Group #28373486 — subgroups

Subgroups — #28373486

Order 13952.

552 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 145 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×#16812728	C₂	1	4	yes	maximal, complemented
2	#20295574	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295417	C₂	1	4	yes	maximal, complemented
5	#20295406	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295405	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295398	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#13182012	C₂	1	1	yes	maximal, characteristic, complemented
9	#492×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13182028	C₂²	1	2	yes	—
11	C₂×Q₈×D₁₀₉	C₂²	1	2	yes	—
12	#16812728	—	2	16	no	—
13	C₂×Dic₄₃₆	C₂²	1	4	yes	—
14	C₂×#13182019	C₂²	1	4	yes	—
15	C₂×#13182017	C₂²	1	4	yes	—
16	C₂×#13182012	C₂²	1	2	yes	—
17	#16812699	C₂²	1	1	yes	characteristic
18	#16812699	C₂²	1	1	yes	characteristic
19	#16812698	—	2	2	no	—
20	#16812697	C₂²	1	2	yes	—
21	#16812686	C₂²	1	2	yes	—
22	#16812670	—	2	2	no	—
23	D₄×Dic₁₀₉	—	2	2	no	—
24	C₄⋊C₄×D₁₀₉	—	2	2	no	—
25	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
26	#16812609	C₂²	1	2	yes	—
27	C₄×#11724784	C₂²	1	1	yes	characteristic
28	C₂×SD₁₆×C₁₀₉	C₂²	1	4	yes	—
29	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
30	#127×C₁₀₉	C₂²	1	1	yes	characteristic
31	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
32	#13182028	—	2	8	no	—
33	C₂×Dic₂₁₈	—	2	2	no	—
34	C₂×Dic₂₁₈	C₂³	1	2	yes	—
35	C₂×Dic₂₁₈	C₂³	1	2	yes	—
36	Q₈×D₁₀₉	—	2	8	no	—
37	C₂×C₂²⋊D₁₀₉	—	2	2	no	—
38	C₂²×Dic₁₀₉	—	2	2	no	—
39	C₂×C₄×D₁₀₉	—	2	2	no	—
40	C₂×C₄×D₁₀₉	D₄	1	2	yes	—
41	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
42	Dic₄₃₆	—	2	8	no	—
43	#13182019	—	2	8	no	—
44	#13182017	—	2	8	no	—
45	#13182012	D₄	1	8	yes	complemented
46	C₂×#11724784	C₂³	1	2	yes	—
47	C₄⋊Dic₁₀₉	—	2	2	no	—
48	C₄⋊Dic₁₀₉	—	2	2	no	—
49	#13182008	—	2	4	no	—
50	C₂²⋊Dic₁₀₉	—	2	4	no	—
51	C₄×Dic₁₀₉	D₄	1	2	yes	—
52	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
53	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
54	C₂×D₄×C₁₀₉	—	2	2	no	—
55	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
56	SD₁₆×C₁₀₉	—	2	8	no	—
57	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
58	C₄⋊C₄×C₁₀₉	—	2	2	no	—
59	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
60	C₂³×C₁₀₉	—	2	2	no	—
61	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
62	Q₈×C₁₀₉	—	2	4	no	—
63	D₄×C₁₀₉	—	2	4	no	—
64	D₄×C₁₀₉	—	4	4	no	—
65	C₂×C₄×C₁₀₉	—	2	2	no	—
66	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
68	C₂²⋊D₁₀₉	—	4	4	no	—
69	C₂×Dic₁₀₉	—	2	2	no	—
70	C₂×Dic₁₀₉	—	2	2	no	—
71	C₂×Dic₁₀₉	—	4	4	no	—
72	C₂×Dic₁₀₉	C₂×D₄	1	2	yes	—
73	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
74	C₄×D₁₀₉	—	2	4	no	—
75	C₄×D₁₀₉	—	4	4	no	—
76	C₄×D₁₀₉	C₂×D₄	1	4	yes	—
77	Dic₂₁₈	—	2	4	no	—
78	Dic₂₁₈	—	2	4	no	—
79	Dic₂₁₈	—	4	4	no	—
80	C₈×C₁₀₉	C₂×D₄	1	4	yes	—
81	#11724784	C₂×D₄	1	4	yes	—
82	C₂²×C₁₀₉	—	4	2	no	—
83	C₂²×C₁₀₉	—	4	4	no	—
84	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
