Group #44400648 — subgroups

Subgroups — #44400648

Order 29056.

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×#16813874	C₂	1	1	yes	maximal, characteristic
2	C₂×#20300223	C₂	1	1	yes	maximal, characteristic
3	#28409040	C₂	1	1	yes	maximal, characteristic
4	C₂×#20300207	C₂	1	1	yes	maximal, characteristic
5	C₂×#20300206	C₂	1	1	yes	maximal, characteristic
6	#28408988	C₂	1	1	yes	maximal, characteristic
7	#419×C₂₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₄₅₄	C₂²	1	1	yes	characteristic
9	C₂×#16813874	—	2	1	no	—
10	C₂×#16813874	—	2	1	no	—
11	C₂×#16813874	—	2	2	no	—
12	C₂×#16813874	C₄	1	4	yes	complemented
13	C₂×#16813868	C₂²	1	1	yes	characteristic
14	C₂×#16813868	C₂²	1	1	yes	characteristic
15	C₂²×C₂₂₇⋊C₈	C₂²	1	1	yes	characteristic
16	#20300223	—	2	2	no	—
17	#20300211	—	2	2	no	—
18	#20300207	—	2	2	no	—
19	#20300206	—	2	2	no	—
20	#20300199	—	2	1	no	—
21	C₄×C₂₂₇⋊C₈	—	2	1	no	—
22	C₂²×Q₈×C₂₂₇	C₂²	1	1	yes	characteristic
23	C₂×M₁₆×C₂₂₇	C₂²	1	1	yes	characteristic
24	#117×C₂₂₇	C₂²	1	1	yes	characteristic
25	#103×C₂₂₇	—	2	2	no	—
26	C₂×Dic₄₅₄	—	2	1	no	—
27	C₂×Dic₄₅₄	—	2	1	no	—
28	C₂×Dic₄₅₄	—	2	1	no	—
29	C₂×Dic₄₅₄	—	2	1	no	—
30	C₂×Dic₄₅₄	—	2	1	no	—
31	C₂×Dic₄₅₄	C₂×C₄	1	2	yes	—
32	C₂²×Dic₂₂₇	—	2	1	no	—
33	#16813874	—	2	4	no	—
34	#16813874	—	2	4	no	—
35	#16813868	—	2	1	no	—
36	#16813868	—	2	1	no	—
37	#16813868	—	2	1	no	—
38	#16813868	D₄	1	2	yes	—
39	C₂×C₂₂₇⋊C₈	—	2	1	no	—
40	C₂×C₂₂₇⋊C₈	—	2	1	no	—
41	C₂×C₂₂₇⋊C₈	C₂×C₄	1	2	yes	—
42	C₂×C₂₂₇⋊C₈	D₄	1	1	yes	characteristic
43	C₂×C₂₂₇⋊C₈	D₄	1	1	yes	characteristic
44	C₂×C₂₂₇⋊C₈	D₄	1	2	yes	—
45	C₂×Q₈×C₂₂₇	—	2	1	no	—
46	C₂×Q₈×C₂₂₇	—	2	1	no	—
47	C₂×Q₈×C₂₂₇	—	2	1	no	—
48	C₂×Q₈×C₂₂₇	—	2	1	no	—
49	C₂×Q₈×C₂₂₇	—	2	1	no	—
50	C₂×Q₈×C₂₂₇	C₂×C₄	1	2	yes	—
51	C₂²×C₄×C₂₂₇	—	2	1	no	—
52	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
53	M₁₆×C₂₂₇	—	2	1	no	—
54	M₁₆×C₂₂₇	—	2	1	no	—
55	C₂×C₈×C₂₂₇	—	2	1	no	—
56	C₄⋊C₄×C₂₂₇	D₄	1	2	yes	—
