Group #28363944 — subgroups

Subgroups — #28363944

Order 13952.

276 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 83 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	C₂²×#13182019	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812634	C₂	1	1	yes	maximal, characteristic
3	#20295036	C₂	1	1	yes	maximal, characteristic
4	#20295035	C₂	1	1	yes	maximal, characteristic
5	C₂×#16812617	C₂	1	1	yes	maximal, characteristic
6	#20294984	C₂	1	1	yes	maximal, characteristic
7	C₂×#101×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₂₁₈	C₂²	1	1	yes	characteristic
9	C₂×#13182019	—	2	1	no	—
10	C₂×#13182019	—	2	1	no	—
11	C₂×#13182019	—	2	2	no	—
12	C₂×#13182019	C₄	1	4	yes	complemented
13	C₂×#13182013	C₂²	1	1	yes	characteristic
14	C₂²×#11724784	C₂²	1	1	yes	characteristic
15	#16812658	C₂²	1	1	yes	characteristic
16	#16812634	—	2	2	no	—
17	#16812624	—	2	2	no	—
18	#16812623	—	2	2	no	—
19	#16812617	—	2	2	no	—
20	#16812609	—	2	1	no	—
21	C₈×Dic₁₀₉	—	2	1	no	—
22	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
24	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
25	#101×C₁₀₉	—	2	2	no	—
26	C₂×Dic₂₁₈	—	2	1	no	—
27	C₂×Dic₂₁₈	—	2	1	no	—
28	C₂×Dic₂₁₈	—	2	1	no	—
29	C₂×Dic₂₁₈	—	2	1	no	—
30	C₂×Dic₂₁₈	—	2	1	no	—
31	C₂×Dic₂₁₈	C₂×C₄	1	2	yes	—
32	C₂²×Dic₁₀₉	—	2	1	no	—
33	#13182019	—	2	4	no	—
34	#13182019	—	2	4	no	—
35	#13182013	—	2	1	no	—
36	#13182013	—	2	1	no	—
37	C₂×#11724784	—	2	1	no	—
38	C₂×#11724784	C₂×C₄	1	2	yes	—
39	C₂×#11724784	D₄	1	1	yes	characteristic
40	C₂×#11724784	D₄	1	1	yes	characteristic
41	C₂×#11724784	D₄	1	2	yes	—
42	C₄⋊Dic₁₀₉	D₄	1	2	yes	—
43	C₂²⋊Dic₁₀₉	—	2	1	no	—
44	C₄×Dic₁₀₉	—	2	1	no	—
45	C₂×Q₈×C₁₀₉	—	2	1	no	—
46	C₂×Q₈×C₁₀₉	—	2	1	no	—
47	C₂×Q₈×C₁₀₉	—	2	1	no	—
48	C₂×Q₈×C₁₀₉	—	2	1	no	—
49	C₂×Q₈×C₁₀₉	—	2	1	no	—
50	C₂×Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
51	C₂²×C₄×C₁₀₉	—	2	1	no	—
52	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
53	M₁₆×C₁₀₉	—	2	1	no	—
54	M₁₆×C₁₀₉	—	2	1	no	—
55	M₁₆×C₁₀₉	—	2	1	no	—
56	M₁₆×C₁₀₉	D₄	1	2	yes	—
57	C₂×C₈×C₁₀₉	—	2	1	no	—
58	C₂×C₈×C₁₀₉	—	2	1	no	—
59	C₂³×C₁₀₉	#41	1	1	yes	characteristic
60	Q₈×C₁₀₉	—	2	2	no	—
61	Q₈×C₁₀₉	—	2	2	no	—
62	Q₈×C₁₀₉	—	4	1	no	—
63	Q₈×C₁₀₉	—	4	1	no	—
64	C₂×C₄×C₁₀₉	—	2	2	no	—
65	C₂×C₄×C₁₀₉	—	4	1	no	—
66	C₂×C₄×C₁₀₉	—	4	1	no	—
67	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
73	C₂×Dic₁₀₉	—	2	2	no	—
74	C₂×Dic₁₀₉	—	2	2	no	—
75	C₂×Dic₁₀₉	—	4	1	no	—
76	C₂×Dic₁₀₉	—	4	1	no	—
77	Dic₂₁₈	—	2	2	no	—
78	Dic₂₁₈	—	2	2	no	—
79	Dic₂₁₈	—	4	1	no	—
80	Dic₂₁₈	—	4	1	no	—
81	C₈×C₁₀₉	—	2	2	no	—
82	C₈×C₁₀₉	—	4	1	no	—
83	#11724784	—	4	1	no	—
84	#11724784	C₂²⋊C₄	1	2	yes	—
85	#11724784	C₂²⋊C₄	1	2	yes	—
86	C₂²×C₁₀₉	—	2	2	no	—
87	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
88	C₂²×C₁₀₉	#121	1	1	yes	characteristic
89	C₂²×C₁₀₉	#124	1	1	yes	characteristic
90	C₄×C₁₀₉	—	4	2	no	—
91	C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
93	C₄×C₁₀₉	#121	1	1	yes	characteristic
94	C₄×C₁₀₉	#127	1	1	yes	characteristic
95	Dic₁₀₉	—	4	2	no	—
