Group #30772169 — subgroups

Subgroups — #30772169

Order 17152.

366 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 127 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	#22561823	C₂	1	1	yes	maximal, characteristic
2	#22560255	C₂	1	1	yes	maximal, characteristic
3	#22558323	C₂	1	2	yes	maximal, complemented
4	#22558306	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2916	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₂²	1	1	yes	characteristic
7	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	#16863434	C₂²	1	2	yes	—
9	#16863370	—	2	1	no	—
10	C₂²⋊(C₆₇⋊C₁₆)	—	2	2	no	—
11	#16863153	—	2	1	no	—
12	#568×C₆₇	—	2	1	no	—
13	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
14	#430×C₆₇	—	2	1	no	—
15	#430×C₆₇	C₄	1	4	yes	complemented
16	C₂×#108×C₆₇	—	2	2	no	—
17	C₂×#108×C₆₇	C₂²	1	2	yes	—
18	#14524002	—	2	2	no	—
19	#14523983	—	2	2	no	—
20	C₂×C₆₇⋊C₁₆	—	2	1	no	—
21	C₂×C₆₇⋊C₁₆	—	2	1	no	—
22	C₂×C₆₇⋊C₁₆	—	2	2	no	—
23	#14523977	—	2	2	no	—
24	#14523975	—	2	2	no	—
25	#14523958	—	4	2	no	—
26	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
27	#131×C₆₇	—	2	2	no	—
28	#131×C₆₇	—	2	2	no	—
29	#131×C₆₇	—	4	4	no	—
30	C₂×M₁₆×C₆₇	—	2	4	no	—
31	C₂×M₁₆×C₆₇	—	2	2	no	—
32	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
33	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	D₄	1	2	yes	—
35	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
36	#108×C₆₇	—	2	4	no	—
37	#108×C₆₇	—	2	2	no	—
38	#108×C₆₇	C₂×C₄	1	4	yes	—
39	C₄⋊Dic₆₇	—	4	2	no	—
40	C₂²⋊Dic₆₇	—	4	2	no	—
41	C₄×Dic₆₇	—	4	2	no	—
42	C₆₇⋊C₁₆	—	4	1	no	—
43	C₆₇⋊C₁₆	—	4	1	no	—
44	#41×C₆₇	—	2	2	no	—
45	#41×C₆₇	—	2	2	no	—
46	C₂×Q₈×C₆₇	Q₁₆	1	1	yes	characteristic
47	C₂×D₄×C₆₇	SD₁₆	1	2	yes	—
48	C₂×D₄×C₆₇	Q₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
51	M₁₆×C₆₇	—	2	2	no	—
52	M₁₆×C₆₇	—	2	4	no	—
53	M₁₆×C₆₇	—	2	4	no	—
54	M₁₆×C₆₇	—	4	4	no	—
55	M₁₆×C₆₇	—	4	2	no	—
56	C₂×C₈×C₆₇	—	2	2	no	—
57	C₂×C₈×C₆₇	—	2	2	no	—
58	C₂×C₈×C₆₇	—	4	4	no	—
59	C₂×C₈×C₆₇	—	4	2	no	—
60	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₆₇	#41	1	2	yes	—
64	C₂×C₈×C₆₇	#41	1	2	yes	—
65	C₂³×C₆₇	#103	1	2	yes	—
66	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
67	Q₈×C₆₇	—	2	2	no	—
68	D₄×C₆₇	—	2	4	no	—
69	D₄×C₆₇	—	2	2	no	—
70	C₂×C₄×C₆₇	—	2	2	no	—
71	C₂×C₄×C₆₇	—	2	2	no	—
72	C₂×C₄×C₆₇	#103	1	2	yes	—
73	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₆₇	#123	1	2	yes	—
75	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×Dic₆₇	—	8	2	no	—
78	C₈×C₆₇	—	2	2	no	—
79	C₈×C₆₇	—	4	4	no	—
80	C₈×C₆₇	#117	1	2	yes	—
81	C₈×C₆₇	#123	1	1	yes	characteristic
82	C₈×C₆₇	#123	1	1	yes	characteristic
83	C₂²×C₆₇	—	2	2	no	—
84	C₂²×C₆₇	—	4	1	no	—
85	C₂²×C₆₇	—	4	2	no	—
86	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
87	C₂²×C₆₇	#481	1	2	yes	—
88	C₄×C₆₇	—	2	2	no	—
89	C₄×C₆₇	#381	1	1	yes	characteristic
90	C₄×C₆₇	#418	1	1	yes	characteristic
91	C₄×C₆₇	#484	1	2	yes	—
92	Dic₆₇	—	16	2	no	—
93	#12877	—	67	1	no	maximal, Sylow
