Group #38769764 — subgroups

Subgroups — #38769764

Order 25472.

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×#15691876	C₂	1	1	yes	maximal, characteristic
2	C₂×#20263890	C₂	1	1	yes	maximal, characteristic
3	#27164026	C₂	1	1	yes	maximal, characteristic
4	C₂×#20263874	C₂	1	1	yes	maximal, characteristic
5	C₂×#20263873	C₂	1	1	yes	maximal, characteristic
6	#27163974	C₂	1	1	yes	maximal, characteristic
7	#419×C₁₉₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₃₉₈	C₂²	1	1	yes	characteristic
9	C₂×#15691876	—	2	1	no	—
10	C₂×#15691876	—	2	1	no	—
11	C₂×#15691876	—	2	2	no	—
12	C₂×#15691876	C₄	1	4	yes	complemented
13	C₂×#15691870	C₂²	1	1	yes	characteristic
14	C₂×#15691870	C₂²	1	1	yes	characteristic
15	C₂²×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
16	#20263890	—	2	2	no	—
17	#20263878	—	2	2	no	—
18	#20263874	—	2	2	no	—
19	#20263873	—	2	2	no	—
20	#20263866	—	2	1	no	—
21	C₄×C₁₉₉⋊C₈	—	2	1	no	—
22	C₂²×Q₈×C₁₉₉	C₂²	1	1	yes	characteristic
23	C₂×M₁₆×C₁₉₉	C₂²	1	1	yes	characteristic
24	#117×C₁₉₉	C₂²	1	1	yes	characteristic
25	#103×C₁₉₉	—	2	2	no	—
26	C₂×Dic₃₉₈	—	2	1	no	—
27	C₂×Dic₃₉₈	—	2	1	no	—
28	C₂×Dic₃₉₈	—	2	1	no	—
29	C₂×Dic₃₉₈	—	2	1	no	—
30	C₂×Dic₃₉₈	—	2	1	no	—
31	C₂×Dic₃₉₈	C₂×C₄	1	2	yes	—
32	C₂²×Dic₁₉₉	—	2	1	no	—
33	#15691876	—	2	4	no	—
34	#15691876	—	2	4	no	—
35	#15691870	—	2	1	no	—
36	#15691870	—	2	1	no	—
37	#15691870	—	2	1	no	—
38	#15691870	D₄	1	2	yes	—
39	C₂×C₁₉₉⋊C₈	—	2	1	no	—
40	C₂×C₁₉₉⋊C₈	—	2	1	no	—
41	C₂×C₁₉₉⋊C₈	C₂×C₄	1	2	yes	—
42	C₂×C₁₉₉⋊C₈	D₄	1	1	yes	characteristic
43	C₂×C₁₉₉⋊C₈	D₄	1	1	yes	characteristic
44	C₂×C₁₉₉⋊C₈	D₄	1	2	yes	—
45	C₂×Q₈×C₁₉₉	—	2	1	no	—
46	C₂×Q₈×C₁₉₉	—	2	1	no	—
47	C₂×Q₈×C₁₉₉	—	2	1	no	—
48	C₂×Q₈×C₁₉₉	—	2	1	no	—
49	C₂×Q₈×C₁₉₉	—	2	1	no	—
50	C₂×Q₈×C₁₉₉	C₂×C₄	1	2	yes	—
51	C₂²×C₄×C₁₉₉	—	2	1	no	—
52	C₂²×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
53	M₁₆×C₁₉₉	—	2	1	no	—
54	M₁₆×C₁₉₉	—	2	1	no	—
55	C₂×C₈×C₁₉₉	—	2	1	no	—
56	C₄⋊C₄×C₁₉₉	D₄	1	2	yes	—
57	C₂²⋊C₄×C₁₉₉	—	2	1	no	—
58	C₄²×C₁₉₉	—	2	1	no	—
59	Dic₃₉₈	—	2	2	no	—
60	Dic₃₉₈	—	2	2	no	—
61	Dic₃₉₈	—	4	1	no	—
62	Dic₃₉₈	—	4	1	no	—
63	C₂×Dic₁₉₉	—	2	2	no	—
64	C₂×Dic₁₉₉	—	4	1	no	—
65	C₂×Dic₁₉₉	—	4	1	no	—
66	C₁₉₉⋊C₈	—	2	2	no	—
67	C₁₉₉⋊C₈	—	4	1	no	—
68	C₁₉₉⋊C₈	C₂²⋊C₄	1	2	yes	—
69	C₁₉₉⋊C₈	C₂²⋊C₄	1	2	yes	—
70	C₂³×C₁₉₉	#41	1	1	yes	characteristic
71	Q₈×C₁₉₉	—	2	2	no	—
72	Q₈×C₁₉₉	—	2	2	no	—
73	Q₈×C₁₉₉	—	4	1	no	—
74	Q₈×C₁₉₉	—	4	1	no	—
75	C₂×C₄×C₁₉₉	—	2	2	no	—
76	C₂×C₄×C₁₉₉	—	2	2	no	—
77	C₂×C₄×C₁₉₉	—	4	1	no	—
78	C₂×C₄×C₁₉₉	—	4	1	no	—
79	C₂×C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×C₁₉₉	#41	1	1	yes	characteristic
85	C₈×C₁₉₉	—	4	1	no	—
86	C₂²×C₁₉₉	—	2	2	no	—
87	C₂²×C₁₉₉	C₄×D₄	1	1	yes	characteristic
88	C₂²×C₁₉₉	#121	1	1	yes	characteristic
89	C₂²×C₁₉₉	#124	1	1	yes	characteristic
90	C₄×C₁₉₉	—	4	2	no	—
91	C₄×C₁₉₉	—	4	2	no	—
92	C₄×C₁₉₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₄×C₁₉₉	C₄×D₄	1	1	yes	characteristic