96	Dic₁₀₉	—	4	2	no	—
97	C₂×C₁₀₉	—	2	1	no	—
98	C₂×C₁₀₉	—	2	1	no	—
99	C₂×C₁₀₉	#390	1	1	yes	characteristic
100	C₂×C₁₀₉	#473	1	2	yes	—
101	#1979	—	109	1	no	maximal, Sylow
102	C₁₀₉	#1979	1	1	yes	characteristic, complemented, Sylow
103	C₂²×Q₁₆	—	109	1	no	—
104	C₂×#108	—	109	1	no	—
105	#419	—	109	1	no	—
106	#419	—	109	1	no	—
107	C₂×#101	—	109	1	no	—
108	C₂×#101	D₁₀₉	1	1	yes	characteristic
109	#405	—	109	1	no	—
110	C₂²×Q₈	—	109	1	no	—
111	C₂²×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
112	C₂×Q₁₆	—	109	4	no	—
113	C₂×Q₁₆	—	218	1	no	—
114	C₂×Q₁₆	—	218	1	no	—
115	C₂×Q₁₆	—	218	2	no	—
116	C₂×M₁₆	—	109	1	no	—
117	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
118	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
119	C₂²×C₈	—	109	1	no	—
120	#117	—	109	1	no	—
121	#108	—	218	2	no	—
122	#103	—	218	2	no	—
123	#103	—	218	2	no	—
124	#101	—	2	2	no	—
125	#101	—	218	2	no	—
126	#97	—	218	1	no	—
127	C₄×C₈	—	218	1	no	—
128	C₂×Q₈	—	2	1	no	—
129	C₂×Q₈	—	2	1	no	—
130	C₂×Q₈	—	2	1	no	—
131	C₂×Q₈	—	2	1	no	—
132	C₂×Q₈	—	2	1	no	—
133	C₂×Q₈	—	109	2	no	—
134	C₂×Q₈	—	218	1	no	—
135	C₂×Q₈	—	218	1	no	—
136	C₂×Q₈	—	218	1	no	—
137	C₂×Q₈	—	218	1	no	—
138	C₂×Q₈	—	218	1	no	—
139	C₂×Q₈	C₄×D₁₀₉	1	2	yes	—
140	C₂²×C₄	—	2	1	no	—
141	C₂²×C₄	—	218	1	no	—
142	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
143	Q₁₆	—	218	4	no	—
144	Q₁₆	—	218	4	no	—
145	M₁₆	—	2	1	no	—
146	M₁₆	—	2	1	no	—
147	M₁₆	—	2	1	no	—
148	M₁₆	—	218	1	no	—
149	M₁₆	—	218	1	no	—
150	M₁₆	C₂²⋊D₁₀₉	1	2	yes	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	109	2	no	—
154	C₂×C₈	—	109	1	no	—
155	C₂×C₈	—	109	1	no	—
156	C₂×C₈	—	109	2	no	—
157	C₂×C₈	—	218	1	no	—
158	C₄⋊C₄	—	109	2	no	—
159	C₂²⋊C₄	—	218	1	no	—
160	C₄²	—	218	1	no	—
161	C₂³	#13182029	1	1	yes	characteristic
162	Q₈	—	2	2	no	—
163	Q₈	—	2	2	no	—
164	Q₈	—	4	1	no	—
165	Q₈	—	4	1	no	—
166	Q₈	—	436	1	no	—
167	Q₈	—	436	1	no	—
168	Q₈	—	218	2	no	—
169	Q₈	—	218	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	436	1	no	—
174	C₂×C₄	—	436	1	no	—
175	C₂×C₄	—	218	2	no	—
176	C₂×C₄	—	218	2	no	—
177	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
178	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
179	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
180	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
181	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
182	C₂×C₄	#13182028	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	—	4	1	no	—
185	C₈	—	109	2	no	—
186	C₈	—	109	2	no	—
187	C₈	—	436	1	no	—
188	C₂²	—	2	2	no	—
189	C₂²	C₄×C₂²⋊D₁₀₉	1	1	yes	characteristic
190	C₂²	#16812672	1	1	yes	characteristic
191	C₂²	#16812685	1	1	yes	characteristic
192	C₄	—	4	2	no	—
193	C₄	—	436	2	no	—
194	C₄	—	436	2	no	—
195	C₄	C₂²⋊C₄×D₁₀₉	1	1	yes	characteristic
196	C₄	#16812659	1	1	yes	characteristic
197	C₄	#16812671	1	1	yes	characteristic
198	C₄	#16812695	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#20294926	1	1	yes	characteristic
202	C₂	#20295300	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#13182019

