Group #30755983 — subgroups

Subgroups — #30755983

Order 17152.

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

normal subgroups only

This group has 438 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	#22560361	C₂	1	1	yes	maximal, characteristic
2	#22560062	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)	—	2	2	no	—
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×#14523954	—	2	1	no	—
9	#16863151	—	2	1	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)	—	4	4	no	—
21	C₄⋊(C₆₇⋊C₈)	Q₈	1	4	yes	—
22	#14523954	—	4	2	no	—
23	#125×C₆₇	—	4	2	no	—
24	#125×C₆₇	—	4	2	no	—
25	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
26	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
27	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
28	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₆₇	D₄	1	2	yes	—
32	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
34	#95×C₆₇	—	2	4	no	—
35	C₂×C₆₇⋊C₈	—	2	4	no	—
36	C₂×C₆₇⋊C₈	—	4	2	no	—
37	C₂×C₆₇⋊C₈	—	4	2	no	—
38	C₂×C₆₇⋊C₈	—	4	2	no	—
39	C₂×C₆₇⋊C₈	—	4	4	no	—
40	C₂²×C₄×C₆₇	—	2	4	no	—
41	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
46	C₄⋊C₄×C₆₇	—	4	4	no	—
47	C₄⋊C₄×C₆₇	—	4	8	no	—
48	C₄⋊C₄×C₆₇	—	4	2	no	—
49	C₄⋊C₄×C₆₇	—	4	2	no	—
50	C₄⋊C₄×C₆₇	—	4	2	no	—
51	C₄⋊C₄×C₆₇	—	4	2	no	—
52	C₄²×C₆₇	—	2	2	no	—
53	C₄²×C₆₇	—	2	2	no	—
54	C₄²×C₆₇	—	2	2	no	—
55	C₄²×C₆₇	—	4	2	no	—
56	C₄²×C₆₇	D₈	1	2	yes	—
57	C₄²×C₆₇	SD₁₆	1	2	yes	—
58	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
59	C₄²×C₆₇	#41	1	2	yes	—
60	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	2	no	—
65	C₂×C₄×C₆₇	—	2	2	no	—
66	C₂×C₄×C₆₇	—	2	2	no	—
67	C₂×C₄×C₆₇	—	4	4	no	—
68	C₂×C₄×C₆₇	—	4	4	no	—
69	C₂×C₄×C₆₇	—	4	4	no	—
70	C₂×C₄×C₆₇	#102	1	4	yes	—
71	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
72	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
73	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
78	C₂×C₄×C₆₇	#137	1	2	yes	—
79	C₂×C₄×C₆₇	#137	1	2	yes	—
80	C₆₇⋊C₈	—	4	4	no	—
81	C₆₇⋊C₈	—	8	2	no	—
82	C₂²×C₆₇	#391	1	1	yes	characteristic
83	C₂²×C₆₇	#413	1	2	yes	—
84	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
85	C₂²×C₆₇	#417	1	2	yes	—
86	C₂²×C₆₇	#419	1	1	yes	characteristic
87	C₄×C₆₇	—	2	4	no	—
88	C₄×C₆₇	—	2	2	no	—
89	C₄×C₆₇	—	4	2	no	—
90	C₄×C₆₇	—	8	4	no	—
91	C₄×C₆₇	#474	1	2	yes	—
92	C₄×C₆₇	#475	1	2	yes	—
93	C₄×C₆₇	#490	1	1	yes	characteristic
94	C₄×C₆₇	#491	1	1	yes	characteristic
95	C₄×C₆₇	#501	1	2	yes	—
96	#9904	—	67	1	no	maximal, Sylow
97	C₂×C₆₇	#1547	1	2	yes	—
98	C₂×C₆₇	#1559	1	2	yes	—
99	C₂×C₆₇	#1988	1	1	yes	characteristic
100	C₂×C₆₇	#1992	1	1	yes	characteristic
101	C₂×C₆₇	#1998	1	1	yes	characteristic
102	#2859	D₆₇	1	1	yes	characteristic
103	#1842	—	67	1	no	—
