Group #44559848 — subgroups

Subgroups — #44559848

Order 32128.

480 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 149 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×#22548746	C₂	1	2	yes	maximal
2	C₄×#16860810	C₂	1	1	yes	maximal, characteristic
3	#30631894	C₂	1	4	yes	maximal, complemented
4	#30631889	C₂	1	4	yes	maximal, complemented
5	#30631711	C₂	1	1	yes	maximal, characteristic, complemented
6	#30631696	C₂	1	2	yes	maximal, complemented
7	C₂×C₄⋊Q₈×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#16860810	C₂²	1	1	yes	characteristic
9	#22548746	—	2	8	no	—
10	C₂×#16860800	C₂²	1	4	yes	—
11	C₂×#16860797	C₂²	1	4	yes	—
12	C₂×#16860794	C₂²	1	2	yes	—
13	C₄×Dic₅₀₂	C₂²	1	2	yes	—
14	C₄×D₁₀₀₄	C₂²	1	2	yes	—
15	C₄×C₂²⋊D₂₅₁	—	2	2	no	—
16	#22548674	C₂²	1	1	yes	characteristic
17	#22548672	—	2	1	no	—
18	C₄²×D₂₅₁	—	2	1	no	—
19	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	4	yes	—
20	#22548639	C₂²	1	8	yes	—
21	C₂²×Q₈×C₂₅₁	C₂²	1	2	yes	—
22	C₄⋊Q₈×C₂₅₁	—	2	1	no	—
23	C₄⋊Q₈×C₂₅₁	—	2	1	no	—
24	C₄⋊Q₈×C₂₅₁	C₂²	1	4	yes	—
25	C₂×C₄⋊C₄×C₂₅₁	—	2	2	no	—
26	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
27	#16860810	—	2	2	no	—
28	#16860810	D₄	1	4	yes	—
29	C₂×Dic₅₀₂	C₂³	1	1	yes	characteristic
30	C₂×D₁₀₀₄	C₂³	1	1	yes	characteristic
31	C₂×C₂²⋊D₂₅₁	—	2	1	no	—
32	C₂×C₄×D₂₅₁	—	2	1	no	—
33	C₂×C₄×D₂₅₁	—	2	2	no	—
34	#16860800	—	2	8	no	—
35	#16860797	—	2	8	no	—
36	#16860794	—	2	4	no	—
37	C₂×C₂₅₁⋊C₈	C₂³	1	4	yes	—
38	C₄⋊Dic₂₅₁	C₂³	1	2	yes	—
39	Dic₂₅₁⋊C₄	—	2	2	no	—
40	C₂²⋊Dic₂₅₁	—	2	1	no	—
41	(C₂×D₂₅₁)⋊C₄	—	2	2	no	—
42	C₄×Dic₂₅₁	—	2	1	no	—
43	C₄×Dic₂₅₁	—	2	2	no	—
44	C₂×Q₈×C₂₅₁	—	2	2	no	—
45	C₂×Q₈×C₂₅₁	—	2	4	no	—
46	C₂×Q₈×C₂₅₁	—	2	4	no	—
47	C₂×Q₈×C₂₅₁	C₂³	1	4	yes	—
48	C₂²×C₄×C₂₅₁	—	2	2	no	—
49	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
50	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₂₅₁	—	2	4	no	—
52	C₄⋊C₄×C₂₅₁	—	2	4	no	—
53	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
54	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
55	C₄²×C₂₅₁	C₂³	1	2	yes	—
