Group #30759964 — subgroups

Subgroups — #30759964

Order 17152.

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

normal subgroups only

This group has 382 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	#22571554	C₂	1	1	yes	maximal, characteristic
2	#22561186	C₂	1	2	yes	maximal
3	#22558897	C₂	1	2	yes	maximal
4	#22558084	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1831	C₂	1	1	yes	maximal, characteristic
6	#16863953	C₂²	1	2	yes	—
7	#16863692	—	2	2	no	—
8	#16863637	—	2	2	no	—
9	#16863607	—	2	2	no	—
10	#16863420	C₂²	1	2	yes	—
11	#16863156	—	2	2	no	—
12	#16863156	—	2	2	no	—
13	#16863144	—	2	2	no	—
14	#16863143	—	2	2	no	—
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₄³×C₆₇	C₂²	1	1	yes	characteristic
18	#336×C₆₇	—	2	2	no	—
19	C₂×Dic₂₆₈	—	2	2	no	—
20	C₂×#11761612	—	2	2	no	—
21	#14524043	D₄	1	2	yes	—
22	#14524043	D₄	1	2	yes	—
23	#14524035	—	2	2	no	—
24	#14524022	—	2	2	no	—
25	#14524022	—	2	2	no	—
26	C₄×Dic₁₃₄	—	2	2	no	—
27	C₄×Dic₁₃₄	—	2	2	no	—
28	#14524002	—	2	2	no	—
29	#14523977	—	2	2	no	—
30	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
31	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
32	#14523968	—	2	4	no	—
33	C₂²×C₈×C₆₇	—	2	1	no	—
34	C₂²×C₈×C₆₇	D₄	1	2	yes	—
35	C₂×C₄²×C₆₇	—	2	1	no	—
36	C₂×C₄²×C₆₇	—	2	1	no	—
37	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₆₇	—	2	2	no	—
41	C₄⋊C₈×C₆₇	—	2	2	no	—
42	C₂×Dic₁₃₄	—	2	2	no	—
43	C₂×Dic₁₃₄	—	2	2	no	—
44	Dic₂₆₈	—	8	2	no	—
45	#11761612	—	4	2	no	—
46	#11761612	—	4	2	no	—
47	C₂×C₆₇⋊C₈	—	4	2	no	—
48	C₄⋊Dic₆₇	—	2	2	no	—
49	C₄⋊Dic₆₇	—	2	2	no	—
50	C₄⋊Dic₆₇	—	2	2	no	—
51	C₄⋊Dic₆₇	—	2	2	no	—
52	Dic₆₇⋊C₄	—	4	2	no	—
53	Dic₆₇⋊C₄	—	4	2	no	—
54	C₂²⋊Dic₆₇	—	4	2	no	—
55	C₄×Dic₆₇	—	4	2	no	—
56	C₂²×C₄×C₆₇	—	2	1	no	—
57	C₂²×C₄×C₆₇	—	2	1	no	—
58	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
61	C₂×C₈×C₆₇	—	2	2	no	—
62	C₂×C₈×C₆₇	—	4	2	no	—
63	C₂×C₈×C₆₇	—	4	2	no	—
64	C₂×C₈×C₆₇	—	4	1	no	—
65	C₄²×C₆₇	—	2	1	no	—
66	C₄²×C₆₇	—	2	2	no	—
67	C₄²×C₆₇	—	4	1	no	—
68	C₄²×C₆₇	—	4	1	no	—
69	C₄²×C₆₇	—	4	2	no	—
70	C₄²×C₆₇	Q₁₆	1	2	yes	complemented
71	C₄²×C₆₇	C₂×D₄	1	2	yes	—
72	C₄²×C₆₇	#41	1	2	yes	—
73	C₂³×C₆₇	#121	1	1	yes	characteristic
74	C₂×C₄×C₆₇	—	2	1	no	—
75	C₂×C₄×C₆₇	—	2	1	no	—
76	C₂×C₄×C₆₇	—	2	2	no	—
77	C₂×C₄×C₆₇	—	2	2	no	—
78	C₂×C₄×C₆₇	—	4	1	no	—
79	C₂×C₄×C₆₇	—	4	1	no	—
80	C₂×C₄×C₆₇	—	4	1	no	—
81	C₂×C₄×C₆₇	—	4	1	no	—
82	C₂×C₄×C₆₇	—	4	1	no	—
83	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
84	C₂×C₄×C₆₇	#121	1	2	yes	—
85	C₂×C₄×C₆₇	#123	1	2	yes	—
86	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
87	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
88	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
89	C₂×C₄×C₆₇	#137	1	2	yes	—
90	C₂×Dic₆₇	—	4	2	no	—
91	C₂×Dic₆₇	—	4	2	no	—
92	Dic₁₃₄	—	4	2	no	—
93	Dic₁₃₄	—	4	2	no	—
94	Dic₁₃₄	—	8	2	no	—
95	C₈×C₆₇	—	4	2	no	—
96	C₈×C₆₇	—	8	1	no	—
97	C₆₇⋊C₈	—	8	2	no	—
98	C₂²×C₆₇	—	2	1	no	—
99	C₂²×C₆₇	—	2	1	no	—
100	C₂²×C₆₇	#394	1	1	yes	characteristic
101	C₂²×C₆₇	#467	1	1	yes	characteristic
102	C₂²×C₆₇	#470	1	1	yes	characteristic
103	C₄×C₆₇	—	2	2	no	—
104	C₄×C₆₇	—	2	1	no	—
105	C₄×C₆₇	—	4	2	no	—
106	C₄×C₆₇	—	4	2	no	—
107	C₄×C₆₇	#454	1	1	yes	characteristic
108	C₄×C₆₇	#455	1	1	yes	characteristic
109	C₄×C₆₇	#462	1	2	yes	—
110	C₄×C₆₇	#464	1	2	yes	—