94	C₂×C₆₇	—	2	2	no	—
95	C₂×C₆₇	—	2	1	no	—
96	C₂×C₆₇	—	4	2	no	—
97	C₂×C₆₇	#1824	1	1	yes	characteristic
98	#2916	D₆₇	1	1	yes	characteristic
99	#2113	—	67	1	no	—
100	#1812	—	67	1	no	—
101	#1389	—	67	2	no	—
102	#1382	—	67	2	no	—
103	C₆₇	#12877	1	1	yes	characteristic, complemented, Sylow
104	#568	—	2	1	no	—
105	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
106	C₂×M₃₂	—	67	1	no	—
107	C₂²×C₁₆	—	67	1	no	—
108	#430	—	2	1	no	—
109	#430	Dic₆₇	1	4	yes	complemented
110	C₂×#108	—	2	2	no	—
111	C₂×#108	C₂×D₆₇	1	2	yes	—
112	#427	—	67	2	no	—
113	#410	—	134	1	no	—
114	C₂²⋊C₁₆	—	134	2	no	—
115	H(3,ℤ/4)	—	134	1	no	—
116	C₂×#41	C₂²⋊D₆₇	1	1	yes	characteristic
117	#131	—	2	2	no	—
118	#131	—	2	2	no	—
119	#131	—	4	4	no	—
120	C₂×M₁₆	—	2	4	no	—
121	C₂×M₁₆	—	2	2	no	—
122	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
123	C₂×M₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
124	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
125	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
126	#117	—	134	2	no	—
127	M₃₂	—	134	2	no	—
128	C₂×C₁₆	—	134	1	no	—
129	C₂×C₁₆	—	134	1	no	—
130	C₂×C₁₆	—	134	2	no	—
131	#108	—	2	4	no	—
132	#108	—	2	2	no	—
133	#108	C₂×Dic₆₇	1	4	yes	—
134	#107	—	134	2	no	—
135	#106	—	134	2	no	—
136	#99	—	268	2	no	—
137	#41	—	2	2	no	—
138	#41	—	2	2	no	—
139	C₂×Q₈	#11761618	1	1	yes	characteristic
140	C₂×D₄	#11761615	1	2	yes	—
141	C₂×D₄	#11761618	1	1	yes	characteristic
142	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
144	M₁₆	—	2	2	no	—
145	M₁₆	—	2	4	no	—
146	M₁₆	—	2	4	no	—
147	M₁₆	—	4	4	no	—
148	M₁₆	—	4	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	4	4	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
154	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
155	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
156	C₂×C₈	#11761628	1	2	yes	—
157	C₂×C₈	#11761628	1	2	yes	—
158	C₄⋊C₄	—	268	2	no	—
159	C₂²⋊C₄	—	268	2	no	—
160	C₄²	—	268	2	no	—
161	C₁₆	—	268	1	no	—
162	C₁₆	—	268	1	no	—
163	C₂³	#14523967	1	2	yes	—
164	C₂³	C₂×#11761615	1	1	yes	characteristic
165	Q₈	—	2	2	no	—
166	D₄	—	2	4	no	—
167	D₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	536	2	no	—
171	C₂×C₄	#14523967	1	2	yes	—
172	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
173	C₂×C₄	#14524026	1	2	yes	—
174	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
175	C₂×C₄	#14524069	1	2	yes	—
176	C₈	—	2	2	no	—
177	C₈	—	4	4	no	—
178	C₈	#14524002	1	2	yes	—
179	C₈	#14524026	1	1	yes	characteristic
180	C₈	#14524026	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	—	4	1	no	—
183	C₂²	—	4	2	no	—
184	C₂²	C₂×#14523967	1	1	yes	characteristic
185	C₂²	#16863714	1	2	yes	—
186	C₄	—	2	2	no	—
187	C₄	—	1072	2	no	—
188	C₄	#16863259	1	1	yes	characteristic
189	C₄	#16863401	1	1	yes	characteristic
190	C₄	#16863731	1	2	yes	—
191	C₂	—	2	2	no	—
192	C₂	—	2	1	no	—
193	C₂	—	4	2	no	—
194	C₂	#22560299	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22561823

