Group #30749593 — subgroups

Subgroups — #30749593

Order 17152.

382 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 163 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	#22559955	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863148	C₂	1	2	yes	maximal
4	C₂×#16863146	C₂	1	2	yes	maximal
5	C₆₇×#1842	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
9	#16863151	—	2	2	no	—
10	#16863151	C₄	1	2	yes	—
11	#16863151	C₄	1	2	yes	—
12	#16863148	—	2	4	no	—
13	#16863146	—	2	4	no	—
14	C₂×C₄⋊C₈×C₆₇	—	2	2	no	—
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
16	C₄³×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×C₆₇⋊C₈	—	2	1	no	—
18	C₂²×C₆₇⋊C₈	—	2	1	no	—
19	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
20	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
21	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
22	C₄×C₆₇⋊C₈	D₄	1	4	yes	—
23	C₄×C₆₇⋊C₈	Q₈	1	4	yes	—
24	C₂²×C₈×C₆₇	—	2	2	no	—
25	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₆₇	D₄	1	2	yes	—
28	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
29	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₆₇	—	2	2	no	—
31	C₄⋊C₈×C₆₇	—	2	2	no	—
32	C₄⋊C₈×C₆₇	—	4	4	no	—
33	C₂×C₆₇⋊C₈	—	2	4	no	—
34	C₂×C₆₇⋊C₈	—	2	4	no	—
35	C₂×C₆₇⋊C₈	—	4	1	no	—
36	C₂×C₆₇⋊C₈	—	4	1	no	—
37	C₂×C₆₇⋊C₈	—	4	2	no	—
38	C₂×C₆₇⋊C₈	—	4	2	no	—
39	C₂×C₆₇⋊C₈	C₄⋊C₄	1	8	yes	—
40	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
41	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₆₇	—	4	2	no	—
46	C₂×C₈×C₆₇	—	4	4	no	—
47	C₄²×C₆₇	—	2	4	no	—
48	C₄²×C₆₇	—	2	2	no	—
49	C₄²×C₆₇	—	2	2	no	—
50	C₄²×C₆₇	—	2	2	no	—
51	C₄²×C₆₇	—	2	2	no	—
52	C₄²×C₆₇	C₂×D₄	1	2	yes	—
53	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
54	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₆₇	—	2	2	no	—
56	C₂×C₄×C₆₇	—	2	2	no	—
57	C₂×C₄×C₆₇	—	2	2	no	—
58	C₂×C₄×C₆₇	—	2	2	no	—
59	C₂×C₄×C₆₇	—	2	2	no	—
60	C₂×C₄×C₆₇	—	2	2	no	—
61	C₂×C₄×C₆₇	#108	1	4	yes	—
62	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
63	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
66	C₂×C₄×C₆₇	#131	1	2	yes	—
67	C₂×C₄×C₆₇	#131	1	2	yes	—
68	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
69	C₂×C₄×C₆₇	#137	1	2	yes	—
70	C₈×C₆₇	—	8	2	no	—
71	C₆₇⋊C₈	—	2	8	no	—
72	C₆₇⋊C₈	—	8	1	no	—
73	C₆₇⋊C₈	—	8	1	no	—
74	C₂²×C₆₇	#384	1	1	yes	characteristic
75	C₂²×C₆₇	#424	1	1	yes	characteristic
76	C₂²×C₆₇	#424	1	1	yes	characteristic
77	C₂²×C₆₇	#428	1	2	yes	—
78	C₂²×C₆₇	C₂×#108	1	2	yes	—
79	C₄×C₆₇	—	2	4	no	—
80	C₄×C₆₇	—	2	2	no	—
81	C₄×C₆₇	—	4	2	no	—
82	C₄×C₆₇	#424	1	4	yes	—
83	C₄×C₆₇	#496	1	1	yes	characteristic
84	C₄×C₆₇	#497	1	1	yes	characteristic
85	C₄×C₆₇	#501	1	2	yes	—
86	#8812	—	67	1	no	maximal, Sylow
87	C₂×C₆₇	#1573	1	2	yes	—
88	C₂×C₆₇	#1574	1	2	yes	—
89	C₂×C₆₇	#1843	1	1	yes	characteristic