111	Dic₆₇	—	8	2	no	—
112	Dic₆₇	—	8	2	no	—
113	#10523	—	67	1	no	maximal, Sylow
114	C₂×C₆₇	—	2	1	no	—
115	C₂×C₆₇	—	2	1	no	—
116	C₂×C₆₇	#2040	1	1	yes	characteristic
117	C₂×C₆₇	#2050	1	1	yes	characteristic
118	C₂×C₆₇	#2056	1	1	yes	characteristic
119	#3249	—	67	1	no	—
120	#1994	—	67	2	no	—
121	#1831	D₆₇	1	1	yes	characteristic
122	#1550	—	67	2	no	—
123	#1292	—	67	1	no	—
124	C₆₇	#10523	1	1	yes	characteristic, complemented, Sylow
125	#533	—	67	2	no	—
126	#477	—	134	2	no	—
127	#467	—	134	2	no	—
128	#462	—	134	2	no	—
129	#423	—	67	2	no	—
130	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
131	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
132	C₄³	C₂×D₆₇	1	1	yes	characteristic
133	#339	—	134	2	no	—
134	#339	—	134	2	no	—
135	#336	—	2	2	no	—
136	#333	—	134	2	no	—
137	#332	—	134	2	no	—
138	C₂×Q₁₆	—	134	2	no	—
139	C₂×M₁₆	—	134	2	no	—
140	C₂²×C₈	—	2	1	no	—
141	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
142	C₄⋊Q₈	—	67	2	no	—
143	C₄⋊Q₈	—	67	2	no	—
144	#125	—	134	2	no	—
145	C₂²⋊Q₈	—	134	2	no	—
146	C₂²⋊Q₈	—	134	2	no	—
147	C₄×Q₈	—	134	2	no	—
148	C₄×Q₈	—	134	2	no	—
149	#117	—	134	2	no	—
150	C₂×C₄²	—	2	1	no	—
151	C₂×C₄²	—	2	1	no	—
152	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
153	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
154	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
155	#107	—	134	2	no	—
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	134	2	no	—
159	C₄⋊C₈	—	134	2	no	—
160	#103	—	134	4	no	—
161	C₂×Q₈	—	134	2	no	—
162	C₂×Q₈	—	134	2	no	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
167	C₂²×C₄	#11761628	1	1	yes	characteristic
168	Q₁₆	—	536	2	no	—
169	M₁₆	—	268	2	no	—
170	M₁₆	—	268	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	—	268	2	no	—
176	C₄⋊C₄	—	268	2	no	—
177	C₄⋊C₄	—	268	2	no	—
178	C₄⋊C₄	—	134	2	no	—
179	C₄⋊C₄	—	134	2	no	—
180	C₄⋊C₄	—	134	2	no	—
181	C₄⋊C₄	—	134	2	no	—
182	C₂²⋊C₄	—	268	2	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	4	1	no	—
186	C₄²	—	4	1	no	—
187	C₄²	—	4	2	no	—
188	C₄²	—	268	2	no	—
189	C₄²	Dic₂₆₈	1	2	yes	complemented
190	C₄²	D₄×D₆₇	1	2	yes	—
191	C₄²	#11761629	1	2	yes	—
192	C₂³	#14524017	1	1	yes	characteristic
193	Q₈	—	268	2	no	—
194	Q₈	—	268	2	no	—
195	Q₈	—	536	2	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	268	2	no	—
206	C₂×C₄	—	268	2	no	—
207	C₂×C₄	#14524011	1	1	yes	characteristic
208	C₂×C₄	#14524018	1	2	yes	—
209	C₂×C₄	#14524028	1	2	yes	—
210	C₂×C₄	#14524032	1	1	yes	characteristic
211	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
212	C₂×C₄	#14524065	1	1	yes	characteristic
213	C₂×C₄	#14524076	1	2	yes	—
214	C₈	—	4	2	no	—
215	C₈	—	8	1	no	—
216	C₈	—	536	2	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	2	1	no	—
219	C₂²	#16863313	1	1	yes	characteristic
220	C₂²	#16863637	1	1	yes	characteristic
221	C₂²	#16863654	1	1	yes	characteristic
222	C₄	—	2	1	no	—
223	C₄	—	2	2	no	—
224	C₄	—	4	2	no	—
225	C₄	—	4	2	no	—
226	C₄	—	536	2	no	—
227	C₄	—	536	2	no	—
228	C₄	#16863568	1	1	yes	characteristic
229	C₄	#16863573	1	1	yes	characteristic
230	C₄	#16863607	1	2	yes	—
231	C₄	#16863619	1	2	yes	—
232	C₂	—	2	1	no	—
233	C₂	—	2	1	no	—
234	C₂	#22561438	1	1	yes	characteristic
235	C₂	#22561496	1	1	yes	characteristic
236	C₂	#22561532	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22571554

