Group #30649184 — subgroups

Subgroups — #30649184

Order 16256.

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×#16861238	C₂	1	4	yes	maximal, complemented
2	#22550424	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Q₈×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#22550267	C₂	1	4	yes	maximal, complemented
5	#22550256	C₂	1	1	yes	maximal, characteristic, complemented
6	#22550255	C₂	1	1	yes	maximal, characteristic, complemented
7	#22550248	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#14523231	C₂	1	1	yes	maximal, characteristic, complemented
9	#492×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14523247	C₂²	1	2	yes	—
11	C₂×Q₈×D₁₂₇	C₂²	1	2	yes	—
12	#16861238	—	2	16	no	—
13	C₂×Dic₅₀₈	C₂²	1	4	yes	—
14	C₂×#14523238	C₂²	1	4	yes	—
15	C₂×#14523236	C₂²	1	4	yes	—
16	C₂×#14523231	C₂²	1	2	yes	—
17	#16861209	C₂²	1	1	yes	characteristic
18	#16861209	C₂²	1	1	yes	characteristic
19	#16861208	—	2	2	no	—
20	#16861207	C₂²	1	2	yes	—
21	#16861196	C₂²	1	2	yes	—
22	#16861180	—	2	2	no	—
23	D₄×Dic₁₂₇	—	2	2	no	—
24	C₄⋊C₄×D₁₂₇	—	2	2	no	—
25	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic
26	#16861119	C₂²	1	2	yes	—
27	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×SD₁₆×C₁₂₇	C₂²	1	4	yes	—
29	C₄⋊Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
30	#127×C₁₂₇	C₂²	1	1	yes	characteristic
31	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
32	#14523247	—	2	8	no	—
33	C₂×Dic₂₅₄	—	2	2	no	—
34	C₂×Dic₂₅₄	C₂³	1	2	yes	—
35	C₂×Dic₂₅₄	C₂³	1	2	yes	—
36	Q₈×D₁₂₇	—	2	8	no	—
37	C₂×C₂²⋊D₁₂₇	—	2	2	no	—
38	C₂²×Dic₁₂₇	—	2	2	no	—
39	C₂×C₄×D₁₂₇	—	2	2	no	—
40	C₂×C₄×D₁₂₇	D₄	1	2	yes	—
41	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
42	Dic₅₀₈	—	2	8	no	—
43	#14523238	—	2	8	no	—
44	#14523236	—	2	8	no	—
45	#14523231	D₄	1	8	yes	complemented
46	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
47	C₄⋊Dic₁₂₇	—	2	2	no	—
48	C₄⋊Dic₁₂₇	—	2	2	no	—
49	Dic₁₂₇⋊C₄	—	2	4	no	—
50	C₂²⋊Dic₁₂₇	—	2	4	no	—
51	C₄×Dic₁₂₇	D₄	1	2	yes	—
