Group #30760283 — subgroups

Subgroups — #30760283

Order 17152.

406 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 127 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	#22565451	C₂	1	1	yes	maximal, characteristic
2	#22560970	C₂	1	2	yes	maximal
3	#22558092	C₂	1	1	yes	maximal, characteristic
4	#22558049	C₂	1	2	yes	maximal
5	C₆₇×#1845	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523974	—	2	2	no	—
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
8	#16863335	—	2	2	no	—
9	#16863295	—	2	2	no	—
10	#16863261	—	2	2	no	—
11	C₄×C₄⋊Dic₆₇	C₂²	1	2	yes	—
12	#16863158	—	2	2	no	—
13	C₂×#107×C₆₇	—	2	2	no	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	#384×C₆₇	C₂²	1	1	yes	characteristic
16	#384×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×C₆₇⋊C₈	—	2	2	no	—
18	C₂×C₄⋊Dic₆₇	—	2	2	no	—
19	C₂×C₄⋊Dic₆₇	D₄	1	2	yes	—
20	C₂×C₄⋊Dic₆₇	D₄	1	2	yes	—
21	C₂×C₄×Dic₆₇	—	2	2	no	—
22	#14523974	—	4	4	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₆₇⋊C₈)	Q₈	1	4	yes	—
25	C₆₇⋊#95	—	2	2	no	—
26	C₆₇⋊#95	—	2	4	no	—
27	C₆₇⋊#95	—	2	2	no	—
28	C₂²×C₈×C₆₇	—	2	1	no	—
29	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
32	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
33	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
34	#107×C₆₇	—	4	4	no	—
35	C₄×C₈×C₆₇	—	2	1	no	—
36	C₄×C₈×C₆₇	Q₈	1	2	yes	—
37	C₂²×Dic₆₇	—	2	2	no	—
38	C₂²×Dic₆₇	—	2	2	no	—
39	C₂×C₆₇⋊C₈	—	2	4	no	—
40	C₂×C₆₇⋊C₈	—	4	2	no	—
41	C₂×C₆₇⋊C₈	—	4	2	no	—
42	C₄⋊Dic₆₇	—	2	2	no	—
43	C₄⋊Dic₆₇	—	2	2	no	—
44	C₄⋊Dic₆₇	—	2	2	no	—
45	C₄⋊Dic₆₇	—	2	2	no	—
46	C₄⋊Dic₆₇	—	8	2	no	—
47	C₄×Dic₆₇	—	4	2	no	—
48	C₄×Dic₆₇	—	4	2	no	—
49	C₂²×C₄×C₆₇	—	2	2	no	—
50	C₂²×C₄×C₆₇	—	4	1	no	—
51	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₆₇	—	2	2	no	—
54	C₂×C₈×C₆₇	—	2	2	no	—
55	C₂×C₈×C₆₇	—	2	2	no	—
56	C₄⋊C₄×C₆₇	—	4	2	no	—
57	C₄⋊C₄×C₆₇	—	4	2	no	—
58	C₄⋊C₄×C₆₇	—	4	2	no	—
59	C₄⋊C₄×C₆₇	—	4	2	no	—
60	C₄⋊C₄×C₆₇	—	4	2	no	—
61	C₄⋊C₄×C₆₇	—	4	2	no	—
62	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