104	#1767	—	67	1	no	—
105	#1296	—	67	4	no	—
106	C₆₇	#9904	1	1	yes	characteristic, complemented, Sylow
107	C₂×#125	—	2	1	no	—
108	C₂×C₄⋊C₈	—	67	2	no	—
109	C₂×C₄⋊C₈	—	67	2	no	—
110	C₂×C₄⋊C₈	—	134	2	no	—
111	C₂×#97	—	134	1	no	—
112	#389	—	2	2	no	—
113	#389	Dic₆₇	1	4	yes	—
114	#384	C₂×D₆₇	1	2	yes	—
115	#382	—	2	1	no	—
116	C₄³	C₂×D₆₇	1	1	yes	characteristic
117	#336	—	134	1	no	—
118	C₂²×C₈	—	134	2	no	—
119	C₂²×C₈	—	134	2	no	—
120	#125	—	4	2	no	—
121	#125	—	4	2	no	—
122	C₂×C₄⋊C₄	—	2	4	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
126	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
127	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
128	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
129	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
130	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
131	C₄⋊C₈	—	268	4	no	—
132	C₄⋊C₈	—	67	4	no	—
133	C₄⋊C₈	—	134	4	no	—
134	C₄⋊C₈	—	134	2	no	—
135	#97	—	268	2	no	—
136	#95	—	2	4	no	—
137	C₂²×C₄	—	2	4	no	—
138	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
139	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
140	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
142	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
143	C₂×C₈	—	268	2	no	—
144	C₂×C₈	—	268	2	no	—
145	C₂×C₈	—	268	2	no	—
146	C₂×C₈	—	268	4	no	—
147	C₂×C₈	—	134	4	no	—
148	C₄⋊C₄	—	4	4	no	—
149	C₄⋊C₄	—	4	8	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	4	2	no	—
158	C₄²	#11761613	1	2	yes	—
159	C₄²	#11761616	1	2	yes	—
160	C₄²	Q₈×D₆₇	1	2	yes	—
161	C₄²	#11761629	1	2	yes	—
162	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	#14523964	1	4	yes	—
173	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
174	C₂×C₄	#14524019	1	2	yes	—
175	C₂×C₄	#14524032	1	1	yes	characteristic
176	C₂×C₄	#14524041	1	1	yes	characteristic
177	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
178	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
179	C₂×C₄	#14524068	1	2	yes	—
180	C₂×C₄	#14524073	1	2	yes	—
181	C₂×C₄	#14524074	1	2	yes	—
182	C₈	—	268	4	no	—
183	C₈	—	536	2	no	—
184	C₂²	#16863303	1	1	yes	characteristic
185	C₂²	#16863380	1	2	yes	—
186	C₂²	C₂×#14523964	1	1	yes	characteristic
187	C₂²	#16863397	1	2	yes	—
188	C₂²	#16863405	1	1	yes	characteristic
189	C₄	—	2	4	no	—
190	C₄	—	2	2	no	—
191	C₄	—	4	2	no	—
192	C₄	—	8	4	no	—
193	C₄	#16863673	1	2	yes	—
194	C₄	#16863678	1	2	yes	—
195	C₄	#16863762	1	1	yes	characteristic
196	C₄	#16863766	1	1	yes	characteristic
197	C₄	#16863806	1	2	yes	—
198	C₂	#22558886	1	2	yes	—
199	C₂	#22558939	1	2	yes	—
200	C₂	#22561162	1	1	yes	characteristic
201	C₂	#22561179	1	1	yes	characteristic
202	C₂	#22561203	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22560361

