Group #16015496 — subgroups

Subgroups — #16015496

Order 3328.

862 conjugacy classes of subgroups; only the 299 normal subgroups are shown.

normal subgroups only

This group has 862 conjugacy classes of subgroups — too many to draw. Only its 299 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×#11718739	C₂	1	2	yes	maximal
2	C₂×#11718341	C₂	1	1	yes	maximal, characteristic
3	#13123213	C₂	1	2	yes	maximal
4	#13114997	C₂	1	8	yes	maximal
5	C₄×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#498	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×#88123	C₂²	1	2	yes	—
9	C₁₃×#498	—	2	4	no	—
10	C₂×C₁₃×#106	C₂²	1	4	yes	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	#11718891	—	2	4	no	—
13	C₂×#88057	C₂²	1	4	yes	—
14	C₂×C₈×Dic₁₃	C₂²	1	4	yes	—
15	#11718739	—	2	8	no	—
16	#11718638	C₂²	1	8	yes	—
17	#11718635	—	2	4	no	—
18	C₂×#88047	C₂²	1	8	yes	—
19	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
21	#11718341	—	2	4	no	—
22	C₂²×Q₈×C₁₃	—	2	2	no	—
23	C₂²×Dic₂₆	—	2	2	no	—
24	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
25	C₁₃×C₄⋊Q₈	—	2	4	no	—
26	C₁₃×C₄⋊Q₈	—	2	4	no	—
27	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
28	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
29	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊Dic₁₃	—	2	2	no	—
31	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
32	C₂×C₄×Dic₁₃	D₄	1	4	yes	—
33	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
34	C₂²×#1055	C₂³	1	2	yes	—
35	#88123	—	2	4	no	—
36	#88123	—	2	4	no	—
37	C₁₃×#106	—	2	8	no	—
38	C₄×C₈×C₁₃	D₄	1	4	yes	—
39	#88071	—	2	16	no	—
40	#88057	—	2	8	no	—
41	C₈×Dic₁₃	Q₈	1	16	yes	complemented
42	#88047	—	2	16	no	—
43	C₄×#1055	D₄	1	4	yes	—
44	#33313	—	13	1	no	maximal, Sylow
45	C₂×Q₈×C₁₃	—	4	4	no	—
46	C₂×Q₈×C₁₃	—	4	8	no	—
47	C₂²×C₄×C₁₃	—	2	4	no	—
48	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
50	C₂²×Dic₁₃	—	2	4	no	—
51	C₂²×Dic₁₃	C₂×D₄	1	4	yes	—
52	C₂×Dic₂₆	—	4	4	no	—
53	C₂×Dic₂₆	—	4	8	no	—
54	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
55	C₂×C₈×C₁₃	C₂×Q₈	1	8	yes	—
56	C₁₃×C₄⋊C₄	—	2	4	no	—
57	C₁₃×C₄⋊C₄	—	2	4	no	—
58	C₁₃×C₄⋊C₄	—	4	4	no	—
59	C₄²×C₁₃	C₂×D₄	1	2	yes	—
60	C₄²×C₁₃	C₂×D₄	1	2	yes	—
61	C₄⋊Dic₁₃	—	2	4	no	—
62	C₄⋊Dic₁₃	—	2	4	no	—
63	C₄⋊Dic₁₃	—	4	4	no	—
64	C₄×Dic₁₃	—	2	8	no	—
65	C₄×Dic₁₃	C₂×Q₈	1	8	yes	—
66	C₂×#1055	C₂×D₄	1	4	yes	—
67	C₂×#1055	C₂×Q₈	1	8	yes	—
68	C₂×#500	—	13	2	no	—
69	C₂×#498	—	13	1	no	—
70	C₂×#498	D₁₃	1	1	yes	characteristic
71	#2862	—	13	2	no	—
72	#1846	—	13	8	no	—
73	C₄²×C₈	—	13	1	no	—
74	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
75	Q₈×C₁₃	—	8	8	no	—
76	C₂×C₄×C₁₃	—	4	4	no	—
77	C₂×C₄×C₁₃	—	4	8	no	—
78	C₂×C₄×C₁₃	#127	1	4	yes	—
79	C₂×C₄×C₁₃	#135	1	8	yes	—
80	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂²×Q₈	1	4	yes	—
83	C₂×Dic₁₃	—	2	8	no	—
84	C₂×Dic₁₃	—	4	4	no	—
85	C₂×Dic₁₃	—	4	8	no	—
86	C₂×Dic₁₃	C₄⋊Q₈	1	8	yes	—
87	Dic₂₆	—	8	8	no	—
88	C₈×C₁₃	C₄⋊Q₈	1	8	yes	—
89	#1055	C₄⋊Q₈	1	8	yes	—
90	C₂×C₄⋊Q₈	—	13	2	no	—
91	C₂×C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