C₂

yes

maximal, characteristic

#22561186

C₂

yes

maximal

#22558897

C₂

yes

maximal

#22558084

C₂

yes

maximal, characteristic

C₆₇×#1831

C₂

yes

maximal, characteristic

#16863953

C₂²

yes

—

—

—

—

C₂²

yes

—

—

—

—

—

C₂×C₄⋊C₈×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₆₇

C₂²

yes

characteristic

C₄³×C₆₇

C₂²

yes

characteristic

#336×C₆₇

—

C₂×Dic₂₆₈

—

C₂×#11761612

—

#14524043

D₄

yes

—

#14524043

D₄

yes

—

#14524035

—

#14524022

—

#14524022

—

C₄×Dic₁₃₄

—

C₄×Dic₁₃₄

—

#14524002

—

#14523977

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

#14523968

—

C₂²×C₈×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

Dic₂₆₈

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

Q₁₆

yes

complemented

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

#121

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

#120

yes

characteristic

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

#123

yes

—

C₂×C₄×C₆₇

#124

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₁₆

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

100

C₂²×C₆₇

#394

yes

characteristic

101

C₂²×C₆₇

#467

yes

characteristic

102

C₂²×C₆₇

#470

yes

characteristic

103

C₄×C₆₇

—

104

C₄×C₆₇

—

105

C₄×C₆₇

—

106

C₄×C₆₇

—

107

C₄×C₆₇

#454

yes

characteristic

108

C₄×C₆₇

#455

yes

characteristic

109

C₄×C₆₇

#462

yes

—

110

C₄×C₆₇

#464

yes

—

111

Dic₆₇

—

112

Dic₆₇

—

113

#10523

—

maximal, Sylow

114

C₂×C₆₇

—

115

C₂×C₆₇

—

116

C₂×C₆₇

#2040

yes

characteristic

117

C₂×C₆₇

#2050

yes

characteristic

118

C₂×C₆₇

#2056

yes

characteristic

119

#3249

—

120

#1994

—

121

#1831

D₆₇

yes

characteristic

122

#1550

—

123

#1292

—

124

C₆₇

#10523

yes

characteristic, complemented, Sylow

125

#533

—

126

#477

—

134

—

127

#467

—

134

—

128

#462

—

134

—

129

#423

—

130

C₂×C₄⋊C₈

C₂×D₆₇

yes

characteristic

131

C₂×C₄⋊C₈

C₂×D₆₇

yes

characteristic

132

C₄³

C₂×D₆₇

yes

characteristic

133

#339

—

134

—

134

#339

—

134

—

135

#336

—

136

#333

—

134

—

137

#332

—

134

—

138

C₂×Q₁₆

—

134

—

139

C₂×M₁₆

—

134

—

140

C₂²×C₈

—

141

C₂²×C₈

C₂²⋊D₆₇

yes

—

142

C₄⋊Q₈

—

143

C₄⋊Q₈

—

144

#125

—

134

—

145

C₂²⋊Q₈

—

134

—

146

C₂²⋊Q₈

—

134

—

147

C₄×Q₈

—

134

—

148

C₄×Q₈

—

134

—

149

#117

—

134

—

150

C₂×C₄²

—

151

C₂×C₄²

—

152

C₂×C₄²

D₂₆₈

yes

characteristic

153

C₂×C₄²

D₂₆₈

yes

characteristic

154

C₂×C₄²

C₂²×D₆₇

yes

characteristic

155

#107

—

134

—

156

C₄⋊C₈

—

157

C₄⋊C₈

—

158

C₄⋊C₈

—

134

—

159

C₄⋊C₈

—

134

—

160

#103

—

134

—

161

C₂×Q₈

—