57	C₂²⋊C₄×C₂₂₇	—	2	1	no	—
58	C₄²×C₂₂₇	—	2	1	no	—
59	Dic₄₅₄	—	2	2	no	—
60	Dic₄₅₄	—	2	2	no	—
61	Dic₄₅₄	—	4	1	no	—
62	Dic₄₅₄	—	4	1	no	—
63	C₂×Dic₂₂₇	—	2	2	no	—
64	C₂×Dic₂₂₇	—	4	1	no	—
65	C₂×Dic₂₂₇	—	4	1	no	—
66	C₂₂₇⋊C₈	—	2	2	no	—
67	C₂₂₇⋊C₈	—	4	1	no	—
68	C₂₂₇⋊C₈	C₂²⋊C₄	1	2	yes	—
69	C₂₂₇⋊C₈	C₂²⋊C₄	1	2	yes	—
70	C₂³×C₂₂₇	#41	1	1	yes	characteristic
71	Q₈×C₂₂₇	—	2	2	no	—
72	Q₈×C₂₂₇	—	2	2	no	—
73	Q₈×C₂₂₇	—	4	1	no	—
74	Q₈×C₂₂₇	—	4	1	no	—
75	C₂×C₄×C₂₂₇	—	2	2	no	—
76	C₂×C₄×C₂₂₇	—	2	2	no	—
77	C₂×C₄×C₂₂₇	—	4	1	no	—
78	C₂×C₄×C₂₂₇	—	4	1	no	—
79	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×C₂₂₇	#41	1	1	yes	characteristic
85	C₈×C₂₂₇	—	4	1	no	—
86	C₂²×C₂₂₇	—	2	2	no	—
87	C₂²×C₂₂₇	C₄×D₄	1	1	yes	characteristic
88	C₂²×C₂₂₇	#121	1	1	yes	characteristic
89	C₂²×C₂₂₇	#124	1	1	yes	characteristic
90	C₄×C₂₂₇	—	4	2	no	—
91	C₄×C₂₂₇	—	4	2	no	—
92	C₄×C₂₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₄×C₂₂₇	C₄×D₄	1	1	yes	characteristic
94	C₄×C₂₂₇	#121	1	1	yes	characteristic
95	C₄×C₂₂₇	#127	1	1	yes	characteristic
96	Dic₂₂₇	—	4	2	no	—
97	C₂×C₂₂₇	—	2	1	no	—
98	C₂×C₂₂₇	—	2	1	no	—
99	C₂×C₂₂₇	#390	1	1	yes	characteristic
100	C₂×C₂₂₇	#473	1	2	yes	—
101	C₂₂₇	#1979	1	1	yes	characteristic, complemented, Sylow
102	#1979	—	227	1	no	maximal, Sylow
103	C₂²×Q₁₆	—	227	1	no	—
104	C₂×#108	—	227	1	no	—
105	#419	—	227	1	no	—
106	#419	D₂₂₇	1	1	yes	characteristic
107	C₂×#101	—	227	1	no	—
108	C₂×#101	—	227	1	no	—
109	#405	—	227	1	no	—
110	C₂²×Q₈	—	227	1	no	—
111	C₂²×Q₈	C₂×D₂₂₇	1	1	yes	characteristic
112	C₂×Q₁₆	—	454	1	no	—
113	C₂×Q₁₆	—	454	1	no	—
114	C₂×Q₁₆	—	454	2	no	—
115	C₂×Q₁₆	—	227	4	no	—
116	C₂×M₁₆	—	227	1	no	—
117	C₂×M₁₆	—	227	1	no	—
118	C₂×M₁₆	C₂×D₂₂₇	1	1	yes	characteristic
119	C₂²×C₈	—	227	1	no	—
120	#117	C₂×D₂₂₇	1	1	yes	characteristic
121	#108	—	454	2	no	—
122	#103	—	2	2	no	—
123	#103	—	454	2	no	—
124	#101	—	454	2	no	—
125	#101	—	454	2	no	—
126	#97	—	454	1	no	—