94	C₄×C₁₉₉	#121	1	1	yes	characteristic
95	C₄×C₁₉₉	#127	1	1	yes	characteristic
96	Dic₁₉₉	—	4	2	no	—
97	C₂×C₁₉₉	—	2	1	no	—
98	C₂×C₁₉₉	—	2	1	no	—
99	C₂×C₁₉₉	#390	1	1	yes	characteristic
100	C₂×C₁₉₉	#473	1	2	yes	—
101	C₁₉₉	#1979	1	1	yes	characteristic, complemented, Sylow
102	#1979	—	199	1	no	maximal, Sylow
103	C₂²×Q₁₆	—	199	1	no	—
104	C₂×#108	—	199	1	no	—
105	#419	—	199	1	no	—
106	#419	D₁₉₉	1	1	yes	characteristic
107	C₂×#101	—	199	1	no	—
108	C₂×#101	—	199	1	no	—
109	#405	—	199	1	no	—
110	C₂²×Q₈	—	199	1	no	—
111	C₂²×Q₈	C₂×D₁₉₉	1	1	yes	characteristic
112	C₂×Q₁₆	—	398	1	no	—
113	C₂×Q₁₆	—	398	1	no	—
114	C₂×Q₁₆	—	398	2	no	—
115	C₂×Q₁₆	—	199	4	no	—
116	C₂×M₁₆	—	199	1	no	—
117	C₂×M₁₆	—	199	1	no	—
118	C₂×M₁₆	C₂×D₁₉₉	1	1	yes	characteristic
119	C₂²×C₈	—	199	1	no	—
120	#117	C₂×D₁₉₉	1	1	yes	characteristic
121	#108	—	398	2	no	—
122	#103	—	2	2	no	—
123	#103	—	398	2	no	—
124	#101	—	398	2	no	—
125	#101	—	398	2	no	—
126	#97	—	398	1	no	—
127	C₄×C₈	—	398	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	—	398	1	no	—
134	C₂×Q₈	—	398	1	no	—
135	C₂×Q₈	—	398	1	no	—
136	C₂×Q₈	—	398	1	no	—
137	C₂×Q₈	—	398	1	no	—
138	C₂×Q₈	—	199	2	no	—
139	C₂×Q₈	C₄×D₁₉₉	1	2	yes	—
140	C₂²×C₄	—	2	1	no	—
141	C₂²×C₄	—	398	1	no	—
142	C₂²×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
143	Q₁₆	—	398	4	no	—
144	Q₁₆	—	398	4	no	—
145	M₁₆	—	2	1	no	—
146	M₁₆	—	2	1	no	—
147	M₁₆	—	398	1	no	—
148	M₁₆	—	398	1	no	—
149	M₁₆	—	398	1	no	—
150	M₁₆	—	199	2	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	398	1	no	—
153	C₂×C₈	—	398	1	no	—
154	C₂×C₈	—	199	2	no	—
155	C₂×C₈	—	199	1	no	—
156	C₂×C₈	—	199	1	no	—
157	C₂×C₈	—	199	2	no	—
158	C₄⋊C₄	C₂²⋊D₁₉₉	1	2	yes	—
159	C₂²⋊C₄	—	2	1	no	—
160	C₄²	—	2	1	no	—
161	C₂³	#15691885	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	—	796	1	no	—
167	Q₈	—	796	1	no	—
168	Q₈	—	398	2	no	—
169	Q₈	—	398	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	—	796	1	no	—
175	C₂×C₄	—	796	1	no	—
176	C₂×C₄	—	398	2	no	—
177	C₂×C₄	C₂×C₄×D₁₉₉	1	1	yes	characteristic
178	C₂×C₄	D₄×D₁₉₉	1	1	yes	characteristic
179	C₂×C₄	D₄×D₁₉₉	1	1	yes	characteristic
180	C₂×C₄	D₄×D₁₉₉	1	1	yes	characteristic
181	C₂×C₄	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
182	C₂×C₄	#15691886	1	1	yes	characteristic
183	C₈	—	4	1	no	—
184	C₈	—	796	1	no	—
185	C₈	—	398	2	no	—
186	C₈	—	199	2	no	—
187	C₈	—	199	2	no	—
188	C₂²	—	2	2	no	—
189	C₂²	(C₂²⋊D₁₉₉)⋊C₄	1	1	yes	characteristic
190	C₂²	#20263927	1	1	yes	characteristic
191	C₂²	#20263941	1	1	yes	characteristic
192	C₄	—	4	2	no	—
193	C₄	—	4	2	no	—
194	C₄	—	796	2	no	—
195	C₄	C₂²⋊C₄×D₁₉₉	1	1	yes	characteristic
196	C₄	C₄×C₂²⋊D₁₉₉	1	1	yes	characteristic
197	C₄	#20263928	1	1	yes	characteristic
198	C₄	#20263951	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#27163918	1	1	yes	characteristic
202	C₂	#27164291	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#15691876