92	#500	—	26	8	no	—
93	#498	—	2	4	no	—
94	#498	—	26	4	no	—
95	C₂×#107	—	13	8	no	—
96	C₂×#106	—	13	4	no	—
97	C₂×#106	C₂×D₁₃	1	4	yes	—
98	C₂×C₄×C₈	—	13	4	no	—
99	C₂×C₄×C₈	—	13	1	no	—
100	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
101	#391	—	26	4	no	—
102	#391	—	26	4	no	—
103	#382	—	13	8	no	—
104	C₄³	—	13	1	no	—
105	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
106	C₂²×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
107	C₂²×C₁₃	C₂×#135	1	4	yes	—
108	C₄×C₁₃	—	8	4	no	—
109	C₄×C₁₃	#495	1	8	yes	—
110	C₄×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
111	C₄×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
112	Dic₁₃	—	2	8	no	—
113	Dic₁₃	—	8	4	no	—
114	C₂²×Q₈	—	2	2	no	—
115	C₂²×Q₈	—	26	2	no	—
116	C₂²×C₈	—	13	2	no	—
117	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
118	C₄⋊Q₈	—	2	4	no	—
119	C₄⋊Q₈	—	2	4	no	—
120	C₄⋊Q₈	—	26	4	no	—
121	C₄⋊Q₈	—	26	4	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	13	4	no	—
124	C₂×C₄⋊C₄	—	26	2	no	—
125	C₂×C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
126	C₂×C₄²	—	13	2	no	—
127	C₂×C₄²	—	13	4	no	—
128	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
129	#107	—	26	16	no	—
130	#106	—	2	8	no	—
131	#106	—	26	8	no	—
132	C₄×C₈	—	13	16	no	—
133	C₄×C₈	—	13	4	no	—
134	C₄×C₈	C₂²⋊D₁₃	1	4	yes	—
135	#95	—	26	16	no	—
136	C₂×C₁₃	#2868	1	1	yes	characteristic
137	C₂×C₁₃	C₂×#495	1	2	yes	—
138	C₂×C₁₃	#3160	1	4	yes	—
139	C₂×Q₈	—	4	8	no	—
140	C₂×Q₈	—	4	4	no	—
141	C₂×Q₈	—	52	8	no	—
142	C₂×Q₈	—	52	4	no	—
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	—	13	4	no	—
145	C₂²×C₄	—	26	4	no	—
146	C₂²×C₄	D₄×D₁₃	1	2	yes	—
147	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
148	C₂×C₈	—	13	4	no	—
149	C₂×C₈	—	13	8	no	—
150	C₂×C₈	Q₈×D₁₃	1	8	yes	—
151	C₂×C₈	C₂×C₂²⋊D₁₃	1	4	yes	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	4	4	no	—
155	C₄⋊C₄	—	26	4	no	—
156	C₄⋊C₄	—	26	4	no	—
157	C₄⋊C₄	—	52	4	no	—
158	C₄²	—	13	8	no	—
159	C₄²	—	26	8	no	—
160	C₄²	D₄×D₁₃	1	2	yes	—
161	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
162	C₁₃	#33313	1	1	yes	characteristic, complemented, Sylow
163	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
164	Q₈	—	8	8	no	—
165	Q₈	—	104	8	no	—
166	C₂×C₄	—	4	8	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	—	13	8	no	—
169	C₂×C₄	—	26	8	no	—
170	C₂×C₄	—	52	8	no	—
171	C₂×C₄	—	52	4	no	—
172	C₂×C₄	#88170	1	8	yes	—
173	C₂×C₄	#88194	1	4	yes	—
174	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
175	C₂×C₄	C₂×Q₈×D₁₃	1	4	yes	—
176	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
177	C₈	—	13	8	no	—
178	C₈	#88198	1	8	yes	—
179	C₂²	D₁₃×C₄⋊Q₈	1	2	yes	—
180	C₂²	C₂×#88170	1	4	yes	—
181	C₂²	C₂×#88194	1	1	yes	characteristic
182	C₄	—	8	4	no	—
183	C₄	—	26	8	no	—
184	C₄	—	104	4	no	—
185	C₄	#11718829	1	8	yes	—
186	C₄	D₁₃×C₄⋊Q₈	1	2	yes	—
187	C₄	C₂×#88198	1	2	yes	—
188	C₂	#13123232	1	1	yes	characteristic
189	C₂	C₂×#11718829	1	2	yes	—
190	C₂	#13125345	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718739

