Group #30775223 — subgroups

Subgroups — #30775223

Order 17152.

436 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 99 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×#16863776	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863211	C₂	1	2	yes	maximal
3	#22561928	C₂	1	2	yes	maximal
4	#22558291	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1945	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₆₈	—	2	1	no	—
7	C₂²×Dic₂₆₈	C₂²	1	2	yes	—
8	C₂×#14524043	—	2	1	no	—
9	C₂×#14523983	C₂²	1	2	yes	—
10	#16863776	—	2	4	no	—
11	#16863211	—	2	4	no	—
12	#16863199	—	2	4	no	—
13	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
14	#427×C₆₇	C₂²	1	1	yes	characteristic
15	#424×C₆₇	—	2	1	no	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	#339×C₆₇	—	2	1	no	—
18	C₂²×Dic₁₃₄	—	2	2	no	—
19	C₂×Dic₂₆₈	—	2	4	no	—
20	C₂×Dic₂₆₈	—	2	2	no	—
21	C₂×Dic₂₆₈	—	2	2	no	—
22	C₂×Dic₂₆₈	—	2	2	no	—
23	C₂×Dic₂₆₈	—	2	2	no	—
24	C₂×Dic₂₆₈	—	2	2	no	—
25	C₂×Dic₂₆₈	—	4	2	no	—
26	C₂×Dic₂₆₈	—	4	2	no	—
27	C₂×Dic₂₆₈	D₄	1	4	yes	—
28	#14524043	—	2	4	no	—
29	#14524043	—	4	2	no	—
30	C₂×C₄⋊Dic₆₇	—	4	2	no	—
31	#14523983	—	2	2	no	—
32	#14523983	—	2	2	no	—
33	C₂×C₆₇⋊C₁₆	—	2	2	no	—
34	C₂×M₁₆×C₆₇	—	2	1	no	—
35	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
37	#117×C₆₇	—	2	1	no	—
38	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
39	#108×C₆₇	—	2	2	no	—
40	#107×C₆₇	D₄	1	2	yes	—
41	#106×C₆₇	—	2	1	no	—
42	C₄⋊C₈×C₆₇	—	4	2	no	—
43	C₄×C₈×C₆₇	—	2	2	no	—
44	C₂×Dic₁₃₄	—	2	4	no	—
45	C₂×Dic₁₃₄	—	4	2	no	—
46	C₂×Dic₁₃₄	—	4	2	no	—
47	C₂×Dic₁₃₄	—	4	2	no	—
48	C₂×Dic₁₃₄	—	4	2	no	—
49	C₂×Dic₁₃₄	—	4	2	no	—
50	C₂²×Dic₆₇	—	4	2	no	—
51	Dic₂₆₈	—	2	4	no	—
52	Dic₂₆₈	—	2	4	no	—
53	Dic₂₆₈	—	4	8	no	—
54	Dic₂₆₈	—	4	2	no	—
55	Dic₂₆₈	—	4	2	no	—
56	C₄⋊Dic₆₇	—	4	8	no	—
57	C₆₇⋊C₁₆	—	4	2	no	—
58	C₂²×C₄×C₆₇	—	2	1	no	—
59	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
60	M₁₆×C₆₇	—	4	1	no	—
61	M₁₆×C₆₇	—	4	1	no	—
62	C₂×C₈×C₆₇	—	2	1	no	—
63	C₂×C₈×C₆₇	—	2	1	no	—
64	C₂×C₈×C₆₇	—	4	1	no	—
65	C₂×C₈×C₆₇	D₈	1	2	yes	—
66	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
69	C₂×C₈×C₆₇	#41	1	2	yes	—
70	C₄⋊C₄×C₆₇	—	2	2	no	—
71	C₂²⋊C₄×C₆₇	—	4	1	no	—
72	C₄²×C₆₇	—	2	1	no	—
73	C₄²×C₆₇	—	4	1	no	—
74	C₄²×C₆₇	SD₁₆	1	2	yes	—
75	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
76	C₂×C₄×C₆₇	—	2	4	no	—
77	C₂×C₄×C₆₇	—	2	2	no	—
78	C₂×C₄×C₆₇	—	4	2	no	—
79	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
80	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
81	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
84	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
85	C₂×Dic₆₇	—	4	4	no	—
86	C₂×Dic₆₇	—	8	2	no	—
87	C₂×Dic₆₇	—	8	2	no	—
88	Dic₁₃₄	—	4	4	no	—
89	Dic₁₃₄	—	4	4	no	—
90	Dic₁₃₄	—	8	2	no	—
91	Dic₁₃₄	—	8	2	no	—
92	C₈×C₆₇	—	2	2	no	—
93	C₈×C₆₇	—	8	1	no	—
94	C₈×C₆₇	#121	1	2	yes	—
95	C₈×C₆₇	#123	1	2	yes	—
96	C₂²×C₆₇	—	2	2	no	—
97	C₂²×C₆₇	#450	1	1	yes	characteristic
98	C₂²×C₆₇	#459	1	1	yes	characteristic
99	C₂²×C₆₇	#490	1	1	yes	characteristic
100	C₄×C₆₇	—	2	4	no	—
101	C₄×C₆₇	—	8	2	no	—
102	C₄×C₆₇	#394	1	1	yes	characteristic
103	C₄×C₆₇	#447	1	1	yes	characteristic
104	C₄×C₆₇	#460	1	1	yes	characteristic
105	C₄×C₆₇	#486	1	1	yes	characteristic
106	Dic₆₇	—	8	4	no	—
107	#13415	—	67	1	no	maximal, Sylow
108	C₂×C₆₇	—	2	1	no	—
109	C₂×C₆₇	—	2	1	no	—
