Group #30754924 — subgroups

Subgroups — #30754924

Order 17152.

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

normal subgroups only

This group has 538 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	#22568607	C₂	1	1	yes	maximal, characteristic
2	#22560062	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863141	C₂	1	4	yes	maximal
4	C₆₇×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×#14524043	—	2	1	no	—
6	C₂×#14524043	—	2	1	no	—
7	C₂×#14524043	C₂²	1	2	yes	—
8	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×#14523954	—	2	1	no	—
10	#16863266	—	2	2	no	—
11	#16863266	C₄	1	4	yes	—
12	#16863151	—	2	1	no	—
13	#16863141	—	2	8	no	—
14	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
15	C₄³×C₆₇	C₂²	1	1	yes	characteristic
16	#336×C₆₇	—	2	2	no	—
17	C₂²×Dic₁₃₄	—	2	2	no	—
18	C₂²×C₆₇⋊C₈	—	2	2	no	—
19	#14524043	—	2	4	no	—
20	#14524043	—	2	4	no	—
21	#14524043	—	4	4	no	—
22	#14524043	—	4	2	no	—
23	#14524043	—	4	2	no	—
24	C₂×C₄⋊Dic₆₇	—	2	4	no	—
25	C₂×C₄⋊Dic₆₇	—	2	4	no	—
26	C₂×C₄⋊Dic₆₇	—	2	2	no	—
27	C₂×C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
28	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
29	C₄⋊(C₆₇⋊C₈)	D₄	1	4	yes	—
30	#14523954	—	4	2	no	—
31	C₂²×C₈×C₆₇	—	2	2	no	—
32	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	D₄	1	2	yes	—
36	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₆₇	—	2	4	no	—
38	C₂×Dic₁₃₄	—	4	4	no	—
39	C₂×Dic₁₃₄	—	4	2	no	—
40	C₂×Dic₁₃₄	—	4	2	no	—
41	C₂×Dic₁₃₄	—	4	4	no	—
42	C₂²×Dic₆₇	—	2	4	no	—
43	C₂×C₆₇⋊C₈	—	2	4	no	—
44	C₂×C₆₇⋊C₈	—	4	2	no	—
45	C₂×C₆₇⋊C₈	—	4	2	no	—
46	C₄⋊Dic₆₇	—	2	8	no	—
47	C₄⋊Dic₆₇	—	4	8	no	—
48	C₄⋊Dic₆₇	—	4	8	no	—
49	C₄⋊Dic₆₇	—	4	2	no	—
50	C₄⋊Dic₆₇	—	4	2	no	—
51	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₆₇	—	4	2	no	—
57	C₂×C₈×C₆₇	—	4	4	no	—
58	C₄²×C₆₇	—	2	4	no	—
59	C₄²×C₆₇	—	2	2	no	—
60	C₄²×C₆₇	—	2	4	no	—
61	C₄²×C₆₇	D₈	1	2	yes	—
62	C₄²×C₆₇	SD₁₆	1	2	yes	—
63	C₄²×C₆₇	C₂×D₄	1	2	yes	—
64	C₄²×C₆₇	#41	1	2	yes	—
65	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₆₇	—	2	2	no	—
67	C₂×C₄×C₆₇	—	2	2	no	—
68	C₂×C₄×C₆₇	—	2	2	no	—
69	C₂×C₄×C₆₇	—	2	2	no	—
70	C₂×C₄×C₆₇	—	2	2	no	—
71	C₂×C₄×C₆₇	#101	1	4	yes	—
72	C₂×C₄×C₆₇	#102	1	4	yes	—
73	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₆₇	#121	1	2	yes	—
75	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
76	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×C₄×C₆₇	#137	1	4	yes	—
81	C₂×Dic₆₇	—	4	4	no	—
82	C₂×Dic₆₇	—	4	4	no	—
83	C₂×Dic₆₇	—	4	4	no	—
84	Dic₁₃₄	—	8	4	no	—
85	Dic₁₃₄	—	8	4	no	—
86	C₈×C₆₇	—	8	2	no	—
87	C₆₇⋊C₈	—	4	4	no	—
88	C₂²×C₆₇	#390	1	1	yes	characteristic
89	C₂²×C₆₇	C₂×#101	1	2	yes	—
90	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
91	C₂²×C₆₇	#418	1	1	yes	characteristic
92	C₂²×C₆₇	#419	1	2	yes	—
93	C₄×C₆₇	—	2	4	no	—
94	C₄×C₆₇	—	2	4	no	—
95	C₄×C₆₇	—	2	2	no	—
96	C₄×C₆₇	#459	1	2	yes	—
97	C₄×C₆₇	#460	1	2	yes	—
98	C₄×C₆₇	#491	1	2	yes	—
99	C₄×C₆₇	#497	1	2	yes	—
100	Dic₆₇	—	8	4	no	—
101	#9705	—	67	1	no	maximal, Sylow
102	C₂×C₆₇	#1546	1	2	yes	—
103	C₂×C₆₇	#1556	1	2	yes	—
104	C₂×C₆₇	#1960	1	1	yes	characteristic
105	C₂×C₆₇	#1966	1	1	yes	characteristic