90	C₂×C₆₇	#1847	1	1	yes	characteristic
91	C₂×C₆₇	#1848	1	1	yes	characteristic
92	#1842	D₆₇	1	1	yes	characteristic
93	#1831	—	67	1	no	—
94	#1744	—	67	1	no	—
95	C₂×#335	—	67	2	no	—
96	C₂×#334	—	67	2	no	—
97	C₆₇	#8812	1	1	yes	characteristic, complemented, Sylow
98	C₂×C₄⋊C₈	—	2	2	no	—
99	C₂×C₄⋊C₈	—	67	1	no	—
100	C₂×C₄⋊C₈	—	67	1	no	—
101	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
102	C₂×C₄×C₈	—	67	2	no	—
103	C₄³	C₂×D₆₇	1	1	yes	characteristic
104	#336	—	67	2	no	—
105	#336	—	67	2	no	—
106	#336	—	134	2	no	—
107	#335	—	134	4	no	—
108	#334	—	134	4	no	—
109	C₂²×C₈	—	2	2	no	—
110	C₂²×C₈	—	67	4	no	—
111	C₂²×C₈	—	134	1	no	—
112	C₂²×C₈	—	134	1	no	—
113	C₂×C₄²	Dic₁₃₄	1	2	yes	—
114	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
115	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
116	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
117	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	4	4	no	—
121	C₄⋊C₈	—	134	2	no	—
122	C₄⋊C₈	—	134	2	no	—
123	C₄×C₈	—	67	4	no	—
124	C₄×C₈	—	67	4	no	—
125	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
126	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
127	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
128	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
129	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
130	C₂×C₈	—	4	2	no	—
131	C₂×C₈	—	4	4	no	—
132	C₂×C₈	—	268	1	no	—
133	C₂×C₈	—	268	1	no	—
134	C₂×C₈	—	268	2	no	—
135	C₂×C₈	—	268	2	no	—
136	C₂×C₈	—	67	8	no	—
137	C₂×C₈	—	134	4	no	—
138	C₂×C₈	—	134	4	no	—
139	C₄²	—	2	4	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	2	no	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	2	no	—
144	C₄²	D₄×D₆₇	1	2	yes	—
145	C₄²	Q₈×D₆₇	1	2	yes	—
146	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	#14523978	1	4	yes	—
154	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
155	C₂×C₄	#14524039	1	1	yes	characteristic
156	C₂×C₄	#14524041	1	1	yes	characteristic
157	C₂×C₄	#14524042	1	2	yes	—
158	C₂×C₄	#14524049	1	2	yes	—
159	C₂×C₄	#14524051	1	2	yes	—
160	C₂×C₄	#14524069	1	2	yes	—
161	C₂×C₄	#14524074	1	2	yes	—
162	C₈	—	8	2	no	—
163	C₈	—	536	1	no	—
164	C₈	—	536	1	no	—
165	C₈	—	134	8	no	—
166	C₂²	#16863265	1	1	yes	characteristic
167	C₂²	#16863422	1	1	yes	characteristic
168	C₂²	#16863424	1	1	yes	characteristic
169	C₂²	#16863436	1	2	yes	—
170	C₂²	C₂×#14523978	1	2	yes	—
171	C₄	—	2	4	no	—
172	C₄	—	2	2	no	—
173	C₄	—	4	2	no	—
174	C₄	#16863421	1	4	yes	—
175	C₄	#16863783	1	1	yes	characteristic
176	C₄	#16863788	1	1	yes	characteristic
177	C₄	#16863802	1	2	yes	—
178	C₂	#22558992	1	2	yes	—
179	C₂	#22559000	1	2	yes	—
180	C₂	#22560362	1	1	yes	characteristic
181	C₂	#22560375	1	1	yes	characteristic
182	C₂	#22560380	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22560326

