Group #30772022 — subgroups

Subgroups — #30772022

Order 17152.

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

normal subgroups only

This group has 354 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	C₂×#16863813	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863225	C₂	1	2	yes	maximal, complemented
3	#22560259	C₂	1	1	yes	maximal, characteristic
4	#22558312	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1813	C₂	1	1	yes	maximal, characteristic
6	#16863813	—	2	4	no	—
7	C₂×#14523983	C₄	1	2	yes	—
8	C₂×#14523983	C₂²	1	1	yes	characteristic
9	C₂²×C₆₇⋊C₁₆	C₄	1	2	yes	—
10	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂×#14523979	C₂²	1	2	yes	—
12	#16863380	—	2	1	no	—
13	#16863225	—	2	4	no	—
14	C₆₇⋊#338	—	2	1	no	—
15	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#108×C₆₇	C₂²	1	2	yes	—
17	#413×C₆₇	—	2	1	no	—
18	#338×C₆₇	—	2	1	no	—
19	C₂×#11761612	—	2	2	no	—
20	#14523983	—	2	4	no	—
21	#14523983	—	2	2	no	—
22	C₂×C₆₇⋊C₁₆	—	2	4	no	—
23	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₆₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂×C₆₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₆₇⋊C₁₆	D₄	1	4	yes	—
27	#14523979	—	2	2	no	—
28	#14523979	—	2	2	no	—
29	#14523962	—	4	1	no	—
30	#14523960	—	4	1	no	—
31	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
32	#131×C₆₇	—	2	2	no	—
33	#131×C₆₇	—	2	2	no	—
34	C₂×M₁₆×C₆₇	—	2	2	no	—
35	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
36	C₂×M₁₆×C₆₇	Q₈	1	2	yes	—
37	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
39	#108×C₆₇	—	2	2	no	—
40	#108×C₆₇	—	2	2	no	—
41	#101×C₆₇	—	4	1	no	—
42	#100×C₆₇	—	4	1	no	—
43	#11761612	—	4	2	no	—
44	#11761612	—	4	2	no	—
45	C₂×C₆₇⋊C₈	—	4	2	no	—
46	C₆₇⋊C₁₆	—	2	2	no	—
47	C₆₇⋊C₁₆	C₂²⋊C₄	1	2	yes	—
48	C₆₇⋊C₁₆	C₂²⋊C₄	1	2	yes	—
49	#41×C₆₇	—	2	2	no	—
50	#41×C₆₇	—	2	2	no	—
51	C₂×Q₈×C₆₇	D₈	1	1	yes	characteristic
52	C₂×D₄×C₆₇	D₈	1	1	yes	characteristic
53	C₂×D₄×C₆₇	Q₁₆	1	2	yes	—
54	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
56	M₁₆×C₆₇	—	2	2	no	—
57	M₁₆×C₆₇	—	2	4	no	—
58	M₁₆×C₆₇	—	4	2	no	—
59	M₁₆×C₆₇	—	4	2	no	—
60	C₂×C₈×C₆₇	—	2	2	no	—
61	C₂×C₈×C₆₇	—	2	2	no	—
62	C₂×C₈×C₆₇	—	4	2	no	—
63	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
64	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₆₇	#41	1	2	yes	—
68	C₂³×C₆₇	#107	1	2	yes	—
69	C₂³×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
70	Q₈×C₆₇	—	2	2	no	—
71	D₄×C₆₇	—	2	4	no	—
72	D₄×C₆₇	—	2	2	no	—
73	C₂×C₄×C₆₇	—	2	2	no	—
74	C₂×C₄×C₆₇	—	2	2	no	—
75	C₂×C₄×C₆₇	#107	1	2	yes	—
76	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₆₇	#121	1	2	yes	—
78	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₆₇	#135	1	2	yes	—
80	C₈×C₆₇	—	2	2	no	—
81	C₈×C₆₇	—	8	2	no	—
82	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₈×C₆₇	#117	1	1	yes	characteristic
84	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
85	C₆₇⋊C₈	—	8	2	no	—
86	C₂²×C₆₇	—	2	2	no	—
87	C₂²×C₆₇	—	4	1	no	—
88	C₂²×C₆₇	—	4	2	no	—
89	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
90	C₂²×C₆₇	#467	1	2	yes	—
91	C₄×C₆₇	—	2	2	no	—
92	C₄×C₆₇	#380	1	1	yes	characteristic
