Group #15761543 — subgroups

Subgroups — #15761543

Order 3328.

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

normal subgroups only

This group has 294 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	C₄×#88059	C₂	1	1	yes	maximal, characteristic
2	#13113586	C₂	1	2	yes	maximal
3	#13113128	C₂	1	1	yes	maximal, characteristic
4	#13112686	C₂	1	2	yes	maximal
5	C₂×C₁₃×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88059	C₄	1	2	yes	—
9	#11718586	C₂²	1	1	yes	characteristic
10	#11718586	C₂²	1	1	yes	characteristic
11	C₄×#6099	C₂²	1	2	yes	—
12	C₁₃×#335	—	2	2	no	—
13	#11718144	C₄	1	1	yes	characteristic
14	#11718144	C₄	1	1	yes	characteristic
15	#11718072	—	2	2	no	—
16	#11718071	—	2	2	no	—
17	#11718046	—	2	2	no	—
18	C₂²×C₈×C₁₃	—	2	2	no	—
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×#6099	C₂×C₄	1	2	yes	—
23	C₂×#6099	C₂×C₄	1	2	yes	—
24	C₁₃×C₄⋊C₈	—	2	2	no	—
25	C₁₃×C₄⋊C₈	—	2	2	no	—
26	C₄×C₈×C₁₃	D₄	1	2	yes	—
27	C₄×C₈×C₁₃	Q₈	1	2	yes	—
28	C₂²⋊#1055	—	2	1	no	—
29	C₂²⋊#1055	Q₈	1	2	yes	—
30	#88059	—	2	4	no	—
31	C₄⋊#1055	—	2	1	no	—
32	C₄⋊#1055	D₄	1	1	yes	characteristic
33	C₄⋊#1055	D₄	1	1	yes	characteristic
34	#88043	—	2	2	no	—
35	C₄×#1055	—	2	1	no	—
36	C₄×#1055	—	2	1	no	—
37	C₄×#1055	—	2	2	no	—
38	#9610	—	13	1	no	maximal, Sylow
39	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
42	C₂×C₈×C₁₃	—	2	2	no	—
43	C₂×C₈×C₁₃	—	2	2	no	—
44	C₂×C₈×C₁₃	—	4	2	no	—
45	C₂×C₈×C₁₃	—	8	2	no	—
46	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
48	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
51	C₄²×C₁₃	#41	1	1	yes	characteristic
52	#6099	—	2	2	no	—
53	#6099	—	2	2	no	—
54	#6099	—	2	2	no	—
55	#6099	M₁₆	1	4	yes	—
56	C₂×#1055	—	2	2	no	—
57	C₂×#1055	—	2	2	no	—
58	C₂×#1055	—	2	1	no	—
59	C₂×#1055	—	2	1	no	—
60	C₂×#1055	—	2	2	no	—
61	C₄×#108	—	13	1	no	—
62	#1565	—	13	2	no	—
63	C₂×#335	D₁₃	1	1	yes	characteristic
64	#1452	—	13	1	no	—
65	#1283	—	13	2	no	—
66	C₂³×C₁₃	C₄×Q₈	1	1	yes	characteristic
67	C₂×C₄×C₁₃	—	2	1	no	—
68	C₂×C₄×C₁₃	—	2	1	no	—
69	C₂×C₄×C₁₃	—	4	1	no	—
70	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
73	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₁₃	#125	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
77	C₂×C₄×C₁₃	#131	1	2	yes	—
78	C₈×C₁₃	—	2	2	no	—
79	C₈×C₁₃	—	2	2	no	—
80	C₈×C₁₃	—	8	2	no	—
81	#1055	—	2	4	no	—
82	#1055	—	4	2	no	—
83	#1055	—	4	2	no	—
84	#1055	—	4	1	no	—
85	#1055	—	4	1	no	—
86	#431	—	13	1	no	—
87	#431	—	13	1	no	—
88	C₂×#108	—	13	2	no	—
89	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
90	C₄×M₁₆	—	13	2	no	—
91	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
92	#341	—	13	1	no	—
93	#341	—	13	1	no	—
94	#335	—	2	2	no	—
95	#335	—	26	2	no	—
96	#334	—	26	2	no	—
97	#322	—	26	2	no	—
98	C₂²×C₁₃	—	2	1	no	—
99	C₂²×C₁₃	—	2	1	no	—
100	C₂²×C₁₃	#389	1	1	yes	characteristic
101	C₂²×C₁₃	#446	1	2	yes	—
