Group #30774314 — subgroups

Subgroups — #30774314

Order 17152.

462 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 135 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×#16863776	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863213	C₂	1	4	yes	maximal
3	C₄×#14523979	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₆₈	—	2	1	no	—
6	C₂²×Dic₂₆₈	C₂²	1	2	yes	—
7	C₂×#14524043	—	2	1	no	—
8	#16863776	—	2	2	no	—
9	#16863776	C₄	1	4	yes	—
10	C₂×#14523979	C₂²	1	2	yes	—
11	C₄×#11761612	—	2	1	no	—
12	#16863213	—	2	8	no	—
13	#16863161	—	2	1	no	—
14	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆×C₆₇	—	2	2	no	—
17	C₂²×Dic₁₃₄	—	2	2	no	—
18	C₂×Dic₂₆₈	—	2	4	no	—
19	C₂×Dic₂₆₈	—	2	4	no	—
20	C₂×Dic₂₆₈	—	2	2	no	—
21	C₂×Dic₂₆₈	—	2	4	no	—
22	C₂×Dic₂₆₈	—	4	4	no	—
23	C₂×Dic₂₆₈	C₂×C₄	1	4	yes	—
24	C₂×#11761612	—	2	2	no	—
25	#14524043	—	2	4	no	—
26	#14524043	—	4	1	no	—
27	#14524043	—	4	1	no	—
28	C₂×C₄⋊Dic₆₇	—	4	2	no	—
29	#14523979	—	2	2	no	—
30	#14523979	D₄	1	4	yes	—
31	#14523954	—	4	1	no	—
32	C₄×C₆₇⋊C₈	—	4	1	no	—
33	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
36	M₃₂×C₆₇	—	2	4	no	—
37	C₂×C₁₆×C₆₇	—	2	2	no	—
38	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₆₇	D₄	1	2	yes	—
40	C₂×Dic₁₃₄	—	2	4	no	—
41	C₂×Dic₁₃₄	—	4	4	no	—
42	C₂×Dic₁₃₄	—	4	4	no	—
43	C₂×Dic₁₃₄	—	4	2	no	—
44	C₂²×Dic₆₇	—	4	2	no	—
45	Dic₂₆₈	—	2	8	no	—
46	Dic₂₆₈	—	4	8	no	—
47	Dic₂₆₈	—	4	2	no	—
48	Dic₂₆₈	—	4	2	no	—
49	#11761612	—	2	4	no	—
50	#11761612	—	4	2	no	—
51	C₂×C₆₇⋊C₈	—	4	2	no	—
52	C₄⋊Dic₆₇	—	4	4	no	—
53	C₄⋊Dic₆₇	—	4	4	no	—
54	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
55	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₆₇	—	2	2	no	—
57	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₆₇	#41	1	2	yes	—
62	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₆₇	D₈	1	1	yes	characteristic
64	C₄²×C₆₇	D₈	1	1	yes	characteristic
65	C₁₆×C₆₇	—	4	2	no	—
66	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
67	C₂×C₄×C₆₇	—	2	4	no	—
68	C₂×C₄×C₆₇	#102	1	2	yes	—
69	C₂×C₄×C₆₇	#102	1	2	yes	—
70	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₆₇	#121	1	2	yes	—
72	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×Dic₆₇	—	4	4	no	—
75	C₂×Dic₆₇	—	8	4	no	—
76	Dic₁₃₄	—	4	4	no	—
77	Dic₁₃₄	—	4	4	no	—
78	Dic₁₃₄	—	8	4	no	—
79	C₈×C₆₇	—	2	2	no	—
80	C₈×C₆₇	C₄×D₄	1	2	yes	—
81	C₈×C₆₇	#124	1	1	yes	characteristic
82	C₈×C₆₇	#127	1	1	yes	characteristic
83	C₆₇⋊C₈	—	4	4	no	—
84	C₂²×C₆₇	—	2	2	no	—
85	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
86	C₂²×C₆₇	#460	1	2	yes	—
87	C₄×C₆₇	—	2	2	no	—
88	C₄×C₆₇	#362	1	4	yes	complemented
89	C₄×C₆₇	#390	1	1	yes	characteristic
90	C₄×C₆₇	#418	1	1	yes	characteristic
91	C₄×C₆₇	#459	1	2	yes	—
92	Dic₆₇	—	8	4	no	—
93	#13256	—	67	1	no	maximal, Sylow
94	C₂×C₆₇	—	2	2	no	—
95	C₂×C₆₇	#1982	1	1	yes	characteristic
96	C₂×C₆₇	C₂×#362	1	2	yes	—
97	C₂×#494	—	67	1	no	—
98	#2150	D₆₇	1	1	yes	characteristic
