Group #26371596 — subgroups

Subgroups — C₂×#20245634 #26371596

Order 12032.

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×#15689525	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689503	C₂	1	1	yes	maximal, characteristic
3	#20245634	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688928	C₂	1	4	yes	maximal
5	C₂×C₄₇×#346	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₈₈	—	2	1	no	—
7	C₂²×Dic₁₈₈	C₂²	1	2	yes	—
8	C₂×#13094024	—	2	1	no	—
9	C₂×#13094016	—	2	1	no	—
10	#15689525	—	2	2	no	—
11	#15689525	C₂²	1	4	yes	—
12	#15689503	—	2	2	no	—
13	#15689503	C₂²	1	4	yes	—
14	C₂×#13093958	C₂²	1	2	yes	—
15	C₂×#13093956	—	2	1	no	—
16	#15688928	C₂²	1	16	yes	—
17	C₂²×C₁₆×C₄₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#346	C₂²	1	4	yes	—
20	C₂²×Dic₉₄	—	2	2	no	—
21	C₂×Dic₁₈₈	—	2	8	no	—
22	C₂×Dic₁₈₈	—	2	2	no	—
23	C₂×Dic₁₈₈	—	2	4	no	—
24	C₂×Dic₁₈₈	—	4	4	no	—
25	C₂×Dic₁₈₈	C₂³	1	4	yes	—
26	#13094024	—	2	4	no	—
27	#13094024	—	4	2	no	—
28	#13094016	—	2	4	no	—
29	#13094016	—	4	2	no	—
30	C₂×C₄⋊Dic₄₇	—	2	2	no	—
31	C₂×C₄⋊Dic₄₇	—	4	2	no	—
32	C₂×Dic₄₇⋊C₄	—	4	2	no	—
33	#13093958	—	2	2	no	—
34	#13093958	C₂³	1	4	yes	—
35	#13093956	—	2	4	no	—
36	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
39	C₂×C₁₆×C₄₇	—	2	4	no	—
40	C₂×C₁₆×C₄₇	C₂³	1	4	yes	—
41	C₄×C₈×C₄₇	D₄	1	2	yes	—
42	C₄×C₈×C₄₇	C₂³	1	2	yes	—
43	C₂²×C₄×C₄₇	D₈	1	2	yes	—
44	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₄₇	—	4	2	no	—
46	C₂²×Dic₄₇	—	4	2	no	—
47	C₂×Dic₉₄	—	2	4	no	—
48	C₂×Dic₉₄	—	4	4	no	—
49	C₂×Dic₉₄	—	4	4	no	—
50	C₂×Dic₉₄	—	4	2	no	—
51	C₂×C₈×C₄₇	SD₁₆	1	4	yes	—
52	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
54	C₂×C₈×C₄₇	#41	1	4	yes	—
55	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
56	C₄²×C₄₇	D₈	1	2	yes	—
57	C₄²×C₄₇	C₂×D₄	1	2	yes	—
58	C₄⋊Dic₄₇	—	2	4	no	—
59	C₄⋊Dic₄₇	—	4	2	no	—
60	C₄⋊Dic₄₇	—	4	8	no	—
61	Dic₄₇⋊C₄	—	4	8	no	—
62	Dic₁₈₈	—	2	8	no	—
63	Dic₁₈₈	—	4	4	no	—
64	Dic₁₈₈	—	4	8	no	—
65	C₁₆×C₄₇	—	2	4	no	—
66	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₄₇	—	2	4	no	—
68	C₂×C₄×C₄₇	#124	1	2	yes	—
69	C₂×C₄×C₄₇	C₂×D₈	1	4	yes	—
70	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
72	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
73	C₂×Dic₄₇	—	4	4	no	—
74	C₂×Dic₄₇	—	4	4	no	—
75	C₂×Dic₄₇	—	8	4	no	—
76	C₂×Dic₄₇	—	8	4	no	—
77	Dic₉₄	—	4	4	no	—
78	Dic₉₄	—	4	4	no	—
79	Dic₉₄	—	8	4	no	—
80	C₈×C₄₇	#124	1	2	yes	—
81	C₈×C₄₇	C₂×SD₁₆	1	4	yes	—
82	C₈×C₄₇	C₂×#41	1	2	yes	—
83	C₂×#2245	—	47	1	no	maximal, Sylow
84	C₂²×C₄₇	#486	1	2	yes	—
85	C₂²×C₄₇	#510	1	4	yes	—
86	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
87	C₄×C₄₇	—	2	4	no	—
88	C₄×C₄₇	#486	1	2	yes	—
89	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
90	C₄×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
91	Dic₄₇	—	8	4	no	—
92	Dic₄₇	—	8	4	no	—
93	C₂×#499	—	47	1	no	—
94	C₂×#494	—	47	1	no	—
95	#2245	—	47	8	no	—