102	C₄×C₁₃	—	2	1	no	—
103	C₄×C₁₃	—	2	1	no	—
104	C₄×C₁₃	—	2	1	no	—
105	C₄×C₁₃	#423	1	1	yes	characteristic
106	C₄×C₁₃	#424	1	1	yes	characteristic
107	C₄×C₁₃	#435	1	2	yes	—
108	C₄×C₁₃	#472	1	1	yes	characteristic
109	C₄×C₁₃	#473	1	1	yes	characteristic
110	C₂×M₁₆	—	13	2	no	—
111	C₂×M₁₆	—	13	2	no	—
112	C₂²×C₈	—	2	2	no	—
113	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
114	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
115	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
116	#108	—	26	4	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	13	1	no	—
120	C₄⋊C₈	—	13	1	no	—
121	C₄⋊C₈	—	26	1	no	—
122	C₂²⋊C₈	—	13	2	no	—
123	C₂²⋊C₈	—	26	1	no	—
124	#97	—	26	2	no	—
125	C₄×C₈	—	26	2	no	—
126	C₄×C₈	—	26	1	no	—
127	C₄×C₈	—	26	1	no	—
128	C₄×C₈	Dic₂₆	1	2	yes	—
129	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
130	C₂×C₁₃	—	2	2	no	—
131	C₂×C₁₃	#1940	1	1	yes	characteristic
132	C₂×C₁₃	#1952	1	1	yes	characteristic
133	C₂×C₁₃	#1953	1	1	yes	characteristic
134	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
135	C₂²×C₄	#6129	1	1	yes	characteristic
136	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
137	M₁₆	—	13	4	no	—
138	M₁₆	—	26	2	no	—
139	M₁₆	—	26	2	no	—
140	M₁₆	—	26	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	4	2	no	—
144	C₂×C₈	—	8	2	no	—
145	C₂×C₈	—	26	2	no	—
146	C₂×C₈	—	26	2	no	—
147	C₂×C₈	—	26	1	no	—
148	C₂×C₈	—	26	1	no	—
149	C₂×C₈	—	26	2	no	—
150	C₂×C₈	#6101	1	2	yes	—
151	C₂×C₈	#6101	1	2	yes	—
152	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
153	C₄²	D₄×D₁₃	1	1	yes	characteristic
154	C₄²	#6131	1	1	yes	characteristic
155	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
156	C₁₃	#9610	1	1	yes	characteristic, complemented, Sylow
157	C₂³	#88141	1	1	yes	characteristic
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	#88142	1	1	yes	characteristic
162	C₂×C₄	#88144	1	1	yes	characteristic
163	C₂×C₄	#88147	1	1	yes	characteristic
164	C₂×C₄	#88155	1	2	yes	—
165	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
166	C₂×C₄	C₂×#6101	1	1	yes	characteristic
167	C₂×C₄	#88192	1	1	yes	characteristic
168	C₂×C₄	#88200	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	—	2	2	no	—
171	C₈	—	8	2	no	—
172	C₈	—	26	4	no	—
173	C₈	—	52	2	no	—
174	C₈	—	52	2	no	—
175	C₈	—	52	1	no	—
176	C₈	—	52	1	no	—
177	C₂²	—	2	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	#11718489	1	2	yes	—
180	C₂²	#11718634	1	1	yes	characteristic
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	#11718497	1	2	yes	—
185	C₄	#11718763	1	1	yes	characteristic
186	C₄	#11718779	1	1	yes	characteristic
187	C₄	#11718821	1	1	yes	characteristic
188	C₄	#11718853	1	1	yes	characteristic
189	C₂	—	2	2	no	—
190	C₂	#13115557	1	1	yes	characteristic
191	C₂	#13115619	1	1	yes	characteristic
192	C₂	#13115624	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#88059