63	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
64	C₄²×C₆₇	#41	1	1	yes	characteristic
65	C₂³×C₆₇	#120	1	1	yes	characteristic
66	C₂×C₄×C₆₇	—	2	1	no	—
67	C₂×C₄×C₆₇	—	2	1	no	—
68	C₂×C₄×C₆₇	—	4	4	no	—
69	C₂×C₄×C₆₇	—	4	2	no	—
70	C₂×C₄×C₆₇	—	8	2	no	—
71	C₂×C₄×C₆₇	—	8	1	no	—
72	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
73	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
76	C₂×C₄×C₆₇	#135	1	2	yes	—
77	C₂×C₄×C₆₇	#135	1	2	yes	—
78	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
79	C₂×C₄×C₆₇	#137	1	2	yes	—
80	C₂×Dic₆₇	—	4	2	no	—
81	C₂×Dic₆₇	—	4	2	no	—
82	C₂×Dic₆₇	—	4	2	no	—
83	C₂×Dic₆₇	—	4	2	no	—
84	C₂×Dic₆₇	—	4	2	no	—
85	C₂×Dic₆₇	—	4	2	no	—
86	C₈×C₆₇	—	2	4	no	—
87	C₆₇⋊C₈	—	4	4	no	—
88	C₂²×C₆₇	#395	1	1	yes	characteristic
89	C₂²×C₆₇	#449	1	2	yes	—
90	C₂²×C₆₇	#452	1	2	yes	—
91	C₂²×C₆₇	#453	1	1	yes	characteristic
92	C₂²×C₆₇	#456	1	1	yes	characteristic
93	C₄×C₆₇	—	2	1	no	—
94	C₄×C₆₇	—	2	1	no	—
95	C₄×C₆₇	—	8	2	no	—
96	C₄×C₆₇	—	16	1	no	—
97	C₄×C₆₇	#478	1	2	yes	—
98	C₄×C₆₇	#479	1	2	yes	—
99	C₄×C₆₇	#499	1	2	yes	—
100	C₄×C₆₇	#501	1	2	yes	—
101	Dic₆₇	—	8	2	no	—
102	Dic₆₇	—	8	2	no	—
103	#10576	—	67	1	no	maximal, Sylow
104	C₂×C₆₇	#1620	1	1	yes	characteristic
105	C₂×C₆₇	#1634	1	1	yes	characteristic
106	C₂×C₆₇	#1650	1	2	yes	—
107	C₂×C₆₇	#2059	1	2	yes	—
108	C₂×C₆₇	#2060	1	1	yes	characteristic
109	#2571	—	67	1	no	—
110	#1948	—	67	2	no	—
111	#1845	D₆₇	1	1	yes	characteristic
112	#1296	—	67	1	no	—
113	#1279	—	67	2	no	—
114	C₆₇	#10576	1	1	yes	characteristic, complemented, Sylow
115	C₂×#107	—	2	2	no	—
116	C₂×#106	—	134	2	no	—
117	C₂×C₄⋊C₈	—	67	2	no	—
118	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
119	#400	—	134	2	no	—
120	#389	—	134	2	no	—
121	#384	C₂×D₆₇	1	1	yes	characteristic
122	#384	C₂×D₆₇	1	1	yes	characteristic
123	#382	—	134	2	no	—
124	C₄×C₄⋊C₄	—	67	2	no	—
125	#340	—	134	2	no	—
126	C₂²×C₈	—	2	1	no	—
127	C₂²×C₈	—	134	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	2	1	no	—
131	C₂×C₄⋊C₄	—	67	2	no	—
132	C₂×C₄⋊C₄	—	67	2	no	—
133	C₂×C₄⋊C₄	—	134	2	no	—
134	C₂×C₄⋊C₄	C₂²⋊D₆₇	1	2	yes	—