56	C₂²×D₂₅₁	—	2	1	no	—
57	Dic₅₀₂	—	2	1	no	—
58	Dic₅₀₂	C₂×D₄	1	2	yes	—
59	D₁₀₀₄	—	2	1	no	—
60	D₁₀₀₄	C₂×D₄	1	2	yes	—
61	C₂²⋊D₂₅₁	—	2	4	no	—
62	C₂×Dic₂₅₁	—	2	1	no	—
63	C₂×Dic₂₅₁	—	2	2	no	—
64	C₄×D₂₅₁	—	2	4	no	—
65	C₄×D₂₅₁	—	4	4	no	—
66	C₂₅₁⋊C₈	—	2	4	no	—
67	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
68	Q₈×C₂₅₁	—	2	4	no	—
69	Q₈×C₂₅₁	—	2	4	no	—
70	Q₈×C₂₅₁	—	4	4	no	—
71	C₂×C₄×C₂₅₁	—	2	4	no	—
72	C₂×C₄×C₂₅₁	—	2	4	no	—
73	C₂×C₄×C₂₅₁	—	4	4	no	—
74	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
75	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
76	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₂₅₁	#41	1	2	yes	—
79	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
80	C₂²×C₂₅₁	—	2	2	no	—
81	C₂²×C₂₅₁	#121	1	2	yes	—
82	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
83	C₂×D₂₅₁	—	2	2	no	—
84	C₂×D₂₅₁	—	4	2	no	—
85	C₄×C₂₅₁	—	2	2	no	—
86	C₄×C₂₅₁	—	4	4	no	—
87	C₄×C₂₅₁	#121	1	2	yes	—
88	C₄×C₂₅₁	#137	1	4	yes	complemented
89	C₄×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
90	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
91	Dic₂₅₁	—	2	2	no	—
92	Dic₂₅₁	—	4	2	no	—
93	C₂×C₂₅₁	—	2	2	no	—
94	C₂×C₂₅₁	C₂×#121	1	1	yes	characteristic
95	C₂×C₂₅₁	C₂×#137	1	2	yes	—
96	D₂₅₁	—	4	2	no	—
97	C₂₅₁	#3039	1	1	yes	characteristic, complemented, Sylow
98	#3039	—	251	1	no	maximal, Sylow
99	C₂×#137	—	251	2	no	—
100	C₂×C₄⋊Q₈	D₂₅₁	1	1	yes	characteristic, complemented
101	C₄×#41	—	251	1	no	—
102	#462	—	251	4	no	—
103	#461	—	251	4	no	—
104	#423	—	251	1	no	—
105	#419	—	251	2	no	—
106	C₂×#41	—	251	1	no	—
107	C₂²×Q₈	C₂×D₂₅₁	1	2	yes	—
108	#137	—	502	8	no	—
109	C₂×Q₁₆	—	251	4	no	—
110	C₂×SD₁₆	—	251	4	no	—
111	C₂×M₁₆	—	251	2	no	—
112	C₄⋊Q₈	—	2	1	no	—
113	C₄⋊Q₈	—	2	1	no	—
114	C₄⋊Q₈	C₂×D₂₅₁	1	4	yes	—
115	C₄×Q₈	—	251	2	no	—
116	C₄×D₄	—	502	2	no	—
117	C₄×D₄	—	251	2	no	—
118	#117	—	502	1	no	—
119	#117	—	251	1	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄²	—	502	1	no	—
122	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
123	C₄⋊C₈	—	251	4	no	—
124	#103	—	251	8	no	—
