Group #16018238 — subgroups

Subgroups — C₂×#13117083 #16018238

Order 3328.

582 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 203 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×#11718360	C₂	1	1	yes	maximal, characteristic
2	#13117083	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718084	C₂	1	4	yes	maximal
4	C₂×#11718053	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#499	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#125	—	2	1	no	—
7	C₂²×Dic₅₂	—	2	1	no	—
8	C₂²×Dic₅₂	C₂²	1	2	yes	—
9	C₂×#88123	—	2	1	no	—
10	C₁₃×#499	—	2	2	no	—
11	C₁₃×#499	C₂²	1	4	yes	—
12	C₂×C₁₃×#107	C₂²	1	2	yes	—
13	C₂×C₁₃×#106	—	2	1	no	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂²×#6090	C₂²	1	2	yes	—
16	#11718360	—	2	2	no	—
17	#11718360	C₂²	1	4	yes	—
18	#11718084	C₂²	1	16	yes	—
19	#11718053	C₂²	1	4	yes	—
20	C₂²×Dic₂₆	—	2	2	no	—
21	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
23	C₁₃×#125	—	2	4	no	—
24	C₁₃×#125	—	4	2	no	—
25	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
26	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
27	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₃	—	4	2	no	—
29	C₂×Dic₅₂	—	2	8	no	—
30	C₂×Dic₅₂	—	2	4	no	—
31	C₂×Dic₅₂	—	2	2	no	—
32	C₂×Dic₅₂	—	4	4	no	—
33	C₂×Dic₅₂	C₂³	1	4	yes	—
34	#88123	—	2	4	no	—
35	#88123	—	4	2	no	—
36	C₁₃×#107	—	2	2	no	—
37	C₁₃×#107	C₂³	1	4	yes	—
38	C₁₃×#106	—	2	4	no	—
39	C₄×C₈×C₁₃	D₄	1	2	yes	—
40	C₄×C₈×C₁₃	C₂³	1	2	yes	—
41	C₂×#6090	—	2	4	no	—
42	C₂×#6090	C₂³	1	4	yes	—
43	C₂×#2245	—	13	1	no	maximal, Sylow
44	C₂²×C₄×C₁₃	—	4	2	no	—
45	C₂²×C₄×C₁₃	D₈	1	2	yes	—
46	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×Dic₁₃	—	4	2	no	—
48	C₂×Dic₂₆	—	2	4	no	—
49	C₂×Dic₂₆	—	4	4	no	—
50	C₂×Dic₂₆	—	4	4	no	—
51	C₂×Dic₂₆	—	4	2	no	—
52	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
53	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
55	C₂×C₈×C₁₃	#41	1	4	yes	—
56	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
57	C₁₃×C₄⋊C₄	—	2	4	no	—
58	C₁₃×C₄⋊C₄	—	4	2	no	—
59	C₁₃×C₄⋊C₄	—	4	8	no	—
60	C₄²×C₁₃	D₈	1	2	yes	—
61	C₄²×C₁₃	C₂×D₄	1	2	yes	—
62	C₄⋊Dic₁₃	—	4	8	no	—
63	Dic₅₂	—	2	8	no	—
64	Dic₅₂	—	4	4	no	—
65	Dic₅₂	—	4	8	no	—
66	#6090	—	2	4	no	—
67	C₂×#499	D₁₃	1	1	yes	characteristic
68	C₂×#494	—	13	1	no	—
69	#2245	—	13	8	no	—
70	C₂×#358	—	13	4	no	—
71	C₂×#346	—	13	1	no	—
72	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₁₃	—	2	4	no	—
74	C₂×C₄×C₁₃	—	4	4	no	—
75	C₂×C₄×C₁₃	—	8	4	no	—
76	C₂×C₄×C₁₃	#124	1	2	yes	—
77	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
78	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
80	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
81	C₂×Dic₁₃	—	4	4	no	—
82	C₂×Dic₁₃	—	8	4	no	—
83	Dic₂₆	—	4	4	no	—
84	Dic₂₆	—	4	4	no	—
85	Dic₂₆	—	8	4	no	—
86	C₈×C₁₃	#124	1	2	yes	—
87	C₈×C₁₃	C₂×SD₁₆	1	4	yes	—
88	C₈×C₁₃	C₂×#41	1	2	yes	—
89	C₂²×Q₁₆	—	13	2	no	—
90	C₂²×Q₁₆	—	26	1	no	—
91	C₂×C₄⋊Q₈	—	26	1	no	—
92	C₂×#125	—	2	1	no	—
93	C₂²×C₁₆	—	13	2	no	—
