Group #28374493 — subgroups

Subgroups — #28374493

Order 13952.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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	#20295647	C₂	1	2	yes	maximal, complemented
2	#20295322	C₂	1	2	yes	maximal, complemented
3	#20295307	C₂	1	2	yes	maximal, complemented
4	#20295272	C₂	1	2	yes	maximal, complemented
5	#20295267	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295262	C₂	1	1	yes	maximal, characteristic
7	#20295249	C₂	1	2	yes	maximal, complemented
8	#20295242	C₂	1	2	yes	maximal, complemented
9	#424×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
11	C₂×#13182018	C₂²	1	2	yes	—
12	C₂×D₈₇₂	C₂²	1	1	yes	characteristic
13	#16812699	C₂²	1	2	yes	—
14	#16812694	—	2	2	no	—
15	#16812687	C₂²	1	2	yes	—
16	#16812685	—	2	2	no	—
17	#16812681	—	2	2	no	—
18	#16812678	C₂²	1	2	yes	—
19	#16812677	—	2	2	no	—
20	#16812673	C₂²	1	2	yes	—
21	C₄×Dic₂₁₈	C₂²	1	2	yes	—
22	C₄×D₄₃₆	C₂²	1	2	yes	—
23	#16812633	C₂²	1	1	yes	characteristic
24	#16812633	C₂²	1	1	yes	characteristic
25	#16812633	C₂²	1	2	yes	—
26	#16812624	C₂²	1	2	yes	—
27	#16812624	C₂²	1	2	yes	—
28	#16812621	C₂²	1	2	yes	—
29	#16812621	C₂²	1	2	yes	—
30	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
31	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
32	#117×C₁₀₉	C₂²	1	1	yes	characteristic
33	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
34	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
35	C₂×Dic₂₁₈	—	2	2	no	—
36	C₂×Dic₂₁₈	C₂³	1	2	yes	—
37	C₂×D₄₃₆	C₂³	1	2	yes	—
38	C₂×C₂²⋊D₁₀₉	—	2	2	no	—
39	C₂×C₄×D₁₀₉	—	2	2	no	—
40	Dic₄₃₆	—	4	1	no	—
41	#13182018	—	4	2	no	—
42	D₈₇₂	—	4	1	no	—
43	C₄⋊Dic₁₀₉	—	2	2	no	—
44	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
45	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
46	#13182008	—	2	2	no	—
47	#13182008	—	2	2	no	—
48	C₂²⋊Dic₁₀₉	—	2	2	no	—
49	#13182006	—	2	2	no	—
50	#13182006	—	2	2	no	—
51	#13182006	—	2	2	no	—
52	#13182006	—	2	2	no	—
53	C₄×Dic₁₀₉	—	2	2	no	—
54	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
55	M₁₆×C₁₀₉	—	4	1	no	—
56	C₂×C₈×C₁₀₉	—	4	1	no	—
57	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
60	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
62	C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
63	C₄²×C₁₀₉	C₂³	1	2	yes	—
64	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×D₁₀₉	—	2	2	no	—
66	C₂×C₄×C₁₀₉	—	2	1	no	—
67	C₂×C₄×C₁₀₉	—	2	1	no	—
68	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
72	C₂²⋊D₁₀₉	—	8	2	no	—
73	C₂×Dic₁₀₉	—	2	2	no	—
74	C₂×Dic₁₀₉	—	2	2	no	—
75	C₂×Dic₁₀₉	—	2	2	no	—
76	D₄₃₆	—	2	2	no	—
77	D₄₃₆	—	2	2	no	—
78	C₄×D₁₀₉	—	8	2	no	—
79	Dic₂₁₈	—	2	2	no	—
80	Dic₂₁₈	—	2	2	no	—
81	Dic₂₁₈	—	8	2	no	—
82	C₈×C₁₀₉	—	2	1	no	—
83	C₈×C₁₀₉	—	2	1	no	—
84	C₈×C₁₀₉	—	2	2	no	—
85	C₂²×C₁₀₉	—	2	1	no	—
86	C₂²×C₁₀₉	—	2	1	no	—
87	C₂²×C₁₀₉	—	2	1	no	—
88	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
89	C₂×D₁₀₉	—	4	2	no	—
90	C₂×D₁₀₉	—	4	2	no	—
91	C₂×D₁₀₉	—	4	2	no	—
92	C₄×C₁₀₉	—	2	1	no	—
93	C₄×C₁₀₉	—	2	2	no	—
94	C₄×C₁₀₉	—	2	2	no	—
95	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
96	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