C₂

yes

maximal, characteristic

#22559955

C₂

yes

maximal, characteristic

C₂×#16863148

C₂

yes

maximal

C₂×#16863146

C₂

yes

maximal

C₆₇×#1842

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

#16863151

—

#16863151

C₄

yes

—

#16863151

C₄

yes

—

#16863148

—

#16863146

—

C₂×C₄⋊C₈×C₆₇

—

C₂×C₄⋊C₈×C₆₇

C₂²

yes

—

C₄³×C₆₇

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

—

C₂²×C₆₇⋊C₈

—

C₂²×C₆₇⋊C₈

C₂×C₄

yes

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₄×C₆₇⋊C₈

Q₈

yes

—

C₂²×C₈×C₆₇

—

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

Q₈

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×Q₈

yes

—

C₂³×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#108

yes

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

—

C₂×C₄×C₆₇

#131

yes

—

C₂×C₄×C₆₇

#131

yes

—

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₆₇

#137

yes

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₂²×C₆₇

C₂×#108

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#424

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

#8812

—

maximal, Sylow

C₂×C₆₇

#1573

yes

—

C₂×C₆₇

#1574

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

yes

—

—

—

—

C₆₇

#8812

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

100

C₂×C₄⋊C₈

—

101

C₂×C₄⋊C₈

C₂×D₆₇

yes

—

102

C₂×C₄×C₈

—

103

C₄³

C₂×D₆₇

yes

characteristic

104

#336

—

105

#336

—

106

#336

—

134

—

107

#335

—

134

—

108

#334

—

134

—

109

C₂²×C₈

—

110

C₂²×C₈

—

111

C₂²×C₈

—

134

—

112

C₂²×C₈

—

134

—

113

C₂×C₄²

Dic₁₃₄

yes

—

114

C₂×C₄²

C₄×D₆₇

yes

characteristic

115

C₂×C₄²

C₄×D₆₇

yes

characteristic

116

C₂×C₄²

C₂²⋊D₆₇

yes

—

117

C₂×C₄²

C₂²×D₆₇

yes

characteristic

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

121

C₄⋊C₈

—

134

—

122

C₄⋊C₈

—

134

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₂²×C₄

Dic₆₇⋊C₄

yes

—

126

C₂²×C₄

Dic₆₇⋊C₄

yes

—

127

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

128

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

129

C₂²×C₄

C₂×Dic₁₃₄

yes

characteristic

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

268

—

133

C₂×C₈

—

268

—

134

C₂×C₈

—

268

—

135

C₂×C₈

—

268

—

136

C₂×C₈

—

137

C₂×C₈

—

134

—

138

C₂×C₈

—

134

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

D₄×D₆₇

yes

—

145

C₄²

Q₈×D₆₇

yes

—

146

C₂³

C₂×Dic₆₇⋊C₄

yes

characteristic

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

#14523978

yes

—

154

C₂×C₄

C₄⋊C₄×D₆₇

yes

—

155

C₂×C₄

#14524039

yes

characteristic

156

C₂×C₄

#14524041

yes

characteristic

157

C₂×C₄

#14524042

yes

—

158

C₂×C₄

#14524049

yes

—

159

C₂×C₄

#14524051

yes

—

160

C₂×C₄

#14524069

yes

—

161

C₂×C₄

#14524074

yes

—

162

C₈

—

163

C₈

—

536

—

164

C₈

—

536

—

165

C₈

—

134

—

166

C₂²

#16863265

yes

characteristic

167

C₂²

#16863422

yes

characteristic

168

C₂²

#16863424

yes

characteristic

169

C₂²

#16863436

yes

—

170

C₂²

C₂×#14523978

yes

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

#16863421

yes

—

175

C₄

#16863783

yes

characteristic

176

C₄

#16863788

yes

characteristic

177

C₄

#16863802

yes

—

178

C₂

#22558992

yes

—

179

C₂

#22559000

yes

—

180

C₂

#22560362

yes

characteristic

181

C₂

#22560375

yes

characteristic

182

C₂

#22560380

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1842	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1842	D₆₇
hypocenter	C₆₇	#8812
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₆₇

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1842

C₂

socle, prime radical

C₂³×C₆₇

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

N	H ≅ G/N	type
97. C₆₇	#8812	semidirect

H ≅ G/N

type

97. C₆₇

#8812

semidirect

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

subgroup	index
#22560326	2
#22559955	2
C₂×#16863148	2
C₂×#16863146	2
C₆₇×#1842	2
#8812	67

subgroup

index

C₆₇×#1842

Subgroups — #30749593

Conjugacy classes of subgroups

Complements

Maximal subgroups