Group #28373526 — subgroups

Subgroups — #28373526

Order 13952.

532 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 165 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×#16812732	C₂	1	2	yes	maximal, complemented
2	C₂×#16812721	C₂	1	1	yes	maximal, characteristic
3	C₂²×Dic₄₃₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295602	C₂	1	2	yes	maximal, complemented
5	#20295420	C₂	1	4	yes	maximal, complemented
6	#20295410	C₂	1	2	yes	maximal, complemented
7	#20295406	C₂	1	2	yes	maximal, complemented
8	#405×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13182029	C₂²	1	2	yes	—
10	C₂²×Dic₂₁₈	C₂²	1	2	yes	—
11	#16812732	—	2	8	no	—
12	#16812721	—	2	4	no	—
13	C₂×Dic₄₃₆	—	2	4	no	—
14	C₂×Dic₄₃₆	C₂²	1	2	yes	—
15	C₂×Dic₄₃₆	C₂²	1	2	yes	—
16	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
17	C₂×Dic₄₃₆	C₂²	1	4	yes	—
18	C₂×#13182018	C₂²	1	2	yes	—
19	C₂×#13182018	C₂²	1	4	yes	—
20	C₂×D₈₇₂	C₂²	1	1	yes	characteristic
21	#16812699	C₂²	1	4	yes	—
22	#16812687	C₂²	1	4	yes	—
23	#16812683	—	2	4	no	—
24	#16812679	—	2	4	no	—
25	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
27	#117×C₁₀₉	C₂²	1	1	yes	characteristic
28	#97×C₁₀₉	C₂²	1	2	yes	—
29	C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
30	#13182029	—	2	4	no	—
31	#13182029	—	2	4	no	—
32	C₂×Dic₂₁₈	—	2	4	no	—
33	C₂×Dic₂₁₈	—	2	4	no	—
34	C₂×Dic₂₁₈	—	2	2	no	—
35	C₂×Dic₂₁₈	C₂³	1	4	yes	—
36	C₂×Dic₂₁₈	C₂³	1	2	yes	—
37	C₂×D₄₃₆	C₂³	1	2	yes	—
38	C₂×C₂²⋊D₁₀₉	—	2	2	no	—
39	C₂²×Dic₁₀₉	—	2	2	no	—
40	C₂×C₄×D₁₀₉	—	2	2	no	—
41	Dic₄₃₆	—	2	4	no	—
42	Dic₄₃₆	—	2	4	no	—
43	Dic₄₃₆	—	2	2	no	—
44	Dic₄₃₆	—	2	8	no	—
45	#13182018	—	2	4	no	—
46	#13182018	—	2	8	no	—
47	D₈₇₂	—	2	2	no	—
48	C₄⋊Dic₁₀₉	—	2	8	no	—
49	#13182006	—	2	8	no	—
50	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
51	M₁₆×C₁₀₉	D₄	1	4	yes	—
52	C₂×C₈×C₁₀₉	D₄	1	4	yes	—
53	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
55	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
56	C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
57	C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
58	C₄²×C₁₀₉	C₂³	1	2	yes	—
59	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×D₁₀₉	—	2	2	no	—
61	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
62	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
66	C₂²⋊D₁₀₉	—	4	4	no	—
67	C₂×Dic₁₀₉	—	2	4	no	—
68	C₂×Dic₁₀₉	—	2	2	no	—
69	C₂×Dic₁₀₉	—	4	4	no	—
70	D₄₃₆	—	2	2	no	—
71	D₄₃₆	—	2	2	no	—
72	C₄×D₁₀₉	—	4	4	no	—
73	Dic₂₁₈	—	2	4	no	—
74	Dic₂₁₈	—	2	2	no	—
75	Dic₂₁₈	—	2	4	no	—
76	Dic₂₁₈	—	2	2	no	—
77	Dic₂₁₈	—	4	4	no	—
78	C₈×C₁₀₉	C₂×D₄	1	2	yes	—
79	C₈×C₁₀₉	C₂×D₄	1	2	yes	—
80	C₈×C₁₀₉	C₂×D₄	1	4	yes	—
81	C₂²×C₁₀₉	—	2	2	no	—
82	C₂²×C₁₀₉	#127	1	2	yes	—
83	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
84	C₂×D₁₀₉	—	4	2	no	—
85	C₂×D₁₀₉	—	4	4	no	—
86	C₄×C₁₀₉	—	2	4	no	—
87	C₄×C₁₀₉	#127	1	2	yes	—
88	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
89	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