106	C₂×C₆₇	#1982	1	1	yes	characteristic
107	#2868	—	67	1	no	—
108	#1831	D₆₇	1	1	yes	characteristic
109	#1767	—	67	1	no	—
110	C₂×#332	—	67	4	no	—
111	C₆₇	#9705	1	1	yes	characteristic, complemented, Sylow
112	C₂×C₄⋊Q₈	—	67	2	no	—
113	C₂×C₄⋊Q₈	—	134	1	no	—
114	C₂×C₄⋊Q₈	—	134	1	no	—
115	C₂×C₄⋊C₈	—	67	2	no	—
116	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
117	C₂×#97	—	134	1	no	—
118	#384	—	67	4	no	—
119	#384	—	134	2	no	—
120	C₄³	C₂×D₆₇	1	1	yes	characteristic
121	#336	—	2	2	no	—
122	#336	—	134	1	no	—
123	#332	—	134	8	no	—
124	C₂²×Q₈	—	134	2	no	—
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	—	134	2	no	—
127	C₄⋊Q₈	—	268	4	no	—
128	C₄⋊Q₈	—	268	2	no	—
129	C₄⋊Q₈	—	268	2	no	—
130	C₄⋊Q₈	—	134	4	no	—
131	C₄⋊Q₈	—	134	4	no	—
132	C₂×C₄⋊C₄	—	67	4	no	—
133	C₂×C₄⋊C₄	—	134	4	no	—
134	C₂×C₄⋊C₄	—	134	4	no	—
135	C₂×C₄⋊C₄	—	134	2	no	—
136	C₂×C₄²	C₄×D₆₇	1	2	yes	—
137	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
138	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
139	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
140	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
141	C₄⋊C₈	—	2	4	no	—
142	C₄⋊C₈	—	67	4	no	—
143	C₄⋊C₈	—	134	2	no	—
144	#97	—	268	2	no	—
145	C₂×Q₈	—	268	4	no	—
146	C₂×Q₈	—	268	2	no	—
147	C₂×Q₈	—	268	2	no	—
148	C₂×Q₈	—	268	4	no	—
149	C₂²×C₄	—	134	4	no	—
150	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
151	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
152	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
154	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	—	4	4	no	—
157	C₂×C₈	—	268	2	no	—
158	C₂×C₈	—	268	2	no	—
159	C₂×C₈	—	134	4	no	—
160	C₄⋊C₄	—	268	8	no	—
161	C₄⋊C₄	—	268	8	no	—
162	C₄⋊C₄	—	268	2	no	—
163	C₄⋊C₄	—	268	2	no	—
164	C₄⋊C₄	—	134	8	no	—
165	C₄²	—	2	4	no	—
166	C₄²	—	2	2	no	—
167	C₄²	—	2	4	no	—
168	C₄²	D₅₃₆	1	2	yes	—
169	C₄²	#11761617	1	2	yes	—
170	C₄²	D₄×D₆₇	1	2	yes	—
171	C₄²	#11761629	1	2	yes	—
172	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
173	Q₈	—	536	4	no	—
174	Q₈	—	536	4	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	268	4	no	—
181	C₂×C₄	—	268	4	no	—
182	C₂×C₄	—	268	4	no	—
183	C₂×C₄	#14523961	1	4	yes	—
184	C₂×C₄	#14523965	1	4	yes	—
185	C₂×C₄	#14524005	1	2	yes	—
186	C₂×C₄	#14524018	1	2	yes	—
187	C₂×C₄	#14524032	1	1	yes	characteristic
188	C₂×C₄	#14524039	1	1	yes	characteristic
189	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
190	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
191	C₂×C₄	#14524071	1	2	yes	—
192	C₂×C₄	#14524073	1	4	yes	—
193	C₈	—	8	2	no	—
194	C₈	—	268	4	no	—
195	C₂²	#16863297	1	1	yes	characteristic
196	C₂²	C₂×#14523961	1	2	yes	—
197	C₂²	C₂×#14523965	1	1	yes	characteristic
198	C₂²	#16863402	1	1	yes	characteristic
199	C₂²	#16863404	1	2	yes	—
200	C₄	—	2	2	no	—
201	C₄	—	2	4	no	—
202	C₄	—	2	4	no	—
203	C₄	—	536	4	no	—
204	C₄	#16863592	1	2	yes	—
205	C₄	#16863597	1	2	yes	—
206	C₄	#16863766	1	2	yes	—
207	C₄	#16863791	1	2	yes	—
208	C₂	#22558880	1	2	yes	—
209	C₂	#22558925	1	2	yes	—
210	C₂	#22561028	1	1	yes	characteristic
211	C₂	#22561052	1	1	yes	characteristic
212	C₂	#22561135	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22568607