85	C₂×D₁₀₉	—	2	2	no	—
86	C₂×D₁₀₉	#127	1	2	yes	—
87	C₄×C₁₀₉	—	4	2	no	—
88	C₄×C₁₀₉	#137	1	4	yes	complemented
89	C₄×C₁₀₉	C₂²×D₄	1	2	yes	—
90	Dic₁₀₉	—	2	2	no	—
91	Dic₁₀₉	—	4	2	no	—
92	Dic₁₀₉	—	4	2	no	—
93	Dic₁₀₉	#127	1	2	yes	—
94	C₂×C₁₀₉	—	8	2	no	—
95	C₂×C₁₀₉	C₂×#127	1	1	yes	characteristic
96	C₂×C₁₀₉	C₂×#137	1	2	yes	—
97	D₁₀₉	—	2	2	no	—
98	#3152	—	109	1	no	maximal, Sylow
99	C₁₀₉	#3152	1	1	yes	characteristic, complemented, Sylow
100	C₂×#137	—	109	4	no	—
101	#532	—	109	1	no	—
102	C₂×C₄⋊Q₈	—	109	1	no	—
103	#497	—	109	4	no	—
104	#494	—	109	1	no	—
105	#494	—	109	1	no	—
106	#492	—	109	1	no	—
107	#492	D₁₀₉	1	1	yes	characteristic, complemented
108	C₄×M₁₆	—	109	1	no	—
109	C₂×#41	—	109	2	no	—
110	C₂²×Q₈	—	109	2	no	—
111	#137	—	218	16	no	—
112	C₂×Q₁₆	—	109	4	no	—
113	C₂×Q₁₆	—	109	4	no	—
114	C₂×SD₁₆	—	109	4	no	—
115	C₂×SD₁₆	C₂×D₁₀₉	1	4	yes	—
116	C₂×M₁₆	—	109	2	no	—
117	C₄⋊Q₈	—	109	2	no	—
118	C₄⋊Q₈	—	109	1	no	—
119	C₄⋊Q₈	—	109	1	no	—
120	C₄⋊Q₈	—	218	2	no	—
121	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
122	#127	C₂×D₁₀₉	1	1	yes	characteristic
123	#124	—	109	2	no	—
124	#121	—	218	2	no	—
125	C₄×D₄	—	218	2	no	—
126	C₂×C₄⋊C₄	—	218	2	no	—
127	C₂×C₄²	—	109	1	no	—
128	#97	—	109	2	no	—
129	C₄×C₈	—	109	1	no	—
130	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
131	#41	—	218	8	no	—
132	C₂×Q₈	—	109	2	no	—
133	C₂×Q₈	—	109	2	no	—
134	C₂×Q₈	—	218	2	no	—
135	C₂×Q₈	—	218	8	no	—
136	C₂×Q₈	C₂²×D₁₀₉	1	2	yes	—
137	C₂×D₄	—	2	2	no	—
138	C₂×D₄	—	218	2	no	—
139	C₂×D₄	C₂²×D₁₀₉	1	2	yes	—
140	C₂²×C₄	—	109	1	no	—
141	C₂²×C₄	—	109	2	no	—
142	C₂²×C₄	—	218	2	no	—
143	C₂²×C₄	—	218	2	no	—
144	Q₁₆	—	218	8	no	—
145	Q₁₆	—	218	8	no	—
146	SD₁₆	—	2	8	no	—
147	SD₁₆	—	218	8	no	—
148	M₁₆	—	109	8	no	—
149	C₂×C₈	—	109	2	no	—
150	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	218	4	no	—
153	C₄⋊C₄	—	218	2	no	—
154	C₄⋊C₄	—	218	2	no	—
155	C₂²⋊C₄	—	218	4	no	—
156	C₄²	—	109	2	no	—
157	C₄²	—	109	1	no	—
158	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
159	C₂³	—	2	2	no	—
160	C₂³	—	109	1	no	—
161	Q₈	—	2	4	no	—
162	Q₈	—	436	4	no	—
163	Q₈	—	218	4	no	—
164	Q₈	—	218	4	no	—
165	D₄	—	2	4	no	—
166	D₄	—	4	4	no	—
167	D₄	—	436	4	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	109	1	no	—
170	C₂×C₄	—	109	4	no	—
171	C₂×C₄	—	109	2	no	—
172	C₂×C₄	—	436	4	no	—
173	C₂×C₄	—	436	4	no	—
174	C₂×C₄	—	218	2	no	—
175	C₂×C₄	—	218	2	no	—
176	C₂×C₄	—	218	4	no	—
177	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
178	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
179	C₈	—	109	4	no	—
180	C₈	D₄×D₁₀₉	1	4	yes	—
181	C₂²	—	4	4	no	—
182	C₂²	—	4	2	no	—
183	C₂²	—	109	2	no	—
184	C₂²	—	218	2	no	—
185	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
186	C₄	—	4	2	no	—
187	C₄	—	109	2	no	—
188	C₄	—	436	2	no	—
189	C₄	—	436	2	no	—
190	C₄	—	218	2	no	—
191	C₄	#16812728	1	4	yes	complemented
192	C₄	C₂×D₄×D₁₀₉	1	2	yes	—
193	C₂	—	8	2	no	—
194	C₂	—	218	2	no	—
195	C₂	#127×D₁₀₉	1	1	yes	characteristic
196	C₂	C₂×#16812728	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812728