52	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
53	C₂×Q₈×C₁₂₇	C₂³	1	2	yes	—
54	C₂×D₄×C₁₂₇	—	2	2	no	—
55	C₂×D₄×C₁₂₇	C₂³	1	2	yes	—
56	SD₁₆×C₁₂₇	—	2	8	no	—
57	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
58	C₄⋊C₄×C₁₂₇	—	2	2	no	—
59	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
60	C₂²×D₁₂₇	C₂×D₄	1	1	yes	characteristic
61	Dic₂₅₄	—	2	4	no	—
62	Dic₂₅₄	—	2	4	no	—
63	Dic₂₅₄	—	4	4	no	—
64	C₂²⋊D₁₂₇	—	4	4	no	—
65	C₂×Dic₁₂₇	—	2	2	no	—
66	C₂×Dic₁₂₇	—	2	2	no	—
67	C₂×Dic₁₂₇	—	4	4	no	—
68	C₂×Dic₁₂₇	C₂×D₄	1	2	yes	—
69	C₂×Dic₁₂₇	C₂×D₄	1	1	yes	characteristic
70	C₄×D₁₂₇	—	2	4	no	—
71	C₄×D₁₂₇	—	4	4	no	—
72	C₄×D₁₂₇	C₂×D₄	1	4	yes	—
73	C₁₂₇⋊C₈	C₂×D₄	1	4	yes	—
74	C₂³×C₁₂₇	—	2	2	no	—
75	Q₈×C₁₂₇	—	2	4	no	—
76	D₄×C₁₂₇	—	2	4	no	—
77	D₄×C₁₂₇	—	4	4	no	—
78	C₂×C₄×C₁₂₇	—	2	2	no	—
79	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
81	C₈×C₁₂₇	C₂×D₄	1	4	yes	—
82	C₂²×C₁₂₇	—	4	2	no	—
83	C₂²×C₁₂₇	—	4	4	no	—
84	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
85	C₂×D₁₂₇	—	2	2	no	—
86	C₂×D₁₂₇	#127	1	2	yes	—
87	C₄×C₁₂₇	—	4	2	no	—
88	C₄×C₁₂₇	#137	1	4	yes	complemented
89	C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
90	Dic₁₂₇	—	2	2	no	—
91	Dic₁₂₇	—	4	2	no	—
92	Dic₁₂₇	—	4	2	no	—
93	Dic₁₂₇	#127	1	2	yes	—
94	C₂×C₁₂₇	—	8	2	no	—
95	C₂×C₁₂₇	C₂×#127	1	1	yes	characteristic
96	C₂×C₁₂₇	C₂×#137	1	2	yes	—
97	D₁₂₇	—	2	2	no	—
98	#3152	—	127	1	no	maximal, Sylow
99	C₁₂₇	#3152	1	1	yes	characteristic, complemented, Sylow
100	C₂×#137	—	127	4	no	—
101	#532	—	127	1	no	—
102	C₂×C₄⋊Q₈	—	127	1	no	—
103	#497	—	127	4	no	—
104	#494	—	127	1	no	—
105	#494	—	127	1	no	—
106	#492	—	127	1	no	—
107	#492	D₁₂₇	1	1	yes	characteristic, complemented
108	C₄×M₁₆	—	127	1	no	—
109	C₂×#41	—	127	2	no	—
110	C₂²×Q₈	—	127	2	no	—
111	#137	—	254	16	no	—
112	C₂×Q₁₆	—	127	4	no	—
113	C₂×Q₁₆	—	127	4	no	—
114	C₂×SD₁₆	—	127	4	no	—
115	C₂×SD₁₆	C₂×D₁₂₇	1	4	yes	—
116	C₂×M₁₆	—	127	2	no	—
117	C₄⋊Q₈	—	127	2	no	—
118	C₄⋊Q₈	—	127	1	no	—
119	C₄⋊Q₈	—	127	1	no	—
120	C₄⋊Q₈	—	254	2	no	—
