Group #30754917 — subgroups

Subgroups — #30754917

Order 17152.

538 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 175 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	#22560326	C₂	1	1	yes	maximal, characteristic
2	#22560062	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863142	C₂	1	4	yes	maximal
4	C₆₇×#2868	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×#14523954	—	2	1	no	—
8	#16863151	—	2	1	no	—
9	#16863151	—	2	2	no	—
10	#16863142	—	2	8	no	—
11	C₂×C₄⋊Q₈×C₆₇	—	2	2	no	—
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
13	#384×C₆₇	—	2	1	no	—
14	#384×C₆₇	—	2	1	no	—
15	#384×C₆₇	C₄	1	4	yes	—
16	C₄³×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×C₆₇⋊C₈	—	2	2	no	—
18	C₂²×C₆₇⋊C₈	—	2	2	no	—
19	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
20	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
21	C₄⋊(C₆₇⋊C₈)	D₄	1	4	yes	—
22	#14523954	—	4	2	no	—
23	C₂²×Q₈×C₆₇	—	2	2	no	—
24	C₄⋊Q₈×C₆₇	—	2	4	no	—
25	C₄⋊Q₈×C₆₇	—	2	4	no	—
26	C₄⋊Q₈×C₆₇	—	4	8	no	—
27	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
28	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
29	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₆₇	D₄	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	C₂³	1	1	yes	characteristic
36	C₂×C₆₇⋊C₈	—	2	4	no	—
37	C₂×C₆₇⋊C₈	—	4	2	no	—
38	C₂×C₆₇⋊C₈	—	4	2	no	—
39	C₂×C₆₇⋊C₈	—	4	2	no	—
40	C₂×C₆₇⋊C₈	—	4	4	no	—
41	C₂×Q₈×C₆₇	—	4	2	no	—
42	C₂×Q₈×C₆₇	—	4	2	no	—
43	C₂×Q₈×C₆₇	—	4	8	no	—
44	C₂²×C₄×C₆₇	—	2	4	no	—
45	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₄⋊C₄×C₆₇	—	2	8	no	—
51	C₄⋊C₄×C₆₇	—	4	8	no	—
52	C₄⋊C₄×C₆₇	—	4	8	no	—
53	C₄⋊C₄×C₆₇	—	4	4	no	—
54	C₄²×C₆₇	—	2	4	no	—
55	C₄²×C₆₇	—	2	4	no	—
56	C₄²×C₆₇	—	2	2	no	—
57	C₄²×C₆₇	SD₁₆	1	2	yes	—
58	C₄²×C₆₇	Q₁₆	1	2	yes	—
59	C₄²×C₆₇	C₂×D₄	1	2	yes	—
60	C₄²×C₆₇	#41	1	2	yes	—
61	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	Q₈×C₆₇	—	8	4	no	—
63	Q₈×C₆₇	—	8	4	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	—	2	2	no	—
68	C₂×C₄×C₆₇	—	2	2	no	—
69	C₂×C₄×C₆₇	—	4	4	no	—
70	C₂×C₄×C₆₇	—	4	8	no	—
71	C₂×C₄×C₆₇	#101	1	4	yes	—
72	C₂×C₄×C₆₇	#103	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×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
80	C₂×C₄×C₆₇	#137	1	2	yes	—
81	C₆₇⋊C₈	—	4	4	no	—
82	C₆₇⋊C₈	—	8	2	no	—
83	C₂²×C₆₇	#390	1	1	yes	characteristic
84	C₂²×C₆₇	C₂×#101	1	2	yes	—
85	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
86	C₂²×C₆₇	#418	1	2	yes	—
87	C₂²×C₆₇	#419	1	1	yes	characteristic
88	C₄×C₆₇	—	2	4	no	—
89	C₄×C₆₇	—	2	4	no	—
90	C₄×C₆₇	—	2	2	no	—
91	C₄×C₆₇	—	8	4	no	—
92	C₄×C₆₇	#461	1	2	yes	—
93	C₄×C₆₇	#462	1	2	yes	—
94	C₄×C₆₇	#490	1	2	yes	—
95	C₄×C₆₇	#496	1	2	yes	—
96	#9703	—	67	1	no	maximal, Sylow
97	C₂×C₆₇	#1549	1	2	yes	—
98	C₂×C₆₇	#1556	1	2	yes	—
99	C₂×C₆₇	#1961	1	1	yes	characteristic
100	C₂×C₆₇	#1966	1	1	yes	characteristic
101	C₂×C₆₇	#1981	1	1	yes	characteristic
102	#2868	D₆₇	1	1	yes	characteristic