135	C₂×C₄²	—	134	2	no	—
136	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
137	#107	—	4	4	no	—
138	#106	—	268	4	no	—
139	C₄⋊C₈	—	67	4	no	—
140	C₄⋊C₈	—	134	2	no	—
141	C₄×C₈	—	2	1	no	—
142	C₄×C₈	Dic₁₃₄	1	2	yes	—
143	#95	—	134	2	no	—
144	#95	—	134	4	no	—
145	#95	—	134	2	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	4	1	no	—
148	C₂²×C₄	—	134	2	no	—
149	C₂²×C₄	—	134	2	no	—
150	C₂²×C₄	D₄×D₆₇	1	2	yes	—
151	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	268	2	no	—
156	C₂×C₈	—	268	2	no	—
157	C₂×C₈	—	134	4	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	—	536	2	no	—
165	C₄⋊C₄	—	134	2	no	—
166	C₄⋊C₄	—	134	2	no	—
167	C₄⋊C₄	—	134	2	no	—
168	C₄⋊C₄	—	134	2	no	—
169	C₄²	—	268	2	no	—
170	C₄²	—	268	2	no	—
171	C₄²	Q₈×D₆₇	1	2	yes	—
172	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
173	C₄²	#11761628	1	1	yes	characteristic
174	C₂³	#14524011	1	1	yes	characteristic
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	—	8	1	no	—
181	C₂×C₄	—	268	2	no	—
182	C₂×C₄	—	268	2	no	—
183	C₂×C₄	—	268	2	no	—
184	C₂×C₄	—	268	2	no	—
185	C₂×C₄	—	268	2	no	—
186	C₂×C₄	—	268	2	no	—
187	C₂×C₄	#14524021	1	2	yes	—
188	C₂×C₄	#14524022	1	1	yes	characteristic
189	C₂×C₄	#14524041	1	1	yes	characteristic
190	C₂×C₄	#14524042	1	2	yes	—
191	C₂×C₄	#14524062	1	2	yes	—
192	C₂×C₄	#14524066	1	2	yes	—
193	C₂×C₄	#14524067	1	2	yes	—
194	C₂×C₄	#14524075	1	2	yes	—
195	C₈	—	2	4	no	—
196	C₈	—	268	4	no	—
197	C₂²	#16863316	1	1	yes	characteristic
198	C₂²	#16863538	1	2	yes	—
199	C₂²	#16863557	1	2	yes	—
200	C₂²	#16863563	1	1	yes	characteristic
201	C₂²	#16863578	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	8	2	no	—
205	C₄	—	16	1	no	—
206	C₄	—	536	2	no	—
207	C₄	—	536	2	no	—
208	C₄	#16863699	1	2	yes	—
209	C₄	#16863702	1	2	yes	—
210	C₄	#16863797	1	2	yes	—
211	C₄	#16863804	1	2	yes	—
212	C₂	#22559266	1	1	yes	characteristic
213	C₂	#22559352	1	1	yes	characteristic
214	C₂	#22559462	1	2	yes	—
215	C₂	#22561545	1	2	yes	—
216	C₂	#22561554	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22565451

