Group #30755961 — subgroups

Subgroups — #30755961

Order 17152.

430 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 167 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	#22560326	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#22558092	C₂	1	4	yes	maximal
4	C₆₇×#2859	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
8	#16863151	—	2	1	no	—
9	#16863151	—	2	2	no	—
10	C₂×#125×C₆₇	—	2	1	no	—
11	#389×C₆₇	—	2	2	no	—
12	#389×C₆₇	C₄	1	4	yes	—
13	#384×C₆₇	C₂²	1	2	yes	—
14	#382×C₆₇	—	2	1	no	—
15	C₄³×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₆₇⋊C₈	—	2	2	no	—
17	C₂²×C₆₇⋊C₈	—	2	2	no	—
18	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
19	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
20	C₄⋊(C₆₇⋊C₈)	Q₈	1	4	yes	—
21	C₄×C₆₇⋊C₈	—	4	2	no	—
22	#125×C₆₇	—	4	2	no	—
23	#125×C₆₇	—	4	2	no	—
24	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
25	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
26	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
27	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₆₇	D₄	1	2	yes	—
31	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
33	#95×C₆₇	—	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	C₂×C₆₇⋊C₈	—	4	2	no	—
38	C₂×C₆₇⋊C₈	—	4	4	no	—
39	C₂²×C₄×C₆₇	—	2	4	no	—
40	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
45	C₄⋊C₄×C₆₇	—	4	4	no	—
46	C₄⋊C₄×C₆₇	—	4	8	no	—
47	C₄⋊C₄×C₆₇	—	4	2	no	—
48	C₄⋊C₄×C₆₇	—	4	2	no	—
49	C₄⋊C₄×C₆₇	—	4	4	no	—
50	C₄²×C₆₇	—	2	4	no	—
51	C₄²×C₆₇	—	2	2	no	—
52	C₄²×C₆₇	—	4	2	no	—
53	C₄²×C₆₇	D₈	1	2	yes	—
54	C₄²×C₆₇	SD₁₆	1	2	yes	—
55	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
56	C₄²×C₆₇	#41	1	2	yes	—
57	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	2	no	—
63	C₂×C₄×C₆₇	—	2	2	no	—
64	C₂×C₄×C₆₇	—	4	4	no	—
65	C₂×C₄×C₆₇	—	4	4	no	—
66	C₂×C₄×C₆₇	—	4	4	no	—
67	C₂×C₄×C₆₇	#102	1	4	yes	—
68	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
69	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
70	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₆₇	#135	1	4	yes	—
75	C₂×C₄×C₆₇	#137	1	2	yes	—
76	C₆₇⋊C₈	—	4	4	no	—
77	C₆₇⋊C₈	—	8	2	no	—
78	C₂²×C₆₇	#391	1	1	yes	characteristic
79	C₂²×C₆₇	#413	1	2	yes	—
80	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
81	C₂²×C₆₇	#416	1	2	yes	—
82	C₂²×C₆₇	#419	1	1	yes	characteristic
83	C₄×C₆₇	—	2	4	no	—
84	C₄×C₆₇	—	2	2	no	—
85	C₄×C₆₇	—	4	2	no	—
86	C₄×C₆₇	—	8	4	no	—
87	C₄×C₆₇	#474	1	2	yes	—
88	C₄×C₆₇	#475	1	2	yes	—
89	C₄×C₆₇	#488	1	2	yes	—
90	C₄×C₆₇	#499	1	2	yes	—
91	#9899	—	67	1	no	maximal, Sylow
92	C₂×C₆₇	#1540	1	2	yes	—
93	C₂×C₆₇	#1554	1	2	yes	—
94	C₂×C₆₇	#1988	1	1	yes	characteristic
95	C₂×C₆₇	#1991	1	1	yes	characteristic
96	C₂×C₆₇	#1998	1	1	yes	characteristic