93	C₄×C₆₇	#427	1	1	yes	characteristic
94	C₄×C₆₇	#466	1	2	yes	—
95	#12840	—	67	1	no	maximal, Sylow
96	C₂×C₆₇	—	2	1	no	—
97	C₂×C₆₇	—	2	2	no	—
98	C₂×C₆₇	—	4	2	no	—
99	C₂×C₆₇	#1808	1	1	yes	characteristic
100	C₂×#504	—	67	1	no	—
101	C₂×#368	—	67	2	no	—
102	#1813	—	67	1	no	—
103	#1813	D₆₇	1	1	yes	characteristic
104	#1385	—	67	2	no	—
105	C₆₇	#12840	1	1	yes	characteristic, complemented, Sylow
106	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
107	#504	—	134	4	no	—
108	C₂×M₃₂	—	67	2	no	—
109	C₂×M₃₂	—	67	1	no	—
110	C₂²×C₁₆	—	67	1	no	—
111	C₂²×C₁₆	—	67	2	no	—
112	C₂×#108	—	67	2	no	—
113	C₂×#108	C₂×D₆₇	1	2	yes	—
114	#413	—	2	1	no	—
115	#413	—	134	1	no	—
116	#368	—	134	4	no	—
117	#338	—	2	1	no	—
118	#338	—	134	1	no	—
119	C₂×#41	C₂²⋊D₆₇	1	1	yes	characteristic
120	#131	—	2	2	no	—
121	#131	—	2	2	no	—
122	C₂×M₁₆	—	2	2	no	—
123	C₂×M₁₆	—	134	2	no	—
124	C₂×M₁₆	Dic₁₃₄	1	2	yes	—
125	C₂×M₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
126	C₂²×C₈	C₄×D₆₇	1	2	yes	—
127	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
128	M₃₂	—	134	4	no	—
129	M₃₂	—	134	2	no	—
130	C₂×C₁₆	—	67	1	no	—
131	C₂×C₁₆	—	67	1	no	—
132	C₂×C₁₆	—	67	4	no	—
133	C₂×C₁₆	—	67	2	no	—
134	C₂×C₁₆	—	134	4	no	—
135	#108	—	2	2	no	—
136	#108	—	2	2	no	—
137	#108	—	134	2	no	—
138	#108	—	134	2	no	—
139	#101	—	4	1	no	—
140	#101	—	268	1	no	—
141	#100	—	4	1	no	—
142	#100	—	268	1	no	—
143	#41	—	2	2	no	—
144	#41	—	2	2	no	—
145	C₂×Q₈	#11761613	1	1	yes	characteristic
146	C₂×D₄	#11761613	1	1	yes	characteristic
147	C₂×D₄	#11761618	1	2	yes	—
148	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
149	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
150	M₁₆	—	2	2	no	—
151	M₁₆	—	2	4	no	—
152	M₁₆	—	4	2	no	—
153	M₁₆	—	4	2	no	—
154	M₁₆	—	268	2	no	—
155	M₁₆	—	268	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	—	268	2	no	—
160	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
161	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
162	C₂×C₈	D₄×D₆₇	1	2	yes	—
163	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
164	C₂×C₈	#11761627	1	2	yes	—
165	C₁₆	—	67	2	no	—
166	C₁₆	—	67	2	no	—
167	C₁₆	—	134	2	no	—
168	C₂³	#14523976	1	2	yes	—
169	C₂³	C₂×#11761618	1	1	yes	characteristic
170	Q₈	—	2	2	no	—
171	D₄	—	2	4	no	—
172	D₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	#14523976	1	2	yes	—
176	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
177	C₂×C₄	#14524015	1	2	yes	—
178	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
179	C₂×C₄	#14524063	1	2	yes	—
180	C₈	—	2	2	no	—
181	C₈	—	8	2	no	—
182	C₈	—	536	2	no	—
183	C₈	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
184	C₈	#14523999	1	1	yes	characteristic
185	C₈	#14524020	1	2	yes	—
186	C₂²	—	2	2	no	—
187	C₂²	—	4	1	no	—
188	C₂²	—	4	2	no	—
189	C₂²	C₂×#14523976	1	1	yes	characteristic
190	C₂²	#16863635	1	2	yes	—
191	C₄	—	2	2	no	—
192	C₄	#16863254	1	1	yes	characteristic
193	C₄	#16863432	1	1	yes	characteristic
194	C₄	#16863630	1	2	yes	—
195	C₂	—	2	1	no	—
196	C₂	—	2	2	no	—
197	C₂	—	4	2	no	—
198	C₂	#22560236	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863813