C₂

yes

maximal, characteristic

#13113586

C₂

yes

maximal

#13113128

C₂

yes

maximal, characteristic

#13112686

C₂

yes

maximal

C₂×C₁₃×#335

C₂

yes

maximal, characteristic

C₂×C₁₃×C₄⋊C₈

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×#88059

C₄

yes

—

#11718586

C₂²

yes

characteristic

#11718586

C₂²

yes

characteristic

C₄×#6099

C₂²

yes

—

C₁₃×#335

—

#11718144

C₄

yes

characteristic

#11718144

C₄

yes

—

—

—

C₂²×C₈×C₁₃

—

C₂²×C₈×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×#6099

C₂×C₄

yes

—

C₂×#6099

C₂×C₄

yes

—

C₁₃×C₄⋊C₈

—

C₁₃×C₄⋊C₈

—

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

Q₈

yes

—

C₂²⋊#1055

—

C₂²⋊#1055

Q₈

yes

—

#88059

—

C₄⋊#1055

—

C₄⋊#1055

D₄

yes

characteristic

C₄⋊#1055

D₄

yes

—

—

—

—

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₁₃

#41

yes

characteristic

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

C₂×Q₈

yes

C₄²×C₁₃

yes

—

—

—

M₁₆

yes

—

—

—

—

—

—

—

—

D₁₃

yes

characteristic

#1452

—

#1283

—

C₂³×C₁₃

C₄×Q₈

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₄×D₄

yes

characteristic

C₂×C₄×C₁₃

#121

yes

characteristic

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

#125

yes

characteristic

C₂×C₄×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

C₂×M₁₆

yes

—

C₂×C₄×C₁₃

#131

yes

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

—

—

—

—

—

—

—

—

C₂×C₄⋊C₈

C₂×D₁₃

yes

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₁₃

yes

—

—

—

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

—

100

C₂²×C₁₃

#389

yes

characteristic

101

C₂²×C₁₃

#446

yes

—

102

C₄×C₁₃

—

103

C₄×C₁₃

—

104

C₄×C₁₃

—

105

C₄×C₁₃

#423

yes

characteristic

106

C₄×C₁₃

#424

yes

characteristic

107

C₄×C₁₃

#435

yes

—

108

C₄×C₁₃

#472

yes

characteristic

109

C₄×C₁₃

#473

yes

characteristic

110

C₂×M₁₆

—

111

C₂×M₁₆

—

112

C₂²×C₈

—

113

C₂²×C₈

C₄×D₁₃

yes

characteristic

114

C₂²×C₈

C₄×D₁₃

yes

characteristic

115

C₂×C₄²

C₂²×D₁₃

yes

characteristic

116

#108

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

121

C₄⋊C₈

—

122

C₂²⋊C₈

—

123

C₂²⋊C₈

—

124

#97

—

125

C₄×C₈

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₄×C₈

Dic₂₆

yes

—

129

C₄×C₈

C₂²⋊D₁₃

yes

—

130

C₂×C₁₃

—

131

C₂×C₁₃

#1940

yes

characteristic

132

C₂×C₁₃

#1952

yes

characteristic

133

C₂×C₁₃

#1953

yes

characteristic

134

C₂²×C₄

C₂×C₄×D₁₃

yes

characteristic

135

C₂²×C₄

#6129

yes

characteristic

136

C₂²×C₄

Q₈×D₁₃

yes

characteristic

137

M₁₆

—

138

M₁₆

—

139

M₁₆

—

140

M₁₆

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

#6101

yes

—

151

C₂×C₈

#6101

yes

—

152

C₄²

C₂×Dic₂₆

yes

characteristic

153

C₄²

D₄×D₁₃

yes

characteristic

154

C₄²

#6131

yes

characteristic

155

C₄²

C₂×C₂²⋊D₁₃

yes

characteristic

156

C₁₃

#9610

yes

characteristic, complemented, Sylow

157

C₂³

#88141

yes

characteristic

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

#88142

yes

characteristic

162

C₂×C₄

#88144

yes

characteristic

163

C₂×C₄

#88147

yes

characteristic

164

C₂×C₄

#88155

yes

—

165

C₂×C₄

M₁₆×D₁₃

yes

—

166

C₂×C₄

C₂×#6101

yes

characteristic

167

C₂×C₄

#88192

yes

characteristic

168

C₂×C₄

#88200

yes

characteristic

169

C₈

—

170

C₈

—

171

C₈

—

172

C₈

—

173

C₈

—

174

C₈

—

175

C₈

—

176

C₈

—

177

C₂²

—

178

C₂²

—

179

C₂²

#11718489

yes

—

180

C₂²

#11718634

yes

characteristic

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

#11718497

yes

—

185

C₄

#11718763

yes

characteristic

186

C₄

#11718779

yes

characteristic

187

C₄

#11718821

yes

characteristic

188

C₄

#11718853

yes

characteristic

189

C₂

—

190

C₂

#13115557

yes

characteristic

191

C₂

#13115619

yes

characteristic

192

C₂

#13115624

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#88192
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#335	C₂
socle	C₂²×C₁₃	#389
prime radical	C₂³×C₁₃	C₄×Q₈
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#335	D₁₃
hypocenter	C₁₃	#9610
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#335

subgroup

G/H

center

C₂×C₄

#88192

commutator

C₂²×C₄×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₂×C₁₃×#335

C₂

socle

C₂²×C₁₃

#389

prime radical

C₂³×C₁₃

C₄×Q₈

cosocle

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#335

N	H ≅ G/N	type
156. C₁₃	#9610	semidirect

H ≅ G/N

type

156. C₁₃

#9610

semidirect

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

subgroup	index
C₄×#88059	2
#13113586	2
#13113128	2
#13112686	2
C₂×C₁₃×#335	2
#9610	13

subgroup

index

C₂×C₁₃×#335

Subgroups — #15761543

Conjugacy classes of subgroups

Complements

Maximal subgroups