C₂

yes

maximal, complemented

#20295574

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₁₀₉

C₂

yes

maximal, characteristic, complemented

#20295417

C₂

yes

maximal, complemented

#20295406

C₂

yes

maximal, characteristic, complemented

#20295405

C₂

yes

maximal, characteristic, complemented

#20295398

C₂

yes

maximal, characteristic, complemented

C₄×#13182012

C₂

yes

maximal, characteristic, complemented

#492×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×#13182028

C₂²

yes

—

C₂×Q₈×D₁₀₉

C₂²

yes

—

#16812728

—

C₂×Dic₄₃₆

C₂²

yes

—

C₂×#13182019

C₂²

yes

—

C₂×#13182017

C₂²

yes

—

C₂×#13182012

C₂²

yes

—

#16812699

C₂²

yes

characteristic

#16812699

C₂²

yes

characteristic

#16812698

—

#16812697

C₂²

yes

—

#16812686

C₂²

yes

—

#16812670

—

D₄×Dic₁₀₉

—

C₄⋊C₄×D₁₀₉

—

C₄²×D₁₀₉

C₂²

yes

characteristic

#16812609

C₂²

yes

—

C₄×#11724784

C₂²

yes

characteristic

C₂×SD₁₆×C₁₀₉

C₂²

yes

—

C₄⋊Q₈×C₁₀₉

C₂²

yes

characteristic

#127×C₁₀₉

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

#13182028

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

C₂³

yes

—

C₂×Dic₂₁₈

C₂³

yes

—

Q₈×D₁₀₉

—

C₂×C₂²⋊D₁₀₉

—

C₂²×Dic₁₀₉

—

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

D₄

yes

—

C₂×C₄×D₁₀₉

C₂³

yes

characteristic

Dic₄₃₆

—

#13182019

—

#13182017

—

#13182012

D₄

yes

complemented

C₂×#11724784

C₂³

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

#13182008

—

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

D₄

yes

—

C₄×Dic₁₀₉

C₂³

yes

characteristic

C₂×Q₈×C₁₀₉

C₂³

yes

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

C₂³

yes

—

SD₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂³

yes

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂³×C₁₀₉

—

C₂²×D₁₀₉

C₂×D₄

yes

characteristic

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×D₄

yes

—

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂×D₄

yes

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

C₈×C₁₀₉

C₂×D₄

yes

—

#11724784

C₂×D₄

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

#127

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#137

yes

complemented

C₄×C₁₀₉

C₂²×D₄

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

#127

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×#127

yes

characteristic

C₂×C₁₀₉

C₂×#137

yes

—

D₁₀₉

—

#3152

—

109

maximal, Sylow

C₁₀₉

#3152

yes

characteristic, complemented, Sylow

100

C₂×#137

—

109

—

101

#532

—

109

—

102

C₂×C₄⋊Q₈

—

109

—

103

#497

—

109

—

104

#494

—

109

—

105

#494

—

109

—

106

#492

—

109

—

107

#492

D₁₀₉

yes

characteristic, complemented

108

C₄×M₁₆

—

109

—

109

C₂×#41

—

109

—

110

C₂²×Q₈

—

109

—

111

#137

—

218

—

112

C₂×Q₁₆

—

109

—

113

C₂×Q₁₆

—

109

—

114

C₂×SD₁₆

—

109

—

115

C₂×SD₁₆

C₂×D₁₀₉

yes

—

116

C₂×M₁₆

—

109

—

117

C₄⋊Q₈

—

109

—

118

C₄⋊Q₈

—

109

—

119

C₄⋊Q₈

—

109

—

120

C₄⋊Q₈

—

218

—

121

C₄⋊Q₈

C₂×D₁₀₉

yes

characteristic

122

#127

C₂×D₁₀₉

yes

characteristic

123

#124

—

109

—

124

#121

—

218

—

125

C₄×D₄

—

218

—

126

