Group #30755975 — subgroups

Subgroups — #30755975

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	4	no	—
24	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
25	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
26	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
27	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₆₇	D₄	1	2	yes	—
31	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
33	#95×C₆₇	—	2	4	no	—
34	C₂×C₆₇⋊C₈	—	2	4	no	—
35	C₂×C₆₇⋊C₈	—	4	2	no	—
36	C₂×C₆₇⋊C₈	—	4	2	no	—
37	C₂×C₆₇⋊C₈	—	4	2	no	—
38	C₂×C₆₇⋊C₈	—	4	4	no	—
39	C₂²×C₄×C₆₇	—	2	4	no	—
40	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
45	C₄⋊C₄×C₆₇	—	4	4	no	—
46	C₄⋊C₄×C₆₇	—	4	8	no	—
47	C₄⋊C₄×C₆₇	—	4	2	no	—
48	C₄⋊C₄×C₆₇	—	4	4	no	—
49	C₄⋊C₄×C₆₇	—	4	2	no	—
50	C₄²×C₆₇	—	2	2	no	—
51	C₄²×C₆₇	—	2	2	no	—
52	C₄²×C₆₇	—	2	2	no	—
53	C₄²×C₆₇	—	4	2	no	—
54	C₄²×C₆₇	SD₁₆	1	2	yes	—
55	C₄²×C₆₇	Q₁₆	1	2	yes	—
56	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
57	C₄²×C₆₇	#41	1	2	yes	—
58	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₆₇	—	2	2	no	—
60	C₂×C₄×C₆₇	—	2	2	no	—
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	—	4	4	no	—
66	C₂×C₄×C₆₇	—	4	4	no	—
67	C₂×C₄×C₆₇	—	4	4	no	—
68	C₂×C₄×C₆₇	#103	1	4	yes	—
69	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
70	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
71	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
76	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×C₄×C₆₇	#137	1	2	yes	—
78	C₆₇⋊C₈	—	4	4	no	—
79	C₆₇⋊C₈	—	8	2	no	—
80	C₂²×C₆₇	#391	1	1	yes	characteristic
81	C₂²×C₆₇	#413	1	2	yes	—
82	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
83	C₂²×C₆₇	#417	1	2	yes	—
84	C₂²×C₆₇	#418	1	1	yes	characteristic
85	C₄×C₆₇	—	2	4	no	—
86	C₄×C₆₇	—	2	2	no	—
87	C₄×C₆₇	—	4	2	no	—
88	C₄×C₆₇	—	8	4	no	—
89	C₄×C₆₇	#476	1	2	yes	—
90	C₄×C₆₇	#477	1	2	yes	—
91	C₄×C₆₇	#490	1	1	yes	characteristic
92	C₄×C₆₇	#491	1	1	yes	characteristic
93	C₄×C₆₇	#501	1	2	yes	—
94	#9902	—	67	1	no	maximal, Sylow
95	C₂×C₆₇	#1550	1	2	yes	—
96	C₂×C₆₇	#1557	1	2	yes	—
97	C₂×C₆₇	#1987	1	1	yes	characteristic
98	C₂×C₆₇	#1992	1	1	yes	characteristic
99	C₂×C₆₇	#1998	1	1	yes	characteristic
100	#2859	D₆₇	1	1	yes	characteristic
101	#1842	—	67	1	no	—
102	#1767	—	67	1	no	—
103	#1296	—	67	4	no	—
104	C₆₇	#9902	1	1	yes	characteristic, complemented, Sylow
105	C₂×#125	—	2	1	no	—
106	C₂×C₄⋊C₈	—	67	2	no	—
107	C₂×C₄⋊C₈	—	67	2	no	—
108	C₂×C₄⋊C₈	—	134	2	no	—
109	C₂×#97	—	134	1	no	—
110	#389	—	2	2	no	—
111	#389	Dic₆₇	1	4	yes	—
112	#384	C₂×D₆₇	1	2	yes	—
113	#382	—	2	1	no	—
114	C₄³	C₂×D₆₇	1	1	yes	characteristic
115	#336	—	134	1	no	—
116	C₂²×C₈	—	134	2	no	—
117	C₂²×C₈	—	134	2	no	—
118	#125	—	4	4	no	—
119	C₂×C₄⋊C₄	—	2	4	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
123	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
124	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
125	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
126	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
127	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
128	C₄⋊C₈	—	268	4	no	—
129	C₄⋊C₈	—	67	4	no	—
130	C₄⋊C₈	—	134	4	no	—
131	C₄⋊C₈	—	134	2	no	—
132	#97	—	268	2	no	—
133	#95	—	2	4	no	—
134	C₂²×C₄	—	2	4	no	—
135	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
136	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
137	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
140	C₂×C₈	—	268	2	no	—
141	C₂×C₈	—	268	2	no	—
142	C₂×C₈	—	268	2	no	—
143	C₂×C₈	—	268	4	no	—
144	C₂×C₈	—	134	4	no	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄⋊C₄	—	4	8	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	4	4	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	—	4	2	no	—
154	C₄²	#11761615	1	2	yes	—
155	C₄²	#11761618	1	2	yes	—
156	C₄²	Q₈×D₆₇	1	2	yes	—
157	C₄²	#11761629	1	2	yes	—
158	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	#14523967	1	4	yes	—
169	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
170	C₂×C₄	#14524019	1	2	yes	—
171	C₂×C₄	#14524032	1	1	yes	characteristic
172	C₂×C₄	#14524041	1	1	yes	characteristic
173	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
174	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
175	C₂×C₄	#14524068	1	2	yes	—
176	C₂×C₄	#14524069	1	2	yes	—
177	C₂×C₄	#14524073	1	2	yes	—
178	C₈	—	268	4	no	—
179	C₈	—	536	2	no	—
180	C₂²	#16863303	1	1	yes	characteristic
181	C₂²	#16863380	1	2	yes	—
182	C₂²	C₂×#14523967	1	1	yes	characteristic
183	C₂²	#16863397	1	2	yes	—
184	C₂²	#16863401	1	1	yes	characteristic
185	C₄	—	2	4	no	—
186	C₄	—	2	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	8	4	no	—
189	C₄	#16863683	1	2	yes	—
190	C₄	#16863688	1	2	yes	—
191	C₄	#16863762	1	1	yes	characteristic
192	C₄	#16863766	1	1	yes	characteristic
193	C₄	#16863806	1	2	yes	—
194	C₂	#22558899	1	2	yes	—
195	C₂	#22558931	1	2	yes	—
196	C₂	#22561158	1	1	yes	characteristic
197	C₂	#22561179	1	1	yes	characteristic
198	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