C₂

yes

maximal, characteristic

#22560062

C₂

yes

maximal, characteristic

C₂×#16863141

C₂

yes

maximal

C₆₇×#1831

C₂

yes

maximal, characteristic

C₂×#14524043

—

C₂×#14524043

—

C₂×#14524043

C₂²

yes

—

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₂×#14523954

—

#16863266

—

#16863266

C₄

yes

—

#16863151

—

#16863141

—

C₂×C₄⋊C₈×C₆₇

C₂²

yes

—

C₄³×C₆₇

C₂²

yes

characteristic

#336×C₆₇

—

C₂²×Dic₁₃₄

—

C₂²×C₆₇⋊C₈

—

—

—

—

—

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

C₂×C₄

yes

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

D₄

yes

—

#14523954

—

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

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

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

D₈

yes

—

C₄²×C₆₇

SD₁₆

yes

—

C₄²×C₆₇

C₂×D₄

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

#101

yes

—

C₂×C₄×C₆₇

#102

yes

—

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₆₇

#137

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

#390

yes

characteristic

C₂²×C₆₇

C₂×#101

yes

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#459

yes

—

C₄×C₆₇

#460

yes

—

C₄×C₆₇

#491

yes

—

C₄×C₆₇

#497

yes

—

100

Dic₆₇

—

101

#9705

—

maximal, Sylow

102

C₂×C₆₇

#1546

yes

—

103

C₂×C₆₇

#1556

yes

—

104

C₂×C₆₇

#1960

yes

characteristic

105

C₂×C₆₇

#1966

yes

characteristic

106

C₂×C₆₇

#1982

yes

characteristic

107

#2868

—

108

#1831

D₆₇

yes

characteristic

109

#1767

—

110

C₂×#332

—

111

C₆₇

#9705

yes

characteristic, complemented, Sylow

112

C₂×C₄⋊Q₈

—

113

C₂×C₄⋊Q₈

—

134

—

114

C₂×C₄⋊Q₈

—

134

—

115

C₂×C₄⋊C₈

—

116

C₂×C₄⋊C₈

C₂×D₆₇

yes

—

117

C₂×#97

—

134

—

118

#384

—

119

#384

—

134

—

120

C₄³

C₂×D₆₇

yes

characteristic

121

#336

—

122

#336

—

134

—

123

#332

—

134

—

124

C₂²×Q₈

—

134

—

125

C₂²×C₈

—

126

C₂²×C₈

—

134

—

127

C₄⋊Q₈

—

268

—

128

C₄⋊Q₈

—

268

—

129

C₄⋊Q₈

—

268

—

130

C₄⋊Q₈

—

134

—

131

C₄⋊Q₈

—

134

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

134

—

134

C₂×C₄⋊C₄

—

134

—

135

C₂×C₄⋊C₄

—

134

—

136

C₂×C₄²

C₄×D₆₇

yes

—

137

C₂×C₄²

D₂₆₈

yes

characteristic

138

C₂×C₄²

D₂₆₈

yes

characteristic

139

C₂×C₄²

C₂²⋊D₆₇

yes

—

140

C₂×C₄²

C₂²×D₆₇

yes