C₂×C₄⋊C₄

—

218

—

127

C₂×C₄²

—

109

—

128

#97

—

109

—

129

C₄×C₈

—

109

—

130

C₄×C₈

C₂×D₁₀₉

yes

characteristic

131

#41

—

218

—

132

C₂×Q₈

—

109

—

133

C₂×Q₈

—

109

—

134

C₂×Q₈

—

218

—

135

C₂×Q₈

—

218

—

136

C₂×Q₈

C₂²×D₁₀₉

yes

—

137

C₂×D₄

—

138

C₂×D₄

—

218

—

139

C₂×D₄

C₂²×D₁₀₉

yes

—

140

C₂²×C₄

—

109

—

141

C₂²×C₄

—

109

—

142

C₂²×C₄

—

218

—

143

C₂²×C₄

—

218

—

144

Q₁₆

—

218

—

145

Q₁₆

—

218

—

146

SD₁₆

—

147

SD₁₆

—

218

—

148

M₁₆

—

109

—

149

C₂×C₈

—

109

—

150

C₂×C₈

C₂²×D₁₀₉

yes

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

218

—

153

C₄⋊C₄

—

218

—

154

C₄⋊C₄

—

218

—

155

C₂²⋊C₄

—

218

—

156

C₄²

—

109

—

157

C₄²

—

109

—

158

C₄²

C₂²×D₁₀₉

yes

characteristic

159

C₂³

—

160

C₂³

—

109

—

161

Q₈

—

162

Q₈

—

436

—

163

Q₈

—

218

—

164

Q₈

—

218

—

165

D₄

—

166

D₄

—

167

D₄

—

436

—

168

C₂×C₄

—

169

C₂×C₄

—

109

—

170

C₂×C₄

—

109

—

171

C₂×C₄

—

109

—

172

C₂×C₄

—

436

—

173

C₂×C₄

—

436

—

174

C₂×C₄

—

218

—

175

C₂×C₄

—

218

—

176

C₂×C₄

—

218

—

177

C₂×C₄

D₄×D₁₀₉

yes

—

178

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

179

C₈

—

109

—

180

C₈

D₄×D₁₀₉

yes

—

181

C₂²

—

182

C₂²

—

183

C₂²

—

109

—

184

C₂²

—

218

—

185

C₂²

C₂×D₄×D₁₀₉

yes

characteristic

186

C₄

—

187

C₄

—

109

—

188

C₄

—

436

—

189

C₄

—

436

—

190

C₄

—

218

—

191

C₄

#16812728

yes

complemented

192

C₄

C₂×D₄×D₁₀₉

yes

—

193

C₂

—

194

C₂

—

218

—

195

C₂

#127×D₁₀₉

yes

characteristic

196

C₂

C₂×#16812728

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#492×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#20295574	C₂
hypercenter	#492	D₁₀₉
hypocenter	C₁₀₉	#3152
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

C₂×D₄×D₁₀₉

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

#492×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂²×D₄

C₂

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. C₂×#16812728	C₂	semidirect
3. C₄⋊Q₈×D₁₀₉	C₂	semidirect
4. #20295417	C₂	semidirect
5. #20295406	C₂	semidirect
6. #20295405	C₂	semidirect
7. #20295398	C₂	semidirect
8. C₄×#13182012	C₂	semidirect
9. #492×C₁₀₉	C₂	semidirect
45. #13182012	D₄	semidirect
88. C₄×C₁₀₉	#137	semidirect
99. C₁₀₉	#3152	semidirect
107. #492	D₁₀₉	semidirect
191. C₄	#16812728	semidirect

H ≅ G/N

type

1. C₂×#16812728

C₂

semidirect

3. C₄⋊Q₈×D₁₀₉

C₂

semidirect

4. #20295417

C₂

semidirect

5. #20295406

C₂

semidirect

6. #20295405

C₂

semidirect

7. #20295398

C₂

semidirect

8. C₄×#13182012

C₂

semidirect

9. #492×C₁₀₉

C₂

semidirect

45. #13182012

D₄

semidirect

88. C₄×C₁₀₉

#137

semidirect

99. C₁₀₉

#3152

semidirect

107. #492

D₁₀₉

semidirect

191. C₄

#16812728

semidirect

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

subgroup	index
C₂×#16812728	2
#20295574	2
C₄⋊Q₈×D₁₀₉	2
#20295417	2
#20295406	2
#20295405	2
#20295398	2
C₄×#13182012	2
#492×C₁₀₉	2
#3152	109

subgroup

index

C₄⋊Q₈×D₁₀₉

#492×C₁₀₉

109

Subgroups — #28373486

Conjugacy classes of subgroups

Complements

Maximal subgroups