96	C₂×#358	—	47	4	no	—
97	C₂×#346	D₄₇	1	1	yes	characteristic
98	C₂×C₄₇	#2245	1	4	yes	complemented
99	C₂×C₄₇	C₂×#486	1	1	yes	characteristic
100	C₂×C₄₇	C₂×#510	1	2	yes	—
101	C₂²×Q₁₆	—	47	2	no	—
102	C₂²×Q₁₆	—	94	1	no	—
103	C₂×C₄⋊Q₈	—	94	1	no	—
104	C₂×#125	—	94	1	no	—
105	C₂²×C₁₆	C₂×D₄₇	1	2	yes	—
106	#499	—	47	4	no	—
107	#499	—	94	2	no	—
108	#494	—	47	4	no	—
109	#494	—	94	2	no	—
110	C₂×#107	—	47	2	no	—
111	C₂×#106	—	94	1	no	—
112	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
113	#358	—	47	16	no	—
114	#346	C₂×D₄₇	1	4	yes	—
115	C₄₇	C₂×#2245	1	1	yes	characteristic, complemented, Sylow
116	C₂²×Q₈	—	94	2	no	—
117	C₂×Q₁₆	—	47	4	no	—
118	C₂×Q₁₆	—	94	2	no	—
119	C₂×Q₁₆	—	94	4	no	—
120	C₂×Q₁₆	—	94	8	no	—
121	C₂×Q₁₆	—	188	4	no	—
122	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
124	C₄⋊Q₈	—	94	4	no	—
125	C₄⋊Q₈	—	188	2	no	—
126	#125	—	94	4	no	—
127	#125	—	188	2	no	—
128	C₂×C₄⋊C₄	—	94	2	no	—
129	C₂×C₄⋊C₄	—	188	2	no	—
130	C₂×C₄⋊C₄	—	188	2	no	—
131	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
132	C₂×C₁₆	—	2	4	no	—
133	C₂×C₁₆	C₂²×D₄₇	1	4	yes	—
134	#107	—	47	4	no	—
135	#107	—	94	2	no	—
136	#106	—	94	4	no	—
137	C₄×C₈	D₁₈₈	1	2	yes	—
138	C₄×C₈	C₂²×D₄₇	1	2	yes	—
139	C₂×Q₈	—	94	4	no	—
140	C₂×Q₈	—	188	4	no	—
141	C₂×Q₈	—	188	4	no	—
142	C₂×Q₈	—	188	2	no	—
143	C₂²×C₄	—	188	2	no	—
144	C₂²×C₄	—	188	2	no	—
145	C₂²×C₄	D₃₇₆	1	2	yes	—
146	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
147	Q₁₆	—	94	8	no	—
148	Q₁₆	—	188	4	no	—
149	Q₁₆	—	188	8	no	—
150	C₂×C₈	#10625325	1	4	yes	—
151	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
152	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
153	C₂×C₈	#10625350	1	4	yes	—
154	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
155	C₄⋊C₄	—	94	4	no	—
156	C₄⋊C₄	—	188	8	no	—
157	C₄⋊C₄	—	188	2	no	—
158	C₄⋊C₄	—	188	8	no	—
159	C₄²	D₃₇₆	1	2	yes	—
160	C₄²	C₂×D₁₈₈	1	2	yes	—
161	C₁₆	—	2	4	no	—
162	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
163	Q₈	—	376	4	no	—
164	Q₈	—	188	4	no	—
165	Q₈	—	188	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	376	4	no	—
168	C₂×C₄	—	376	4	no	—
169	C₂×C₄	—	188	4	no	—
170	C₂×C₄	—	188	4	no	—
171	C₂×C₄	#13094012	1	2	yes	—
172	C₂×C₄	C₂×D₃₇₆	1	4	yes	—
173	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
174	C₂×C₄	C₂×#10625325	1	2	yes	—
175	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
176	C₈	#13094012	1	2	yes	—
177	C₈	C₂×#10625325	1	4	yes	—
178	C₈	C₂×#10625350	1	2	yes	—
179	C₂²	#15689470	1	2	yes	—
180	C₂²	#15689566	1	4	yes	—
181	C₂²	C₂²×D₃₇₆	1	1	yes	characteristic
182	C₄	—	2	4	no	—
183	C₄	—	376	4	no	—
184	C₄	—	376	4	no	—
185	C₄	#15689470	1	2	yes	—
186	C₄	C₂×#13094012	1	1	yes	characteristic
187	C₄	C₂²×#10625325	1	1	yes	characteristic
188	C₂	#20245634	1	4	yes	direct product factor
189	C₂	C₂×#15689470	1	1	yes	characteristic
190	C₂	C₂×#15689566	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15689525