121	C₄⋊Q₈	C₂×D₁₂₇	1	1	yes	characteristic
122	#127	C₂×D₁₂₇	1	1	yes	characteristic
123	#124	—	127	2	no	—
124	#121	—	254	2	no	—
125	C₄×D₄	—	254	2	no	—
126	C₂×C₄⋊C₄	—	254	2	no	—
127	C₂×C₄²	—	127	1	no	—
128	#97	—	127	2	no	—
129	C₄×C₈	—	127	1	no	—
130	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
131	#41	—	254	8	no	—
132	C₂×Q₈	—	127	2	no	—
133	C₂×Q₈	—	127	2	no	—
134	C₂×Q₈	—	254	2	no	—
135	C₂×Q₈	—	254	8	no	—
136	C₂×Q₈	C₂²×D₁₂₇	1	2	yes	—
137	C₂×D₄	—	2	2	no	—
138	C₂×D₄	—	254	2	no	—
139	C₂×D₄	C₂²×D₁₂₇	1	2	yes	—
140	C₂²×C₄	—	127	1	no	—
141	C₂²×C₄	—	127	2	no	—
142	C₂²×C₄	—	254	2	no	—
143	C₂²×C₄	—	254	2	no	—
144	Q₁₆	—	254	8	no	—
145	Q₁₆	—	254	8	no	—
146	SD₁₆	—	2	8	no	—
147	SD₁₆	—	254	8	no	—
148	M₁₆	—	127	8	no	—
149	C₂×C₈	—	127	2	no	—
150	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	254	4	no	—
153	C₄⋊C₄	—	254	2	no	—
154	C₄⋊C₄	—	254	2	no	—
155	C₂²⋊C₄	—	254	4	no	—
156	C₄²	—	127	2	no	—
157	C₄²	—	127	1	no	—
158	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
159	C₂³	—	2	2	no	—
160	C₂³	—	127	1	no	—
161	Q₈	—	2	4	no	—
162	Q₈	—	508	4	no	—
163	Q₈	—	254	4	no	—
164	Q₈	—	254	4	no	—
165	D₄	—	2	4	no	—
166	D₄	—	4	4	no	—
167	D₄	—	508	4	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	127	1	no	—
170	C₂×C₄	—	127	4	no	—
171	C₂×C₄	—	127	2	no	—
172	C₂×C₄	—	508	4	no	—
173	C₂×C₄	—	508	4	no	—
174	C₂×C₄	—	254	2	no	—
175	C₂×C₄	—	254	2	no	—
176	C₂×C₄	—	254	4	no	—
177	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
178	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
179	C₈	—	127	4	no	—
180	C₈	D₄×D₁₂₇	1	4	yes	—
181	C₂²	—	4	4	no	—
182	C₂²	—	4	2	no	—
183	C₂²	—	127	2	no	—
184	C₂²	—	254	2	no	—
185	C₂²	C₂×D₄×D₁₂₇	1	1	yes	characteristic
186	C₄	—	4	2	no	—
187	C₄	—	127	2	no	—
188	C₄	—	508	2	no	—
189	C₄	—	508	2	no	—
190	C₄	—	254	2	no	—
191	C₄	#16861238	1	4	yes	complemented
192	C₄	C₂×D₄×D₁₂₇	1	2	yes	—
193	C₂	—	8	2	no	—
194	C₂	—	254	2	no	—
195	C₂	#127×D₁₂₇	1	1	yes	characteristic
196	C₂	C₂×#16861238	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861238