C₂

yes

maximal

C₂×#11718341

C₂

yes

maximal, characteristic

#13123213

C₂

yes

maximal

#13114997

C₂

yes

maximal

C₄×C₈×Dic₁₃

C₂

yes

maximal, characteristic

C₂×C₁₃×#498

C₂

yes

maximal, characteristic

C₂×C₁₃×C₄⋊Q₈

C₂²

yes

—

C₂×#88123

C₂²

yes

—

C₁₃×#498

—

C₂×C₁₃×#106

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

#11718891

—

C₂×#88057

C₂²

yes

—

C₂×C₈×Dic₁₃

C₂²

yes

—

#11718739

—

#11718638

C₂²

yes

—

#11718635

—

C₂×#88047

C₂²

yes

—

C₄²×Dic₁₃

C₂²

yes

characteristic

C₂×C₄×#1055

C₂²

yes

characteristic

#11718341

—

C₂²×Q₈×C₁₃

—

C₂²×Dic₂₆

—

C₂²×C₈×C₁₃

C₂³

yes

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

C₂³

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

C₂³

yes

—

C₂×C₄×Dic₁₃

D₄

yes

—

C₂×C₄×Dic₁₃

C₂³

yes

—

C₂²×#1055

C₂³

yes

—

#88123

—

#88123

—

C₁₃×#106

—

C₄×C₈×C₁₃

D₄

yes

—

#88071

—

#88057

—

C₈×Dic₁₃

Q₈

yes

complemented

#88047

—

C₄×#1055

D₄

yes

—

#33313

—

maximal, Sylow

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

C₂⁴

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

C₂×D₄

yes

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₂×C₈×C₁₃

C₂×Q₈

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂×D₄

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

C₂×Q₈

yes

—

C₂×#1055

C₂×D₄

yes

—

C₂×#1055

C₂×Q₈

yes

—

C₂×#500

—

C₂×#498

—

C₂×#498

D₁₃

yes

characteristic

#2862

—

#1846

—

C₄²×C₈

—

C₂³×C₁₃

C₂²×D₄

yes

characteristic

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#127

yes

—

C₂×C₄×C₁₃

#135

yes

—

C₂×C₄×C₁₃

C₂²×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×Q₈

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₄⋊Q₈

yes

—

Dic₂₆

—

C₈×C₁₃

C₄⋊Q₈

yes

—

#1055

C₄⋊Q₈

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

C₂×D₁₃

yes

—

—

—

—

—

—

C₂×D₁₃

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

100

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

101

#391

—

102

#391

—

103

#382

—

104

C₄³

—

105

C₂²×C₁₃

C₂×#127

yes

characteristic

106

C₂²×C₁₃

C₂×C₄⋊Q₈

yes

—

107

C₂²×C₁₃

C₂×#135

yes

—

108

C₄×C₁₃

—

109

C₄×C₁₃

#495

yes

—

110

C₄×C₁₃

C₂×C₄⋊Q₈

yes

—

111

C₄×C₁₃

C₂×C₄⋊Q₈

yes

—

112

Dic₁₃

—

113

Dic₁₃

—

114

C₂²×Q₈

—

115

C₂²×Q₈

—

116

C₂²×C₈

—

117

C₂²×C₈

C₂²×D₁₃

yes

—

118

C₄⋊Q₈

—

119