127	C₄×C₈	—	454	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	—	454	1	no	—
134	C₂×Q₈	—	454	1	no	—
135	C₂×Q₈	—	454	1	no	—
136	C₂×Q₈	—	454	1	no	—
137	C₂×Q₈	—	454	1	no	—
138	C₂×Q₈	—	227	2	no	—
139	C₂×Q₈	C₄×D₂₂₇	1	2	yes	—
140	C₂²×C₄	—	2	1	no	—
141	C₂²×C₄	—	454	1	no	—
142	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
143	Q₁₆	—	454	4	no	—
144	Q₁₆	—	454	4	no	—
145	M₁₆	—	2	1	no	—
146	M₁₆	—	2	1	no	—
147	M₁₆	—	454	1	no	—
148	M₁₆	—	454	1	no	—
149	M₁₆	—	454	1	no	—
150	M₁₆	—	227	2	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	454	1	no	—
153	C₂×C₈	—	454	1	no	—
154	C₂×C₈	—	227	2	no	—
155	C₂×C₈	—	227	1	no	—
156	C₂×C₈	—	227	1	no	—
157	C₂×C₈	—	227	2	no	—
158	C₄⋊C₄	C₂²⋊D₂₂₇	1	2	yes	—
159	C₂²⋊C₄	—	2	1	no	—
160	C₄²	—	2	1	no	—
161	C₂³	#16813883	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	—	908	1	no	—
167	Q₈	—	908	1	no	—
168	Q₈	—	454	2	no	—
169	Q₈	—	454	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	—	908	1	no	—
175	C₂×C₄	—	908	1	no	—
176	C₂×C₄	—	454	2	no	—
177	C₂×C₄	C₂×C₄×D₂₂₇	1	1	yes	characteristic
178	C₂×C₄	D₄×D₂₂₇	1	1	yes	characteristic
179	C₂×C₄	D₄×D₂₂₇	1	1	yes	characteristic
180	C₂×C₄	D₄×D₂₂₇	1	1	yes	characteristic
181	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	1	yes	characteristic
182	C₂×C₄	#16813884	1	1	yes	characteristic
183	C₈	—	4	1	no	—
184	C₈	—	908	1	no	—
185	C₈	—	454	2	no	—
186	C₈	—	227	2	no	—
187	C₈	—	227	2	no	—
188	C₂²	—	2	2	no	—
189	C₂²	(C₂²⋊D₂₂₇)⋊C₄	1	1	yes	characteristic
190	C₂²	#20300260	1	1	yes	characteristic
191	C₂²	#20300274	1	1	yes	characteristic
192	C₄	—	4	2	no	—
193	C₄	—	4	2	no	—
194	C₄	—	908	2	no	—
195	C₄	C₂²⋊C₄×D₂₂₇	1	1	yes	characteristic
196	C₄	C₄×C₂²⋊D₂₂₇	1	1	yes	characteristic
197	C₄	#20300261	1	1	yes	characteristic
198	C₄	#20300284	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#28408932	1	1	yes	characteristic
202	C₂	#28409305	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#16813874

