Group #30755951 — subgroups

Subgroups — #30755951

Order 17152.

430 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 159 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	#22568572	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#22558093	C₂	1	4	yes	maximal
4	C₆₇×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×#14524035	—	2	1	no	—
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
8	#16863294	—	2	2	no	—
9	#16863294	C₄	1	4	yes	—
10	#16863266	C₂²	1	2	yes	—
11	#16863262	—	2	1	no	—
12	#16863151	—	2	1	no	—
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
14	C₄³×C₆₇	C₂²	1	1	yes	characteristic
15	#336×C₆₇	—	2	2	no	—
16	C₂²×C₆₇⋊C₈	—	2	2	no	—
17	#14524035	—	4	4	no	—
18	C₂×C₄⋊Dic₆₇	—	2	2	no	—
19	C₂×C₄⋊Dic₆₇	—	2	2	no	—
20	C₂×C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
21	C₂×Dic₆₇⋊C₄	—	2	4	no	—
22	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₆₇⋊C₈)	Q₈	1	4	yes	—
24	C₄×C₆₇⋊C₈	—	4	2	no	—
25	C₆₇⋊#95	—	2	4	no	—
26	C₂²×C₈×C₆₇	—	2	2	no	—
27	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₆₇	D₄	1	2	yes	—
31	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₆₇	—	2	4	no	—
33	C₂²×Dic₆₇	—	2	4	no	—
34	C₂×C₆₇⋊C₈	—	2	4	no	—
35	C₂×C₆₇⋊C₈	—	4	2	no	—
36	C₂×C₆₇⋊C₈	—	4	2	no	—
37	C₄⋊Dic₆₇	—	4	4	no	—
38	C₄⋊Dic₆₇	—	4	4	no	—
39	C₄⋊Dic₆₇	—	4	4	no	—
40	Dic₆₇⋊C₄	—	4	8	no	—
41	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₆₇	—	4	2	no	—
47	C₂×C₈×C₆₇	—	4	4	no	—
48	C₄²×C₆₇	—	2	4	no	—
49	C₄²×C₆₇	—	2	2	no	—
50	C₄²×C₆₇	—	4	2	no	—
51	C₄²×C₆₇	SD₁₆	1	2	yes	—
52	C₄²×C₆₇	Q₁₆	1	2	yes	—
53	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
54	C₄²×C₆₇	#41	1	2	yes	—
55	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₆₇	—	2	2	no	—
57	C₂×C₄×C₆₇	—	2	2	no	—
58	C₂×C₄×C₆₇	—	2	2	no	—
59	C₂×C₄×C₆₇	—	2	2	no	—
60	C₂×C₄×C₆₇	—	2	2	no	—
61	C₂×C₄×C₆₇	—	2	2	no	—
62	C₂×C₄×C₆₇	#103	1	4	yes	—
63	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
64	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
65	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
68	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₆₇	#135	1	4	yes	—
70	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
71	C₂×Dic₆₇	—	4	4	no	—
72	C₂×Dic₆₇	—	4	4	no	—
73	C₂×Dic₆₇	—	4	4	no	—
74	C₈×C₆₇	—	8	2	no	—
75	C₆₇⋊C₈	—	4	4	no	—
76	C₂²×C₆₇	#391	1	1	yes	characteristic
77	C₂²×C₆₇	#413	1	2	yes	—
78	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
79	C₂²×C₆₇	#416	1	2	yes	—
80	C₂²×C₆₇	#418	1	1	yes	characteristic
81	C₄×C₆₇	—	2	4	no	—
82	C₄×C₆₇	—	2	2	no	—
83	C₄×C₆₇	—	4	2	no	—
84	C₄×C₆₇	#476	1	2	yes	—
85	C₄×C₆₇	#477	1	2	yes	—
86	C₄×C₆₇	#488	1	2	yes	—
87	C₄×C₆₇	#499	1	2	yes	—
88	Dic₆₇	—	8	4	no	—
89	#9897	—	67	1	no	maximal, Sylow
90	C₂×C₆₇	#1542	1	2	yes	—
91	C₂×C₆₇	#1553	1	2	yes	—
92	C₂×C₆₇	#1987	1	1	yes	characteristic