110	C₂×C₆₇	#2041	1	1	yes	characteristic
111	C₂×C₆₇	#2214	1	2	yes	—
112	C₂×#494	—	67	1	no	—
113	C₂×#362	—	67	2	no	—
114	#2141	—	67	2	no	—
115	#1945	D₆₇	1	1	yes	characteristic
116	#1376	—	67	1	no	—
117	C₆₇	#13415	1	1	yes	characteristic, complemented, Sylow
118	C₂²×Q₁₆	—	67	2	no	—
119	C₂²×Q₁₆	—	134	1	no	—
120	C₂×C₄⋊Q₈	—	134	1	no	—
121	C₂×M₃₂	—	67	2	no	—
122	#494	—	134	4	no	—
123	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
124	#427	C₂×D₆₇	1	1	yes	characteristic
125	#424	—	2	1	no	—
126	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
127	#362	—	134	4	no	—
128	#358	—	134	4	no	—
129	#339	—	2	1	no	—
130	C₂²×Q₈	—	134	2	no	—
131	C₂×Q₁₆	—	268	2	no	—
132	C₂×Q₁₆	—	268	2	no	—
133	C₂×Q₁₆	—	67	4	no	—
134	C₂×Q₁₆	—	134	4	no	—
135	C₂×Q₁₆	—	134	2	no	—
136	C₂×Q₁₆	—	134	2	no	—
137	C₂×Q₁₆	—	134	2	no	—
138	C₂×Q₁₆	—	134	2	no	—
139	C₂×Q₁₆	—	134	2	no	—
140	C₂×M₁₆	—	2	1	no	—
141	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
142	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
143	C₄⋊Q₈	—	268	2	no	—
144	C₄⋊Q₈	—	134	4	no	—
145	#117	—	2	1	no	—
146	C₂×C₄⋊C₄	—	268	2	no	—
147	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
148	M₃₂	—	134	2	no	—
149	M₃₂	—	134	2	no	—
150	C₂×C₁₆	—	134	2	no	—
151	#108	—	2	2	no	—
152	#107	D₂₆₈	1	2	yes	—
153	#106	—	2	1	no	—
154	C₄⋊C₈	—	4	2	no	—
155	C₄×C₈	—	2	2	no	—
156	C₂×Q₈	—	268	2	no	—
157	C₂×Q₈	—	268	2	no	—
158	C₂×Q₈	—	268	2	no	—
159	C₂×Q₈	—	268	2	no	—
160	C₂×Q₈	—	268	2	no	—
161	C₂×Q₈	—	134	4	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	268	2	no	—
164	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
165	Q₁₆	—	268	8	no	—
166	Q₁₆	—	268	2	no	—
167	Q₁₆	—	268	2	no	—
168	Q₁₆	—	134	4	no	—
169	Q₁₆	—	134	4	no	—
170	M₁₆	—	4	1	no	—
171	M₁₆	—	4	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	#11761613	1	2	yes	—
176	C₂×C₈	D₄×D₆₇	1	2	yes	—
177	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
178	C₂×C₈	#11761628	1	1	yes	characteristic
179	C₂×C₈	#11761629	1	2	yes	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	268	8	no	—
182	C₂²⋊C₄	—	4	1	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	4	1	no	—
185	C₄²	#11761615	1	2	yes	—
186	C₁₆	—	268	2	no	—
187	C₂³	#14524068	1	1	yes	characteristic
188	Q₈	—	268	4	no	—
189	Q₈	—	268	4	no	—
190	Q₈	—	536	2	no	—
191	Q₈	—	536	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	268	4	no	—
196	C₂×C₄	—	536	2	no	—
197	C₂×C₄	—	536	2	no	—
198	C₂×C₄	#14524011	1	1	yes	characteristic
199	C₂×C₄	#14524017	1	1	yes	characteristic
200	C₂×C₄	#14524032	1	1	yes	characteristic
201	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
202	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
203	C₂×C₄	#14524068	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	8	1	no	—
206	C₈	#14524018	1	2	yes	—
207	C₈	#14524028	1	2	yes	—
208	C₂²	—	2	2	no	—
209	C₂²	#16863546	1	1	yes	characteristic
210	C₂²	#16863591	1	1	yes	characteristic
211	C₂²	#16863762	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	8	2	no	—
214	C₄	—	536	4	no	—
215	C₄	#16863313	1	1	yes	characteristic
216	C₄	#16863529	1	1	yes	characteristic
217	C₄	#16863596	1	1	yes	characteristic
218	C₄	#16863744	1	1	yes	characteristic
219	C₂	—	2	1	no	—
220	C₂	—	2	1	no	—
221	C₂	#22561443	1	1	yes	characteristic
222	C₂	#22562290	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863776

