Group #28364557 — subgroups

Subgroups — #28364557

Order 13952.

290 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 87 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×#16812632	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16812621	C₂	1	1	yes	maximal, characteristic
3	C₂×#16812619	C₂	1	1	yes	maximal, characteristic
4	#20294927	C₂	1	1	yes	maximal, characteristic
5	#20294895	C₂	1	1	yes	maximal, characteristic, complemented
6	#20294780	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#107×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄₃₆	C₂²	1	1	yes	characteristic
9	C₂²×#11724784	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×#13182008	—	2	2	no	—
12	C₂×#13182006	—	2	2	no	—
13	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
14	#16812632	—	2	2	no	—
15	#16812621	—	2	2	no	—
16	#16812619	—	2	2	no	—
17	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
20	#107×C₁₀₉	—	2	2	no	—
21	C₂³×D₁₀₉	—	2	1	no	—
22	C₂×D₄₃₆	—	2	1	no	—
23	C₂×D₄₃₆	—	2	1	no	—
24	C₂×D₄₃₆	—	2	1	no	—
25	C₂×D₄₃₆	—	2	1	no	—
26	C₂×D₄₃₆	—	2	1	no	—
27	C₂×D₄₃₆	Q₈	1	2	yes	—
28	C₂²×Dic₁₀₉	—	2	1	no	—
29	C₂²×Dic₁₀₉	D₄	1	2	yes	—
30	C₂×#11724784	—	2	1	no	—
31	C₂×#11724784	—	2	1	no	—
32	C₂×#11724784	D₄	1	2	yes	—
33	C₄⋊Dic₁₀₉	—	2	1	no	—
34	C₄⋊Dic₁₀₉	—	2	1	no	—
35	#13182008	—	4	4	no	—
36	#13182006	—	4	4	no	—
37	C₄×Dic₁₀₉	—	2	1	no	—
38	C₄×Dic₁₀₉	—	2	1	no	—
39	C₂²×C₄×C₁₀₉	—	2	1	no	—
40	C₂²×C₄×C₁₀₉	—	2	1	no	—
41	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
42	C₂×C₈×C₁₀₉	—	2	1	no	—
43	C₂×C₈×C₁₀₉	—	2	1	no	—
44	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
45	C₄⋊C₄×C₁₀₉	—	2	1	no	—
46	C₄⋊C₄×C₁₀₉	—	2	1	no	—
47	C₄⋊C₄×C₁₀₉	—	2	1	no	—
48	C₄⋊C₄×C₁₀₉	—	2	1	no	—
49	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×D₁₀₉	—	2	2	no	—
51	C₂²×D₁₀₉	—	4	1	no	—
52	C₂²×D₁₀₉	—	4	1	no	—
53	C₂²×D₁₀₉	—	4	2	no	—
54	C₂²×D₁₀₉	—	4	2	no	—
55	C₂×C₄×C₁₀₉	—	4	1	no	—
56	C₂×C₄×C₁₀₉	—	4	1	no	—
57	C₂×C₄×C₁₀₉	—	4	2	no	—
58	C₂×C₄×C₁₀₉	—	4	2	no	—
59	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
63	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
64	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
65	C₂×Dic₁₀₉	—	2	2	no	—
66	C₂×Dic₁₀₉	—	2	2	no	—
67	C₂×Dic₁₀₉	—	4	1	no	—
68	C₂×Dic₁₀₉	—	4	2	no	—
69	C₂×Dic₁₀₉	D₈	1	4	yes	—
70	D₄₃₆	—	2	2	no	—
71	D₄₃₆	—	2	2	no	—
72	D₄₃₆	—	4	1	no	—
73	D₄₃₆	—	4	1	no	—
74	C₈×C₁₀₉	—	2	2	no	—
75	#11724784	—	2	2	no	—
76	C₂²×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
77	C₂²×C₁₀₉	#123	1	1	yes	characteristic
78	C₂²×C₁₀₉	#127	1	1	yes	characteristic
79	C₂²×C₁₀₉	C₂×D₈	1	2	yes	—
80	C₂²×C₁₀₉	#137	1	2	yes	—
81	C₂×D₁₀₉	—	4	2	no	—
82	C₂×D₁₀₉	—	4	4	no	—
83	C₂×D₁₀₉	—	8	2	no	—
84	C₂×D₁₀₉	—	8	1	no	—
85	C₂×D₁₀₉	—	8	1	no	—
86	C₄×C₁₀₉	—	8	1	no	—
87	C₄×C₁₀₉	—	8	1	no	—
88	C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
89	C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
90	C₄×C₁₀₉	#123	1	1	yes	characteristic
91	C₄×C₁₀₉	#123	1	1	yes	characteristic
92	Dic₁₀₉	—	2	4	no	—
93	Dic₁₀₉	—	8	1	no	—
94	C₂×C₁₀₉	#399	1	1	yes	characteristic
95	C₂×C₁₀₉	#474	1	2	yes	—