C₂

yes

maximal, characteristic

C₂×#15689503

C₂

yes

maximal, characteristic

#20245634

C₂

yes

maximal, direct product factor

C₂×#15688928

C₂

yes

maximal

C₂×C₄₇×#346

C₂

yes

maximal, characteristic

C₂²×Dic₁₈₈

—

C₂²×Dic₁₈₈

C₂²

yes

—

—

—

—

C₂²

yes

—

#15689503

—

#15689503

C₂²

yes

—

C₂×#13093958

C₂²

yes

—

C₂×#13093956

—

#15688928

C₂²

yes

—

C₂²×C₁₆×C₄₇

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#346

C₂²

yes

—

C₂²×Dic₉₄

—

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

C₂³

yes

—

—

—

—

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×Dic₄₇⋊C₄

—

#13093958

—

#13093958

C₂³

yes

—

#13093956

—

C₂²×C₈×C₄₇

D₄

yes

characteristic

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

C₂³

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

D₈

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₂×C₈×C₄₇

#41

yes

—

C₂×C₈×C₄₇

C₂⁴

yes

characteristic

C₄²×C₄₇

D₈

yes

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

Dic₁₈₈

—

Dic₁₈₈

—

Dic₁₈₈

—

C₁₆×C₄₇

—

C₂³×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#124

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

C₂²×D₄

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

#124

yes

—

C₈×C₄₇

C₂×SD₁₆

yes

—

C₈×C₄₇

C₂×#41

yes

—

C₂×#2245

—

maximal, Sylow

C₂²×C₄₇

#486

yes

—

C₂²×C₄₇

#510

yes

—

C₂²×C₄₇

C₂²×D₈

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

#486

yes

—

C₄×C₄₇

C₂×#124

yes

characteristic

C₄×C₄₇

C₂²×SD₁₆

yes

characteristic

Dic₄₇

—

Dic₄₇

—

—

—

—

—

D₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

100

C₂×C₄₇

C₂×#510

yes

—

101

C₂²×Q₁₆

—

102

C₂²×Q₁₆

—

103

C₂×C₄⋊Q₈

—

104

C₂×#125

—

105

C₂²×C₁₆

C₂×D₄₇

yes

—

106

#499

—

107

#499

—

108

#494

—

109

#494

—

110

C₂×#107

—

111

C₂×#106

—

112

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

113

#358

—

114

#346

C₂×D₄₇

yes

—

115

C₄₇

C₂×#2245

yes

characteristic, complemented, Sylow

116

C₂²×Q₈

—

117

C₂×Q₁₆

—

118

C₂×Q₁₆

—

119

C₂×Q₁₆

—

120

C₂×Q₁₆

—

121

C₂×Q₁₆

—

188

—

122

C₂²×C₈

D₁₈₈

yes

characteristic

123

C₂²×C₈

C₂²×D₄₇

yes

characteristic

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

188

—

126

#125

—

127

#125

—

188

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

188

—

130

C₂×C₄⋊C₄

—

188

—

131

C₂×C₄²

D₁₈₈

yes

characteristic

132

C₂×C₁₆

—

133

C₂×C₁₆

C₂²×D₄₇

yes

—

134

#107

—

135

#107

—

136

#106

—

137

C₄×C₈

D₁₈₈

yes

—