C₄⋊Q₈

—

120

C₄⋊Q₈

—

121

C₄⋊Q₈

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

C₂²×D₁₃

yes

—

126

C₂×C₄²

—

127

C₂×C₄²

—

128

C₂×C₄²

C₂²×D₁₃

yes

characteristic

129

#107

—

130

#106

—

131

#106

—

132

C₄×C₈

—

133

C₄×C₈

—

134

C₄×C₈

C₂²⋊D₁₃

yes

—

135

#95

—

136

C₂×C₁₃

#2868

yes

characteristic

137

C₂×C₁₃

C₂×#495

yes

—

138

C₂×C₁₃

#3160

yes

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

D₄×D₁₃

yes

—

147

C₂²×C₄

C₂³×D₁₃

yes

characteristic

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

Q₈×D₁₃

yes

—

151

C₂×C₈

C₂×C₂²⋊D₁₃

yes

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄²

—

159

C₄²

—

160

C₄²

D₄×D₁₃

yes

—

161

C₄²

C₂×C₂²⋊D₁₃

yes

—

162

C₁₃

#33313

yes

characteristic, complemented, Sylow

163

C₂³

C₂×D₄×D₁₃

yes

characteristic

164

Q₈

—

165

Q₈

—

104

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

#88170

yes

—

173

C₂×C₄

#88194

yes

—

174

C₂×C₄

C₂×D₄×D₁₃

yes

characteristic

175

C₂×C₄

C₂×Q₈×D₁₃

yes

—

176

C₂×C₄

C₂²×C₂²⋊D₁₃

yes

characteristic

177

C₈

—

178

C₈

#88198

yes

—

179

C₂²

D₁₃×C₄⋊Q₈

yes

—

180

C₂²

C₂×#88170

yes

—

181

C₂²

C₂×#88194

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

104

—

185

C₄

#11718829

yes

—

186

C₄

D₁₃×C₄⋊Q₈

yes

—

187

C₄

C₂×#88198

yes

—

188

C₂

#13123232

yes

characteristic

189

C₂

C₂×#11718829

yes

—

190

C₂

#13125345

yes

—

	subgroup	G/H
center	C₂³	C₂×D₄×D₁₃
commutator, cosocle	C₂²×C₄×C₁₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#498	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
hypercenter	C₂×#498	D₁₃
hypocenter	C₁₃	#33313
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

subgroup

G/H

center

C₂³

C₂×D₄×D₁₃

commutator, cosocle

C₂²×C₄×C₁₃

C₂⁴

Frattini

C₂²×C₄

C₂³×D₁₃

Fitting

C₂×C₁₃×#498

C₂

socle, prime radical

C₂³×C₁₃

C₂²×D₄

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

N	H ≅ G/N	type
41. C₈×Dic₁₃	Q₈	semidirect
162. C₁₃	#33313	semidirect

H ≅ G/N

type

41. C₈×Dic₁₃

Q₈

semidirect

162. C₁₃

#33313

semidirect

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

subgroup	index
C₂×#11718739	2
C₂×#11718341	2
#13123213	2
#13114997	2
C₄×C₈×Dic₁₃	2
C₂×C₁₃×#498	2
#33313	13

subgroup

index

C₄×C₈×Dic₁₃

C₂×C₁₃×#498

#33313

Subgroups — #16015496

Conjugacy classes of subgroups

Complements

Maximal subgroups