94	#499	—	2	2	no	—
95	#499	C₂×D₁₃	1	4	yes	—
96	#494	—	13	4	no	—
97	#494	—	26	2	no	—
98	C₂×#107	C₂×D₁₃	1	2	yes	—
99	C₂×#106	—	2	1	no	—
100	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
101	#358	—	13	16	no	—
102	#346	—	13	4	no	—
103	C₂²×C₁₃	#486	1	2	yes	—
104	C₂²×C₁₃	#510	1	4	yes	—
105	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
106	C₄×C₁₃	—	2	4	no	—
107	C₄×C₁₃	—	8	4	no	—
108	C₄×C₁₃	#486	1	2	yes	—
109	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
110	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
111	Dic₁₃	—	8	4	no	—
112	C₂²×Q₈	—	26	2	no	—
113	C₂×Q₁₆	—	13	4	no	—
114	C₂×Q₁₆	—	26	8	no	—
115	C₂×Q₁₆	—	26	2	no	—
116	C₂×Q₁₆	—	26	4	no	—
117	C₂×Q₁₆	—	52	4	no	—
118	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
120	C₄⋊Q₈	—	26	4	no	—
121	C₄⋊Q₈	—	52	2	no	—
122	#125	—	2	4	no	—
123	#125	—	4	2	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄⋊C₄	—	4	2	no	—
126	C₂×C₄⋊C₄	—	52	2	no	—
127	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
128	C₂×C₁₆	—	13	4	no	—
129	C₂×C₁₆	—	26	4	no	—
130	#107	—	2	2	no	—
131	#107	C₂²×D₁₃	1	4	yes	—
132	#106	—	2	4	no	—
133	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
134	C₄×C₈	C₂²×D₁₃	1	2	yes	—
135	C₂×C₁₃	#2245	1	4	yes	complemented
136	C₂×C₁₃	C₂×#486	1	1	yes	characteristic
137	C₂×C₁₃	C₂×#510	1	2	yes	—
138	C₂×Q₈	—	26	4	no	—
139	C₂×Q₈	—	52	2	no	—
140	C₂×Q₈	—	52	4	no	—
141	C₂×Q₈	—	52	4	no	—
142	C₂²×C₄	—	4	2	no	—
143	C₂²×C₄	—	52	2	no	—
144	C₂²×C₄	#6104	1	2	yes	—
145	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
146	Q₁₆	—	26	8	no	—
147	Q₁₆	—	52	8	no	—
148	Q₁₆	—	52	4	no	—
149	C₂×C₈	#6106	1	4	yes	—
150	C₂×C₈	#6131	1	4	yes	—
151	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
152	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
153	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	4	8	no	—
157	C₄⋊C₄	—	52	8	no	—
158	C₄²	#6104	1	2	yes	—
159	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
160	C₁₆	—	26	4	no	—
161	C₁₃	C₂×#2245	1	1	yes	characteristic, complemented, Sylow
162	C₂³	C₂×#6104	1	1	yes	characteristic
163	Q₈	—	52	4	no	—
164	Q₈	—	52	4	no	—
165	Q₈	—	104	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	—	8	4	no	—
169	C₂×C₄	—	52	4	no	—
170	C₂×C₄	—	104	4	no	—
171	C₂×C₄	C₂×#6104	1	4	yes	—
172	C₂×C₄	C₂×#6104	1	1	yes	characteristic
173	C₂×C₄	C₂×#6106	1	2	yes	—
174	C₂×C₄	#88201	1	2	yes	—
175	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
176	C₈	C₂×#6106	1	4	yes	—
177	C₈	#88201	1	2	yes	—
178	C₈	C₂×#6131	1	2	yes	—
179	C₂²	#11718749	1	2	yes	—
180	C₂²	#11718856	1	4	yes	—
181	C₂²	C₂²×#6104	1	1	yes	characteristic
182	C₄	—	2	4	no	—
183	C₄	—	8	4	no	—
184	C₄	—	104	4	no	—
185	C₄	#11718749	1	2	yes	—
186	C₄	C₂²×#6106	1	1	yes	characteristic
187	C₄	C₂×#88201	1	1	yes	characteristic
188	C₂	#13117083	1	4	yes	direct product factor
189	C₂	C₂×#11718749	1	1	yes	characteristic
190	C₂	C₂×#11718856	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718360