97	Dic₁₀₉	—	4	2	no	—
98	Dic₁₀₉	—	4	2	no	—
99	Dic₁₀₉	—	4	2	no	—
100	C₂×C₁₀₉	—	4	1	no	—
101	C₂×C₁₀₉	#537	1	1	yes	characteristic
102	C₂×C₁₀₉	#576	1	1	yes	characteristic
103	C₂×C₁₀₉	#578	1	1	yes	characteristic
104	D₁₀₉	—	8	2	no	—
105	#3262	—	109	1	no	maximal, Sylow
106	C₁₀₉	#3262	1	1	yes	characteristic, complemented, Sylow
107	#542	—	109	2	no	—
108	#477	—	109	2	no	—
109	#474	—	109	2	no	—
110	#468	—	109	2	no	—
111	#467	—	109	1	no	—
112	#466	—	109	1	no	—
113	#464	—	109	2	no	—
114	#463	—	109	2	no	—
115	#424	D₁₀₉	1	1	yes	characteristic, complemented
116	C₂×Q₁₆	—	109	1	no	—
117	C₂×SD₁₆	—	109	2	no	—
118	C₂×D₈	—	109	1	no	—
119	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
120	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
121	C₄⋊Q₈	—	109	2	no	—
122	#126	—	218	2	no	—
123	#124	—	109	2	no	—
124	#124	—	218	2	no	—
125	#123	—	218	2	no	—
126	C₂²⋊Q₈	—	109	2	no	—
127	C₂²⋊Q₈	—	218	2	no	—
128	#121	—	109	2	no	—
129	C₄×Q₈	—	109	2	no	—
130	C₄×D₄	—	109	2	no	—
131	#117	C₂×D₁₀₉	1	1	yes	characteristic
132	#107	—	109	1	no	—
133	#107	—	109	1	no	—
134	#107	—	109	2	no	—
135	C₄⋊C₈	C₂×D₁₀₉	1	2	yes	—
136	C₄⋊C₈	C₂×D₁₀₉	1	2	yes	—
137	#103	—	109	2	no	—
138	#103	—	109	2	no	—
139	#102	—	109	2	no	—
140	#102	—	109	2	no	—
141	C₂×Q₈	—	109	2	no	—
142	C₂×Q₈	—	218	2	no	—
143	C₂×D₄	—	109	2	no	—
144	C₂×D₄	—	218	2	no	—
145	C₂²×C₄	—	218	2	no	—
146	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
147	Q₁₆	—	436	1	no	—
148	SD₁₆	—	436	2	no	—
149	D₈	—	436	1	no	—
150	M₁₆	—	4	1	no	—
151	C₂×C₈	—	4	1	no	—
152	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
153	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
154	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
155	C₄⋊C₄	—	109	2	no	—
156	C₄⋊C₄	—	109	2	no	—
157	C₄⋊C₄	—	218	2	no	—
158	C₄⋊C₄	—	218	2	no	—
159	C₄⋊C₄	—	218	2	no	—
160	C₄⋊C₄	D₄₃₆	1	1	yes	characteristic
161	C₄⋊C₄	D₄₃₆	1	1	yes	characteristic
162	C₂²⋊C₄	—	218	2	no	—
163	C₂²⋊C₄	—	218	2	no	—
164	C₂²⋊C₄	—	218	2	no	—
165	C₂²⋊C₄	—	218	2	no	—
166	C₂²⋊C₄	—	218	2	no	—
167	C₂²⋊C₄	D₄₃₆	1	2	yes	—
168	C₄²	—	218	2	no	—
169	C₄²	C₂²×D₁₀₉	1	2	yes	—
170	C₂³	—	218	2	no	—
171	C₂³	C₂×D₄₃₆	1	1	yes	characteristic
172	Q₈	—	872	2	no	—
173	Q₈	—	218	2	no	—
174	Q₈	—	218	2	no	—
175	D₄	—	872	2	no	—
176	D₄	—	218	2	no	—
177	D₄	—	218	2	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	872	2	no	—
181	C₂×C₄	—	218	2	no	—
182	C₂×C₄	—	218	2	no	—
183	C₂×C₄	—	218	2	no	—
184	C₂×C₄	C₂×D₄₃₆	1	2	yes	—
185	C₂×C₄	C₂×D₄₃₆	1	2	yes	—
186	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
187	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	2	2	no	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	436	2	no	—
195	C₂²	—	436	2	no	—
196	C₂²	—	436	2	no	—
197	C₂²	C₂²×D₄₃₆	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	2	2	no	—
200	C₄	—	2	1	no	—
201	C₄	—	436	2	no	—
202	C₄	—	436	2	no	—
203	C₄	—	436	2	no	—
204	C₄	#16812744	1	1	yes	characteristic
205	C₄	#16812747	1	1	yes	characteristic
206	C₂	—	4	1	no	—
207	C₂	—	872	2	no	—
208	C₂	#20295607	1	1	yes	characteristic
209	C₂	#20295831	1	1	yes	characteristic
210	C₂	#20295843	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295647