C₂

yes

maximal, characteristic

C₂×#16863225

C₂

yes

maximal, complemented

#22560259

C₂

yes

maximal, characteristic

#22558312

C₂

yes

maximal, complemented

C₆₇×#1813

C₂

yes

maximal, characteristic

#16863813

—

C₂×#14523983

C₄

yes

—

C₂×#14523983

C₂²

yes

characteristic

C₂²×C₆₇⋊C₁₆

C₄

yes

—

C₂²×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₂×#14523979

C₂²

yes

—

#16863380

—

#16863225

—

C₆₇⋊#338

—

C₂×#131×C₆₇

C₂²

yes

characteristic

C₂×#108×C₆₇

C₂²

yes

—

#413×C₆₇

—

#338×C₆₇

—

C₂×#11761612

—

#14523983

—

#14523983

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

—

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₁₆

D₄

yes

—

—

—

—

—

C₂×#41×C₆₇

D₄

yes

characteristic

#131×C₆₇

—

#131×C₆₇

—

C₂×M₁₆×C₆₇

—

C₂×M₁₆×C₆₇

D₄

yes

characteristic

C₂×M₁₆×C₆₇

Q₈

yes

—

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

#108×C₆₇

—

#108×C₆₇

—

#101×C₆₇

—

#100×C₆₇

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₆₇⋊C₁₆

—

C₆₇⋊C₁₆

C₂²⋊C₄

yes

—

C₆₇⋊C₁₆

C₂²⋊C₄

yes

—

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

D₈

yes

characteristic

C₂×D₄×C₆₇

D₈

yes

characteristic

C₂×D₄×C₆₇

Q₁₆

yes

—

C₂²×C₄×C₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₂³×C₆₇

#107

yes

—

C₂³×C₆₇

C₂×Q₁₆

yes

characteristic

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#107

yes

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

#135

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₆₇

#117

yes

characteristic

C₈×C₆₇

C₂²⋊Q₈

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×#107

yes

characteristic

C₂²×C₆₇

#467

yes

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

#12840

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#1808

yes

characteristic

100

C₂×#504

—

101

C₂×#368

—

102

#1813

—

103

#1813

D₆₇

yes

characteristic

104

#1385

—

105

C₆₇

#12840

yes

characteristic, complemented, Sylow

106

C₂×#131

C₂×D₆₇

yes

characteristic

107

#504

—

134

—

108

C₂×M₃₂

—

109

C₂×M₃₂

—

110

C₂²×C₁₆

—

111

C₂²×C₁₆

—

112

C₂×#108

—

113

C₂×#108

C₂×D₆₇

yes

—

114

#413

—

115

#413

—

134

—

116

#368

—

134

—

117

#338

—

118

#338

—

134

—

119

C₂×#41

C₂²⋊D₆₇

yes

characteristic

120

#131

—

121

#131

—

122

C₂×M₁₆

—

123

C₂×M₁₆

—

134

—

124

C₂×M₁₆

Dic₁₃₄

yes

—

125

C₂×M₁₆

C₂²⋊D₆₇

yes

characteristic

126

C₂²×C₈

C₄×D₆₇

yes

—

127

C₂²×C₈

C₂²×D₆₇

yes

characteristic

128

M₃₂

—

134

—

129

M₃₂

—

134

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

132

C₂×C₁₆

—