C₂

yes

maximal, characteristic

#13117083

C₂

yes

maximal, direct product factor

C₂×#11718084

C₂

yes

maximal

C₂×#11718053

C₂

yes

maximal, characteristic

C₂×C₁₃×#499

C₂

yes

maximal, characteristic

C₂×C₁₃×#125

—

C₂²×Dic₅₂

—

C₂²×Dic₅₂

C₂²

yes

—

C₂×#88123

—

C₁₃×#499

—

C₁₃×#499

C₂²

yes

—

C₂×C₁₃×#107

C₂²

yes

—

C₂×C₁₃×#106

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂²×#6090

C₂²

yes

—

#11718360

—

#11718360

C₂²

yes

—

#11718084

C₂²

yes

—

#11718053

C₂²

yes

—

C₂²×Dic₂₆

—

C₂²×C₈×C₁₃

D₄

yes

characteristic

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₁₃×#125

—

C₁₃×#125

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₄²×C₁₃

D₄

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×Dic₅₂

—

C₂×Dic₅₂

—

C₂×Dic₅₂

—

C₂×Dic₅₂

—

C₂×Dic₅₂

C₂³

yes

—

#88123

—

#88123

—

C₁₃×#107

—

C₁₃×#107

C₂³

yes

—

C₁₃×#106

—

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

C₂³

yes

—

C₂×#6090

—

C₂×#6090

C₂³

yes

—

C₂×#2245

—

maximal, Sylow

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

D₈

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

SD₁₆

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₂×C₈×C₁₃

#41

yes

—

C₂×C₈×C₁₃

C₂⁴

yes

characteristic

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

D₈

yes

—

C₄²×C₁₃

C₂×D₄

yes

—

C₄⋊Dic₁₃

—

Dic₅₂

—

Dic₅₂

—

Dic₅₂

—

#6090

—

C₂×#499

D₁₃

yes

—

—

—

—

C₂³×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#124

yes

—

C₂×C₄×C₁₃

C₂×D₈

yes

—

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₂×C₄×C₁₃

C₂²×D₄

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

#124

yes

—

C₈×C₁₃

C₂×SD₁₆

yes

—

C₈×C₁₃

C₂×#41

yes

—

C₂²×Q₁₆

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×#125

—

C₂²×C₁₆

—

#499

—

#499

C₂×D₁₃

yes

—

#494

—

#494

—

C₂×#107

C₂×D₁₃

yes

—

C₂×#106

—

100

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

101

#358

—

102

#346

—

103

C₂²×C₁₃

#486

yes

—

104

C₂²×C₁₃

#510

yes

—

105

C₂²×C₁₃

C₂²×D₈

yes

characteristic

106

C₄×C₁₃

—

107

C₄×C₁₃

—

108

C₄×C₁₃

#486

yes

—

109

C₄×C₁₃

C₂×#124

yes

characteristic

110

C₄×C₁₃

C₂²×SD₁₆

yes

characteristic

111

Dic₁₃

—

112

C₂²×Q₈

—

113

C₂×Q₁₆

—

114

C₂×Q₁₆

—

115

C₂×Q₁₆

—

116

C₂×Q₁₆

—

117

C₂×Q₁₆

—

118

C₂²×C₈

C₂²⋊D₁₃

yes

characteristic

119

C₂²×C₈

C₂²×D₁₃

yes

characteristic

120

C₄⋊Q₈

—

121

C₄⋊Q₈

—

122

#125

—

123

#125

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄²

C₂²⋊D₁₃

yes

characteristic

128

C₂×C₁₆

—

129

C₂×C₁₆

—

130

#107

—