C₂

yes

maximal, characteristic

C₂×#16812634

C₂

yes

maximal, characteristic

#20295036

C₂

yes

maximal, characteristic

#20295035

C₂

yes

maximal, characteristic

C₂×#16812617

C₂

yes

maximal, characteristic

#20294984

C₂

yes

maximal, characteristic

C₂×#101×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₈

C₂²

yes

—

—

—

C₄

yes

complemented

C₂×#13182013

C₂²

yes

characteristic

C₂²×#11724784

C₂²

yes

characteristic

#16812658

C₂²

yes

—

—

—

—

—

C₈×Dic₁₀₉

—

C₂²×Q₈×C₁₀₉

C₂²

yes

characteristic

C₂×M₁₆×C₁₀₉

C₂²

yes

characteristic

C₂×M₁₆×C₁₀₉

C₂²

yes

characteristic

#101×C₁₀₉

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

C₂×C₄

yes

—

C₂²×Dic₁₀₉

—

—

—

—

—

—

C₂×C₄

yes

—

C₂×#11724784

D₄

yes

characteristic

C₂×#11724784

D₄

yes

characteristic

C₂×#11724784

D₄

yes

—

C₄⋊Dic₁₀₉

D₄

yes

—

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

D₄

yes

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂³×C₁₀₉

#41

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

#11724784

C₂²⋊C₄

yes

—

#11724784

C₂²⋊C₄

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₄×D₄

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₀₉

C₄×D₄

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#390

yes

characteristic

100

C₂×C₁₀₉

#473

yes

—

101

#1979

—

109

maximal, Sylow

102

C₁₀₉

#1979

yes

characteristic, complemented, Sylow

103

C₂²×Q₁₆

—

109

—

104

C₂×#108

—

109

—

105

#419

—

109

—

106

#419

—

109

—

107

C₂×#101

—

109

—

108

C₂×#101

D₁₀₉

yes

characteristic

109

#405

—

109

—

110

C₂²×Q₈

—

109

—

111

C₂²×Q₈

C₂×D₁₀₉

yes

characteristic

112

C₂×Q₁₆

—

109

—

113

C₂×Q₁₆

—

218

—

114

C₂×Q₁₆

—

218

—

115

C₂×Q₁₆

—

218

—

116

C₂×M₁₆

—

109

—

117

C₂×M₁₆

C₂×D₁₀₉

yes

characteristic

118

C₂×M₁₆

C₂×D₁₀₉

yes

characteristic

119

C₂²×C₈

—

109

—

120

#117

—

109

—

121

#108

—

218

—

122

#103

—

218

—

123

#103

—

218

—

124

#101

—

125

#101

—

218

—

126

#97

—

218

—

127

C₄×C₈

—

218

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

109

—

134

C₂×Q₈

—

218

—

135

C₂×Q₈

—

218

—

136

C₂×Q₈

—

218

—

137

C₂×Q₈

—

218

—

138

C₂×Q₈

—

218

—

139

C₂×Q₈

C₄×D₁₀₉

yes

—

140

C₂²×C₄

—

141

C₂²×C₄

—

218

—

142

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

