Group #30755441 — subgroups

Subgroups — #30755441

Order 17152.

474 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 131 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	#22561197	C₂	1	2	yes	maximal
3	#22558891	C₂	1	2	yes	maximal
4	C₈×#11761612	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1567	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₆₈	C₄	1	4	yes	—
7	C₂×#14524043	C₂²	1	2	yes	—
8	#16863776	—	2	1	no	—
9	#16863776	—	2	1	no	—
10	#16863776	—	2	2	no	—
11	#16863449	C₂²	1	1	yes	characteristic
12	C₄×#11761612	C₂²	1	1	yes	characteristic
13	#16863164	—	2	4	no	—
14	#16863144	—	2	4	no	—
15	#16863118	—	2	1	no	—
16	C₈×C₆₇⋊C₈	—	2	1	no	—
17	#423×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	#335×C₆₇	—	2	2	no	—
20	C₂²×Dic₁₃₄	C₂×C₄	1	4	yes	—
21	C₂×Dic₂₆₈	—	2	4	no	—
22	C₂×Dic₂₆₈	—	2	4	no	—
23	C₂×Dic₂₆₈	—	4	8	no	—
24	C₂×#11761612	D₄	1	2	yes	—
25	#14524043	—	2	2	no	—
26	#14524043	—	2	2	no	—
27	#14524043	—	2	2	no	—
28	#14524043	—	2	2	no	—
29	C₂×C₄⋊Dic₆₇	—	2	2	no	—
30	C₂×C₄⋊Dic₆₇	—	2	2	no	—
31	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
32	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
33	#14523954	—	2	1	no	—
34	C₄×C₆₇⋊C₈	—	2	1	no	—
35	C₄×C₆₇⋊C₈	—	2	1	no	—
36	C₂×M₁₆×C₆₇	—	2	2	no	—
37	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₆₇	—	2	2	no	—
40	C₄⋊C₈×C₆₇	—	2	2	no	—
41	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
42	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
43	C₄×C₈×C₆₇	D₄	1	2	yes	—
44	C₂×Dic₁₃₄	—	2	4	no	—
45	C₂×Dic₁₃₄	—	2	4	no	—
46	C₂×Dic₁₃₄	—	4	8	no	—
47	C₂²×Dic₆₇	—	2	4	no	—
48	Dic₂₆₈	—	4	8	no	—
49	Dic₂₆₈	—	4	8	no	—
50	#11761612	—	2	2	no	—
51	#11761612	Q₁₆	1	4	yes	complemented
52	C₂×C₆₇⋊C₈	—	2	2	no	—
53	C₂×C₆₇⋊C₈	—	2	2	no	—
54	C₄⋊Dic₆₇	—	4	2	no	—
55	C₄⋊Dic₆₇	—	4	2	no	—
56	C₄⋊Dic₆₇	—	4	2	no	—
57	C₄⋊Dic₆₇	—	4	2	no	—
58	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
61	M₁₆×C₆₇	—	4	4	no	—
62	C₂×C₈×C₆₇	—	2	4	no	—
63	C₂×C₈×C₆₇	—	4	2	no	—
64	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₆₇	#41	1	1	yes	characteristic
70	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₆₇	—	2	2	no	—
72	C₂×C₄×C₆₇	—	2	2	no	—
73	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
76	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
77	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
78	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
79	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
80	C₂×C₄×C₆₇	#135	1	2	yes	—
81	C₂×Dic₆₇	—	4	4	no	—
82	C₂×Dic₆₇	—	8	4	no	—
83	Dic₁₃₄	—	4	8	no	—
84	Dic₁₃₄	—	8	4	no	—
85	C₈×C₆₇	—	2	2	no	—
86	C₈×C₆₇	—	8	2	no	—
87	C₈×C₆₇	#100	1	4	yes	—
88	C₆₇⋊C₈	—	2	4	no	—
89	C₆₇⋊C₈	—	4	2	no	—
90	C₂²×C₆₇	—	2	1	no	—
91	C₂²×C₆₇	—	2	1	no	—
92	C₂²×C₆₇	#390	1	1	yes	characteristic
93	C₂²×C₆₇	C₄×Q₁₆	1	2	yes	—
94	C₄×C₆₇	—	2	1	no	—
95	C₄×C₆₇	—	2	1	no	—
96	C₄×C₆₇	C₂×#100	1	2	yes	—
97	C₄×C₆₇	#467	1	2	yes	—
98	C₄×C₆₇	#473	1	2	yes	—
99	C₄×C₆₇	#494	1	1	yes	characteristic
100	C₄×C₆₇	#495	1	1	yes	characteristic
101	Dic₆₇	—	8	4	no	—
102	#9798	—	67	1	no	maximal, Sylow
103	C₂×C₆₇	—	2	2	no	—
104	C₂×C₆₇	#1965	1	1	yes	characteristic
105	C₂×C₆₇	#1969	1	1	yes	characteristic
106	C₂×C₆₇	#1979	1	1	yes	characteristic