131

#107

C₂²×D₁₃

yes

—

132

#106

—

133

C₄×C₈

C₂²⋊D₁₃

yes

—

134

C₄×C₈

C₂²×D₁₃

yes

—

135

C₂×C₁₃

#2245

yes

complemented

136

C₂×C₁₃

C₂×#486

yes

characteristic

137

C₂×C₁₃

C₂×#510

yes

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

#6104

yes

—

145

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

146

Q₁₆

—

147

Q₁₆

—

148

Q₁₆

—

149

C₂×C₈

#6106

yes

—

150

C₂×C₈

#6131

yes

—

151

C₂×C₈

C₂×C₂²⋊D₁₃

yes

—

152

C₂×C₈

C₂×C₂²⋊D₁₃

yes

characteristic

153

C₂×C₈

C₂³×D₁₃

yes

characteristic

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄²

#6104

yes

—

159

C₄²

C₂×C₂²⋊D₁₃

yes

—

160

C₁₆

—

161

C₁₃

C₂×#2245

yes

characteristic, complemented, Sylow

162

C₂³

C₂×#6104

yes

characteristic

163

Q₈

—

164

Q₈

—

165

Q₈

—

104

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

104

—

171

C₂×C₄

C₂×#6104

yes

—

172

C₂×C₄

C₂×#6104

yes

characteristic

173

C₂×C₄

C₂×#6106

yes

—

174

C₂×C₄

#88201

yes

—

175

C₂×C₄

C₂²×C₂²⋊D₁₃

yes

characteristic

176

C₈

C₂×#6106

yes

—

177

C₈

#88201

yes

—

178

C₈

C₂×#6131

yes

—

179

C₂²

#11718749

yes

—

180

C₂²

#11718856

yes

—

181

C₂²

C₂²×#6104

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

104

—

185

C₄

#11718749

yes

—

186

C₄

C₂²×#6106

yes

characteristic

187

C₄

C₂×#88201

yes

characteristic

188

C₂

#13117083

yes

direct product factor

189

C₂

C₂×#11718749

yes

characteristic

190

C₂

C₂×#11718856

yes

—

	subgroup	G/H
center	C₂³	C₂×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#499	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₈
hypercenter	C₂×#499	D₁₃
hypocenter	C₁₃	C₂×#2245
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

subgroup

G/H

center

C₂³

C₂×#6104

commutator, cosocle

C₂×C₈×C₁₃

C₂⁴

Frattini

C₂×C₈

C₂³×D₁₃

Fitting

C₂×C₁₃×#499

C₂

socle, prime radical

C₂³×C₁₃

C₂×D₈

D₁₃

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

N	H ≅ G/N	type
2. #13117083	C₂	direct (N × H)
135. C₂×C₁₃	#2245	semidirect
161. C₁₃	C₂×#2245	semidirect
188. C₂	#13117083	direct (N × H)

H ≅ G/N

type

2. #13117083

C₂

direct (N × H)

135. C₂×C₁₃

#2245

semidirect

161. C₁₃

C₂×#2245

semidirect

188. C₂

#13117083

direct (N × H)

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

subgroup	index
C₂×#11718360	2
#13117083	2
C₂×#11718084	2
C₂×#11718053	2
C₂×C₁₃×#499	2
C₂×#2245	13

subgroup

index

C₂×C₁₃×#499

Subgroups — C₂×#13117083 #16018238

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#13117083 #16018238

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13117083 #16018238