99	C₂×#362	—	67	4	no	—
100	C₄×#108	—	67	1	no	—
101	C₆₇	#13256	1	1	yes	characteristic, complemented, Sylow
102	C₂²×Q₁₆	—	67	2	no	—
103	C₂²×Q₁₆	—	134	1	no	—
104	C₂×C₄⋊Q₈	—	134	1	no	—
105	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
106	#494	—	67	4	no	—
107	#494	—	134	2	no	—
108	C₂×#108	—	67	2	no	—
109	C₄×M₁₆	—	134	1	no	—
110	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
111	C₄⋊C₁₆	—	2	2	no	—
112	#362	—	134	8	no	—
113	#341	—	134	1	no	—
114	C₂²×Q₈	—	134	2	no	—
115	C₂×Q₁₆	—	268	4	no	—
116	C₂×Q₁₆	—	67	4	no	—
117	C₂×Q₁₆	—	134	4	no	—
118	C₂×Q₁₆	—	134	4	no	—
119	C₂×Q₁₆	—	134	2	no	—
120	C₂×Q₁₆	—	134	4	no	—
121	C₂×M₁₆	—	134	2	no	—
122	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
124	C₄⋊Q₈	—	268	1	no	—
125	C₄⋊Q₈	—	268	1	no	—
126	C₄⋊Q₈	—	134	4	no	—
127	C₂×C₄⋊C₄	—	268	2	no	—
128	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
129	M₃₂	—	2	4	no	—
130	C₂×C₁₆	—	2	2	no	—
131	#108	—	67	4	no	—
132	#108	—	134	2	no	—
133	#97	—	268	1	no	—
134	C₄×C₈	—	268	1	no	—
135	C₄×C₈	C₄×D₆₇	1	2	yes	—
136	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
137	C₂×Q₈	—	268	2	no	—
138	C₂×Q₈	—	268	4	no	—
139	C₂×Q₈	—	268	4	no	—
140	C₂×Q₈	—	134	4	no	—
141	C₂²×C₄	—	268	2	no	—
142	C₂²×C₄	#11761617	1	2	yes	—
143	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
144	Q₁₆	—	268	8	no	—
145	Q₁₆	—	268	2	no	—
146	Q₁₆	—	268	2	no	—
147	Q₁₆	—	134	8	no	—
148	M₁₆	—	268	2	no	—
149	M₁₆	—	134	4	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	268	2	no	—
152	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
153	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
154	C₂×C₈	D₄×D₆₇	1	2	yes	—
155	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
156	C₂×C₈	#11761629	1	2	yes	—
157	C₄⋊C₄	—	268	4	no	—
158	C₄⋊C₄	—	268	4	no	—
159	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
160	C₄²	D₅₃₆	1	1	yes	characteristic
161	C₄²	D₅₃₆	1	1	yes	characteristic
162	C₁₆	—	4	2	no	—
163	C₂³	C₂×#11761617	1	1	yes	characteristic
164	Q₈	—	268	4	no	—
165	Q₈	—	268	4	no	—
166	Q₈	—	536	4	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	268	4	no	—
169	C₂×C₄	—	536	4	no	—
170	C₂×C₄	#14523965	1	2	yes	—
171	C₂×C₄	#14523965	1	2	yes	—
172	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
173	C₂×C₄	#14524018	1	2	yes	—
174	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
175	C₂×C₄	#14524071	1	2	yes	—
176	C₈	—	2	2	no	—
177	C₈	—	268	4	no	—
178	C₈	#14524005	1	2	yes	—
179	C₈	#14524032	1	1	yes	characteristic
180	C₈	#14524039	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	C₂×#14523965	1	1	yes	characteristic
183	C₂²	#16863597	1	2	yes	—
184	C₄	—	2	2	no	—
185	C₄	—	536	4	no	—
186	C₄	#16863213	1	4	yes	complemented
187	C₄	#16863297	1	1	yes	characteristic
188	C₄	#16863402	1	1	yes	characteristic
189	C₄	#16863592	1	2	yes	—
190	C₂	—	2	2	no	—
191	C₂	#22561135	1	1	yes	characteristic
192	C₂	C₂×#16863213	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863776