133

C₂×C₁₆

—

134

C₂×C₁₆

—

134

—

135

#108

—

136

#108

—

137

#108

—

134

—

138

#108

—

134

—

139

#101

—

140

#101

—

268

—

141

#100

—

142

#100

—

268

—

143

#41

—

144

#41

—

145

C₂×Q₈

#11761613

yes

characteristic

146

C₂×D₄

#11761613

yes

characteristic

147

C₂×D₄

#11761618

yes

—

148

C₂²×C₄

Dic₆₇⋊C₄

yes

—

149

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

150

M₁₆

—

151

M₁₆

—

152

M₁₆

—

153

M₁₆

—

154

M₁₆

—

268

—

155

M₁₆

—

268

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

268

—

160

C₂×C₈

Dic₆₇⋊C₄

yes

—

161

C₂×C₈

C₂×C₄×D₆₇

yes

characteristic

162

C₂×C₈

D₄×D₆₇

yes

—

163

C₂×C₈

C₂×Dic₁₃₄

yes

characteristic

164

C₂×C₈

#11761627

yes

—

165

C₁₆

—

166

C₁₆

—

167

C₁₆

—

134

—

168

C₂³

#14523976

yes

—

169

C₂³

C₂×#11761618

yes

characteristic

170

Q₈

—

171

D₄

—

172

D₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

#14523976

yes

—

176

C₂×C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

177

C₂×C₄

#14524015

yes

—

178

C₂×C₄

C₂×#11761613

yes

characteristic

179

C₂×C₄

#14524063

yes

—

180

C₈

—

181

C₈

—

182

C₈

—

536

—

183

C₈

C₂²⋊C₄×D₆₇

yes

characteristic

184

C₈

#14523999

yes

characteristic

185

C₈

#14524020

yes

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

C₂×#14523976

yes

characteristic

190

C₂²

#16863635

yes

—

191

C₄

—

192

C₄

#16863254

yes

characteristic

193

C₄

#16863432

yes

characteristic

194

C₄

#16863630

yes

—

195

C₂

—

196

C₂

—

197

C₂

—

198

C₂

#22560236

yes

characteristic

	subgroup	G/H
center	C₄	#16863432
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1813	C₂
socle	C₂×C₆₇	#1808
prime radical	C₂×#41×C₆₇	D₄
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1813	D₆₇
hypocenter	C₆₇	#12840
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ #1813

subgroup

G/H

center

C₄

#16863432

commutator

C₂×C₈×C₆₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#1813

C₂

socle

C₂×C₆₇

#1808

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 ◁ C₂×M₁₆ ◁ C₂×#131 ◁ #1813

N	H ≅ G/N	type
2. C₂×#16863225	C₂	semidirect
4. #22558312	C₂	semidirect
105. C₆₇	#12840	semidirect

H ≅ G/N

type

2. C₂×#16863225

C₂

semidirect

4. #22558312

C₂

semidirect

105. C₆₇

#12840

semidirect

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

subgroup	index
C₂×#16863813	2
C₂×#16863225	2
#22560259	2
#22558312	2
C₆₇×#1813	2
#12840	67

subgroup

index

C₆₇×#1813

Subgroups — #30772022

Conjugacy classes of subgroups

Complements

Maximal subgroups