characteristic

141

C₄⋊C₈

—

142

C₄⋊C₈

—

143

C₄⋊C₈

—

134

—

144

#97

—

268

—

145

C₂×Q₈

—

268

—

146

C₂×Q₈

—

268

—

147

C₂×Q₈

—

268

—

148

C₂×Q₈

—

268

—

149

C₂²×C₄

—

134

—

150

C₂²×C₄

(C₂×D₆₇)⋊C₄

yes

—

151

C₂²×C₄

(C₂×D₆₇)⋊C₄

yes

—

152

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

153

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

154

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

268

—

158

C₂×C₈

—

268

—

159

C₂×C₈

—

134

—

160

C₄⋊C₄

—

268

—

161

C₄⋊C₄

—

268

—

162

C₄⋊C₄

—

268

—

163

C₄⋊C₄

—

268

—

164

C₄⋊C₄

—

134

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₄²

D₅₃₆

yes

—

169

C₄²

#11761617

yes

—

170

C₄²

D₄×D₆₇

yes

—

171

C₄²

#11761629

yes

—

172

C₂³

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

173

Q₈

—

536

—

174

Q₈

—

536

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

268

—

181

C₂×C₄

—

268

—

182

C₂×C₄

—

268

—

183

C₂×C₄

#14523961

yes

—

184

C₂×C₄

#14523965

yes

—

185

C₂×C₄

#14524005

yes

—

186

C₂×C₄

#14524018

yes

—

187

C₂×C₄

#14524032

yes

characteristic

188

C₂×C₄

#14524039

yes

characteristic

189

C₂×C₄

C₂×D₅₃₆

yes

characteristic

190

C₂×C₄

C₂×#11761617

yes

characteristic

191

C₂×C₄

#14524071

yes

—

192

C₂×C₄

#14524073

yes

—

193

C₈

—

194

C₈

—

268

—

195

C₂²

#16863297

yes

characteristic

196

C₂²

C₂×#14523961

yes

—

197

C₂²

C₂×#14523965

yes

characteristic

198

C₂²

#16863402

yes

characteristic

199

C₂²

#16863404

yes

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

536

—

204

C₄

#16863592

yes

—

205

C₄

#16863597

yes

—

206

C₄

#16863766

yes

—

207

C₄

#16863791

yes

—

208

C₂

#22558880

yes

—

209

C₂

#22558925

yes

—

210

C₂

#22561028

yes

characteristic

211

C₂

#22561052

yes

characteristic

212

C₂

#22561135

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	#9705
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
111. C₆₇	#9705	semidirect

H ≅ G/N

type

111. C₆₇

#9705

semidirect

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

subgroup	index
#22568607	2
#22560062	2
C₂×#16863141	2
C₆₇×#1831	2
#9705	67

subgroup

index

C₆₇×#1831

Subgroups — #30754924

Conjugacy classes of subgroups

Complements

Maximal subgroups