C₂

yes

maximal, characteristic

#22560255

C₂

yes

maximal, characteristic

#22558323

C₂

yes

maximal, complemented

#22558306

C₂

yes

maximal, complemented

C₆₇×#2916

C₂

yes

maximal, characteristic

C₂×#14523983

C₂²

yes

characteristic

C₂²×C₆₇⋊C₁₆

C₂²

yes

characteristic

#16863434

C₂²

yes

—

#16863370

—

C₂²⋊(C₆₇⋊C₁₆)

—

#16863153

—

#568×C₆₇

—

C₂×#131×C₆₇

C₂²

yes

characteristic

#430×C₆₇

—

#430×C₆₇

C₄

yes

complemented

C₂×#108×C₆₇

—

C₂×#108×C₆₇

C₂²

yes

—

#14524002

—

#14523983

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

—

#14523977

—

#14523975

—

#14523958

—

C₂×#41×C₆₇

D₄

yes

characteristic

#131×C₆₇

—

#131×C₆₇

—

#131×C₆₇

—

C₂×M₁₆×C₆₇

—

C₂×M₁₆×C₆₇

—

C₂×M₁₆×C₆₇

C₂×C₄

yes

—

C₂×M₁₆×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

D₄

yes

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

#108×C₆₇

—

#108×C₆₇

—

#108×C₆₇

C₂×C₄

yes

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

—

C₆₇⋊C₁₆

—

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

Q₁₆

yes

characteristic

C₂×D₄×C₆₇

SD₁₆

yes

—

C₂×D₄×C₆₇

Q₁₆

yes

characteristic

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₂×C₈×C₆₇

#41

yes

—

C₂³×C₆₇

#103

yes

—

C₂³×C₆₇

C₂×SD₁₆

yes

characteristic

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#103

yes

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#123

yes

—

C₂×C₄×C₆₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#117

yes

—

C₈×C₆₇

yes

C₈×C₆₇

yes

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×#103

yes

characteristic

C₂²×C₆₇

#481

yes

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

#12877

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#1824

yes

characteristic

#2916

D₆₇

yes

characteristic

#2113

—

100

#1812

—

101

#1389

—

102

#1382

—

103

C₆₇

#12877

yes

characteristic, complemented, Sylow

104

#568

—

105

C₂×#131

C₂×D₆₇

yes

characteristic

106

C₂×M₃₂

—

107

C₂²×C₁₆

—

108

#430

—

109

#430

Dic₆₇

yes

complemented

110

C₂×#108

—

111

C₂×#108

C₂×D₆₇

yes

—

112

#427

—

113

#410

—

134

—

114

C₂²⋊C₁₆

—

134

—

115

H(3,ℤ/4)