—

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

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

SD₁₆

yes

—

C₄²×C₆₇

Q₁₆

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

#103

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×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₆₇

#137

yes

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

#391

yes

characteristic

C₂²×C₆₇

#413

yes

—

C₂²×C₆₇

C₂×#103

yes

characteristic

C₂²×C₆₇

#417

yes

—

C₂²×C₆₇

#418

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#476

yes

—

C₄×C₆₇

#477

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

#9902

—

maximal, Sylow

C₂×C₆₇

#1550

yes

—

C₂×C₆₇

#1557

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

100

D₆₇

yes

characteristic

101

#1842

—

102

#1767

—

103

#1296

—

104

C₆₇

#9902

yes

characteristic, complemented, Sylow

105

C₂×#125

—

106

C₂×C₄⋊C₈

—

107

C₂×C₄⋊C₈

—

108

C₂×C₄⋊C₈

—

134

—

109

C₂×#97

—

134

—

110

#389

—

111

#389

Dic₆₇

yes

—

112

#384

C₂×D₆₇

yes

—

113

#382

—

114

C₄³

C₂×D₆₇

yes

characteristic

115

#336

—

134

—

116

C₂²×C₈

—

134

—

117

C₂²×C₈

—

134

—

118

#125

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

C₂×Dic₆₇

yes

—

123

C₂×C₄²

C₂×Dic₆₇

yes

—

124

C₂×C₄²

C₂²⋊D₆₇

yes

characteristic

125

C₂×C₄²

C₂²⋊D₆₇

yes

—

126

C₂×C₄²

C₂²⋊D₆₇

yes

characteristic

127

C₂×C₄²

C₂²×D₆₇

yes

characteristic

128

C₄⋊C₈

—

268

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

134

—

131

C₄⋊C₈

—

134

—

132

#97

—

268

—

133

#95

—

134

C₂²×C₄

—

135

C₂²×C₄

C₂²⋊Dic₆₇

yes

—

136

C₂²×C₄

C₂²⋊Dic₆₇

yes

—

137

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

138

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

139

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

140

C₂×C₈

—

268

—

141

C₂×C₈

—

268

—

142

C₂×C₈

—

268

—

143

C₂×C₈

—

268

—

144

C₂×C₈

—

134

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄²

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

#11761615

yes

—

155

C₄²

#11761618

yes

—

156

C₄²

Q₈×D₆₇

yes

—

157

C₄²

#11761629

yes

—

158

C₂³

C₂×C₂²⋊Dic₆₇

yes

characteristic

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

#14523967

yes

—

169

C₂×C₄

Q₈×Dic₆₇

yes

—

170

C₂×C₄

#14524019

yes

—

171

C₂×C₄

#14524032

yes

characteristic

172

C₂×C₄

#14524041

yes

characteristic

173

C₂×C₄

C₂×#11761615

yes

characteristic

174

C₂×C₄

C₂×#11761618

yes

characteristic

175

C₂×C₄

#14524068

yes

—

176

C₂×C₄

#14524069

yes

—

177

C₂×C₄

#14524073

yes

—

178

C₈

—

268

—

179

C₈

—

536

—

180

C₂²

#16863303

yes

characteristic

181

C₂²

#16863380

yes

—

182

C₂²

C₂×#14523967

yes

characteristic

183

C₂²

#16863397

yes

—

184

C₂²

#16863401

yes

characteristic

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

#16863683

yes

—

190

C₄

#16863688

yes

—

191

C₄

#16863762

yes

characteristic

192

C₄

#16863766

yes

characteristic

193

C₄

#16863806

yes

—

194

C₂

#22558899

yes

—

195

C₂

#22558931

yes

—

196

C₂

#22561158

yes

characteristic

197

C₂

#22561179

yes

characteristic

198

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	#9902
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
104. C₆₇	#9902	semidirect

H ≅ G/N

type

104. C₆₇

#9902

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
#9902	67

subgroup

index

C₆₇×#2859

Subgroups — #30755975

Conjugacy classes of subgroups

Complements

Maximal subgroups