C₂

yes

maximal, characteristic

C₂×#16863211

C₂

yes

maximal

#22561928

C₂

yes

maximal

#22558291

C₂

yes

maximal, characteristic

C₆₇×#1945

C₂

yes

maximal, characteristic

C₂²×Dic₂₆₈

—

C₂²×Dic₂₆₈

C₂²

yes

—

C₂×#14524043

—

C₂×#14523983

C₂²

yes

—

#16863776

—

#16863211

—

#16863199

—

C₂×#108×C₆₇

C₂²

yes

characteristic

#427×C₆₇

C₂²

yes

characteristic

#424×C₆₇

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#339×C₆₇

—

C₂²×Dic₁₃₄

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

D₄

yes

—

#14524043

—

#14524043

—

C₂×C₄⋊Dic₆₇

—

#14523983

—

#14523983

—

C₂×C₆₇⋊C₁₆

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

#117×C₆₇

—

C₂×C₄²×C₆₇

D₄

yes

characteristic

#108×C₆₇

—

#107×C₆₇

D₄

yes

—

#106×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄×C₈×C₆₇

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

Dic₂₆₈

—

Dic₂₆₈

—

Dic₂₆₈

—

Dic₂₆₈

—

Dic₂₆₈

—

C₄⋊Dic₆₇

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

C₂×C₈×C₆₇

yes

C₂×C₈×C₆₇

yes

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

SD₁₆

yes

—

C₂³×C₆₇

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#121

yes

—

C₈×C₆₇

#123

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

100

C₄×C₆₇

—

101

C₄×C₆₇

—

102

C₄×C₆₇

#394

yes

characteristic

103

C₄×C₆₇

#447

yes

characteristic

104

C₄×C₆₇

#460

yes

characteristic

105

C₄×C₆₇

#486

yes

characteristic

106

Dic₆₇

—

107

#13415

—

maximal, Sylow

108

C₂×C₆₇

—

109

C₂×C₆₇

—

110

C₂×C₆₇

#2041

yes

characteristic

111

C₂×C₆₇

#2214

yes

—

112

C₂×#494

—

113

C₂×#362

—

114

#2141

—

115

#1945

D₆₇

yes

characteristic

116

#1376

—

117

C₆₇

#13415

yes

characteristic, complemented, Sylow

118

C₂²×Q₁₆

—

119

C₂²×Q₁₆

—

134

—

120

C₂×C₄⋊Q₈

—

134

—

121

C₂×M₃₂

—

122

#494

—

134

—

123

C₂×#108

C₂×D₆₇

yes

characteristic

124

#427

C₂×D₆₇

yes

characteristic

125

#424

—

126

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

127

#362

—

134

—

128

#358

—

134

—

129

#339

—

130

C₂²×Q₈

—

134

—

131

C₂×Q₁₆

—

268

—

132

C₂×Q₁₆

—

268

—

133

C₂×Q₁₆

—

134

C₂×Q₁₆

—

134

—

135

C₂×Q₁₆

—

134

—

136

C₂×Q₁₆

—

134

—

137

C₂×Q₁₆

—

134

—

138

C₂×Q₁₆

—

134

—

139

C₂×Q₁₆

—

134

—

140

C₂×M₁₆

—

141

C₂²×C₈

C₂²⋊D₆₇

yes

characteristic

142

C₂²×C₈

C₂²×D₆₇

yes

characteristic

143

C₄⋊Q₈

—

268

—

144

C₄⋊Q₈

—

134

—

145

#117

—

146

C₂×C₄⋊C₄

—

268

—

147

C₂×C₄²

C₂²⋊D₆₇

yes

characteristic

148

M₃₂

—

134

—

149

M₃₂

—

134

—

150

C₂×C₁₆

—

134

—

151

#108

—

152

#107

D₂₆₈

yes

—

153

#106