C₂

yes

maximal, complemented

#20295322

C₂

yes

maximal, complemented

#20295307

C₂

yes

maximal, complemented

#20295272

C₂

yes

maximal, complemented

#20295267

C₂

yes

maximal, characteristic, complemented

#20295262

C₂

yes

maximal, characteristic

#20295249

C₂

yes

maximal, complemented

#20295242

C₂

yes

maximal, complemented

#424×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₄₃₆

C₂²

yes

characteristic

C₂×#13182018

C₂²

yes

—

C₂×D₈₇₂

C₂²

yes

characteristic

#16812699

C₂²

yes

—

#16812694

—

#16812687

C₂²

yes

—

#16812685

—

#16812681

—

#16812678

C₂²

yes

—

#16812677

—

#16812673

C₂²

yes

—

C₄×Dic₂₁₈

C₂²

yes

—

C₄×D₄₃₆

C₂²

yes

—

#16812633

C₂²

yes

characteristic

#16812633

C₂²

yes

characteristic

#16812633

C₂²

yes

—

#16812624

C₂²

yes

—

#16812624

C₂²

yes

—

#16812621

C₂²

yes

—

#16812621

C₂²

yes

—

C₂×M₁₆×C₁₀₉

C₂²

yes

characteristic

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

#117×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

C₂²

yes

—

C₄⋊C₈×C₁₀₉

C₂²

yes

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

C₂³

yes

—

C₂×D₄₃₆

C₂³

yes

—

C₂×C₂²⋊D₁₀₉

—

C₂×C₄×D₁₀₉

—

Dic₄₃₆

—

#13182018

—

D₈₇₂

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂³

yes

—

C₄⋊Dic₁₀₉

C₂³

yes

—

#13182008

—

#13182008

—

C₂²⋊Dic₁₀₉

—

—

—

—

—

C₄×Dic₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

—

C₄⋊C₄×C₁₀₉

D₄

yes

characteristic

C₄⋊C₄×C₁₀₉

D₄

yes

characteristic

C₂²⋊C₄×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂²×D₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

D₄₃₆

—

D₄₃₆

—

C₄×D₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

H(5,𝔽₂)