C₂

yes

maximal, complemented

#22550424

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₁₂₇

C₂

yes

maximal, characteristic, complemented

#22550267

C₂

yes

maximal, complemented

#22550256

C₂

yes

maximal, characteristic, complemented

#22550255

C₂

yes

maximal, characteristic, complemented

#22550248

C₂

yes

maximal, characteristic, complemented

C₄×#14523231

C₂

yes

maximal, characteristic, complemented

#492×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×#14523247

C₂²

yes

—

C₂×Q₈×D₁₂₇

C₂²

yes

—

#16861238

—

C₂×Dic₅₀₈

C₂²

yes

—

C₂×#14523238

C₂²

yes

—

C₂×#14523236

C₂²

yes

—

C₂×#14523231

C₂²

yes

—

#16861209

C₂²

yes

characteristic

#16861209

C₂²

yes

characteristic

#16861208

—

#16861207

C₂²

yes

—

#16861196

C₂²

yes

—

#16861180

—

D₄×Dic₁₂₇

—

C₄⋊C₄×D₁₂₇

—

C₄²×D₁₂₇

C₂²

yes

characteristic

#16861119

C₂²

yes

—

C₄×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂×SD₁₆×C₁₂₇

C₂²

yes

—

C₄⋊Q₈×C₁₂₇

C₂²

yes

characteristic

#127×C₁₂₇

C₂²

yes

characteristic

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

#14523247

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

C₂³

yes

—

C₂×Dic₂₅₄

C₂³

yes

—

Q₈×D₁₂₇

—

C₂×C₂²⋊D₁₂₇

—

C₂²×Dic₁₂₇

—

C₂×C₄×D₁₂₇

—

C₂×C₄×D₁₂₇

D₄

yes

—

C₂×C₄×D₁₂₇

C₂³

yes

characteristic

Dic₅₀₈

—

#14523238

—

#14523236

—

#14523231

D₄

yes

complemented

C₂×C₁₂₇⋊C₈

C₂³

yes

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

Dic₁₂₇⋊C₄

—

C₂²⋊Dic₁₂₇

—

C₄×Dic₁₂₇

D₄

yes

—

C₄×Dic₁₂₇

C₂³

yes

characteristic

C₂×Q₈×C₁₂₇

C₂³

yes

—

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

C₂³

yes

—

SD₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂³

yes

—

C₄⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

C₂³

yes

characteristic

C₂²×D₁₂₇

C₂×D₄

yes

characteristic

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₂×D₄

yes

—

C₂×Dic₁₂₇

C₂×D₄

yes

characteristic

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₄×D₁₂₇

C₂×D₄

yes

—

C₁₂₇⋊C₈

C₂×D₄

yes

—

C₂³×C₁₂₇

—

Q₈×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₈×C₁₂₇

C₂×D₄

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂²×D₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