96	C₂×C₁₀₉	#480	1	2	yes	—
97	C₂×C₁₀₉	#493	1	1	yes	characteristic
98	C₂×C₁₀₉	#497	1	1	yes	characteristic
99	D₁₀₉	—	8	2	no	—
100	#2093	—	109	1	no	maximal, Sylow
101	C₁₀₉	#2093	1	1	yes	characteristic, complemented, Sylow
102	C₂×#107	—	109	1	no	—
103	C₂×#107	D₁₀₉	1	1	yes	characteristic, complemented
104	C₂×#102	—	109	1	no	—
105	C₂×#102	—	109	1	no	—
106	#390	—	109	1	no	—
107	#384	—	109	1	no	—
108	#336	—	109	1	no	—
109	C₂²×D₄	—	109	1	no	—
110	C₂²×C₈	—	109	1	no	—
111	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
112	C₂×C₄⋊C₄	—	109	1	no	—
113	C₂×C₄⋊C₄	—	218	2	no	—
114	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
115	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
116	C₂×C₂²⋊C₄	—	218	2	no	—
117	C₂×C₄²	—	109	1	no	—
118	#107	—	2	2	no	—
119	#107	—	218	2	no	—
120	#102	—	218	2	no	—
121	#102	—	218	2	no	—
122	C₂⁴	—	218	1	no	—
123	C₂×D₄	—	109	2	no	—
124	C₂×D₄	—	218	1	no	—
125	C₂×D₄	—	218	1	no	—
126	C₂×D₄	—	218	1	no	—
127	C₂×D₄	—	218	1	no	—
128	C₂×D₄	—	218	1	no	—
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	—	109	2	no	—
132	C₂²×C₄	—	218	1	no	—
133	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	109	2	no	—
137	C₂×C₈	—	218	1	no	—
138	C₂×C₈	—	218	1	no	—
139	C₂×C₈	C₂²⋊D₁₀₉	1	2	yes	—
140	C₄⋊C₄	—	2	1	no	—
141	C₄⋊C₄	—	2	1	no	—
142	C₄⋊C₄	—	2	1	no	—
143	C₄⋊C₄	—	2	1	no	—
144	C₄⋊C₄	—	436	4	no	—
145	C₄⋊C₄	—	218	1	no	—
146	C₄⋊C₄	—	218	1	no	—
147	C₂²⋊C₄	—	436	4	no	—
148	C₄²	—	218	1	no	—
149	C₄²	—	218	1	no	—
150	C₂³	—	436	1	no	—
151	C₂³	—	436	1	no	—
152	C₂³	—	436	2	no	—
153	C₂³	—	436	2	no	—
154	C₂³	—	218	2	no	—
155	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
156	D₄	—	436	1	no	—
157	D₄	—	436	1	no	—
158	D₄	—	218	2	no	—
159	D₄	—	218	2	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	—	4	1	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	109	4	no	—
165	C₂×C₄	—	436	1	no	—
166	C₂×C₄	—	436	2	no	—
167	C₂×C₄	—	218	2	no	—
168	C₂×C₄	—	218	2	no	—
169	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
170	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
171	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
172	C₂×C₄	#13182029	1	1	yes	characteristic
173	C₂×C₄	#13182029	1	1	yes	characteristic
174	C₂×C₄	#13182030	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	—	218	2	no	—
177	C₂²	—	872	2	no	—
178	C₂²	—	872	1	no	—
179	C₂²	—	872	1	no	—
180	C₂²	—	436	2	no	—
181	C₂²	—	436	4	no	—
182	C₂²	#16812677	1	1	yes	characteristic
183	C₂²	#16812684	1	1	yes	characteristic
184	C₂²	#16812695	1	1	yes	characteristic
185	C₂²	D₈×D₁₀₉	1	2	yes	—
186	C₂²	#16812731	1	2	yes	—
187	C₄	—	8	1	no	—
188	C₄	—	8	1	no	—
189	C₄	—	872	1	no	—
190	C₄	—	218	4	no	—
191	C₄	#16812675	1	1	yes	characteristic
192	C₄	#16812677	1	1	yes	characteristic
193	C₄	#16812682	1	1	yes	characteristic
194	C₄	#16812684	1	1	yes	characteristic
195	C₂	—	872	2	no	—
196	C₂	#20294961	1	1	yes	characteristic
197	C₂	#20295305	1	2	yes	—
198	C₂	#20295341	1	2	yes	—
199	C₂	#20295401	1	1	yes	characteristic
200	C₂	#20295419	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812632