107	C₂×#494	—	67	1	no	—
108	#1997	—	67	2	no	—
109	#1567	D₆₇	1	1	yes	characteristic
110	#1548	—	67	2	no	—
111	C₈×M₁₆	—	67	1	no	—
112	C₆₇	#9798	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₁₆	—	67	4	no	—
114	C₂×C₄⋊Q₈	—	67	2	no	—
115	#494	—	134	1	no	—
116	#494	—	134	1	no	—
117	#494	—	134	2	no	—
118	#431	—	67	1	no	—
119	#423	C₂×D₆₇	1	2	yes	—
120	C₄×M₁₆	—	67	1	no	—
121	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
122	#343	—	134	4	no	—
123	#335	—	2	2	no	—
124	#333	—	134	4	no	—
125	#322	—	134	1	no	—
126	C₈²	—	134	1	no	—
127	C₂²×Q₈	—	67	4	no	—
128	C₂×Q₁₆	—	268	8	no	—
129	C₂×Q₁₆	—	134	4	no	—
130	C₂×Q₁₆	—	134	4	no	—
131	C₂×M₁₆	—	2	2	no	—
132	C₂×M₁₆	—	67	2	no	—
133	C₂²×C₈	C₄×D₆₇	1	2	yes	—
134	C₄⋊Q₈	—	134	2	no	—
135	C₄⋊Q₈	—	134	2	no	—
136	C₄⋊Q₈	—	134	2	no	—
137	C₄⋊Q₈	—	134	2	no	—
138	C₂×C₄⋊C₄	—	134	2	no	—
139	C₂×C₄⋊C₄	—	134	2	no	—
140	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	2	2	no	—
143	C₄⋊C₈	—	67	2	no	—
144	C₂²⋊C₈	—	134	1	no	—
145	#97	—	134	1	no	—
146	C₄×C₈	—	134	1	no	—
147	C₄×C₈	—	134	1	no	—
148	C₄×C₈	D₂₆₈	1	1	yes	characteristic
149	C₄×C₈	D₂₆₈	1	1	yes	characteristic
150	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
151	C₂×Q₈	—	268	8	no	—
152	C₂×Q₈	—	134	4	no	—
153	C₂×Q₈	—	134	4	no	—
154	C₂²×C₄	—	134	4	no	—
155	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
156	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
157	C₂²×C₄	#11761629	1	1	yes	characteristic
158	Q₁₆	—	268	8	no	—
159	Q₁₆	—	268	8	no	—
160	M₁₆	—	4	4	no	—
161	M₁₆	—	67	4	no	—
162	M₁₆	—	134	2	no	—
163	C₂×C₈	—	2	4	no	—
164	C₂×C₈	—	4	2	no	—
165	C₂×C₈	—	134	2	no	—
166	C₂×C₈	—	134	2	no	—
167	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
168	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
169	C₄⋊C₄	—	268	2	no	—
170	C₄⋊C₄	—	268	2	no	—
171	C₄⋊C₄	—	268	2	no	—
172	C₄⋊C₄	—	268	2	no	—
173	C₄²	D₄×D₆₇	1	1	yes	characteristic
174	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
175	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
176	C₄²	#11761629	1	1	yes	characteristic
177	C₂³	#14524005	1	1	yes	characteristic
178	Q₈	—	268	8	no	—
179	Q₈	—	536	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	268	4	no	—
183	C₂×C₄	—	536	4	no	—
184	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
185	C₂×C₄	#14524005	1	1	yes	characteristic
186	C₂×C₄	#14524018	1	1	yes	characteristic
187	C₂×C₄	#14524018	1	1	yes	characteristic
188	C₂×C₄	#14524032	1	1	yes	characteristic
189	C₂×C₄	#14524039	1	1	yes	characteristic
190	C₂×C₄	Q₁₆×D₆₇	1	2	yes	—
191	C₂×C₄	#14524062	1	2	yes	—
192	C₈	—	2	2	no	—
193	C₈	—	8	2	no	—
194	C₈	—	268	2	no	—
195	C₈	—	134	4	no	—
196	C₈	#14523959	1	4	yes	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	#16863297	1	1	yes	characteristic
200	C₂²	#16863490	1	2	yes	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	536	4	no	—
204	C₄	C₂×#14523959	1	2	yes	—
205	C₄	#16863638	1	2	yes	—
206	C₄	#16863672	1	2	yes	—
207	C₄	#16863774	1	1	yes	characteristic
208	C₄	#16863779	1	1	yes	characteristic
209	C₂	—	2	2	no	—
210	C₂	#22561048	1	1	yes	characteristic
211	C₂	#22561068	1	1	yes	characteristic
212	C₂	#22561122	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863776

