Group #30755323 — subgroups

Subgroups — #30755323

Order 17152.

506 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 127 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	#22560347	C₂	1	1	yes	maximal, characteristic, complemented
2	#22558881	C₂	1	2	yes	maximal, complemented
3	#22558877	C₂	1	2	yes	maximal
4	#22558501	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#492×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863449	C₂²	1	2	yes	—
7	#16863420	C₂²	1	2	yes	—
8	#16863161	—	2	2	no	—
9	#16863142	—	2	4	no	—
10	#16863140	—	2	4	no	—
11	#16863118	—	2	2	no	—
12	C₂²×SD₁₆×C₆₇	—	2	2	no	—
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
15	#492×C₆₇	—	2	2	no	—
16	#492×C₆₇	C₄	1	4	yes	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#11761612	—	2	2	no	—
19	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
20	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
21	C₄⋊(C₆₇⋊C₈)	D₄	1	4	yes	—
22	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
23	C₄×C₆₇⋊C₈	—	2	2	no	—
24	C₂²×Q₈×C₆₇	—	2	1	no	—
25	C₂²×D₄×C₆₇	—	2	1	no	—
26	C₂²×D₄×C₆₇	—	2	2	no	—
27	C₂×SD₁₆×C₆₇	—	2	8	no	—
28	C₂×SD₁₆×C₆₇	—	4	8	no	—
29	C₂²×C₈×C₆₇	D₄	1	2	yes	—
30	C₄⋊Q₈×C₆₇	—	2	2	no	—
31	C₄⋊Q₈×C₆₇	—	2	1	no	—
32	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
33	#127×C₆₇	—	2	1	no	—
34	#127×C₆₇	—	2	2	no	—
35	#127×C₆₇	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
37	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
38	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₆₇	D₄	1	2	yes	—
40	#11761612	—	4	4	no	—
41	C₂×C₆₇⋊C₈	—	2	4	no	—
42	C₂×C₆₇⋊C₈	—	4	2	no	—
43	C₂⁴×C₆₇	—	4	1	no	—
44	C₂×Q₈×C₆₇	—	2	2	no	—
45	C₂×Q₈×C₆₇	—	4	1	no	—
46	C₂×Q₈×C₆₇	—	4	4	no	—
47	C₂×D₄×C₆₇	—	2	2	no	—
48	C₂×D₄×C₆₇	—	2	4	no	—
49	C₂×D₄×C₆₇	—	4	2	no	—
50	C₂×D₄×C₆₇	—	4	1	no	—
51	C₂×D₄×C₆₇	—	4	4	no	—
52	C₂×D₄×C₆₇	—	8	4	no	—
53	C₂²×C₄×C₆₇	—	4	1	no	—
54	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	#41	1	2	yes	—
56	SD₁₆×C₆₇	—	4	16	no	—
57	C₂×C₈×C₆₇	—	2	4	no	—
58	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
59	C₄⋊C₄×C₆₇	—	2	4	no	—
60	C₄⋊C₄×C₆₇	—	4	2	no	—
61	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₆₇	C₂×D₄	1	2	yes	—
63	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₂³×C₆₇	—	4	2	no	—
65	C₂³×C₆₇	—	8	4	no	—
66	C₂³×C₆₇	—	8	2	no	—
67	C₂³×C₆₇	#124	1	1	yes	characteristic
68	Q₈×C₆₇	—	4	4	no	—
69	Q₈×C₆₇	—	8	2	no	—
70	D₄×C₆₇	—	4	4	no	—
71	D₄×C₆₇	—	8	4	no	—
72	D₄×C₆₇	—	8	4	no	—
73	D₄×C₆₇	—	8	2	no	—
74	C₂×C₄×C₆₇	—	2	4	no	—
75	C₂×C₄×C₆₇	—	4	2	no	—
76	C₂×C₄×C₆₇	—	8	2	no	—
77	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₆₇	#121	1	2	yes	—
80	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
81	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
82	C₈×C₆₇	—	2	4	no	—
83	C₆₇⋊C₈	—	4	4	no	—
84	C₆₇⋊C₈	—	8	2	no	—
85	C₂²×C₆₇	—	2	2	no	—
86	C₂²×C₆₇	—	8	2	no	—
87	C₂²×C₆₇	—	8	4	no	—
88	C₂²×C₆₇	—	16	4	no	—
89	C₂²×C₆₇	#390	1	1	yes	characteristic
90	C₂²×C₆₇	#490	1	2	yes	—