C₂

yes

maximal, characteristic, complemented

C₂×#16812621

C₂

yes

maximal, characteristic

C₂×#16812619

C₂

yes

maximal, characteristic

#20294927

C₂

yes

maximal, characteristic

#20294895

C₂

yes

maximal, characteristic, complemented

#20294780

C₂

yes

maximal, characteristic, complemented

C₂×#107×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₄₃₆

C₂²

yes

characteristic

C₂²×#11724784

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₀₉

C₂²

yes

characteristic

C₂×#13182008

—

C₂×#13182006

—

C₂×C₄×Dic₁₀₉

C₂²

yes

—

—

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

characteristic

#107×C₁₀₉

—

C₂³×D₁₀₉

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂×D₄₃₆

Q₈

yes

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

D₄

yes

—

C₂×#11724784

—

C₂×#11724784

—

C₂×#11724784

D₄

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

#13182008

—

#13182006

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

D₄

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

C₂×C₄×C₁₀₉

yes

C₂×C₄×C₁₀₉

yes

C₂×C₄×C₁₀₉

yes

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

D₈

yes

—

D₄₃₆

—

D₄₃₆

—

D₄₃₆

—

D₄₃₆

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

C₂²⋊Q₈

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

C₂×D₈

yes

—

C₂²×C₁₀₉

#137

yes

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²⋊Q₈

yes

characteristic

C₄×C₁₀₉

C₂²⋊Q₈

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

#399

yes

characteristic

C₂×C₁₀₉

#474

yes

—

C₂×C₁₀₉

#480

yes

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

100

#2093

—

109

maximal, Sylow

101

C₁₀₉

#2093

yes

characteristic, complemented, Sylow

102

C₂×#107

—

109

—

103

C₂×#107

D₁₀₉

yes

characteristic, complemented

104

C₂×#102

—

109

—

105

C₂×#102

—

109

—

106

#390

—

109

—

107

#384

—

109

—

108

#336

—

109

—

109

C₂²×D₄

—

109

—

110

C₂²×C₈

—

109

—

111

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

112

C₂×C₄⋊C₄

—

109

—

113

C₂×C₄⋊C₄

—

218

—

114

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

characteristic

115

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

characteristic

116

C₂×C₂²⋊C₄

—

218

—

117

C₂×C₄²

—

109

—

118

#107

—

119

#107

—

218

—

120

#102

—

218

—

121

#102

—

218

—

122

C₂⁴

—

218

—

123

C₂×D₄

—

109

—

124

C₂×D₄

—

218

—

125

C₂×D₄

—

218

—

126

C₂×D₄

—

218

—

127

C₂×D₄

—

218

—

128

C₂×D₄

—

218

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

109

—

132

C₂²×C₄

—

218

—

133

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

109

—

137

C₂×C₈

—

218

—

138

C₂×C₈

—

218

—

139

C₂×C₈

C₂²⋊D₁₀₉

yes

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