143

Q₁₆

—

218

—

144

Q₁₆

—

218

—

145

M₁₆

—

146

M₁₆

—

147

M₁₆

—

148

M₁₆

—

218

—

149

M₁₆

—

218

—

150

M₁₆

C₂²⋊D₁₀₉

yes

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

109

—

154

C₂×C₈

—

109

—

155

C₂×C₈

—

109

—

156

C₂×C₈

—

109

—

157

C₂×C₈

—

218

—

158

C₄⋊C₄

—

109

—

159

C₂²⋊C₄

—

218

—

160

C₄²

—

218

—

161

C₂³

#13182029

yes

characteristic

162

Q₈

—

163

Q₈

—

164

Q₈

—

165

Q₈

—

166

Q₈

—

436

—

167

Q₈

—

436

—

168

Q₈

—

218

—

169

Q₈

—

218

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

436

—

174

C₂×C₄

—

436

—

175

C₂×C₄

—

218

—

176

C₂×C₄

—

218

—

177

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

178

C₂×C₄

D₄×D₁₀₉

yes

characteristic

179

C₂×C₄

D₄×D₁₀₉

yes

characteristic

180

C₂×C₄

D₄×D₁₀₉

yes

characteristic

181

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

182

C₂×C₄

#13182028

yes

characteristic

183

C₈

—

184

C₈

—

185

C₈

—

109

—

186

C₈

—

109

—

187

C₈

—

436

—

188

C₂²

—

189

C₂²

C₄×C₂²⋊D₁₀₉

yes

characteristic

190

C₂²

#16812672

yes

characteristic

191

C₂²

#16812685

yes

characteristic

192

C₄

—

193

C₄

—

436

—

194

C₄

—

436

—

195

C₄

C₂²⋊C₄×D₁₀₉

yes

characteristic

196

C₄

#16812659

yes

characteristic

197

C₄

#16812671

yes

characteristic

198

C₄

#16812695

yes

characteristic

199

C₂

—

200

C₂

—

201

C₂

#20294926

yes

characteristic

202

C₂

#20295300

yes

—

	subgroup	G/H
center	C₂²	#16812672
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂×#101×C₁₀₉	C₂
socle	C₂²×C₁₀₉	#121
prime radical	C₂³×C₁₀₉	#41
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂×#101	D₁₀₉
hypocenter	C₁₀₉	#1979
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂²

#16812672

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₂×#101×C₁₀₉

C₂

C₂²×C₁₀₉

C₂³×C₁₀₉

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
12. C₂×#13182019	C₄	semidirect
102. C₁₀₉	#1979	semidirect

H ≅ G/N

type

12. C₂×#13182019

C₄

semidirect

102. C₁₀₉

#1979

semidirect

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

subgroup	index
C₂²×#13182019	2
C₂×#16812634	2
#20295036	2
#20295035	2
C₂×#16812617	2
#20294984	2
C₂×#101×C₁₀₉	2
#1979	109

subgroup

index

C₂×#101×C₁₀₉

109

Subgroups — #28363944

Conjugacy classes of subgroups

Complements

Maximal subgroups