C₂

yes

maximal, characteristic

#22560970

C₂

yes

maximal

#22558092

C₂

yes

maximal, characteristic

#22558049

C₂

yes

maximal

C₆₇×#1845

C₂

yes

maximal, characteristic

C₂×#14523974

—

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

#16863335

—

#16863295

—

#16863261

—

C₄×C₄⋊Dic₆₇

C₂²

yes

—

#16863158

—

C₂×#107×C₆₇

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#384×C₆₇

C₂²

yes

characteristic

#384×C₆₇

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

D₄

yes

—

C₂×C₄⋊Dic₆₇

D₄

yes

—

C₂×C₄×Dic₆₇

—

#14523974

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

Q₈

yes

—

C₆₇⋊#95

—

C₆₇⋊#95

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#107×C₆₇

—

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

Q₈

yes

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂×Q₈

yes

—

C₄²×C₆₇

C₂×Q₈

yes

C₄²×C₆₇

yes

C₂³×C₆₇

yes

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

—

C₂×C₄×C₆₇

#135

yes

—

C₂×C₄×C₆₇

#135

yes

—

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₆₇

#137

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

#395

yes

characteristic

C₂²×C₆₇

#449

yes

—

C₂²×C₆₇

#452

yes

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#478

yes

—

C₄×C₆₇

#479

yes

—

C₄×C₆₇

#499

yes

—

100

C₄×C₆₇

#501

yes

—

101

Dic₆₇

—

102

Dic₆₇

—

103

#10576

—

maximal, Sylow

104

C₂×C₆₇

#1620

yes

characteristic

105

C₂×C₆₇

#1634

yes

characteristic

106

C₂×C₆₇

#1650

yes

—

107

C₂×C₆₇

#2059

yes

—

108

C₂×C₆₇

#2060

yes

characteristic

109

#2571

—

110

#1948

—

111

#1845

D₆₇

yes

characteristic

112

#1296

—

113

#1279

—

114

C₆₇

#10576

yes

characteristic, complemented, Sylow

115

C₂×#107

—

116

C₂×#106

—

134

—

117

C₂×C₄⋊C₈

—

118

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

119

#400

—

134

—

120

#389

—

134

—

121

#384

C₂×D₆₇

yes

characteristic

122

#384

C₂×D₆₇

yes

characteristic

123

#382

—

134

—

124

C₄×C₄⋊C₄

—

125

#340

—

134

—

126

C₂²×C₈

—

127

C₂²×C₈

—

134

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

134

—

134

C₂×C₄⋊C₄

C₂²⋊D₆₇

yes

—

135

C₂×C₄²

—

134

—

136

C₂×C₄²

C₂²×D₆₇

yes

characteristic

137

#107

—

138

#106

—

268

—

139

C₄⋊C₈

—

140

C₄⋊C₈

—

134

—

141

C₄×C₈

—

142

C₄×C₈

Dic₁₃₄

yes

—

143

#95

—

134

—

144

#95

—

134

—

145

#95

—

134

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

134

—

149

C₂²×C₄

—

134

—

150

C₂²×C₄

D₄×D₆₇

yes

—

151

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

268

—

156

C₂×C₈

—

268

—

157

C₂×C₈

—

134

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

536

—

165

C₄⋊C₄

—

134

—

166

C₄⋊C₄

—

134

—

167

C₄⋊C₄

—

134

—

168

C₄⋊C₄

—

134

—

169

C₄²

—

268

—

170

C₄²

—

268

—

171

C₄²

Q₈×D₆₇

yes

—

172

C₄²

C₂×Dic₁₃₄

yes

characteristic

173

C₄²

#11761628

yes

characteristic

174

C₂³

#14524011

yes

characteristic

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

268

—

182

C₂×C₄

—

268

—

183

C₂×C₄

—

268

—

184

C₂×C₄

—

268

—

185

C₂×C₄

—

268

—

186

C₂×C₄

—

268

—

187

C₂×C₄

#14524021

yes

—

188

C₂×C₄

#14524022

yes

characteristic

189

C₂×C₄

#14524041

yes

characteristic

190

C₂×C₄

#14524042

yes

—

191

C₂×C₄

#14524062

yes

—

192

C₂×C₄

#14524066

yes

—

193

C₂×C₄

#14524067

yes

—

194

C₂×C₄

#14524075

yes

—

195

C₈

—

196

C₈

—

268

—

197

C₂²

#16863316

yes

characteristic

198

C₂²

#16863538

yes

—

199

C₂²

#16863557

yes

—

200

C₂²

#16863563

yes

characteristic

201

C₂²

#16863578

yes

characteristic

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

—

536

—

207

C₄

—

536

—

208

C₄

#16863699

yes

—

209

C₄

#16863702

yes

—

210

C₄

#16863797

yes

—

211

C₄

#16863804

yes

—

212

C₂

#22559266

yes

characteristic

213

C₂

#22559352

yes

characteristic

214

C₂

#22559462

yes

—

215

C₂

#22561545

yes

—

216

C₂

#22561554

yes

characteristic

	subgroup	G/H
center	C₂³	#14524011
commutator, cosocle	C₂×C₄²×C₆₇	C₂³
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1845	C₂
socle, prime radical	C₂³×C₆₇	#120
hypercenter	#1845	D₆₇
hypocenter	C₆₇	#10576
LCS	G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

subgroup

G/H

center

C₂³

#14524011

commutator, cosocle

C₂×C₄²×C₆₇

C₂³

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1845

C₂

C₂³×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

N	H ≅ G/N	type
114. C₆₇	#10576	semidirect

H ≅ G/N

type

114. C₆₇

#10576

semidirect

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

subgroup	index
#22565451	2
#22560970	2
#22558092	2
#22558049	2
C₆₇×#1845	2
#10576	67

subgroup

index

C₆₇×#1845

Subgroups — #30760283

Conjugacy classes of subgroups

Complements

Maximal subgroups