138

C₄×C₈

C₂²×D₄₇

yes

—

139

C₂×Q₈

—

140

C₂×Q₈

—

188

—

141

C₂×Q₈

—

188

—

142

C₂×Q₈

—

188

—

143

C₂²×C₄

—

188

—

144

C₂²×C₄

—

188

—

145

C₂²×C₄

D₃₇₆

yes

—

146

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

147

Q₁₆

—

148

Q₁₆

—

188

—

149

Q₁₆

—

188

—

150

C₂×C₈

#10625325

yes

—

151

C₂×C₈

C₂×D₁₈₈

yes

characteristic

152

C₂×C₈

C₂×D₁₈₈

yes

—

153

C₂×C₈

#10625350

yes

—

154

C₂×C₈

C₂³×D₄₇

yes

characteristic

155

C₄⋊C₄

—

156

C₄⋊C₄

—

188

—

157

C₄⋊C₄

—

188

—

158

C₄⋊C₄

—

188

—

159

C₄²

D₃₇₆

yes

—

160

C₄²

C₂×D₁₈₈

yes

—

161

C₁₆

—

162

C₂³

C₂×D₃₇₆

yes

characteristic

163

Q₈

—

376

—

164

Q₈

—

188

—

165

Q₈

—

188

—

166

C₂×C₄

—

167

C₂×C₄

—

376

—

168

C₂×C₄

—

376

—

169

C₂×C₄

—

188

—

170

C₂×C₄

—

188

—

171

C₂×C₄

#13094012

yes

—

172

C₂×C₄

C₂×D₃₇₆

yes

—

173

C₂×C₄

C₂×D₃₇₆

yes

characteristic

174

C₂×C₄

C₂×#10625325

yes

—

175

C₂×C₄

C₂²×D₁₈₈

yes

characteristic

176

C₈

#13094012

yes

—

177

C₈

C₂×#10625325

yes

—

178

C₈

C₂×#10625350

yes

—

179

C₂²

#15689470

yes

—

180

C₂²

#15689566

yes

—

181

C₂²

C₂²×D₃₇₆

yes

characteristic

182

C₄

—

183

C₄

—

376

—

184

C₄

—

376

—

185

C₄

#15689470

yes

—

186

C₄

C₂×#13094012

yes

characteristic

187

C₄

C₂²×#10625325

yes

characteristic

188

C₂

#20245634

yes

direct product factor

189

C₂

C₂×#15689470

yes

characteristic

190

C₂

C₂×#15689566

yes

—

	subgroup	G/H
center	C₂³	C₂×D₃₇₆
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#346	C₂
socle, prime radical	C₂³×C₄₇	C₂×D₈
hypercenter	C₂×#346	D₄₇
hypocenter	C₄₇	C₂×#2245
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂³

C₂×D₃₇₆

commutator, cosocle

C₂×C₈×C₄₇

C₂⁴

Frattini

C₂×C₈

C₂³×D₄₇

Fitting

C₂×C₄₇×#346

C₂

socle, prime radical

C₂³×C₄₇

C₂×D₈

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
3. #20245634	C₂	direct (N × H)
98. C₂×C₄₇	#2245	semidirect
115. C₄₇	C₂×#2245	semidirect
188. C₂	#20245634	direct (N × H)

H ≅ G/N

type

3. #20245634

C₂

direct (N × H)

98. C₂×C₄₇

#2245

semidirect

115. C₄₇

C₂×#2245

semidirect

188. C₂

#20245634

direct (N × H)

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

subgroup	index
C₂×#15689525	2
C₂×#15689503	2
#20245634	2
C₂×#15688928	2
C₂×C₄₇×#346	2
C₂×#2245	47

subgroup

index

C₂×C₄₇×#346

Subgroups — C₂×#20245634 #26371596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20245634 #26371596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245634 #26371596