91	C₄×C₆₇	—	2	2	no	—
92	C₄×C₆₇	—	8	2	no	—
93	C₄×C₆₇	#418	1	2	yes	—
94	C₄×C₆₇	#471	1	4	yes	—
95	C₄×C₆₇	#496	1	2	yes	—
96	#9777	—	67	1	no	maximal, Sylow
97	C₂×C₆₇	—	2	2	no	—
98	C₂×C₆₇	—	16	2	no	—
99	C₂×C₆₇	#1961	1	1	yes	characteristic
100	C₂×C₆₇	#1977	1	2	yes	—
101	C₂×#492	D₆₇	1	1	yes	characteristic
102	#1837	—	67	1	no	—
103	#1546	—	67	2	no	—
104	#1545	—	67	2	no	—
105	#1452	—	67	1	no	—
106	C₆₇	#9777	1	1	yes	characteristic, complemented, Sylow
107	C₂²×SD₁₆	—	2	2	no	—
108	C₂×C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
109	C₂×#127	C₂×D₆₇	1	1	yes	characteristic
110	#492	—	2	2	no	—
111	#492	Dic₆₇	1	4	yes	—
112	#431	—	67	2	no	—
113	#423	—	67	2	no	—
114	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
115	#341	—	134	2	no	—
116	#332	—	134	4	no	—
117	#331	—	134	4	no	—
118	#322	—	134	2	no	—
119	C₂²×Q₈	—	2	1	no	—
120	C₂²×D₄	—	2	1	no	—
121	C₂²×D₄	—	2	2	no	—
122	C₂×SD₁₆	—	2	8	no	—
123	C₂×SD₁₆	—	4	8	no	—
124	C₂×M₁₆	—	134	2	no	—
125	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	—	2	1	no	—
128	C₄⋊Q₈	C₂×Dic₆₇	1	2	yes	—
129	#127	—	2	2	no	—
130	#127	—	2	1	no	—
131	#127	C₂×Dic₆₇	1	2	yes	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
134	C₄⋊C₈	—	67	4	no	—
135	C₄⋊C₈	—	134	2	no	—
136	C₄⋊C₈	—	134	2	no	—
137	C₂²⋊C₈	—	134	2	no	—
138	C₄×C₈	—	134	2	no	—
139	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
140	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
141	C₂⁴	—	4	1	no	—
142	C₂×Q₈	—	2	2	no	—
143	C₂×Q₈	—	4	1	no	—
144	C₂×Q₈	—	4	4	no	—
145	C₂×D₄	—	2	2	no	—
146	C₂×D₄	—	2	4	no	—
147	C₂×D₄	—	4	2	no	—
148	C₂×D₄	—	4	1	no	—
149	C₂×D₄	—	4	4	no	—
150	C₂×D₄	—	8	4	no	—
151	C₂²×C₄	—	4	1	no	—
152	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
153	C₂²×C₄	#11761627	1	2	yes	—
154	SD₁₆	—	4	16	no	—
155	M₁₆	—	268	4	no	—
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	—	268	2	no	—
158	C₂×C₈	—	134	4	no	—
159	C₂×C₈	C₂²⋊Dic₆₇	1	4	yes	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
163	C₄²	D₄×D₆₇	1	2	yes	—
164	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
165	C₂³	—	4	2	no	—
166	C₂³	—	8	4	no	—
167	C₂³	—	8	2	no	—
168	C₂³	#14524030	1	1	yes	characteristic
169	Q₈	—	4	4	no	—
170	Q₈	—	8	2	no	—
171	D₄	—	4	4	no	—
172	D₄	—	8	4	no	—
173	D₄	—	8	2	no	—
174	D₄	—	8	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
179	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
180	C₂×C₄	#14524014	1	2	yes	—
181	C₂×C₄	#14524039	1	1	yes	characteristic
182	C₂×C₄	#14524069	1	4	yes	—
183	C₈	—	2	4	no	—
184	C₈	—	268	4	no	—
185	C₈	—	536	2	no	—
186	C₂²	—	2	2	no	—
187	C₂²	—	8	2	no	—
188	C₂²	—	8	4	no	—
189	C₂²	—	16	4	no	—
190	C₂²	#16863299	1	1	yes	characteristic
191	C₂²	#16863760	1	2	yes	—
192	C₄	—	2	2	no	—
193	C₄	—	8	2	no	—
194	C₄	#16863401	1	2	yes	—
195	C₄	#16863656	1	4	yes	—
196	C₄	#16863783	1	2	yes	—
197	C₂	—	2	2	no	—
198	C₂	—	16	2	no	—
199	C₂	#22561034	1	1	yes	characteristic
200	C₂	#22561114	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22560347

