Group #30765161 — subgroups

Subgroups — #30765161

Order 17152.

356 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 119 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	#22562012	C₂	1	2	yes	maximal, complemented
2	#22558135	C₂	1	4	yes	maximal, complemented
3	C₆₇×#2931	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₆₇⋊C₁₆)	—	2	2	no	—
5	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	4	yes	—
6	#16863172	—	2	2	no	—
7	C₂×#131×C₆₇	C₄	1	2	yes	—
8	#532×C₆₇	C₄	1	1	yes	characteristic
9	#436×C₆₇	—	2	2	no	—
10	C₈×D₄×C₆₇	—	2	2	no	—
11	#423×C₆₇	C₄	1	1	yes	characteristic
12	#408×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₆₇⋊C₁₆	—	2	4	no	—
15	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
16	#131×C₆₇	—	4	4	no	—
17	C₂×M₁₆×C₆₇	—	2	2	no	—
18	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₆₇	—	2	2	no	—
20	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
22	C₄⋊Q₈×C₆₇	C₈	1	1	yes	characteristic
23	#127×C₆₇	C₈	1	1	yes	characteristic
24	#124×C₆₇	C₈	1	2	yes	—
25	#121×C₆₇	—	2	2	no	—
26	C₄×D₄×C₆₇	—	2	2	no	—
27	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₄⋊C₈×C₆₇	—	2	2	no	—
29	C₂²⋊C₈×C₆₇	—	2	4	no	—
30	C₄×C₈×C₆₇	D₄	1	2	yes	—
31	C₄×C₈×C₆₇	D₄	1	2	yes	—
32	C₆₇⋊C₁₆	—	4	4	no	—
33	#41×C₆₇	—	4	2	no	—
34	#41×C₆₇	—	4	2	no	—
35	C₂×Q₈×C₆₇	C₂×C₈	1	2	yes	—
36	C₂×D₄×C₆₇	—	2	2	no	—
37	C₂×D₄×C₆₇	—	2	2	no	—
38	C₂×D₄×C₆₇	C₂×C₈	1	2	yes	—
39	C₂²×C₄×C₆₇	—	2	2	no	—
40	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
41	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
42	M₁₆×C₆₇	—	4	4	no	—
43	M₁₆×C₆₇	—	4	2	no	—
44	C₂×C₈×C₆₇	—	2	2	no	—
45	C₂×C₈×C₆₇	—	2	2	no	—
46	C₂×C₈×C₆₇	—	4	1	no	—
47	C₂×C₈×C₆₇	—	4	4	no	—
48	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
51	C₄⋊C₄×C₆₇	—	2	2	no	—
52	C₂²⋊C₄×C₆₇	—	2	4	no	—
53	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
54	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
55	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
56	C₂³×C₆₇	—	2	2	no	—
57	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
58	Q₈×C₆₇	—	4	2	no	—
59	D₄×C₆₇	—	4	2	no	—
60	D₄×C₆₇	—	4	4	no	—
61	D₄×C₆₇	—	4	4	no	—
62	C₂×C₄×C₆₇	—	2	1	no	—
63	C₂×C₄×C₆₇	—	2	1	no	—
64	C₂×C₄×C₆₇	—	2	2	no	—
65	C₂×C₄×C₆₇	—	4	2	no	—
66	C₂×C₄×C₆₇	—	4	2	no	—
67	C₂×C₄×C₆₇	—	4	2	no	—
68	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
69	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
70	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
72	C₈×C₆₇	—	2	2	no	—
73	C₈×C₆₇	—	2	1	no	—
74	C₈×C₆₇	—	4	2	no	—
75	C₈×C₆₇	#100	1	2	yes	—
76	C₂²×C₆₇	—	2	2	no	—
77	C₂²×C₆₇	—	2	1	no	—
78	C₂²×C₆₇	—	4	4	no	—
79	C₂²×C₆₇	—	4	2	no	—
80	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
81	C₄×C₆₇	—	2	2	no	—
82	C₄×C₆₇	—	2	2	no	—
83	C₄×C₆₇	—	2	1	no	—
84	C₄×C₆₇	—	4	2	no	—
85	C₄×C₆₇	#410	1	1	yes	characteristic
86	C₄×C₆₇	C₂×#100	1	1	yes	characteristic
87	#11491	—	67	1	no	maximal, Sylow
