

—

104

C₄×C₁₃

#442

yes

—

105

C₄×C₁₃

#469

yes

—

106

C₄×C₁₃

#486

yes

—

107

C₂²×D₄

—

108

C₂²×D₄

—

109

C₂×D₈

—

110

C₂×D₈

—

111

C₂×D₈

—

112

C₂²×C₈

—

113

C₂²×C₈

C₂²⋊D₁₃

yes

—

114

#127

—

115

#127

—

116

#127

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄²

C₂²×D₁₃

yes

characteristic

120

#106

—

121

#102

—

122

C₄×C₈

—

123

C₄×C₈

—

124

C₄×C₈

C₄×D₁₃

yes

—

125

C₄×C₈

D₅₂

yes

—

126

C₂×C₁₃

#1715

yes

—

127

C₂×C₁₃

#1721

yes

—

128

C₂×C₁₃

#1957

yes

characteristic

129

C₂×C₁₃

#1972

yes

—

130

D₁₃

—

131

C₂⁴

—

132

C₂×D₄

—

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

—

136

C₂×D₄

—

137

C₂×D₄

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂²×C₄

—

141

C₂²×C₄

D₄×D₁₃

yes

—

142

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

143

D₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

#6103

yes

—

147

C₂×C₈

#6104

yes

—

148

C₂×C₈

#6106

yes

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄²

C₂×C₄×D₁₃

yes

characteristic

152

C₄²

C₂×D₅₂

yes

characteristic

153

C₄²

#6131

yes

—

154

C₁₃

#9837

yes

characteristic, complemented, Sylow

155

C₂³

—

156

C₂³

—

104

—

157

C₂³

—

104

—

158

C₂³

—

104

—

159

C₂³

—

104

—

160

C₂³

#88194

yes

characteristic

161

D₄

—

162

D₄

—

163

D₄

—

104

—

164

D₄

—

104

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

#88147

yes

—

168

C₂×C₄

#88149

yes

—

169

C₂×C₄

#88168

yes

—

170

C₂×C₄

C₂×#6103

yes

characteristic

171

C₂×C₄

C₂×#6104

yes

—

172

C₂×C₄

C₂×#6106

yes

—

173

C₂×C₄

#88201

yes

characteristic

174

C₈

—

175

C₈

#88049

yes

complemented

176

C₂²

—

104

—

177

C₂²

—

104

—

178

C₂²

—

208

—

179

C₂²

—

208

—

180

C₂²

—

208

—

181

C₂²

#11718649

yes

characteristic

182

C₂²

#11718734

yes

characteristic

183

C₂²

#11718746

yes

characteristic

184

C₂²

#11718837

yes

—

185

C₄

—

186

C₄

#11718526

yes

—

187

C₄

#11718530

yes

—

188

C₄

C₂×#88049

yes

—

189

C₄

#11718749

yes

—

190

C₂

—

208

—

191

C₂

#13114465

yes

—

192

C₂

#13114500

yes

—

193

C₂

#13115641

yes

characteristic

194

C₂

#13115710

yes

—

	subgroup	G/H
center	C₂³	#88194
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1844	C₂
socle	C₂³×C₁₃	#127
prime radical	C₂×#11718356	C₂
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1844	D₁₃
hypocenter	C₁₃	#9837
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

subgroup

G/H

center

C₂³

#88194

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1844

C₂

C₂³×C₁₃

C₂

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

N	H ≅ G/N	type
2. #13115641	C₂	semidirect
3. C₂×#11718047	C₂	semidirect
4. C₁₃×#1844	C₂	semidirect
13. #11718356	C₄	semidirect
60. #1844	D₁₃	semidirect
81. C₈×C₁₃	#102	semidirect
154. C₁₃	#9837	semidirect
175. C₈	#88049	semidirect

H ≅ G/N

type

2. #13115641

C₂

semidirect

3. C₂×#11718047

C₂

semidirect

4. C₁₃×#1844

C₂

semidirect

13. #11718356

C₄

semidirect

60. #1844

D₁₃

semidirect

81. C₈×C₁₃

#102

semidirect

154. C₁₃

#9837

semidirect

175. C₈

#88049

semidirect

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

subgroup	index
C₂×#11718356	2
#13115641	2
C₂×#11718047	2
C₁₃×#1844	2
#9837	13

subgroup

index

C₁₃×#1844

Subgroups — #15762756

Conjugacy classes of subgroups

Complements

Maximal subgroups