93	C₂×C₆₇	#1991	1	1	yes	characteristic
94	C₂×C₆₇	#1998	1	1	yes	characteristic
95	#2859	—	67	1	no	—
96	#1831	D₆₇	1	1	yes	characteristic
97	C₄×C₄⋊C₈	—	67	1	no	—
98	#1296	—	67	4	no	—
99	C₆₇	#9897	1	1	yes	characteristic, complemented, Sylow
100	C₂×#125	—	134	1	no	—
101	C₂×C₄⋊C₈	—	67	2	no	—
102	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
103	C₂×C₄×C₈	—	134	1	no	—
104	#389	—	67	4	no	—
105	#389	—	134	2	no	—
106	#384	—	67	2	no	—
107	#382	—	134	1	no	—
108	C₄³	C₂×D₆₇	1	1	yes	characteristic
109	#336	—	2	2	no	—
110	#336	—	134	1	no	—
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	—	134	2	no	—
113	#125	—	268	4	no	—
114	C₂×C₄⋊C₄	—	67	4	no	—
115	C₂×C₄⋊C₄	—	134	4	no	—
116	C₂×C₄⋊C₄	—	134	2	no	—
117	C₂×C₄⋊C₄	—	134	2	no	—
118	C₂×C₄²	C₄×D₆₇	1	2	yes	—
119	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
120	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
121	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
122	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
123	C₄⋊C₈	—	2	4	no	—
124	C₄⋊C₈	—	67	4	no	—
125	C₄⋊C₈	—	134	2	no	—
126	C₄×C₈	—	268	2	no	—
127	#95	—	134	4	no	—
128	C₂²×C₄	—	134	4	no	—
129	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
130	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
131	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
133	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	—	4	4	no	—
136	C₂×C₈	—	268	2	no	—
137	C₂×C₈	—	268	2	no	—
138	C₂×C₈	—	134	4	no	—
139	C₄⋊C₄	—	268	4	no	—
140	C₄⋊C₄	—	268	8	no	—
141	C₄⋊C₄	—	268	4	no	—
142	C₄⋊C₄	—	268	4	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	#11761617	1	2	yes	—
147	C₄²	Dic₂₆₈	1	2	yes	—
148	C₄²	Q₈×D₆₇	1	2	yes	—
149	C₄²	#11761627	1	2	yes	—
150	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	268	4	no	—
158	C₂×C₄	—	268	4	no	—
159	C₂×C₄	—	268	4	no	—
160	C₂×C₄	#14523968	1	4	yes	—
161	C₂×C₄	#14524008	1	2	yes	—
162	C₂×C₄	#14524023	1	2	yes	—
163	C₂×C₄	#14524030	1	1	yes	characteristic
164	C₂×C₄	#14524041	1	1	yes	characteristic
165	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
166	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
167	C₂×C₄	#14524063	1	4	yes	—
168	C₂×C₄	#14524071	1	2	yes	—
169	C₈	—	8	2	no	—
170	C₈	—	268	4	no	—
171	C₂²	#16863301	1	1	yes	characteristic
172	C₂²	#16863378	1	2	yes	—
173	C₂²	C₂×#14523968	1	1	yes	characteristic
174	C₂²	#16863390	1	2	yes	—
175	C₂²	#16863402	1	1	yes	characteristic
176	C₄	—	2	4	no	—
177	C₄	—	2	2	no	—
178	C₄	—	4	2	no	—
179	C₄	—	536	4	no	—
180	C₄	#16863687	1	2	yes	—
181	C₄	#16863692	1	2	yes	—
182	C₄	#16863750	1	2	yes	—
183	C₄	#16863795	1	2	yes	—
184	C₂	#22558864	1	2	yes	—
185	C₂	#22558915	1	2	yes	—
186	C₂	#22561156	1	1	yes	characteristic
187	C₂	#22561173	1	1	yes	characteristic
188	C₂	#22561201	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22568572