125	#41	—	502	2	no	—
126	#41	—	251	4	no	—
127	C₂×Q₈	—	2	2	no	—
128	C₂×Q₈	—	2	4	no	—
129	C₂×Q₈	—	2	4	no	—
130	C₂×Q₈	—	251	1	no	—
131	C₂×Q₈	C₂²×D₂₅₁	1	4	yes	—
132	C₂×D₄	—	502	1	no	—
133	C₂×D₄	—	251	1	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	502	2	no	—
136	C₂²×C₄	—	502	1	no	—
137	C₂²×C₄	C₂²⋊D₂₅₁	1	2	yes	—
138	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
139	Q₁₆	—	502	8	no	—
140	SD₁₆	—	502	8	no	—
141	M₁₆	—	502	4	no	—
142	C₂×C₈	—	251	4	no	—
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	2	4	no	—
145	C₄⋊C₄	—	502	2	no	—
146	C₄⋊C₄	—	251	2	no	—
147	C₂²⋊C₄	—	502	1	no	—
148	C₂²⋊C₄	—	502	2	no	—
149	C₄²	—	502	1	no	—
150	C₄²	—	502	2	no	—
151	C₄²	C₂²⋊D₂₅₁	1	1	yes	characteristic
152	C₄²	C₂²⋊D₂₅₁	1	1	yes	characteristic
153	C₄²	C₂²×D₂₅₁	1	2	yes	—
154	C₂³	—	502	1	no	—
155	C₂³	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
156	Q₈	—	2	4	no	—
157	Q₈	—	2	4	no	—
158	Q₈	—	4	4	no	—
159	Q₈	—	502	1	no	—
160	Q₈	—	251	2	no	—
161	D₄	—	502	4	no	—
162	D₄	—	502	1	no	—
163	D₄	—	251	2	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	1004	4	no	—
168	C₂×C₄	—	502	2	no	—
169	C₂×C₄	—	502	1	no	—
170	C₂×C₄	—	502	4	no	—
171	C₂×C₄	D₄×D₂₅₁	1	2	yes	—
172	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	2	yes	—
173	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	2	yes	—
174	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
175	C₂×C₄	#16860809	1	2	yes	—
176	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
177	C₈	—	502	4	no	—
178	C₂²	—	2	2	no	—
179	C₂²	—	1004	2	no	—
180	C₂²	—	502	2	no	—
181	C₂²	#22548686	1	2	yes	—
182	C₂²	C₂²×C₂²⋊D₂₅₁	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	4	4	no	—
185	C₄	—	1004	2	no	—
186	C₄	—	502	2	no	—
187	C₄	#22548686	1	2	yes	—
188	C₄	#22548746	1	4	yes	complemented
189	C₄	C₂×D₄×D₂₅₁	1	1	yes	characteristic
190	C₄	C₂×#16860809	1	1	yes	characteristic
191	C₂	—	2	2	no	—
192	C₂	—	1004	2	no	—
193	C₂	C₂×#22548686	1	1	yes	characteristic
194	C₂	C₂×#22548746	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22548746