88	C₂×C₆₇	—	4	1	no	—
89	C₂×C₆₇	—	8	2	no	—
90	C₂×C₆₇	#1460	1	1	yes	characteristic
91	C₂×C₆₇	#2117	1	2	yes	—
92	#2931	D₆₇	1	1	yes	characteristic
93	#2163	—	67	2	no	—
94	#1315	—	67	4	no	—
95	C₆₇	#11491	1	1	yes	characteristic, complemented, Sylow
96	C₂×#131	Dic₆₇	1	2	yes	—
97	#532	Dic₆₇	1	1	yes	characteristic
98	#436	—	2	2	no	—
99	C₈×D₄	—	2	2	no	—
100	#423	Dic₆₇	1	1	yes	characteristic
101	#408	C₂×D₆₇	1	2	yes	—
102	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
103	C₄⋊C₁₆	—	134	2	no	—
104	C₂²⋊C₁₆	—	67	4	no	—
105	#346	—	134	2	no	—
106	C₂×#41	C₂×Dic₆₇	1	2	yes	—
107	#131	—	4	4	no	—
108	C₂×M₁₆	—	2	2	no	—
109	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
110	C₂²×C₈	—	2	2	no	—
111	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
112	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
113	C₄⋊Q₈	C₆₇⋊C₈	1	1	yes	characteristic
114	#127	C₆₇⋊C₈	1	1	yes	characteristic
115	#124	C₆₇⋊C₈	1	2	yes	—
116	#121	—	2	2	no	—
117	C₄×D₄	—	2	2	no	—
118	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
119	C₂×C₁₆	—	134	4	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₂²⋊C₈	—	2	4	no	—
122	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
123	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
124	#41	—	4	2	no	—
125	#41	—	4	2	no	—
126	C₂×Q₈	C₂×C₆₇⋊C₈	1	2	yes	—
127	C₂×D₄	—	2	2	no	—
128	C₂×D₄	—	2	2	no	—
129	C₂×D₄	C₂×C₆₇⋊C₈	1	2	yes	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	#11761612	1	2	yes	—
132	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
133	M₁₆	—	4	4	no	—
134	M₁₆	—	4	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	4	1	no	—
138	C₂×C₈	—	4	4	no	—
139	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
140	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
141	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
142	C₄⋊C₄	—	2	2	no	—
143	C₂²⋊C₄	—	2	4	no	—
144	C₄²	C₂²⋊Dic₆₇	1	2	yes	—
145	C₄²	C₂×C₆₇⋊C₈	1	1	yes	characteristic
146	C₄²	C₂×C₆₇⋊C₈	1	1	yes	characteristic
147	C₁₆	—	268	4	no	—
148	C₂³	—	2	2	no	—
149	C₂³	C₂×#11761612	1	1	yes	characteristic
150	Q₈	—	4	2	no	—
151	D₄	—	4	2	no	—
152	D₄	—	4	4	no	—
153	D₄	—	4	4	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
161	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
162	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
163	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
164	C₈	—	2	2	no	—
165	C₈	—	2	1	no	—
166	C₈	—	4	2	no	—
167	C₈	#14523960	1	2	yes	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	4	4	no	—
171	C₂²	—	4	2	no	—
172	C₂²	C₂×C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
173	C₄	—	2	2	no	—
174	C₄	—	2	2	no	—
175	C₄	—	2	1	no	—
176	C₄	—	4	2	no	—
177	C₄	#16863370	1	1	yes	characteristic
178	C₄	C₂×#14523960	1	1	yes	characteristic
179	C₂	—	4	1	no	—
180	C₂	—	8	2	no	—
181	C₂	#22558530	1	1	yes	characteristic
182	C₂	#22561837	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22562012