103	#1831	—	67	1	no	—
104	#1767	—	67	1	no	—
105	C₂×#332	—	67	4	no	—
106	C₆₇	#9703	1	1	yes	characteristic, complemented, Sylow
107	C₂×C₄⋊Q₈	—	2	2	no	—
108	C₂×C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
109	C₂×C₄⋊C₈	—	67	2	no	—
110	C₂×C₄⋊C₈	—	67	2	no	—
111	C₂×#97	—	134	1	no	—
112	#384	—	2	1	no	—
113	#384	—	2	1	no	—
114	#384	Dic₆₇	1	4	yes	—
115	C₄³	C₂×D₆₇	1	1	yes	characteristic
116	#336	—	134	2	no	—
117	#336	—	134	1	no	—
118	#332	—	134	8	no	—
119	C₂²×Q₈	—	2	2	no	—
120	C₂²×C₈	—	134	2	no	—
121	C₂²×C₈	—	134	2	no	—
122	C₄⋊Q₈	—	2	4	no	—
123	C₄⋊Q₈	—	2	4	no	—
124	C₄⋊Q₈	—	4	8	no	—
125	C₂×C₄⋊C₄	—	2	4	no	—
126	C₂×C₄⋊C₄	—	2	4	no	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
129	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
130	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
131	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
132	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
133	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
134	C₄⋊C₈	—	67	4	no	—
135	C₄⋊C₈	—	134	4	no	—
136	C₄⋊C₈	—	134	2	no	—
137	#97	—	268	2	no	—
138	C₂×Q₈	—	4	2	no	—
139	C₂×Q₈	—	4	2	no	—
140	C₂×Q₈	—	4	8	no	—
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
143	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
144	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
147	C₂×C₈	—	268	2	no	—
148	C₂×C₈	—	268	2	no	—
149	C₂×C₈	—	268	2	no	—
150	C₂×C₈	—	268	4	no	—
151	C₂×C₈	—	134	4	no	—
152	C₄⋊C₄	—	2	8	no	—
153	C₄⋊C₄	—	4	8	no	—
154	C₄⋊C₄	—	4	8	no	—
155	C₄⋊C₄	—	4	4	no	—
156	C₄²	—	2	4	no	—
157	C₄²	—	2	4	no	—
158	C₄²	—	2	2	no	—
159	C₄²	#11761615	1	2	yes	—
160	C₄²	#11761618	1	2	yes	—
161	C₄²	D₄×D₆₇	1	2	yes	—
162	C₄²	#11761627	1	2	yes	—
163	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
164	Q₈	—	8	4	no	—
165	Q₈	—	8	4	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	4	8	no	—
173	C₂×C₄	#14523962	1	4	yes	—
174	C₂×C₄	#14523967	1	4	yes	—
175	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
176	C₂×C₄	#14524014	1	2	yes	—
177	C₂×C₄	#14524030	1	1	yes	characteristic
178	C₂×C₄	#14524039	1	1	yes	characteristic
179	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
180	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
181	C₂×C₄	#14524069	1	4	yes	—
182	C₂×C₄	#14524074	1	2	yes	—
183	C₈	—	268	4	no	—
184	C₈	—	536	2	no	—
185	C₂²	#16863299	1	1	yes	characteristic
186	C₂²	C₂×#14523962	1	2	yes	—
187	C₂²	C₂×#14523967	1	1	yes	characteristic
188	C₂²	#16863401	1	2	yes	—
189	C₂²	#16863405	1	1	yes	characteristic
190	C₄	—	2	2	no	—
191	C₄	—	2	4	no	—
192	C₄	—	2	4	no	—
193	C₄	—	8	4	no	—
194	C₄	#16863598	1	2	yes	—
195	C₄	#16863603	1	2	yes	—
196	C₄	#16863760	1	2	yes	—
197	C₄	#16863783	1	2	yes	—
198	C₂	#22558894	1	2	yes	—
199	C₂	#22558926	1	2	yes	—
200	C₂	#22561034	1	1	yes	characteristic
201	C₂	#22561054	1	1	yes	characteristic
202	C₂	#22561133	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22560326