C₂

yes

maximal, characteristic

C₂×#20263890

C₂

yes

maximal, characteristic

#27164026

C₂

yes

maximal, characteristic

C₂×#20263874

C₂

yes

maximal, characteristic

C₂×#20263873

C₂

yes

maximal, characteristic

#27163974

C₂

yes

maximal, characteristic

#419×C₁₉₉

C₂

yes

maximal, characteristic

C₂²×Dic₃₉₈

C₂²

yes

—

—

—

C₄

yes

complemented

C₂×#15691870

C₂²

yes

characteristic

C₂×#15691870

C₂²

yes

characteristic

C₂²×C₁₉₉⋊C₈

C₂²

yes

—

—

—

—

—

C₄×C₁₉₉⋊C₈

—

C₂²×Q₈×C₁₉₉

C₂²

yes

characteristic

C₂×M₁₆×C₁₉₉

C₂²

yes

characteristic

#117×C₁₉₉

C₂²

yes

characteristic

#103×C₁₉₉

—

C₂×Dic₃₉₈

—

C₂×Dic₃₉₈

—

C₂×Dic₃₉₈

—

C₂×Dic₃₉₈

—

C₂×Dic₃₉₈

—

C₂×Dic₃₉₈

C₂×C₄

yes

—

C₂²×Dic₁₉₉

—

—

—

—

—

—

D₄

yes

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

C₂×C₄

yes

—

C₂×C₁₉₉⋊C₈

D₄

yes

characteristic

C₂×C₁₉₉⋊C₈

D₄

yes

characteristic

C₂×C₁₉₉⋊C₈

D₄

yes

—

C₂×Q₈×C₁₉₉

—

C₂×Q₈×C₁₉₉

—

C₂×Q₈×C₁₉₉

—

C₂×Q₈×C₁₉₉

—

C₂×Q₈×C₁₉₉

—

C₂×Q₈×C₁₉₉

C₂×C₄

yes

—

C₂²×C₄×C₁₉₉

—

C₂²×C₄×C₁₉₉

C₂³

yes

characteristic

M₁₆×C₁₉₉

—

M₁₆×C₁₉₉

—

C₂×C₈×C₁₉₉

—

C₄⋊C₄×C₁₉₉

D₄

yes

—

C₂²⋊C₄×C₁₉₉

—

C₄²×C₁₉₉

—

Dic₃₉₈

—

Dic₃₉₈

—

Dic₃₉₈

—

Dic₃₉₈

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₁₉₉⋊C₈

—

C₁₉₉⋊C₈

—

C₁₉₉⋊C₈

C₂²⋊C₄

yes

—

C₁₉₉⋊C₈

C₂²⋊C₄

yes

—

C₂³×C₁₉₉

#41

yes

characteristic

Q₈×C₁₉₉

—

Q₈×C₁₉₉

—

Q₈×C₁₉₉

—

Q₈×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

#41

yes

characteristic

C₈×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

C₄×D₄

yes

C₂²×C₁₉₉

yes

C₂²×C₁₉₉

yes

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₉₉

C₄×D₄

yes

C₄×C₁₉₉

yes

C₄×C₁₉₉

yes

Dic₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

#390

yes

characteristic

100

C₂×C₁₉₉

#473

yes

—

101

C₁₉₉

#1979

yes

characteristic, complemented, Sylow

102

#1979

—

199

maximal, Sylow

103

C₂²×Q₁₆

—

199

—

104

C₂×#108

—

199

—

105

#419

—

199

—

106

#419

D₁₉₉

yes

characteristic

107

C₂×#101

—

199

—

108

C₂×#101

—

199

—

109

#405

—

199

—

110

C₂²×Q₈

—

199

—

111

C₂²×Q₈

C₂×D₁₉₉

yes

characteristic

112

C₂×Q₁₆

—

398

—

113

C₂×Q₁₆

—

398

—

114

C₂×Q₁₆

—

398

—

115

C₂×Q₁₆

—

199

—

116

C₂×M₁₆

—

199

—

117

C₂×M₁₆

—

199

—

118

C₂×M₁₆

C₂×D₁₉₉

yes

characteristic

119

C₂²×C₈

—

199

—

120

#117

C₂×D₁₉₉

yes

characteristic

121

#108

—

398

—

122

#103

—

123

#103

—

398

—

124

#101

—

398

—

125

#101

—

398

—

126

#97

—

398

—

127

C₄×C₈

—

398

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

398

—

134

C₂×Q₈

—

398

—