C₂

yes

maximal, complemented

#22558135

C₂

yes

maximal, complemented

C₆₇×#2931

C₂

yes

maximal, characteristic

C₄⋊(C₆₇⋊C₁₆)

—

C₂²⋊(C₆₇⋊C₁₆)

C₂²

yes

—

#16863172

—

C₂×#131×C₆₇

C₄

yes

—

#532×C₆₇

C₄

yes

characteristic

#436×C₆₇

—

C₈×D₄×C₆₇

—

#423×C₆₇

C₄

yes

characteristic

#408×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₆₇⋊C₁₆

—

C₂×#41×C₆₇

C₂×C₄

yes

—

#131×C₆₇

—

C₂×M₁₆×C₆₇

—

C₂×M₁₆×C₆₇

C₂×C₄

yes

—

C₂²×C₈×C₆₇

—

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊Q₈×C₆₇

C₈

yes

characteristic

#127×C₆₇

C₈

yes

characteristic

#124×C₆₇

C₈

yes

—

#121×C₆₇

—

C₄×D₄×C₆₇

—

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₆₇⋊C₁₆

—

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

C₂×C₈

yes

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

C₂×C₈

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

M₁₆

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

C₂²⋊C₄

yes

—

C₄²×C₆₇

C₂×C₈

yes

characteristic

C₄²×C₆₇

C₂×C₈

yes

characteristic

C₂³×C₆₇

—

C₂³×C₆₇

C₂×M₁₆

yes

characteristic

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊C₈

yes

—

C₂×C₄×C₆₇

C₂²⋊C₈

yes

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₈

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#100

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₂²⋊C₈

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#1460

yes

characteristic

C₂×C₆₇

#2117

yes

—

#2931

D₆₇

yes

characteristic

#2163

—

#1315

—

C₆₇

#11491

yes

characteristic, complemented, Sylow

C₂×#131

Dic₆₇

yes

—

#532

Dic₆₇

yes

characteristic

#436

—

C₈×D₄

—

100

#423

Dic₆₇

yes

characteristic

101

#408

C₂×D₆₇

yes

—

102

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

103

C₄⋊C₁₆

—

134

—

104

C₂²⋊C₁₆

—

105

#346

—

134

—

106

C₂×#41

C₂×Dic₆₇

yes

—

107

#131

—

108

C₂×M₁₆

—

109

C₂×M₁₆

C₂×Dic₆₇

yes

—

110

C₂²×C₈

—

111

C₂²×C₈

C₂×Dic₆₇

yes

characteristic

112

C₂²×C₈

C₂²×D₆₇

yes

characteristic

113

C₄⋊Q₈

C₆₇⋊C₈

yes

characteristic

114

#127

C₆₇⋊C₈

yes

characteristic

115

#124

C₆₇⋊C₈

yes

—

116

#121

—

117

C₄×D₄

—

118

C₂×C₄²

C₂×Dic₆₇

yes

characteristic

119

C₂×C₁₆

—

134

—

120

C₄⋊C₈

—

121

C₂²⋊C₈

—

122

C₄×C₈

C₂²⋊D₆₇

yes

—

123

C₄×C₈

C₂²⋊D₆₇

yes

—

124

#41

—

125

#41

—

126

C₂×Q₈

C₂×C₆₇⋊C₈

yes

—

127

C₂×D₄

—

128

C₂×D₄

—

129

C₂×D₄

C₂×C₆₇⋊C₈

yes

—

130

C₂²×C₄

—

131

C₂²×C₄

#11761612

yes

—

132

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

133

M₁₆

—

134

M₁₆

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

C₂²⋊Dic₆₇

yes

characteristic

140

C₂×C₈

C₂²⋊Dic₆₇

yes

characteristic

141

C₂×C₈

C₂×C₂²⋊D₆₇

yes

—

142

C₄⋊C₄

—

143

C₂²⋊C₄

—

144

C₄²

C₂²⋊Dic₆₇

yes

—

145

C₄²

C₂×C₆₇⋊C₈

yes

characteristic

146

C₄²

C₂×C₆₇⋊C₈

yes

characteristic

147

C₁₆

—

268

—

148

C₂³

—

149

C₂³

C₂×#11761612

yes

characteristic

150

Q₈

—

151

D₄

—

152

D₄

—

153

D₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

C₂²⋊(C₆₇⋊C₈)

yes

—

161

C₂×C₄

C₂²⋊(C₆₇⋊C₈)

yes

—

162

C₂×C₄

C₂×C₂²⋊Dic₆₇

yes

characteristic

163

C₂×C₄

C₂²×C₆₇⋊C₈

yes

characteristic

164

C₈

—

165

C₈

—

166

C₈

—

167

C₈

#14523960

yes

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

C₂×C₂²⋊(C₆₇⋊C₈)

yes

characteristic

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

#16863370

yes

characteristic

178

C₄

C₂×#14523960

yes

characteristic

179

C₂

—

180

C₂

—

181

C₂

#22558530

yes

characteristic

182

C₂

#22561837

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#2931	C₂
socle	C₂²×C₆₇	C₂×C₂²⋊C₈
prime radical	#532×C₆₇	C₄
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#2931	D₆₇
hypocenter	C₆₇	#11491
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2931

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊Dic₆₇

commutator

C₂×C₄×C₆₇

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#2931

C₂

socle

C₂²×C₆₇

C₂×C₂²⋊C₈

prime radical

#532×C₆₇

C₄

cosocle

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2931

N	H ≅ G/N	type
1. #22562012	C₂	semidirect
2. #22558135	C₂	semidirect
95. C₆₇	#11491	semidirect

H ≅ G/N

type

1. #22562012

C₂

semidirect

2. #22558135

C₂

semidirect

95. C₆₇

#11491

semidirect

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

subgroup	index
#22562012	2
#22558135	2
C₆₇×#2931	2
#11491	67

subgroup

index

#22562012

#22558135

C₆₇×#2931

#11491

Subgroups — #30765161

Conjugacy classes of subgroups

Complements

Maximal subgroups