C₂

yes

maximal, characteristic

#22560062

C₂

yes

maximal, characteristic

C₂×#16863142

C₂

yes

maximal

C₆₇×#2868

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Q₈×C₆₇

—

C₂×C₄⋊Q₈×C₆₇

C₂²

yes

—

#384×C₆₇

—

#384×C₆₇

—

#384×C₆₇

C₄

yes

—

C₄³×C₆₇

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

—

C₂²×C₆₇⋊C₈

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

D₄

yes

—

#14523954

—

C₂²×Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×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

—

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

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

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

SD₁₆

yes

—

C₄²×C₆₇

Q₁₆

yes

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

Q₈×C₆₇

—

Q₈×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

#101

yes

—

C₂×C₄×C₆₇

#103

yes

—

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

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

#137

yes

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

#390

yes

characteristic

C₂²×C₆₇

C₂×#101

yes

—

C₂²×C₆₇

C₂×#103

yes

characteristic

C₂²×C₆₇

#418

yes

—

C₂²×C₆₇

#419

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#461

yes

—

C₄×C₆₇

#462

yes

—

C₄×C₆₇

#490

yes

—

C₄×C₆₇

#496

yes

—

#9703

—

maximal, Sylow

C₂×C₆₇

#1549

yes

—

C₂×C₆₇

#1556

yes

—

C₂×C₆₇

#1961

yes

characteristic

100

C₂×C₆₇

#1966

yes

characteristic

101

C₂×C₆₇

#1981

yes

characteristic

102

#2868

D₆₇

yes

characteristic

103

#1831

—

104

#1767

—

105

C₂×#332

—

106

C₆₇

#9703

yes

characteristic, complemented, Sylow

107

C₂×C₄⋊Q₈

—

108

C₂×C₄⋊Q₈

C₂×D₆₇

yes

—

109

C₂×C₄⋊C₈

—

110

C₂×C₄⋊C₈

—

111

C₂×#97

—

134

—

112

#384

—

113

#384

—

114

#384

Dic₆₇

yes

—

115

C₄³

C₂×D₆₇

yes

characteristic

116

#336

—

134

—

117

#336

—

134

—

118

#332

—

134

—

119

C₂²×Q₈

—

120

C₂²×C₈

—

134

—

121

C₂²×C₈

—

134

—

122

C₄⋊Q₈

—

123

C₄⋊Q₈

—

124

C₄⋊Q₈

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

C₂×Dic₆₇

yes

—

129

C₂×C₄²

C₂×Dic₆₇

yes

—

130

C₂×C₄²

C₂²⋊D₆₇

yes

—

131

C₂×C₄²

C₂²⋊D₆₇

yes

characteristic

132

C₂×C₄²

C₂²⋊D₆₇

yes

characteristic

133

C₂×C₄²

C₂²×D₆₇

yes

characteristic

134

C₄⋊C₈

—

135

C₄⋊C₈

—

134

—

136

C₄⋊C₈

—

134

—

137

#97

—

268

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂²×C₄

—

142

C₂²×C₄

C₂²⋊Dic₆₇

yes

—

143

C₂²×C₄

C₂²⋊Dic₆₇

yes

—

144

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

145

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

146

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

147

C₂×C₈

—

268

—

148

C₂×C₈

—

268

—

149

C₂×C₈

—

268

—

150

C₂×C₈

—

268

—

151

C₂×C₈

—

134

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄²

—

157

C₄²

—

158

C₄²

—

159

C₄²

#11761615

yes

—

160

C₄²

#11761618

yes

—

161

C₄²

D₄×D₆₇

yes

—

162

C₄²

#11761627

yes

—

163

C₂³

C₂×C₂²⋊Dic₆₇

yes

characteristic

164

Q₈

—

165

Q₈

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

#14523962

yes

—

174

C₂×C₄

#14523967

yes

—

175

C₂×C₄

D₄×Dic₆₇

yes

—

176

C₂×C₄

#14524014

yes

—

177

C₂×C₄

#14524030

yes

characteristic

178

C₂×C₄

#14524039

yes

characteristic

179

C₂×C₄

C₂×#11761615

yes

characteristic

180

C₂×C₄

C₂×#11761618

yes

characteristic

181

C₂×C₄

#14524069

yes

—

182

C₂×C₄

#14524074

yes

—

183

C₈

—

268

—

184

C₈

—

536

—

185

C₂²

#16863299

yes

characteristic

186

C₂²

C₂×#14523962

yes

—

187

C₂²

C₂×#14523967

yes

characteristic

188

C₂²

#16863401

yes

—

189

C₂²

#16863405

yes

characteristic

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

#16863598

yes

—

195

C₄

#16863603

yes

—

196

C₄

#16863760

yes

—

197

C₄

#16863783

yes

—

198

C₂

#22558894

yes

—

199

C₂

#22558926

yes

—

200

C₂

#22561034

yes

characteristic

201

C₂

#22561054

yes

characteristic

202

C₂

#22561133

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2868	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#2868	D₆₇
hypocenter	C₆₇	#9703
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₆₇

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#2868

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 ◁ #2868

N	H ≅ G/N	type
106. C₆₇	#9703	semidirect

H ≅ G/N

type

106. C₆₇

#9703

semidirect

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

subgroup	index
#22560326	2
#22560062	2
C₂×#16863142	2
C₆₇×#2868	2
#9703	67

subgroup

index

C₆₇×#2868

Subgroups — #30754917

Conjugacy classes of subgroups

Complements

Maximal subgroups