90	Dic₁₀₉	—	4	4	no	—
91	Dic₁₀₉	—	4	2	no	—
92	C₂×C₁₀₉	—	2	2	no	—
93	C₂×C₁₀₉	C₂×#127	1	1	yes	characteristic
94	C₂×C₁₀₉	#578	1	2	yes	—
95	D₁₀₉	—	8	2	no	—
96	#3156	—	109	1	no	maximal, Sylow
97	C₁₀₉	#3156	1	1	yes	characteristic, complemented, Sylow
98	C₂×#137	—	109	2	no	—
99	C₂×#135	—	109	1	no	—
100	C₂²×Q₁₆	—	109	1	no	—
101	#536	—	109	2	no	—
102	#497	—	109	4	no	—
103	#495	—	109	2	no	—
104	#494	—	109	2	no	—
105	#405	D₁₀₉	1	1	yes	characteristic, complemented
106	C₂×#41	—	109	2	no	—
107	C₂²×Q₈	—	109	2	no	—
108	#137	—	218	8	no	—
109	#135	—	218	4	no	—
110	C₂×Q₁₆	—	109	2	no	—
111	C₂×Q₁₆	—	109	2	no	—
112	C₂×Q₁₆	—	109	1	no	—
113	C₂×Q₁₆	—	109	4	no	—
114	C₂×Q₁₆	—	218	4	no	—
115	C₂×SD₁₆	—	109	2	no	—
116	C₂×SD₁₆	—	109	4	no	—
117	C₂×D₈	—	109	1	no	—
118	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
119	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
120	C₄⋊Q₈	—	109	4	no	—
121	#124	—	109	4	no	—
122	#123	—	218	4	no	—
123	C₂²⋊Q₈	—	218	4	no	—
124	#117	C₂×D₁₀₉	1	1	yes	characteristic
125	#97	C₂×D₁₀₉	1	2	yes	—
126	C₄×C₈	C₂×D₁₀₉	1	2	yes	—
127	#41	—	218	4	no	—
128	#41	—	218	4	no	—
129	C₂×Q₈	—	109	4	no	—
130	C₂×Q₈	—	109	2	no	—
131	C₂×Q₈	—	218	4	no	—
132	C₂×Q₈	—	218	4	no	—
133	C₂×Q₈	—	218	2	no	—
134	C₂×D₄	—	109	2	no	—
135	C₂×D₄	—	218	2	no	—
136	C₂²×C₄	—	218	2	no	—
137	C₂²×C₄	—	218	2	no	—
138	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
139	Q₁₆	—	218	4	no	—
140	Q₁₆	—	218	4	no	—
141	Q₁₆	—	218	2	no	—
142	Q₁₆	—	218	8	no	—
143	SD₁₆	—	218	4	no	—
144	SD₁₆	—	218	8	no	—
145	D₈	—	218	2	no	—
146	M₁₆	D₄₃₆	1	4	yes	—
147	C₂×C₈	D₄₃₆	1	4	yes	—
148	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
149	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
150	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
151	C₄⋊C₄	—	218	8	no	—
152	C₄⋊C₄	D₄₃₆	1	2	yes	—
153	C₂²⋊C₄	—	218	8	no	—
154	C₂²⋊C₄	D₄₃₆	1	2	yes	—
155	C₄²	C₂²×D₁₀₉	1	2	yes	—
156	C₂³	—	218	2	no	—
157	C₂³	C₂×D₄₃₆	1	1	yes	characteristic
158	Q₈	—	436	4	no	—
159	Q₈	—	218	4	no	—
160	Q₈	—	218	2	no	—
161	Q₈	—	218	4	no	—
162	Q₈	—	218	2	no	—
163	D₄	—	436	4	no	—
164	D₄	—	218	2	no	—
165	D₄	—	218	2	no	—
166	C₂×C₄	—	436	4	no	—
167	C₂×C₄	—	436	4	no	—
168	C₂×C₄	—	218	4	no	—
169	C₂×C₄	—	218	2	no	—
170	C₂×C₄	C₂×D₄₃₆	1	4	yes	—
171	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
172	C₂×C₄	C₂×D₄₃₆	1	2	yes	—
173	C₂×C₄	C₂×D₄₃₆	1	2	yes	—
174	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
175	C₈	C₂×D₄₃₆	1	2	yes	—
176	C₈	C₂×D₄₃₆	1	2	yes	—
177	C₈	C₂×D₄₃₆	1	4	yes	—
178	C₂²	—	2	2	no	—
179	C₂²	—	436	2	no	—
180	C₂²	—	436	4	no	—
181	C₂²	#16812696	1	2	yes	—
182	C₂²	C₂²×D₄₃₆	1	1	yes	characteristic
183	C₄	—	2	4	no	—
184	C₄	—	436	2	no	—
185	C₄	—	436	4	no	—
186	C₄	#16812696	1	2	yes	—
187	C₄	C₂²×D₄₃₆	1	1	yes	characteristic
188	C₄	C₂²×D₄₃₆	1	1	yes	characteristic
189	C₂	—	2	2	no	—
190	C₂	—	872	2	no	—
191	C₂	C₂×#16812696	1	1	yes	characteristic
192	C₂	#20295843	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812732