C₂

yes

maximal, characteristic

C₄×C₄⋊(C₆₇⋊C₈)

C₂

yes

maximal, characteristic

#22558093

C₂

yes

maximal

C₆₇×#1831

C₂

yes

maximal, characteristic

C₂×#14524035

—

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₂×C₄×C₆₇⋊C₈

—

#16863294

—

#16863294

C₄

yes

—

#16863266

C₂²

yes

—

#16863262

—

#16863151

—

C₂×C₄⋊C₈×C₆₇

C₂²

yes

—

C₄³×C₆₇

C₂²

yes

characteristic

#336×C₆₇

—

C₂²×C₆₇⋊C₈

—

#14524035

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

C₂×C₄

yes

—

C₂×Dic₆₇⋊C₄

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

Q₈

yes

—

C₄×C₆₇⋊C₈

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

—

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

SD₁₆

yes

—

C₄²×C₆₇

Q₁₆

yes

—

C₄²×C₆₇

C₂×Q₈

yes

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#103

yes

—

C₂×C₄×C₆₇

C₄×Q₈

yes

—

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₇

#124

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₇

#135

yes

—

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

#391

yes

characteristic

C₂²×C₆₇

#413

yes

—

C₂²×C₆₇

C₂×#103

yes

characteristic

C₂²×C₆₇

#416

yes

—

C₂²×C₆₇

#418

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#476

yes

—

C₄×C₆₇

#477

yes

—

C₄×C₆₇

#488

yes

—

C₄×C₆₇

#499

yes

—

Dic₆₇

—

#9897

—

maximal, Sylow

C₂×C₆₇

#1542

yes

—

C₂×C₆₇

#1553

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

D₆₇

yes

characteristic

C₄×C₄⋊C₈

—

#1296

—

C₆₇

#9897

yes

characteristic, complemented, Sylow

100

C₂×#125

—

134

—

101

C₂×C₄⋊C₈

—

102

C₂×C₄⋊C₈

C₂×D₆₇

yes

—

103

C₂×C₄×C₈

—

134

—

104

#389

—

105

#389

—

134

—

106

#384

—

107

#382

—

134

—

108

C₄³

C₂×D₆₇

yes

characteristic

109

#336

—

110

#336

—

134

—

111

C₂²×C₈

—

112

C₂²×C₈

—

134

—

113

#125

—

268

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

134

—

116

C₂×C₄⋊C₄

—

134

—

117

C₂×C₄⋊C₄

—

134

—

118

C₂×C₄²

C₄×D₆₇

yes

—

119

C₂×C₄²

D₂₆₈

yes

characteristic

120

C₂×C₄²

D₂₆₈

yes

characteristic

121

C₂×C₄²

C₂²⋊D₆₇

yes

—

122

C₂×C₄²

C₂²×D₆₇

yes

characteristic

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

C₄⋊C₈

—

134

—

126

C₄×C₈

—

268

—

127

#95

—

134

—

128

C₂²×C₄

—

134

—

129

C₂²×C₄

(C₂×D₆₇)⋊C₄

yes

—

130

C₂²×C₄

(C₂×D₆₇)⋊C₄

yes

—

131

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

132

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

133

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

268

—

137

C₂×C₈

—

268

—

138

C₂×C₈

—

134

—

139

C₄⋊C₄

—

268

—

140

C₄⋊C₄

—

268

—

141

C₄⋊C₄

—

268

—

142

C₄⋊C₄

—

268

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

#11761617

yes

—

147

C₄²

Dic₂₆₈

yes

—

148

C₄²

Q₈×D₆₇

yes

—

149

C₄²

#11761627

yes

—

150

C₂³

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

268

—

158

C₂×C₄

—

268

—

159

C₂×C₄

—

268

—

160

C₂×C₄

#14523968

yes

—

161

C₂×C₄

#14524008

yes

—

162

C₂×C₄

#14524023

yes

—

163

C₂×C₄

#14524030

yes

characteristic

164

C₂×C₄

#14524041

yes

characteristic

165

C₂×C₄

C₂×#11761617

yes

characteristic

166

C₂×C₄

C₂×Dic₂₆₈

yes

characteristic

167

C₂×C₄

#14524063

yes

—

168

C₂×C₄

#14524071

yes

—

169

C₈

—

170

C₈

—

268

—

171

C₂²

#16863301

yes

characteristic

172

C₂²

#16863378

yes

—

173

C₂²

C₂×#14523968

yes

characteristic

174

C₂²

#16863390

yes

—

175

C₂²

#16863402

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

536

—

180

C₄

#16863687

yes

—

181

C₄

#16863692

yes

—

182

C₄

#16863750

yes

—

183

C₄

#16863795

yes

—

184

C₂

#22558864

yes

—

185

C₂

#22558915

yes

—

186

C₂

#22561156

yes

characteristic

187

C₂

#22561173

yes

characteristic

188

C₂

#22561201

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₆₇)⋊C₄
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1831	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1831	D₆₇
hypocenter	C₆₇	#9897
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂³

C₂×(C₂×D₆₇)⋊C₄

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1831

C₂

socle, prime radical

C₂³×C₆₇

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

N	H ≅ G/N	type
99. C₆₇	#9897	semidirect

H ≅ G/N

type

99. C₆₇

#9897

semidirect

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

subgroup	index
#22568572	2
C₄×C₄⋊(C₆₇⋊C₈)	2
#22558093	2
C₆₇×#1831	2
#9897	67

subgroup

index

#22568572

C₄×C₄⋊(C₆₇⋊C₈)

#22558093

C₆₇×#1831

#9897

Subgroups — #30755951

Conjugacy classes of subgroups

Complements

Maximal subgroups