C₂

yes

maximal, characteristic, complemented

#22558881

C₂

yes

maximal, complemented

#22558877

C₂

yes

maximal

#22558501

C₂

yes

maximal, characteristic, complemented

C₂×#492×C₆₇

C₂

yes

maximal, characteristic

#16863449

C₂²

yes

—

#16863420

C₂²

yes

—

—

—

—

—

C₂²×SD₁₆×C₆₇

—

C₂×C₄⋊Q₈×C₆₇

C₂²

yes

characteristic

C₂×#127×C₆₇

C₂²

yes

characteristic

#492×C₆₇

—

#492×C₆₇

C₄

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×#11761612

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

D₄

yes

—

C₂²⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

—

C₂²×Q₈×C₆₇

—

C₂²×D₄×C₆₇

—

C₂²×D₄×C₆₇

—

C₂×SD₁₆×C₆₇

—

C₂×SD₁₆×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

C₂×C₄

yes

—

#127×C₆₇

—

#127×C₆₇

—

#127×C₆₇

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂⁴×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

SD₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

#124

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#390

yes

characteristic

C₂²×C₆₇

#490

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#418

yes

—

C₄×C₆₇

#471

yes

—

C₄×C₆₇

#496

yes

—

#9777

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#1961

yes

characteristic

100

C₂×C₆₇

#1977

yes

—

101

C₂×#492

D₆₇

yes

characteristic

102

#1837

—

103

#1546

—

104

#1545

—

105

#1452

—

106

C₆₇

#9777

yes

characteristic, complemented, Sylow

107

C₂²×SD₁₆

—

108

C₂×C₄⋊Q₈

C₂×D₆₇

yes

characteristic

109

C₂×#127

C₂×D₆₇

yes

characteristic

110

#492

—

111

#492

Dic₆₇

yes

—

112

#431

—

113

#423

—

114

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

115

#341

—

134

—

116

#332

—

134

—

117

#331

—

134

—

118

#322

—

134

—

119

C₂²×Q₈

—

120

C₂²×D₄

—

121

C₂²×D₄

—

122

C₂×SD₁₆

—

123

C₂×SD₁₆

—

124

C₂×M₁₆

—

134

—

125

C₂²×C₈

C₂²⋊D₆₇

yes

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

C₂×Dic₆₇

yes

—

129

#127

—

130

#127

—

131

#127

C₂×Dic₆₇

yes

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄²

C₂²×D₆₇

yes

characteristic

134

C₄⋊C₈

—

135

C₄⋊C₈

—

134

—

136

C₄⋊C₈

—

134

—

137

C₂²⋊C₈

—

134

—

138

C₄×C₈

—

134

—

139

C₄×C₈

C₂×Dic₆₇

yes

—

140

C₄×C₈

C₂²⋊D₆₇

yes

—

141

C₂⁴

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

—

150

C₂×D₄

—

151

C₂²×C₄

—

152

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

153

C₂²×C₄

#11761627

yes

—

154

SD₁₆

—

155

M₁₆

—

268

—

156

C₂×C₈

—

157

C₂×C₈

—

268

—

158

C₂×C₈

—

134

—

159

C₂×C₈

C₂²⋊Dic₆₇

yes

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄²

C₂²×Dic₆₇

yes

characteristic

163

C₄²

D₄×D₆₇

yes

—

164

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

165

C₂³

—

166

C₂³

—

167

C₂³

—

168

C₂³

#14524030

yes

characteristic

169

Q₈

—

170

Q₈

—

171

D₄

—

172

D₄

—

173

D₄

—

174

D₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

C₂×C₂²⋊Dic₆₇

yes

characteristic

179

C₂×C₄

D₄×Dic₆₇

yes

—

180

C₂×C₄

#14524014

yes

—

181

C₂×C₄

#14524039

yes

characteristic

182

C₂×C₄

#14524069

yes

—

183

C₈

—

184

C₈

—

268

—

185

C₈

—

536

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

#16863299

yes

characteristic

191

C₂²

#16863760

yes

—

192

C₄

—

193

C₄

—

194

C₄

#16863401

yes

—

195

C₄

#16863656

yes

—

196

C₄

#16863783

yes

—

197

C₂

—

198

C₂

—

199

C₂

#22561034

yes

characteristic

200

C₂

#22561114

yes

—

	subgroup	G/H
center	C₂²	#16863299
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#492×C₆₇	C₂
socle	C₂²×C₆₇	#390
prime radical	C₂×#127×C₆₇	C₂²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#492	D₆₇
hypocenter	C₆₇	#9777
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#492

subgroup

G/H

center

C₂²

#16863299

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₂×#492×C₆₇

C₂

socle

C₂²×C₆₇

#390

prime radical

C₂×#127×C₆₇

C₂²

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#492

N	H ≅ G/N	type
1. #22560347	C₂	semidirect
2. #22558881	C₂	semidirect
4. #22558501	C₂	semidirect
106. C₆₇	#9777	semidirect

H ≅ G/N

type

1. #22560347

C₂

semidirect

2. #22558881

C₂

semidirect

4. #22558501

C₂

semidirect

106. C₆₇

#9777

semidirect

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

subgroup	index
#22560347	2
#22558881	2
#22558877	2
#22558501	2
C₂×#492×C₆₇	2
#9777	67

subgroup

index

C₂×#492×C₆₇

Subgroups — #30755323

Conjugacy classes of subgroups

Complements

Maximal subgroups