C₂

yes

maximal, complemented

C₂×#16812721

C₂

yes

maximal, characteristic

C₂²×Dic₄₃₆

C₂

yes

maximal, characteristic, complemented

#20295602

C₂

yes

maximal, complemented

#20295420

C₂

yes

maximal, complemented

#20295410

C₂

yes

maximal, complemented

#20295406

C₂

yes

maximal, complemented

#405×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×#13182029

C₂²

yes

—

C₂²×Dic₂₁₈

C₂²

yes

—

#16812732

—

#16812721

—

C₂×Dic₄₃₆

—

C₂×Dic₄₃₆

C₂²

yes

—

C₂×Dic₄₃₆

C₂²

yes

—

C₂×Dic₄₃₆

C₂²

yes

characteristic

C₂×Dic₄₃₆

C₂²

yes

—

C₂×#13182018

C₂²

yes

—

C₂×#13182018

C₂²

yes

—

C₂×D₈₇₂

C₂²

yes

characteristic

#16812699

C₂²

yes

—

#16812687

C₂²

yes

—

#16812683

—

#16812679

—

C₂×M₁₆×C₁₀₉

C₂²

yes

characteristic

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

#117×C₁₀₉

C₂²

yes

characteristic

#97×C₁₀₉

C₂²

yes

—

C₄×C₈×C₁₀₉

C₂²

yes

—

#13182029

—

#13182029

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

C₂³

yes

—

C₂×Dic₂₁₈

C₂³

yes

—

C₂×D₄₃₆

C₂³

yes

—

C₂×C₂²⋊D₁₀₉

—

C₂²×Dic₁₀₉

—

C₂×C₄×D₁₀₉

—

Dic₄₃₆

—

Dic₄₃₆

—

Dic₄₃₆

—

Dic₄₃₆

—

#13182018

—

#13182018

—

D₈₇₂

—

C₄⋊Dic₁₀₉

—

#13182006

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

D₄

yes

—

C₂×C₈×C₁₀₉

D₄

yes

—

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

—

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

D₄

yes

—

C₂²⋊C₄×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂²×D₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