C₂

yes

maximal, characteristic

#22561197

C₂

yes

maximal

#22558891

C₂

yes

maximal

C₈×#11761612

C₂

yes

maximal, characteristic

C₆₇×#1567

C₂

yes

maximal, characteristic

C₂²×Dic₂₆₈

C₄

yes

—

C₂×#14524043

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄×#11761612

C₂²

yes

—

—

—

C₈×C₆₇⋊C₈

—

#423×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#335×C₆₇

—

C₂²×Dic₁₃₄

C₂×C₄

yes

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×#11761612

D₄

yes

—

—

—

—

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₄⋊(C₆₇⋊C₈)

D₄

yes

—

C₂²⋊(C₆₇⋊C₈)

—

#14523954

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

Dic₂₆₈

—

Dic₂₆₈

—

#11761612

—

#11761612

Q₁₆

yes

complemented

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

characteristic

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

#41

yes

characteristic

C₂³×C₆₇

C₄×D₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄×D₄

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

C₂×Q₁₆

yes

—

C₂×C₄×C₆₇

#135

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#100

yes

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#390

yes

characteristic

C₂²×C₆₇

C₄×Q₁₆

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×#100

yes

—

C₄×C₆₇

#467

yes

—

C₄×C₆₇

#473

yes

—

C₄×C₆₇

#494

yes

characteristic

100

C₄×C₆₇

#495

yes

characteristic

101

Dic₆₇

—

102

#9798

—

maximal, Sylow

103

C₂×C₆₇

—

104

C₂×C₆₇

#1965

yes

characteristic

105

C₂×C₆₇

#1969

yes

characteristic

106

C₂×C₆₇

#1979

yes

characteristic

107

C₂×#494

—

108

#1997

—

109

#1567

D₆₇

yes

characteristic

110

#1548

—

111

C₈×M₁₆

—

112

C₆₇

#9798

yes

characteristic, complemented, Sylow

113

C₂²×Q₁₆

—

114

C₂×C₄⋊Q₈

—

115

#494

—

134

—

116

#494

—

134

—

117

#494

—

134

—

118

#431

—

119

#423

C₂×D₆₇

yes

—

120

C₄×M₁₆

—

121

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

122

#343

—

134

—

123

#335

—

124

#333

—

134

—

125

#322

—

134

—

126

C₈²

—

134

—

127

C₂²×Q₈

—

128

C₂×Q₁₆

—

268

—

129

C₂×Q₁₆

—

134

—

130

C₂×Q₁₆

—

134

—

131

C₂×M₁₆

—

132

C₂×M₁₆

—

133

C₂²×C₈

C₄×D₆₇

yes

—

134

C₄⋊Q₈

—

134

—

135

C₄⋊Q₈

—

134

—

136

C₄⋊Q₈

—

134

—

137

C₄⋊Q₈

—

134

—

138

C₂×C₄⋊C₄

—

134

—

139

C₂×C₄⋊C₄

—

134

—

140

C₂×C₄²

C₂²×D₆₇

yes

characteristic

141

C₄⋊C₈

—

142

C₄⋊C₈

—

143

C₄⋊C₈

—

144

C₂²⋊C₈

—

134

—

145

#97

—

134

—

146

C₄×C₈

—

134

—

147

C₄×C₈

—

134

—

148

C₄×C₈

D₂₆₈

yes

characteristic

149

C₄×C₈

D₂₆₈

yes

characteristic

150

C₄×C₈

C₂²⋊D₆₇

yes

—

151

C₂×Q₈

—

268

—

152

C₂×Q₈

—

134

—