C₂

yes

maximal, characteristic

#22560062

C₂

yes

maximal, characteristic

#22558092

C₂

yes

maximal

C₆₇×#2859

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₆₇⋊C₈)

—

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₂×#14523954

—

#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)

—

C₄⋊(C₆₇⋊C₈)

Q₈

yes

—

#14523954

—

#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

—

C₄⋊C₄×C₆₇

—

C₄²×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

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₆₇

#137

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

#417

yes

—

C₂²×C₆₇

#419

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#474

yes

—

C₄×C₆₇

#475

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

#9904

—

maximal, Sylow

C₂×C₆₇

#1547

yes

—

C₂×C₆₇

#1559

yes

—

C₂×C₆₇

#1988

yes

characteristic

100

C₂×C₆₇

#1992

yes

characteristic

101

C₂×C₆₇

#1998

yes

characteristic

102

#2859

D₆₇

yes

characteristic

103

#1842

—

104

#1767

—

105

#1296

—

106

C₆₇

#9904

yes

characteristic, complemented, Sylow

107

C₂×#125

—

108

C₂×C₄⋊C₈

—

109

C₂×C₄⋊C₈

—

110

C₂×C₄⋊C₈

—

134

—

111

C₂×#97

—

134

—

112

#389

—

113

#389

Dic₆₇

yes

—

114

#384

C₂×D₆₇

yes

—

115

#382

—

116

C₄³

C₂×D₆₇

yes

characteristic

117

#336

—

134

—

118

C₂²×C₈

—

134

—

119

C₂²×C₈

—

134

—

120

#125

—

121

#125

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

C₂×Dic₆₇

yes

—

126

C₂×C₄²

C₂×Dic₆₇

yes

—

127

C₂×C₄²

C₂²⋊D₆₇

yes

characteristic

128

C₂×C₄²

C₂²⋊D₆₇

yes

—

129

C₂×C₄²

C₂²⋊D₆₇

yes

characteristic

130

C₂×C₄²

C₂²×D₆₇

yes

characteristic

131

C₄⋊C₈

—

268

—

132

C₄⋊C₈

—

133

C₄⋊C₈

—

134

—

134

C₄⋊C₈

—

134

—

135

#97

—

268

—

136

#95

—

137

C₂²×C₄

—

138

C₂²×C₄

C₂²⋊Dic₆₇

yes

—

139

C₂²×C₄

C₂²⋊Dic₆₇

yes

—

140

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

141

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

142

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

143

C₂×C₈

—

268

—

144

C₂×C₈

—

268

—

145

C₂×C₈

—

268

—

146

C₂×C₈

—

268

—

147

C₂×C₈

—

134

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄²

—

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

#11761613

yes

—

159

C₄²

#11761616

yes

—

160

C₄²

Q₈×D₆₇

yes

—

161

C₄²

#11761629

yes

—

162

C₂³

C₂×C₂²⋊Dic₆₇

yes

characteristic

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

#14523964

yes

—

173

C₂×C₄

Q₈×Dic₆₇

yes

—

174

C₂×C₄

#14524019

yes

—

175

C₂×C₄

#14524032

yes

characteristic

176

C₂×C₄

#14524041

yes

characteristic

177

C₂×C₄

C₂×#11761613

yes

characteristic

178

C₂×C₄

C₂×#11761616

yes

characteristic

179

C₂×C₄

#14524068

yes

—

180

C₂×C₄

#14524073

yes

—

181

C₂×C₄

#14524074

yes

—

182

C₈

—

268

—

183

C₈

—

536

—

184

C₂²

#16863303

yes

characteristic

185

C₂²

#16863380

yes

—

186

C₂²

C₂×#14523964

yes

characteristic

187

C₂²

#16863397

yes

—

188

C₂²

#16863405

yes

characteristic

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

#16863673

yes

—

194

C₄

#16863678

yes

—

195

C₄

#16863762

yes

characteristic

196

C₄

#16863766

yes

characteristic

197

C₄

#16863806

yes

—

198

C₂

#22558886

yes

—

199

C₂

#22558939

yes

—

200

C₂

#22561162

yes

characteristic

201

C₂

#22561179

yes

characteristic

202

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	#9904
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
106. C₆₇	#9904	semidirect

H ≅ G/N

type

106. C₆₇

#9904

semidirect

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

subgroup	index
#22560361	2
#22560062	2
#22558092	2
C₆₇×#2859	2
#9904	67

subgroup

index

C₆₇×#2859

Subgroups — #30755983

Conjugacy classes of subgroups

Complements

Maximal subgroups