D₄₃₆

—

D₄₃₆

—

C₄×D₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

C₈×C₁₀₉

C₂×D₄

yes

—

C₈×C₁₀₉

C₂×D₄

yes

—

C₈×C₁₀₉

C₂×D₄

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#127

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#127

yes

—

C₄×C₁₀₉

C₂²×D₄

yes

characteristic

C₄×C₁₀₉

C₂²×D₄

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×#127

yes

characteristic

C₂×C₁₀₉

#578

yes

—

D₁₀₉

—

#3156

—

109

maximal, Sylow

C₁₀₉

#3156

yes

characteristic, complemented, Sylow

C₂×#137

—

109

—

C₂×#135

—

109

—

100

C₂²×Q₁₆

—

109

—

101

#536

—

109

—

102

#497

—

109

—

103

#495

—

109

—

104

#494

—

109

—

105

#405

D₁₀₉

yes

characteristic, complemented

106

C₂×#41

—

109

—

107

C₂²×Q₈

—

109

—

108

#137

—

218

—

109

#135

—

218

—

110

C₂×Q₁₆

—

109

—

111

C₂×Q₁₆

—

109

—

112

C₂×Q₁₆

—

109

—

113

C₂×Q₁₆

—

109

—

114

C₂×Q₁₆

—

218

—

115

C₂×SD₁₆

—

109

—

116

C₂×SD₁₆

—

109

—

117

C₂×D₈

—

109

—

118

C₂×M₁₆

C₂×D₁₀₉

yes

characteristic

119

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

120

C₄⋊Q₈

—

109

—

121

#124

—

109

—

122

#123

—

218

—

123

C₂²⋊Q₈

—

218

—

124

#117

C₂×D₁₀₉

yes

characteristic

125

#97

C₂×D₁₀₉

yes

—

126

C₄×C₈

C₂×D₁₀₉

yes

—

127

#41

—

218

—

128

#41

—

218

—

129

C₂×Q₈

—

109

—

130

C₂×Q₈

—

109

—

131

C₂×Q₈

—

218

—

132

C₂×Q₈

—

218

—

133

C₂×Q₈

—

218

—

134

C₂×D₄

—

109

—

135

C₂×D₄

—

218

—

136

C₂²×C₄

—

218

—

137

C₂²×C₄

—

218

—

138

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

139

Q₁₆

—

218

—

140

Q₁₆

—

218

—

141

Q₁₆

—

218

—

142

Q₁₆

—

218

—

143

SD₁₆

—

218

—

144

SD₁₆

—

218

—

145

D₈

—

218

—

146

M₁₆

D₄₃₆

yes

—

147

C₂×C₈

D₄₃₆

yes

—

148

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

149

C₂×C₈

C₂²×D₁₀₉

yes

—

150

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

151

C₄⋊C₄

—

218

—

152

C₄⋊C₄

D₄₃₆

yes

—

153

C₂²⋊C₄

—

218

—

154

C₂²⋊C₄

D₄₃₆

yes

—

155

C₄²

C₂²×D₁₀₉

yes

—

156

C₂³

—

218

—

157

C₂³

C₂×D₄₃₆

yes

characteristic

158

Q₈

—

436

—

159

Q₈

—

218

—

160

Q₈

—

218

—

161

Q₈

—

218

—

162

Q₈

—

218

—

163

D₄

—

436

—

164

D₄

—

218

—

165

D₄

—

218

—

166

C₂×C₄

—

436

—

167

C₂×C₄

—

436

—

168

C₂×C₄

—

218

—

169

C₂×C₄

—

218

—

170

C₂×C₄

C₂×D₄₃₆

yes

—

171

C₂×C₄

C₂×D₄₃₆

yes

characteristic

172

C₂×C₄

C₂×D₄₃₆

yes

—

173

C₂×C₄

C₂×D₄₃₆

yes

—

174

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

175

C₈

C₂×D₄₃₆

yes

—

176

C₈

C₂×D₄₃₆

yes

—

177

C₈

C₂×D₄₃₆

yes

—

178

C₂²

—

179

C₂²

—

436

—

180

C₂²

—

436

—

181

C₂²

#16812696

yes

—

182

C₂²

C₂²×D₄₃₆

yes

characteristic

183

C₄

—

184

C₄

—

436

—

185

C₄

—

436

—

186

C₄

#16812696

yes

—

187

C₄

C₂²×D₄₃₆

yes

characteristic

188

C₄

C₂²×D₄₃₆

yes

characteristic

189

C₂

—

190

C₂

—

872

—

191

C₂

C₂×#16812696

yes

characteristic

192

C₂

#20295843

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₄₃₆
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#405×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×#16812721	C₂
hypercenter	#405	D₁₀₉
hypocenter	C₁₀₉	#3156
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

subgroup

G/H

center

C₂²

C₂²×D₄₃₆

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

#405×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂²×D₄

C₂

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ #117 ◁ #405

N	H ≅ G/N	type
1. C₂×#16812732	C₂	semidirect
3. C₂²×Dic₄₃₆	C₂	semidirect
4. #20295602	C₂	semidirect
5. #20295420	C₂	semidirect
6. #20295410	C₂	semidirect
7. #20295406	C₂	semidirect
8. #405×C₁₀₉	C₂	semidirect
97. C₁₀₉	#3156	semidirect
105. #405	D₁₀₉	semidirect

H ≅ G/N

type

1. C₂×#16812732

C₂

semidirect

3. C₂²×Dic₄₃₆

C₂

semidirect

4. #20295602

C₂

semidirect

5. #20295420

C₂

semidirect

6. #20295410

C₂

semidirect

7. #20295406

C₂

semidirect

8. #405×C₁₀₉

C₂

semidirect

97. C₁₀₉

#3156

semidirect

105. #405

D₁₀₉

semidirect

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

subgroup	index
C₂×#16812732	2
C₂×#16812721	2
C₂²×Dic₄₃₆	2
#20295602	2
#20295420	2
#20295410	2
#20295406	2
#405×C₁₀₉	2
#3156	109

subgroup

index

C₂²×Dic₄₃₆

#405×C₁₀₉

109

Subgroups — #28373526

Conjugacy classes of subgroups

Complements

Maximal subgroups