134

—

162

C₂×Q₈

—

134

—

163

C₂²×C₄

—

164

C₂²×C₄

—

165

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

166

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

167

C₂²×C₄

#11761628

yes

characteristic

168

Q₁₆

—

536

—

169

M₁₆

—

268

—

170

M₁₆

—

268

—

171

C₂×C₈

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

268

—

176

C₄⋊C₄

—

268

—

177

C₄⋊C₄

—

268

—

178

C₄⋊C₄

—

134

—

179

C₄⋊C₄

—

134

—

180

C₄⋊C₄

—

134

—

181

C₄⋊C₄

—

134

—

182

C₂²⋊C₄

—

268

—

183

C₄²

—

184

C₄²

—

185

C₄²

—

186

C₄²

—

187

C₄²

—

188

C₄²

—

268

—

189

C₄²

Dic₂₆₈

yes

complemented

190

C₄²

D₄×D₆₇

yes

—

191

C₄²

#11761629

yes

—

192

C₂³

#14524017

yes

characteristic

193

Q₈

—

268

—

194

Q₈

—

268

—

195

Q₈

—

536

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

—

268

—

206

C₂×C₄

—

268

—

207

C₂×C₄

#14524011

yes

characteristic

208

C₂×C₄

#14524018

yes

—

209

C₂×C₄

#14524028

yes

—

210

C₂×C₄

#14524032

yes

characteristic

211

C₂×C₄

C₂×Dic₂₆₈

yes

characteristic

212

C₂×C₄

#14524065

yes

characteristic

213

C₂×C₄

#14524076

yes

—

214

C₈

—

215

C₈

—

216

C₈

—

536

—

217

C₂²

—

218

C₂²

—

219

C₂²

#16863313

yes

characteristic

220

C₂²

#16863637

yes

characteristic

221

C₂²

#16863654

yes

characteristic

222

C₄

—

223

C₄

—

224

C₄

—

225

C₄

—

226

C₄

—

536

—

227

C₄

—

536

—

228

C₄

#16863568

yes

characteristic

229

C₄

#16863573

yes

characteristic

230

C₄

#16863607

yes

—

231

C₄

#16863619

yes

—

232

C₂

—

233

C₂

—

234

C₂

#22561438

yes

characteristic

235

C₂

#22561496

yes

characteristic

236

C₂

#22561532

yes

characteristic

	subgroup	G/H
center	C₂²	#16863313
commutator, cosocle	C₂×C₄²×C₆₇	C₂³
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1831	C₂
socle	C₂²×C₆₇	#394
prime radical	C₂³×C₆₇	#121
hypercenter	#1831	D₆₇
hypocenter	C₆₇	#10523
LCS	G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1831

subgroup

G/H

center

C₂²

#16863313

commutator, cosocle

C₂×C₄²×C₆₇

C₂³

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1831

C₂

C₂²×C₆₇

C₂³×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1831

N	H ≅ G/N	type
70. C₄²×C₆₇	Q₁₆	semidirect
124. C₆₇	#10523	semidirect
189. C₄²	Dic₂₆₈	semidirect

H ≅ G/N

type

70. C₄²×C₆₇

Q₁₆

semidirect

124. C₆₇

#10523

semidirect

189. C₄²

Dic₂₆₈

semidirect

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

subgroup	index
#22571554	2
#22561186	2
#22558897	2
#22558084	2
C₆₇×#1831	2
#10523	67

subgroup

index

C₆₇×#1831

Subgroups — #30759964

Conjugacy classes of subgroups

Complements

Maximal subgroups