C₂

yes

maximal

C₄×#16860810

C₂

yes

maximal, characteristic

#30631894

C₂

yes

maximal, complemented

#30631889

C₂

yes

maximal, complemented

#30631711

C₂

yes

maximal, characteristic, complemented

#30631696

C₂

yes

maximal, complemented

C₂×C₄⋊Q₈×C₂₅₁

C₂

yes

maximal, characteristic, complemented

C₂×#16860810

C₂²

yes

characteristic

#22548746

—

C₂×#16860800

C₂²

yes

—

C₂×#16860797

C₂²

yes

—

C₂×#16860794

C₂²

yes

—

C₄×Dic₅₀₂

C₂²

yes

—

C₄×D₁₀₀₄

C₂²

yes

—

C₄×C₂²⋊D₂₅₁

—

#22548674

C₂²

yes

characteristic

#22548672

—

C₄²×D₂₅₁

—

C₄⋊(C₂₅₁⋊C₈)

C₂²

yes

—

#22548639

C₂²

yes

—

C₂²×Q₈×C₂₅₁

C₂²

yes

—

C₄⋊Q₈×C₂₅₁

—

C₄⋊Q₈×C₂₅₁

—

C₄⋊Q₈×C₂₅₁

C₂²

yes

—

C₂×C₄⋊C₄×C₂₅₁

—

C₂×C₄²×C₂₅₁

C₂²

yes

characteristic

#16860810

—

#16860810

D₄

yes

—

C₂×Dic₅₀₂

C₂³

yes

characteristic

C₂×D₁₀₀₄

C₂³

yes

characteristic

C₂×C₂²⋊D₂₅₁

—

C₂×C₄×D₂₅₁

—

C₂×C₄×D₂₅₁

—

#16860800

—

#16860797

—

#16860794

—

C₂×C₂₅₁⋊C₈

C₂³

yes

—

C₄⋊Dic₂₅₁

C₂³

yes

—

Dic₂₅₁⋊C₄

—

C₂²⋊Dic₂₅₁

—

(C₂×D₂₅₁)⋊C₄

—

C₄×Dic₂₅₁

—

C₄×Dic₂₅₁

—

C₂×Q₈×C₂₅₁

—

C₂×Q₈×C₂₅₁

—

C₂×Q₈×C₂₅₁

—

C₂×Q₈×C₂₅₁

C₂³

yes

—

C₂²×C₄×C₂₅₁

—

C₂²×C₄×C₂₅₁

D₄

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄²×C₂₅₁

D₄

yes

characteristic

C₄²×C₂₅₁

D₄

yes

characteristic

C₄²×C₂₅₁

C₂³

yes

—

C₂²×D₂₅₁

—

Dic₅₀₂

—

Dic₅₀₂

C₂×D₄

yes

—

D₁₀₀₄

—

D₁₀₀₄

C₂×D₄

yes

—

C₂²⋊D₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₄×D₂₅₁

—

C₄×D₂₅₁

—

C₂₅₁⋊C₈

—

C₂³×C₂₅₁

C₂×D₄

yes

characteristic

Q₈×C₂₅₁

—

Q₈×C₂₅₁

—

Q₈×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂×C₄×C₂₅₁

#41

yes

—

C₂×C₄×C₂₅₁

C₂⁴

yes

characteristic

C₂²×C₂₅₁

—

C₂²×C₂₅₁

#121

yes

—

C₂²×C₂₅₁

C₂²×D₄

yes

characteristic

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

#121

yes

—

C₄×C₂₅₁

#137

yes

complemented

C₄×C₂₅₁

C₂²×D₄

yes

characteristic

C₄×C₂₅₁

C₂×#41

yes

characteristic

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

C₂×#121

yes

characteristic

C₂×C₂₅₁

C₂×#137

yes

—

D₂₅₁

—

C₂₅₁

#3039

yes

characteristic, complemented, Sylow

#3039

—

251

maximal, Sylow

C₂×#137

—

251

—

100

C₂×C₄⋊Q₈

D₂₅₁

yes

characteristic, complemented

101

C₄×#41

—

251

—

102

#462

—

251

—

103

#461

—

251

—

104

#423

—

251

—

105

#419

—

251

—

106

C₂×#41

—

251

—

107

C₂²×Q₈

C₂×D₂₅₁

yes

—

108

#137

—

502

—

109

C₂×Q₁₆

—

251

—

110

C₂×SD₁₆

—

251

—

111

C₂×M₁₆

—

251

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

—

114

C₄⋊Q₈

C₂×D₂₅₁

yes

—

115

C₄×Q₈

—

251

—

116

C₄×D₄

—

502

—

117

C₄×D₄

—

251

—

118

#117

—

502

—

119

#117

—

251

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄²

—

502

—

122

C₂×C₄²

C₂×D₂₅₁

yes

characteristic

123

C₄⋊C₈

—

251

—

124

#103

—

251

—

125

#41

—

502

—

126

#41

—

251

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

251

—

131

C₂×Q₈

C₂²×D₂₅₁

yes

—

132

C₂×D₄

—

502

—

133

C₂×D₄

—

251

—

134

C₂²×C₄

—

135

C₂²×C₄

—

502

—