97	#2859	D₆₇	1	1	yes	characteristic
98	#1831	—	67	1	no	—
99	C₄×C₄⋊C₈	—	67	1	no	—
100	#1296	—	67	4	no	—
101	C₆₇	#9899	1	1	yes	characteristic, complemented, Sylow
102	C₂×#125	—	2	1	no	—
103	C₂×C₄⋊C₈	—	67	2	no	—
104	C₂×C₄⋊C₈	—	67	2	no	—
105	C₂×C₄×C₈	—	134	1	no	—
106	#389	—	2	2	no	—
107	#389	Dic₆₇	1	4	yes	—
108	#384	C₂×D₆₇	1	2	yes	—
109	#382	—	2	1	no	—
110	C₄³	C₂×D₆₇	1	1	yes	characteristic
111	#336	—	134	2	no	—
112	#336	—	134	1	no	—
113	C₂²×C₈	—	134	2	no	—
114	C₂²×C₈	—	134	2	no	—
115	#125	—	4	2	no	—
116	#125	—	4	2	no	—
117	C₂×C₄⋊C₄	—	2	4	no	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
121	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
122	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
123	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
124	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
125	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
126	C₄⋊C₈	—	67	4	no	—
127	C₄⋊C₈	—	134	4	no	—
128	C₄⋊C₈	—	134	2	no	—
129	C₄×C₈	—	268	2	no	—
130	#95	—	2	4	no	—
131	C₂²×C₄	—	2	4	no	—
132	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
133	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
134	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
137	C₂×C₈	—	268	2	no	—
138	C₂×C₈	—	268	2	no	—
139	C₂×C₈	—	268	2	no	—
140	C₂×C₈	—	268	4	no	—
141	C₂×C₈	—	134	4	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₄⋊C₄	—	4	8	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	4	4	no	—
147	C₄²	—	2	4	no	—
148	C₄²	—	2	2	no	—
149	C₄²	—	4	2	no	—
150	C₄²	#11761613	1	2	yes	—
151	C₄²	#11761616	1	2	yes	—
152	C₄²	Q₈×D₆₇	1	2	yes	—
153	C₄²	#11761629	1	2	yes	—
154	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	#14523964	1	4	yes	—
165	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
166	C₂×C₄	#14524019	1	2	yes	—
167	C₂×C₄	#14524032	1	1	yes	characteristic
168	C₂×C₄	#14524041	1	1	yes	characteristic
169	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
170	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
171	C₂×C₄	#14524063	1	4	yes	—
172	C₂×C₄	#14524074	1	2	yes	—
173	C₈	—	268	4	no	—
174	C₈	—	536	2	no	—
175	C₂²	#16863303	1	1	yes	characteristic
176	C₂²	#16863380	1	2	yes	—
177	C₂²	C₂×#14523964	1	1	yes	characteristic
178	C₂²	#16863392	1	2	yes	—
179	C₂²	#16863405	1	1	yes	characteristic
180	C₄	—	2	4	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	8	4	no	—
184	C₄	#16863673	1	2	yes	—
185	C₄	#16863678	1	2	yes	—
186	C₄	#16863753	1	2	yes	—
187	C₄	#16863795	1	2	yes	—
188	C₂	#22558858	1	2	yes	—
189	C₂	#22558919	1	2	yes	—
190	C₂	#22561162	1	1	yes	characteristic
191	C₂	#22561175	1	1	yes	characteristic
192	C₂	#22561203	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22560326