—

154

C₄⋊C₈

—

155

C₄×C₈

—

156

C₂×Q₈

—

268

—

157

C₂×Q₈

—

268

—

158

C₂×Q₈

—

268

—

159

C₂×Q₈

—

268

—

160

C₂×Q₈

—

268

—

161

C₂×Q₈

—

134

—

162

C₂²×C₄

—

163

C₂²×C₄

—

268

—

164

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

165

Q₁₆

—

268

—

166

Q₁₆

—

268

—

167

Q₁₆

—

268

—

168

Q₁₆

—

134

—

169

Q₁₆

—

134

—

170

M₁₆

—

171

M₁₆

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

#11761613

yes

—

176

C₂×C₈

D₄×D₆₇

yes

—

177

C₂×C₈

C₂×D₂₆₈

yes

characteristic

178

C₂×C₈

#11761628

yes

characteristic

179

C₂×C₈

#11761629

yes

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

268

—

182

C₂²⋊C₄

—

183

C₄²

—

184

C₄²

—

185

C₄²

#11761615

yes

—

186

C₁₆

—

268

—

187

C₂³

#14524068

yes

characteristic

188

Q₈

—

268

—

189

Q₈

—

268

—

190

Q₈

—

536

—

191

Q₈

—

536

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

268

—

196

C₂×C₄

—

536

—

197

C₂×C₄

—

536

—

198

C₂×C₄

#14524011

yes

characteristic

199

C₂×C₄

#14524017

yes

characteristic

200

C₂×C₄

#14524032

yes

characteristic

201

C₂×C₄

C₂×#11761613

yes

characteristic

202

C₂×C₄

C₂×#11761615

yes

characteristic

203

C₂×C₄

#14524068

yes

characteristic

204

C₈

—

205

C₈

—

206

C₈

#14524018

yes

—

207

C₈

#14524028

yes

—

208

C₂²

—

209

C₂²

#16863546

yes

characteristic

210

C₂²

#16863591

yes

characteristic

211

C₂²

#16863762

yes

characteristic

212

C₄

—

213

C₄

—

214

C₄

—

536

—

215

C₄

#16863313

yes

characteristic

216

C₄

#16863529

yes

characteristic

217

C₄

#16863596

yes

characteristic

218

C₄

#16863744

yes

characteristic

219

C₂

—

220

C₂

—

221

C₂

#22561443

yes

characteristic

222

C₂

#22562290

yes

—

	subgroup	G/H
center	C₂²	#16863591
commutator, cosocle	C₂²×C₈×C₆₇	C₂³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1945	C₂
socle	C₂²×C₆₇	#459
prime radical	C₂³×C₆₇	AGL(1,ℤ/8)
hypercenter	#1945	D₆₇
hypocenter	C₆₇	#13415
LCS	G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1945

subgroup

G/H

center

C₂²

#16863591

commutator, cosocle

C₂²×C₈×C₆₇

C₂³

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#1945

C₂

C₂²×C₆₇

C₂³×C₆₇

D₆₇

C₆₇

G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1945

N	H ≅ G/N	type
117. C₆₇	#13415	semidirect

H ≅ G/N

type

117. C₆₇

#13415

semidirect

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

subgroup	index
C₂×#16863776	2
C₂×#16863211	2
#22561928	2
#22558291	2
C₆₇×#1945	2
#13415	67

subgroup

index

C₆₇×#1945

Subgroups — #30775223

Conjugacy classes of subgroups

Complements

Maximal subgroups