C₂

yes

maximal, characteristic

C₂×#20300223

C₂

yes

maximal, characteristic

#28409040

C₂

yes

maximal, characteristic

C₂×#20300207

C₂

yes

maximal, characteristic

C₂×#20300206

C₂

yes

maximal, characteristic

#28408988

C₂

yes

maximal, characteristic

#419×C₂₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₄₅₄

C₂²

yes

—

—

—

C₄

yes

complemented

C₂×#16813868

C₂²

yes

characteristic

C₂×#16813868

C₂²

yes

characteristic

C₂²×C₂₂₇⋊C₈

C₂²

yes

—

—

—

—

—

C₄×C₂₂₇⋊C₈

—

C₂²×Q₈×C₂₂₇

C₂²

yes

characteristic

C₂×M₁₆×C₂₂₇

C₂²

yes

characteristic

#117×C₂₂₇

C₂²

yes

characteristic

#103×C₂₂₇

—

C₂×Dic₄₅₄

—

C₂×Dic₄₅₄

—

C₂×Dic₄₅₄

—

C₂×Dic₄₅₄

—

C₂×Dic₄₅₄

—

C₂×Dic₄₅₄

C₂×C₄

yes

—

C₂²×Dic₂₂₇

—

—

—

—

—

—

D₄

yes

—

C₂×C₂₂₇⋊C₈

—

C₂×C₂₂₇⋊C₈

—

C₂×C₂₂₇⋊C₈

C₂×C₄

yes

—

C₂×C₂₂₇⋊C₈

D₄

yes

characteristic

C₂×C₂₂₇⋊C₈

D₄

yes

characteristic

C₂×C₂₂₇⋊C₈

D₄

yes

—

C₂×Q₈×C₂₂₇

—

C₂×Q₈×C₂₂₇

—

C₂×Q₈×C₂₂₇

—

C₂×Q₈×C₂₂₇

—

C₂×Q₈×C₂₂₇

—

C₂×Q₈×C₂₂₇

C₂×C₄

yes

—

C₂²×C₄×C₂₂₇

—

C₂²×C₄×C₂₂₇

C₂³

yes

characteristic

M₁₆×C₂₂₇

—

M₁₆×C₂₂₇

—

C₂×C₈×C₂₂₇

—

C₄⋊C₄×C₂₂₇

D₄

yes

—

C₂²⋊C₄×C₂₂₇

—

C₄²×C₂₂₇

—

Dic₄₅₄

—

Dic₄₅₄

—

Dic₄₅₄

—

Dic₄₅₄

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂₂₇⋊C₈

—

C₂₂₇⋊C₈

—

C₂₂₇⋊C₈

C₂²⋊C₄

yes

—

C₂₂₇⋊C₈

C₂²⋊C₄

yes

—

C₂³×C₂₂₇

#41

yes

characteristic

Q₈×C₂₂₇

—

Q₈×C₂₂₇

—

Q₈×C₂₂₇

—

Q₈×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₇

#41

yes

characteristic

C₈×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

C₄×D₄

yes

C₂²×C₂₂₇

yes

C₂²×C₂₂₇

yes

C₄×C₂₂₇

—

C₄×C₂₂₇

—

C₄×C₂₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₂₇

C₄×D₄

yes

C₄×C₂₂₇

yes

C₄×C₂₂₇

yes

Dic₂₂₇

—

C₂×C₂₂₇

—

C₂×C₂₂₇

—

C₂×C₂₂₇

#390

yes

characteristic

100

C₂×C₂₂₇

#473

yes

—

101

C₂₂₇

#1979

yes

characteristic, complemented, Sylow

102

#1979

—

227

maximal, Sylow

103

C₂²×Q₁₆

—

227

—

104

C₂×#108

—

227

—

105

#419

—

227

—

106

#419

D₂₂₇

yes

characteristic

107

C₂×#101

—

227

—

108

C₂×#101

—

227

—

109

#405

—

227

—

110

C₂²×Q₈

—

227

—

111

C₂²×Q₈

C₂×D₂₂₇

yes

characteristic

112

C₂×Q₁₆

—

454

—

113

C₂×Q₁₆

—

454

—

114

C₂×Q₁₆

—

454

—

115

C₂×Q₁₆

—

227

—

116

C₂×M₁₆

—

227

—

117

C₂×M₁₆

—

227

—

118

C₂×M₁₆

C₂×D₂₂₇

yes

characteristic

119

C₂²×C₈

—

227

—

120

#117

C₂×D₂₂₇

yes

characteristic

121

#108

—

454

—

122

#103

—

123

#103

—

454

—

124

#101

—

454

—

125

#101

—

454

—

126

#97

—

454

—

127

C₄×C₈

—

454

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

454

—

134

C₂×Q₈

—

454

—