—

134

—

116

C₂×#41

C₂²⋊D₆₇

yes

characteristic

117

#131

—

118

#131

—

119

#131

—

120

C₂×M₁₆

—

121

C₂×M₁₆

—

122

C₂×M₁₆

C₂×Dic₆₇

yes

—

123

C₂×M₁₆

C₂²⋊D₆₇

yes

characteristic

124

C₂²×C₈

C₂²⋊D₆₇

yes

—

125

C₂²×C₈

C₂²×D₆₇

yes

characteristic

126

#117

—

134

—

127

M₃₂

—

134

—

128

C₂×C₁₆

—

134

—

129

C₂×C₁₆

—

134

—

130

C₂×C₁₆

—

134

—

131

#108

—

132

#108

—

133

#108

C₂×Dic₆₇

yes

—

134

#107

—

134

—

135

#106

—

134

—

136

#99

—

268

—

137

#41

—

138

#41

—

139

C₂×Q₈

#11761618

yes

characteristic

140

C₂×D₄

#11761615

yes

—

141

C₂×D₄

#11761618

yes

characteristic

142

C₂²×C₄

C₂²⋊Dic₆₇

yes

—

143

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

144

M₁₆

—

145

M₁₆

—

146

M₁₆

—

147

M₁₆

—

148

M₁₆

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

C₂²⋊Dic₆₇

yes

—

154

C₂×C₈

C₂²×Dic₆₇

yes

characteristic

155

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

156

C₂×C₈

#11761628

yes

—

157

C₂×C₈

#11761628

yes

—

158

C₄⋊C₄

—

268

—

159

C₂²⋊C₄

—

268

—

160

C₄²

—

268

—

161

C₁₆

—

268

—

162

C₁₆

—

268

—

163

C₂³

#14523967

yes

—

164

C₂³

C₂×#11761615

yes

characteristic

165

Q₈

—

166

D₄

—

167

D₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

536

—

171

C₂×C₄

#14523967

yes

—

172

C₂×C₄

C₂×C₂²⋊Dic₆₇

yes

characteristic

173

C₂×C₄

#14524026

yes

—

174

C₂×C₄

C₂×#11761618

yes

characteristic

175

C₂×C₄

#14524069

yes

—

176

C₈

—

177

C₈

—

178

C₈

#14524002

yes

—

179

C₈

#14524026

yes

characteristic

180

C₈

#14524026

yes

characteristic

181

C₂²

—

182

C₂²

—

183

C₂²

—

184

C₂²

C₂×#14523967

yes

characteristic

185

C₂²

#16863714

yes

—

186

C₄

—

187

C₄

—

1072

—

188

C₄

#16863259

yes

characteristic

189

C₄

#16863401

yes

characteristic

190

C₄

#16863731

yes

—

191

C₂

—

192

C₂

—

193

C₂

—

194

C₂

#22560299

yes

characteristic

	subgroup	G/H
center	C₄	#16863401
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#2916	C₂
socle	C₂×C₆₇	#1824
prime radical	C₂×#41×C₆₇	D₄
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#2916	D₆₇
hypocenter	C₆₇	#12877
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2916

subgroup

G/H

center

C₄

#16863401

commutator

C₂×C₈×C₆₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#2916

C₂

socle

C₂×C₆₇

#1824

prime radical

C₂×#41×C₆₇

D₄

cosocle

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2916

N	H ≅ G/N	type
3. #22558323	C₂	semidirect
4. #22558306	C₂	semidirect
15. #430×C₆₇	C₄	semidirect
103. C₆₇	#12877	semidirect
109. #430	Dic₆₇	semidirect

H ≅ G/N

type

3. #22558323

C₂

semidirect

4. #22558306

C₂

semidirect

15. #430×C₆₇

C₄

semidirect

103. C₆₇

#12877

semidirect

109. #430

Dic₆₇

semidirect

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

subgroup	index
#22561823	2
#22560255	2
#22558323	2
#22558306	2
C₆₇×#2916	2
#12877	67

subgroup

index

C₆₇×#2916

Subgroups — #30772169

Conjugacy classes of subgroups

Complements

Maximal subgroups