C₂

yes

maximal, characteristic

C₂×#16863213

C₂

yes

maximal

C₄×#14523979

C₂

yes

maximal, characteristic

C₆₇×#2150

C₂

yes

maximal, characteristic

C₂²×Dic₂₆₈

—

C₂²×Dic₂₆₈

C₂²

yes

—

C₂×#14524043

—

#16863776

—

#16863776

C₄

yes

—

C₂×#14523979

C₂²

yes

—

C₄×#11761612

—

#16863213

—

#16863161

—

C₂×M₃₂×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₁₆×C₆₇

—

C₂²×Dic₁₃₄

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

C₂×C₄

yes

—

—

—

—

—

C₂×C₄⋊Dic₆₇

—

#14523979

—

#14523979

D₄

yes

—

#14523954

—

C₄×C₆₇⋊C₈

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

characteristic

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

Dic₂₆₈

—

Dic₂₆₈

—

Dic₂₆₈

—

Dic₂₆₈

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

SD₁₆

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₄²×C₆₇

C₂²⋊C₄

yes

—

C₄²×C₆₇

D₈

yes

characteristic

C₄²×C₆₇

D₈

yes

characteristic

C₁₆×C₆₇

—

C₂³×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#102

yes

—

C₂×C₄×C₆₇

#102

yes

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

C₄×D₄

yes

—

C₈×C₆₇

yes

C₈×C₆₇

yes

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×#102

yes

characteristic

C₂²×C₆₇

#460

yes

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

#13256

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1982

yes

characteristic

C₂×C₆₇

C₂×#362

yes

—

C₂×#494

—

#2150

D₆₇

yes

characteristic

C₂×#362

—

100

C₄×#108

—

101

C₆₇

#13256

yes

characteristic, complemented, Sylow

102

C₂²×Q₁₆

—

103

C₂²×Q₁₆

—

134

—

104

C₂×C₄⋊Q₈

—

134

—

105

C₂×M₃₂

C₂×D₆₇

yes

—

106

#494

—

107

#494

—

134

—

108

C₂×#108

—

109

C₄×M₁₆

—

134

—

110

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

111

C₄⋊C₁₆

—

112

#362

—

134

—

113

#341

—

134

—

114

C₂²×Q₈

—

134

—

115

C₂×Q₁₆

—

268

—

116

C₂×Q₁₆

—

117

C₂×Q₁₆

—

134

—

118

C₂×Q₁₆

—

134

—

119

C₂×Q₁₆

—

134

—

120

C₂×Q₁₆

—

134

—

121

C₂×M₁₆

—

134

—

122

C₂²×C₈

D₂₆₈

yes

characteristic

123

C₂²×C₈

C₂²×D₆₇

yes

characteristic

124

C₄⋊Q₈

—

268

—

125

C₄⋊Q₈

—

268

—

126

C₄⋊Q₈

—

134

—

127

C₂×C₄⋊C₄

—

268

—

128

C₂×C₄²

D₂₆₈

yes

characteristic

129

M₃₂

—

130

C₂×C₁₆

—

131

#108

—

132

#108

—

134

—

133

#97

—

268

—

134

C₄×C₈

—

268

—

135

C₄×C₈

C₄×D₆₇

yes

—

136

C₄×C₈

C₂²⋊D₆₇

yes

—

137

C₂×Q₈

—

268

—

138

C₂×Q₈

—

268

—

139

C₂×Q₈

—

268

—

140

C₂×Q₈

—

134

—

141