136

C₂²×C₄

—

502

—

137

C₂²×C₄

C₂²⋊D₂₅₁

yes

—

138

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

139

Q₁₆

—

502

—

140

SD₁₆

—

502

—

141

M₁₆

—

502

—

142

C₂×C₈

—

251

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

502

—

146

C₄⋊C₄

—

251

—

147

C₂²⋊C₄

—

502

—

148

C₂²⋊C₄

—

502

—

149

C₄²

—

502

—

150

C₄²

—

502

—

151

C₄²

C₂²⋊D₂₅₁

yes

characteristic

152

C₄²

C₂²⋊D₂₅₁

yes

characteristic

153

C₄²

C₂²×D₂₅₁

yes

—

154

C₂³

—

502

—

155

C₂³

C₂×C₂²⋊D₂₅₁

yes

characteristic

156

Q₈

—

157

Q₈

—

158

Q₈

—

159

Q₈

—

502

—

160

Q₈

—

251

—

161

D₄

—

502

—

162

D₄

—

502

—

163

D₄

—

251

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

1004

—

168

C₂×C₄

—

502

—

169

C₂×C₄

—

502

—

170

C₂×C₄

—

502

—

171

C₂×C₄

D₄×D₂₅₁

yes

—

172

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

—

173

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

—

174

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

characteristic

175

C₂×C₄

#16860809

yes

—

176

C₂×C₄

C₂³×D₂₅₁

yes

characteristic

177

C₈

—

502

—

178

C₂²

—

179

C₂²

—

1004

—

180

C₂²

—

502

—

181

C₂²

#22548686

yes

—

182

C₂²

C₂²×C₂²⋊D₂₅₁

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

1004

—

186

C₄

—

502

—

187

C₄

#22548686

yes

—

188

C₄

#22548746

yes

complemented

189

C₄

C₂×D₄×D₂₅₁

yes

characteristic

190

C₄

C₂×#16860809

yes

characteristic

191

C₂

—

192

C₂

—

1004

—

193

C₂

C₂×#22548686

yes

characteristic

194

C₂

C₂×#22548746

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₂₅₁
commutator, cosocle	C₂×C₄×C₂₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	C₂×C₄⋊Q₈×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
prime radical	C₂×#16860810	C₂²
hypercenter	C₂×C₄⋊Q₈	D₂₅₁
hypocenter	C₂₅₁	#3039
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₂₅₁

commutator, cosocle

C₂×C₄×C₂₅₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₅₁

Fitting

C₂×C₄⋊Q₈×C₂₅₁

C₂

socle

C₂²×C₂₅₁

C₂²×D₄

prime radical

C₂×#16860810

C₂²

hypercenter

C₂×C₄⋊Q₈

D₂₅₁

hypocenter

C₂₅₁

#3039

LCS

G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
3. #30631894	C₂	semidirect
4. #30631889	C₂	semidirect
5. #30631711	C₂	semidirect
6. #30631696	C₂	semidirect
7. C₂×C₄⋊Q₈×C₂₅₁	C₂	semidirect
88. C₄×C₂₅₁	#137	semidirect
97. C₂₅₁	#3039	semidirect
100. C₂×C₄⋊Q₈	D₂₅₁	semidirect
188. C₄	#22548746	semidirect

H ≅ G/N

type

3. #30631894

C₂

semidirect

4. #30631889

C₂

semidirect

5. #30631711

C₂

semidirect

6. #30631696

C₂

semidirect

7. C₂×C₄⋊Q₈×C₂₅₁

C₂

semidirect

88. C₄×C₂₅₁

#137

semidirect

97. C₂₅₁

#3039

semidirect

100. C₂×C₄⋊Q₈

D₂₅₁

semidirect

188. C₄

#22548746

semidirect

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

subgroup	index
C₂×#22548746	2
C₄×#16860810	2
#30631894	2
#30631889	2
#30631711	2
#30631696	2
C₂×C₄⋊Q₈×C₂₅₁	2
#3039	251

subgroup

index

C₂×C₄⋊Q₈×C₂₅₁

251

Subgroups — #44559848

Conjugacy classes of subgroups

Complements

Maximal subgroups