C₂

yes

maximal, characteristic

C₄×C₄⋊(C₆₇⋊C₈)

C₂

yes

maximal, characteristic

#22558092

C₂

yes

maximal

C₆₇×#2859

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₂×C₄×C₆₇⋊C₈

—

#16863151

—

#16863151

—

C₂×#125×C₆₇

—

#389×C₆₇

—

#389×C₆₇

C₄

yes

—

#384×C₆₇

C₂²

yes

—

#382×C₆₇

—

C₄³×C₆₇

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

—

C₂²×C₆₇⋊C₈

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

Q₈

yes

—

C₄×C₆₇⋊C₈

—

#125×C₆₇

—

#125×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#95×C₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

—

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

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

D₈

yes

—

C₄²×C₆₇

SD₁₆

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

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#102

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×D₈

yes

characteristic

C₂×C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

#135

yes

—

C₂×C₄×C₆₇

#137

yes

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

#391

yes

characteristic

C₂²×C₆₇

#413

yes

—

C₂²×C₆₇

C₂×#102

yes

characteristic

C₂²×C₆₇

#416

yes

—

C₂²×C₆₇

#419

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#474

yes

—

C₄×C₆₇

#475

yes

—

C₄×C₆₇

#488

yes

—

C₄×C₆₇

#499

yes

—

#9899

—

maximal, Sylow

C₂×C₆₇

#1540

yes

—

C₂×C₆₇

#1554

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

yes

characteristic

#1831

—

C₄×C₄⋊C₈

—

100

#1296

—

101

C₆₇

#9899

yes

characteristic, complemented, Sylow

102

C₂×#125

—

103

C₂×C₄⋊C₈

—

104

C₂×C₄⋊C₈

—

105

C₂×C₄×C₈

—

134

—

106

#389

—

107

#389

Dic₆₇

yes

—

108

#384

C₂×D₆₇

yes

—

109

#382

—

110

C₄³

C₂×D₆₇

yes

characteristic

111

#336

—

134

—

112

#336

—

134

—

113

C₂²×C₈

—

134

—

114

C₂²×C₈

—

134

—

115

#125

—

116

#125

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

C₂×Dic₆₇

yes

—

121

C₂×C₄²

C₂×Dic₆₇

yes

—

122

C₂×C₄²

C₂²⋊D₆₇

yes

characteristic

123

C₂×C₄²

C₂²⋊D₆₇

yes

—

124

C₂×C₄²

C₂²⋊D₆₇

yes

characteristic

125

C₂×C₄²

C₂²×D₆₇

yes

characteristic

126

C₄⋊C₈

—

127

C₄⋊C₈

—

134

—

128

C₄⋊C₈

—

134

—

129

C₄×C₈

—

268

—

130

#95

—

131

C₂²×C₄

—

132

C₂²×C₄

C₂²⋊Dic₆₇

yes

—

133

C₂²×C₄

C₂²⋊Dic₆₇

yes

—

134

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

135

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

136

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

137

C₂×C₈

—

268

—

138

C₂×C₈

—

268

—

139

C₂×C₈

—

268

—

140

C₂×C₈

—

268

—

141

C₂×C₈

—

134

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄²

—

148

C₄²

—

149

C₄²

—

150

C₄²

#11761613

yes

—

151

C₄²

#11761616

yes

—

152

C₄²

Q₈×D₆₇

yes

—

153

C₄²

#11761629

yes

—

154

C₂³

C₂×C₂²⋊Dic₆₇

yes

characteristic

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

#14523964

yes

—

165

C₂×C₄

Q₈×Dic₆₇

yes

—

166

C₂×C₄

#14524019

yes

—

167

C₂×C₄

#14524032

yes

characteristic

168

C₂×C₄

#14524041

yes

characteristic

169

C₂×C₄

C₂×#11761613

yes

characteristic

170

C₂×C₄

C₂×#11761616

yes

characteristic

171

C₂×C₄

#14524063

yes

—

172

C₂×C₄

#14524074

yes

—

173

C₈

—

268

—

174

C₈

—

536

—

175

C₂²

#16863303

yes

characteristic

176

C₂²

#16863380

yes

—

177

C₂²

C₂×#14523964

yes

characteristic

178

C₂²

#16863392

yes

—

179

C₂²

#16863405

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

#16863673

yes

—

185

C₄

#16863678

yes

—

186

C₄

#16863753

yes

—

187

C₄

#16863795

yes

—

188

C₂

#22558858

yes

—

189

C₂

#22558919

yes

—

190

C₂

#22561162

yes

characteristic

191

C₂

#22561175

yes

characteristic

192

C₂

#22561203

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2859	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#2859	D₆₇
hypocenter	C₆₇	#9899
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₆₇

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#2859

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 ◁ #2859

N	H ≅ G/N	type
101. C₆₇	#9899	semidirect

H ≅ G/N

type

101. C₆₇

#9899

semidirect

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

subgroup	index
#22560326	2
C₄×C₄⋊(C₆₇⋊C₈)	2
#22558092	2
C₆₇×#2859	2
#9899	67

subgroup

index

#22560326

C₄×C₄⋊(C₆₇⋊C₈)

#22558092

C₆₇×#2859

#9899

Subgroups — #30755961

Conjugacy classes of subgroups

Complements

Maximal subgroups