436

—

145

C₄⋊C₄

—

218

—

146

C₄⋊C₄

—

218

—

147

C₂²⋊C₄

—

436

—

148

C₄²

—

218

—

149

C₄²

—

218

—

150

C₂³

—

436

—

151

C₂³

—

436

—

152

C₂³

—

436

—

153

C₂³

—

436

—

154

C₂³

—

218

—

155

C₂³

D₄×D₁₀₉

yes

characteristic

156

D₄

—

436

—

157

D₄

—

436

—

158

D₄

—

218

—

159

D₄

—

218

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

109

—

165

C₂×C₄

—

436

—

166

C₂×C₄

—

436

—

167

C₂×C₄

—

218

—

168

C₂×C₄

—

218

—

169

C₂×C₄

D₄×D₁₀₉

yes

characteristic

170

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

171

C₂×C₄

Q₈×D₁₀₉

yes

characteristic

172

C₂×C₄

#13182029

yes

characteristic

173

C₂×C₄

#13182029

yes

characteristic

174

C₂×C₄

#13182030

yes

characteristic

175

C₈

—

176

C₈

—

218

—

177

C₂²

—

872

—

178

C₂²

—

872

—

179

C₂²

—

872

—

180

C₂²

—

436

—

181

C₂²

—

436

—

182

C₂²

#16812677

yes

characteristic

183

C₂²

#16812684

yes

characteristic

184

C₂²

#16812695

yes

characteristic

185

C₂²

D₈×D₁₀₉

yes

—

186

C₂²

#16812731

yes

—

187

C₄

—

188

C₄

—

189

C₄

—

872

—

190

C₄

—

218

—

191

C₄

#16812675

yes

characteristic

192

C₄

#16812677

yes

characteristic

193

C₄

#16812682

yes

characteristic

194

C₄

#16812684

yes

characteristic

195

C₂

—

872

—

196

C₂

#20294961

yes

characteristic

197

C₂

#20295305

yes

—

198

C₂

#20295341

yes

—

199

C₂

#20295401

yes

characteristic

200

C₂

#20295419

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₁₀₉
commutator, cosocle	C₂²×C₄×C₁₀₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂×#107×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂×D₄
prime radical	C₂²×D₄₃₆	C₂²
hypercenter	C₂×#107	D₁₀₉
hypocenter	C₁₀₉	#2093
LCS	G ▷ C₂²×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₁₀₉

commutator, cosocle

C₂²×C₄×C₁₀₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₂×#107×C₁₀₉

C₂

socle

C₂³×C₁₀₉

C₂×D₄

prime radical

C₂²×D₄₃₆

C₂²

D₁₀₉

C₁₀₉

G ▷ C₂²×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
1. C₂×#16812632	C₂	semidirect
5. #20294895	C₂	semidirect
6. #20294780	C₂	semidirect
7. C₂×#107×C₁₀₉	C₂	semidirect
101. C₁₀₉	#2093	semidirect
103. C₂×#107	D₁₀₉	semidirect

H ≅ G/N

type

1. C₂×#16812632

C₂

semidirect

5. #20294895

C₂

semidirect

6. #20294780

C₂

semidirect

7. C₂×#107×C₁₀₉

C₂

semidirect

101. C₁₀₉

#2093

semidirect

103. C₂×#107

D₁₀₉

semidirect

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

subgroup	index
C₂×#16812632	2
C₂×#16812621	2
C₂×#16812619	2
#20294927	2
#20294895	2
#20294780	2
C₂×#107×C₁₀₉	2
#2093	109

subgroup

index

C₂×#107×C₁₀₉

109

Subgroups — #28364557

Conjugacy classes of subgroups

Complements

Maximal subgroups