#127

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#137

yes

complemented

C₄×C₁₂₇

C₂²×D₄

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

#127

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×#127

yes

characteristic

C₂×C₁₂₇

C₂×#137

yes

—

D₁₂₇

—

#3152

—

127

maximal, Sylow

C₁₂₇

#3152

yes

characteristic, complemented, Sylow

100

C₂×#137

—

127

—

101

#532

—

127

—

102

C₂×C₄⋊Q₈

—

127

—

103

#497

—

127

—

104

#494

—

127

—

105

#494

—

127

—

106

#492

—

127

—

107

#492

D₁₂₇

yes

characteristic, complemented

108

C₄×M₁₆

—

127

—

109

C₂×#41

—

127

—

110

C₂²×Q₈

—

127

—

111

#137

—

254

—

112

C₂×Q₁₆

—

127

—

113

C₂×Q₁₆

—

127

—

114

C₂×SD₁₆

—

127

—

115

C₂×SD₁₆

C₂×D₁₂₇

yes

—

116

C₂×M₁₆

—

127

—

117

C₄⋊Q₈

—

127

—

118

C₄⋊Q₈

—

127

—

119

C₄⋊Q₈

—

127

—

120

C₄⋊Q₈

—

254

—

121

C₄⋊Q₈

C₂×D₁₂₇

yes

characteristic

122

#127

C₂×D₁₂₇

yes

characteristic

123

#124

—

127

—

124

#121

—

254

—

125

C₄×D₄

—

254

—

126

C₂×C₄⋊C₄

—

254

—

127

C₂×C₄²

—

127

—

128

#97

—

127

—

129

C₄×C₈

—

127

—

130

C₄×C₈

C₂×D₁₂₇

yes

characteristic

131

#41

—

254

—

132

C₂×Q₈

—

127

—

133

C₂×Q₈

—

127

—

134

C₂×Q₈

—

254

—

135

C₂×Q₈

—

254

—

136

C₂×Q₈

C₂²×D₁₂₇

yes

—

137

C₂×D₄

—

138

C₂×D₄

—

254

—

139

C₂×D₄

C₂²×D₁₂₇

yes

—

140

C₂²×C₄

—

127

—

141

C₂²×C₄

—

127

—

142

C₂²×C₄

—

254

—

143

C₂²×C₄

—

254

—

144

Q₁₆

—

254

—

145

Q₁₆

—

254

—

146

SD₁₆

—

147

SD₁₆

—

254

—

148

M₁₆

—

127

—

149

C₂×C₈

—

127

—

150

C₂×C₈

C₂²×D₁₂₇

yes

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

254

—

153

C₄⋊C₄

—

254

—

154

C₄⋊C₄

—

254

—

155

C₂²⋊C₄

—

254

—

156

C₄²

—

127

—

157

C₄²

—

127

—

158

C₄²

C₂²×D₁₂₇

yes

characteristic

159

C₂³

—

160

C₂³

—

127

—

161

Q₈

—

162

Q₈

—

508

—

163

Q₈

—

254

—

164

Q₈

—

254

—

165

D₄

—

166

D₄

—

167

D₄

—

508

—

168

C₂×C₄

—

169

C₂×C₄

—

127

—

170

C₂×C₄

—

127

—

171

C₂×C₄

—

127

—

172

C₂×C₄

—

508

—

173

C₂×C₄

—

508

—

174

C₂×C₄

—

254

—

175

C₂×C₄

—

254

—

176

C₂×C₄

—

254

—

177

C₂×C₄

D₄×D₁₂₇

yes

—

178

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

179

C₈

—

127

—

180

C₈

D₄×D₁₂₇

yes

—

181

C₂²

—

182

C₂²

—

183

C₂²

—

127

—

184

C₂²

—

254

—

185

C₂²

C₂×D₄×D₁₂₇

yes

characteristic

186

C₄

—

187

C₄

—

127

—

188

C₄

—

508

—

189

C₄

—

508

—

190

C₄

—

254

—

191

C₄

#16861238

yes

complemented

192

C₄

C₂×D₄×D₁₂₇

yes

—

193

C₂

—

194

C₂

—

254

—

195

C₂

#127×D₁₂₇

yes

characteristic

196

C₂

C₂×#16861238

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₂₇
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#492×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	#22550424	C₂
hypercenter	#492	D₁₂₇
hypocenter	C₁₂₇	#3152
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

C₂×D₄×D₁₂₇

commutator, cosocle

C₂×C₄×C₁₂₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₂₇

Fitting

#492×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂²×D₄

C₂

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. C₂×#16861238	C₂	semidirect
3. C₄⋊Q₈×D₁₂₇	C₂	semidirect
4. #22550267	C₂	semidirect
5. #22550256	C₂	semidirect
6. #22550255	C₂	semidirect
7. #22550248	C₂	semidirect
8. C₄×#14523231	C₂	semidirect
9. #492×C₁₂₇	C₂	semidirect
45. #14523231	D₄	semidirect
88. C₄×C₁₂₇	#137	semidirect
99. C₁₂₇	#3152	semidirect
107. #492	D₁₂₇	semidirect
191. C₄	#16861238	semidirect

H ≅ G/N

type

1. C₂×#16861238

C₂

semidirect

3. C₄⋊Q₈×D₁₂₇

C₂

semidirect

4. #22550267

C₂

semidirect

5. #22550256

C₂

semidirect

6. #22550255

C₂

semidirect

7. #22550248

C₂

semidirect

8. C₄×#14523231

C₂

semidirect

9. #492×C₁₂₇

C₂

semidirect

45. #14523231

D₄

semidirect

88. C₄×C₁₂₇

#137

semidirect

99. C₁₂₇

#3152

semidirect

107. #492

D₁₂₇

semidirect

191. C₄

#16861238

semidirect

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

subgroup	index
C₂×#16861238	2
#22550424	2
C₄⋊Q₈×D₁₂₇	2
#22550267	2
#22550256	2
#22550255	2
#22550248	2
C₄×#14523231	2
#492×C₁₂₇	2
#3152	127

subgroup

index

C₄⋊Q₈×D₁₂₇

#492×C₁₂₇

127

Subgroups — #30649184

Conjugacy classes of subgroups

Complements

Maximal subgroups