C₂²×C₄

—

268

—

142

C₂²×C₄

#11761617

yes

—

143

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

144

Q₁₆

—

268

—

145

Q₁₆

—

268

—

146

Q₁₆

—

268

—

147

Q₁₆

—

134

—

148

M₁₆

—

268

—

149

M₁₆

—

134

—

150

C₂×C₈

—

151

C₂×C₈

—

268

—

152

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

153

C₂×C₈

C₂×C₄×D₆₇

yes

characteristic

154

C₂×C₈

D₄×D₆₇

yes

—

155

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

156

C₂×C₈

#11761629

yes

—

157

C₄⋊C₄

—

268

—

158

C₄⋊C₄

—

268

—

159

C₄²

(C₂×D₆₇)⋊C₄

yes

—

160

C₄²

D₅₃₆

yes

characteristic

161

C₄²

D₅₃₆

yes

characteristic

162

C₁₆

—

163

C₂³

C₂×#11761617

yes

characteristic

164

Q₈

—

268

—

165

Q₈

—

268

—

166

Q₈

—

536

—

167

C₂×C₄

—

168

C₂×C₄

—

268

—

169

C₂×C₄

—

536

—

170

C₂×C₄

#14523965

yes

—

171

C₂×C₄

#14523965

yes

—

172

C₂×C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

173

C₂×C₄

#14524018

yes

—

174

C₂×C₄

C₂×D₅₃₆

yes

characteristic

175

C₂×C₄

#14524071

yes

—

176

C₈

—

177

C₈

—

268

—

178

C₈

#14524005

yes

—

179

C₈

#14524032

yes

characteristic

180

C₈

#14524039

yes

characteristic

181

C₂²

—

182

C₂²

C₂×#14523965

yes

characteristic

183

C₂²

#16863597

yes

—

184

C₄

—

185

C₄

—

536

—

186

C₄

#16863213

yes

complemented

187

C₄

#16863297

yes

characteristic

188

C₄

#16863402

yes

characteristic

189

C₄

#16863592

yes

—

190

C₂

—

191

C₂

#22561135

yes

characteristic

192

C₂

C₂×#16863213

yes

—

	subgroup	G/H
center	C₂²	C₂×#14523965
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#2150	C₂
socle	C₂²×C₆₇	C₂×#102
prime radical	C₂³×C₆₇	C₂×SD₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#2150	D₆₇
hypocenter	C₆₇	#13256
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂²

C₂×#14523965

commutator

C₂×C₈×C₆₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#2150

C₂

socle

C₂²×C₆₇

C₂×#102

prime radical

C₂³×C₆₇

C₂×SD₁₆

cosocle

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
88. C₄×C₆₇	#362	semidirect
101. C₆₇	#13256	semidirect
186. C₄	#16863213	semidirect

H ≅ G/N

type

88. C₄×C₆₇

#362

semidirect

101. C₆₇

#13256

semidirect

186. C₄

#16863213

semidirect

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

subgroup	index
C₂×#16863776	2
C₂×#16863213	2
C₄×#14523979	2
C₆₇×#2150	2
#13256	67

subgroup

index

C₆₇×#2150

Subgroups — #30774314

Conjugacy classes of subgroups

Complements

Maximal subgroups