153

C₂×Q₈

—

134

—

154

C₂²×C₄

—

134

—

155

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

156

C₂²×C₄

D₄×D₆₇

yes

characteristic

157

C₂²×C₄

#11761629

yes

characteristic

158

Q₁₆

—

268

—

159

Q₁₆

—

268

—

160

M₁₆

—

161

M₁₆

—

162

M₁₆

—

134

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

134

—

166

C₂×C₈

—

134

—

167

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

168

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

169

C₄⋊C₄

—

268

—

170

C₄⋊C₄

—

268

—

171

C₄⋊C₄

—

268

—

172

C₄⋊C₄

—

268

—

173

C₄²

D₄×D₆₇

yes

characteristic

174

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

175

C₄²

C₂×D₂₆₈

yes

characteristic

176

C₄²

#11761629

yes

characteristic

177

C₂³

#14524005

yes

characteristic

178

Q₈

—

268

—

179

Q₈

—

536

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

268

—

183

C₂×C₄

—

536

—

184

C₂×C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

185

C₂×C₄

#14524005

yes

characteristic

186

C₂×C₄

#14524018

yes

characteristic

187

C₂×C₄

#14524018

yes

characteristic

188

C₂×C₄

#14524032

yes

characteristic

189

C₂×C₄

#14524039

yes

characteristic

190

C₂×C₄

Q₁₆×D₆₇

yes

—

191

C₂×C₄

#14524062

yes

—

192

C₈

—

193

C₈

—

194

C₈

—

268

—

195

C₈

—

134

—

196

C₈

#14523959

yes

—

197

C₂²

—

198

C₂²

—

199

C₂²

#16863297

yes

characteristic

200

C₂²

#16863490

yes

—

201

C₄

—

202

C₄

—

203

C₄

—

536

—

204

C₄

C₂×#14523959

yes

—

205

C₄

#16863638

yes

—

206

C₄

#16863672

yes

—

207

C₄

#16863774

yes

characteristic

208

C₄

#16863779

yes

characteristic

209

C₂

—

210

C₂

#22561048

yes

characteristic

211

C₂

#22561068

yes

characteristic

212

C₂

#22561122

yes

characteristic

	subgroup	G/H
center	C₂²	#16863297
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1567	C₂
socle	C₂²×C₆₇	#390
prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1567	D₆₇
hypocenter	C₆₇	#9798
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1567

subgroup

G/H

center

C₂²

#16863297

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1567

C₂

socle

C₂²×C₆₇

#390

prime radical

C₂³×C₆₇

C₄×D₄

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1567

N	H ≅ G/N	type
51. #11761612	Q₁₆	semidirect
112. C₆₇	#9798	semidirect

H ≅ G/N

type

51. #11761612

Q₁₆

semidirect

112. C₆₇

#9798

semidirect

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

subgroup	index
C₂×#16863776	2
#22561197	2
#22558891	2
C₈×#11761612	2
C₆₇×#1567	2
#9798	67

subgroup

index

C₆₇×#1567

Subgroups — #30755441

Conjugacy classes of subgroups

Complements

Maximal subgroups