135

C₂×Q₈

—

454

—

136

C₂×Q₈

—

454

—

137

C₂×Q₈

—

454

—

138

C₂×Q₈

—

227

—

139

C₂×Q₈

C₄×D₂₂₇

yes

—

140

C₂²×C₄

—

141

C₂²×C₄

—

454

—

142

C₂²×C₄

C₂²×D₂₂₇

yes

characteristic

143

Q₁₆

—

454

—

144

Q₁₆

—

454

—

145

M₁₆

—

146

M₁₆

—

147

M₁₆

—

454

—

148

M₁₆

—

454

—

149

M₁₆

—

454

—

150

M₁₆

—

227

—

151

C₂×C₈

—

152

C₂×C₈

—

454

—

153

C₂×C₈

—

454

—

154

C₂×C₈

—

227

—

155

C₂×C₈

—

227

—

156

C₂×C₈

—

227

—

157

C₂×C₈

—

227

—

158

C₄⋊C₄

C₂²⋊D₂₂₇

yes

—

159

C₂²⋊C₄

—

160

C₄²

—

161

C₂³

#16813883

yes

characteristic

162

Q₈

—

163

Q₈

—

164

Q₈

—

165

Q₈

—

166

Q₈

—

908

—

167

Q₈

—

908

—

168

Q₈

—

454

—

169

Q₈

—

454

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

908

—

175

C₂×C₄

—

908

—

176

C₂×C₄

—

454

—

177

C₂×C₄

C₂×C₄×D₂₂₇

yes

characteristic

178

C₂×C₄

D₄×D₂₂₇

yes

characteristic

179

C₂×C₄

D₄×D₂₂₇

yes

characteristic

180

C₂×C₄

D₄×D₂₂₇

yes

characteristic

181

C₂×C₄

C₂×C₂²⋊D₂₂₇

yes

characteristic

182

C₂×C₄

#16813884

yes

characteristic

183

C₈

—

184

C₈

—

908

—

185

C₈

—

454

—

186

C₈

—

227

—

187

C₈

—

227

—

188

C₂²

—

189

C₂²

(C₂²⋊D₂₂₇)⋊C₄

yes

characteristic

190

C₂²

#20300260

yes

characteristic

191

C₂²

#20300274

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

908

—

195

C₄

C₂²⋊C₄×D₂₂₇

yes

characteristic

196

C₄

C₄×C₂²⋊D₂₂₇

yes

characteristic

197

C₄

#20300261

yes

characteristic

198

C₄

#20300284

yes

characteristic

199

C₂

—

200

C₂

—

201

C₂

#28408932

yes

characteristic

202

C₂

#28409305

yes

—

	subgroup	G/H
center	C₂²	#20300260
commutator	C₂×C₄×C₂₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₂₇
Fitting	#419×C₂₂₇	C₂
socle	C₂²×C₂₂₇	#121
prime radical	C₂³×C₂₂₇	#41
cosocle	C₂²×C₄×C₂₂₇	C₂³
hypercenter	#419	D₂₂₇
hypocenter	C₂₂₇	#1979
LCS	G ▷ C₂×C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

subgroup

G/H

center

C₂²

#20300260

commutator

C₂×C₄×C₂₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₂₇

Fitting

#419×C₂₂₇

C₂

C₂²×C₂₂₇

C₂³×C₂₂₇

C₂²×C₄×C₂₂₇

C₂³

D₂₂₇

C₂₂₇

G ▷ C₂×C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

N	H ≅ G/N	type
12. C₂×#16813874	C₄	semidirect
101. C₂₂₇	#1979	semidirect

H ≅ G/N

type

12. C₂×#16813874

C₄

semidirect

101. C₂₂₇

#1979

semidirect

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

subgroup	index
C₂²×#16813874	2
C₂×#20300223	2
#28409040	2
C₂×#20300207	2
C₂×#20300206	2
#28408988	2
#419×C₂₂₇	2
#1979	227

subgroup

index

#419×C₂₂₇

227

Subgroups — #44400648

Conjugacy classes of subgroups

Complements

Maximal subgroups