yes

characteristic

C₄×C₁₀₉

2₋¹⁺⁴

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

100

C₂×C₁₀₉

—

101

C₂×C₁₀₉

#537

yes

characteristic

102

C₂×C₁₀₉

#576

yes

characteristic

103

C₂×C₁₀₉

#578

yes

characteristic

104

D₁₀₉

—

105

#3262

—

109

maximal, Sylow

106

C₁₀₉

#3262

yes

characteristic, complemented, Sylow

107

#542

—

109

—

108

#477

—

109

—

109

#474

—

109

—

110

#468

—

109

—

111

#467

—

109

—

112

#466

—

109

—

113

#464

—

109

—

114

#463

—

109

—

115

#424

D₁₀₉

yes

characteristic, complemented

116

C₂×Q₁₆

—

109

—

117

C₂×SD₁₆

—

109

—

118

C₂×D₈

—

109

—

119

C₂×M₁₆

C₂×D₁₀₉

yes

characteristic

120

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

121

C₄⋊Q₈

—

109

—

122

#126

—

218

—

123

#124

—

109

—

124

#124

—

218

—

125

#123

—

218

—

126

C₂²⋊Q₈

—

109

—

127

C₂²⋊Q₈

—

218

—

128

#121

—

109

—

129

C₄×Q₈

—

109

—

130

C₄×D₄

—

109

—

131

#117

C₂×D₁₀₉

yes

characteristic

132

#107

—

109

—

133

#107

—

109

—

134

#107

—

109

—

135

C₄⋊C₈

C₂×D₁₀₉

yes

—

136

C₄⋊C₈

C₂×D₁₀₉

yes

—

137

#103

—

109

—

138

#103

—

109

—

139

#102

—

109

—

140

#102

—

109

—

141

C₂×Q₈

—

109

—

142

C₂×Q₈

—

218

—

143

C₂×D₄

—

109

—

144

C₂×D₄

—

218

—

145

C₂²×C₄

—

218

—

146

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

147

Q₁₆

—

436

—

148

SD₁₆

—

436

—

149

D₈

—

436

—

150

M₁₆

—

151

C₂×C₈

—

152

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

153

C₂×C₈

C₂²×D₁₀₉

yes

—

154

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

155

C₄⋊C₄

—

109

—

156

C₄⋊C₄

—

109

—

157

C₄⋊C₄

—

218

—

158

C₄⋊C₄

—

218

—

159

C₄⋊C₄

—

218

—

160

C₄⋊C₄

D₄₃₆

yes

characteristic

161

C₄⋊C₄

D₄₃₆

yes

characteristic

162

C₂²⋊C₄

—

218

—

163

C₂²⋊C₄

—

218

—

164

C₂²⋊C₄

—

218

—

165

C₂²⋊C₄

—

218

—

166

C₂²⋊C₄

—

218

—

167

C₂²⋊C₄

D₄₃₆

yes

—

168

C₄²

—

218

—

169

C₄²

C₂²×D₁₀₉

yes

—

170

C₂³

—

218

—

171

C₂³

C₂×D₄₃₆

yes

characteristic

172

Q₈

—

872

—

173

Q₈

—

218

—

174

Q₈

—

218

—

175

D₄

—

872

—

176

D₄

—

218

—

177

D₄

—

218

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

872

—

181

C₂×C₄

—

218

—

182

C₂×C₄

—

218

—

183

C₂×C₄

—

218

—

184

C₂×C₄

C₂×D₄₃₆

yes

—

185

C₂×C₄

C₂×D₄₃₆

yes

—

186

C₂×C₄

C₂×D₄₃₆

yes

characteristic

187

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

188

C₈

—

189

C₈

—

190

C₈

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

194

C₂²

—

436

—

195

C₂²

—

436

—

196

C₂²

—

436

—

197

C₂²

C₂²×D₄₃₆

yes

characteristic

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

—

436

—

202

C₄

—

436

—

203

C₄

—

436

—

204

C₄

#16812744

yes

characteristic

205

C₄

#16812747

yes

characteristic

206

C₂

—

207

C₂

—

872

—

208

C₂

#20295607

yes

characteristic

209

C₂

#20295831

yes

characteristic

210

C₂

#20295843

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₃₆
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#424×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#20295262	C₂
hypercenter	#424	D₁₀₉
hypocenter	C₁₀₉	#3262
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #424

subgroup

G/H

center

C₂²

C₂²×D₄₃₆

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

#424×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂²×D₄

C₂

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ #117 ◁ #424

N	H ≅ G/N	type
1. #20295647	C₂	semidirect
2. #20295322	C₂	semidirect
3. #20295307	C₂	semidirect
4. #20295272	C₂	semidirect
5. #20295267	C₂	semidirect
7. #20295249	C₂	semidirect
8. #20295242	C₂	semidirect
9. #424×C₁₀₉	C₂	semidirect
106. C₁₀₉	#3262	semidirect
115. #424	D₁₀₉	semidirect

H ≅ G/N

type

1. #20295647

C₂

semidirect

2. #20295322

C₂

semidirect

3. #20295307

C₂

semidirect

4. #20295272

C₂

semidirect

5. #20295267

C₂

semidirect

7. #20295249

C₂

semidirect

8. #20295242

C₂

semidirect

9. #424×C₁₀₉

C₂

semidirect

106. C₁₀₉

#3262

semidirect

115. #424

D₁₀₉

semidirect

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

subgroup	index
#20295647	2
#20295322	2
#20295307	2
#20295272	2
#20295267	2
#20295262	2
#20295249	2
#20295242	2
#424×C₁₀₉	2
#3262	109

subgroup

index

#424×C₁₀₉

109

Subgroups — #28374493

Conjugacy classes of subgroups

Complements

Maximal subgroups