135

C₂×Q₈

—

398

—

136

C₂×Q₈

—

398

—

137

C₂×Q₈

—

398

—

138

C₂×Q₈

—

199

—

139

C₂×Q₈

C₄×D₁₉₉

yes

—

140

C₂²×C₄

—

141

C₂²×C₄

—

398

—

142

C₂²×C₄

C₂²×D₁₉₉

yes

characteristic

143

Q₁₆

—

398

—

144

Q₁₆

—

398

—

145

M₁₆

—

146

M₁₆

—

147

M₁₆

—

398

—

148

M₁₆

—

398

—

149

M₁₆

—

398

—

150

M₁₆

—

199

—

151

C₂×C₈

—

152

C₂×C₈

—

398

—

153

C₂×C₈

—

398

—

154

C₂×C₈

—

199

—

155

C₂×C₈

—

199

—

156

C₂×C₈

—

199

—

157

C₂×C₈

—

199

—

158

C₄⋊C₄

C₂²⋊D₁₉₉

yes

—

159

C₂²⋊C₄

—

160

C₄²

—

161

C₂³

#15691885

yes

characteristic

162

Q₈

—

163

Q₈

—

164

Q₈

—

165

Q₈

—

166

Q₈

—

796

—

167

Q₈

—

796

—

168

Q₈

—

398

—

169

Q₈

—

398

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

796

—

175

C₂×C₄

—

796

—

176

C₂×C₄

—

398

—

177

C₂×C₄

C₂×C₄×D₁₉₉

yes

characteristic

178

C₂×C₄

D₄×D₁₉₉

yes

characteristic

179

C₂×C₄

D₄×D₁₉₉

yes

characteristic

180

C₂×C₄

D₄×D₁₉₉

yes

characteristic

181

C₂×C₄

C₂×C₂²⋊D₁₉₉

yes

characteristic

182

C₂×C₄

#15691886

yes

characteristic

183

C₈

—

184

C₈

—

796

—

185

C₈

—

398

—

186

C₈

—

199

—

187

C₈

—

199

—

188

C₂²

—

189

C₂²

(C₂²⋊D₁₉₉)⋊C₄

yes

characteristic

190

C₂²

#20263927

yes

characteristic

191

C₂²

#20263941

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

796

—

195

C₄

C₂²⋊C₄×D₁₉₉

yes

characteristic

196

C₄

C₄×C₂²⋊D₁₉₉

yes

characteristic

197

C₄

#20263928

yes

characteristic

198

C₄

#20263951

yes

characteristic

199

C₂

—

200

C₂

—

201

C₂

#27163918

yes

characteristic

202

C₂

#27164291

yes

—

	subgroup	G/H
center	C₂²	#20263927
commutator	C₂×C₄×C₁₉₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₉
Fitting	#419×C₁₉₉	C₂
socle	C₂²×C₁₉₉	#121
prime radical	C₂³×C₁₉₉	#41
cosocle	C₂²×C₄×C₁₉₉	C₂³
hypercenter	#419	D₁₉₉
hypocenter	C₁₉₉	#1979
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

subgroup

G/H

center

C₂²

#20263927

commutator

C₂×C₄×C₁₉₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₉

Fitting

#419×C₁₉₉

C₂

C₂²×C₁₉₉

C₂³×C₁₉₉

C₂²×C₄×C₁₉₉

C₂³

D₁₉₉

C₁₉₉

G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

N	H ≅ G/N	type
12. C₂×#15691876	C₄	semidirect
101. C₁₉₉	#1979	semidirect

H ≅ G/N

type

12. C₂×#15691876

C₄

semidirect

101. C₁₉₉

#1979

semidirect

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

subgroup	index
C₂²×#15691876	2
C₂×#20263890	2
#27164026	2
C₂×#20263874	2
C₂×#20263873	2
#27163974	2
#419×C₁₉₉	2
#1979	199

subgroup

index

#419×C₁₉₉

199

Subgroups — #38769764

Conjugacy classes of subgroups

Complements

Maximal subgroups