Group #30775794 — subgroups

Subgroups — #30775794

Order 17152.

270 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 97 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×#16863220	C₂	1	2	yes	maximal
2	#22560952	C₂	1	1	yes	maximal, characteristic
3	#22558319	C₂	1	2	yes	maximal
4	#22558293	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1849	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₂²	1	2	yes	—
7	C₂×#14523977	C₂²	1	1	yes	characteristic
8	C₂×#14523975	C₂²	1	1	yes	characteristic
9	#16863294	—	2	1	no	—
10	#16863220	—	2	4	no	—
11	#16863159	—	2	1	no	—
12	C₂×#108×C₆₇	—	2	1	no	—
13	C₂×#108×C₆₇	C₂²	1	2	yes	—
14	#423×C₆₇	—	2	1	no	—
15	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₆₇	—	2	1	no	—
17	C₂×C₄⋊Dic₆₇	—	2	1	no	—
18	C₂×Dic₆₇⋊C₄	—	4	1	no	—
19	#14523983	—	2	2	no	—
20	#14523983	Q₈	1	4	yes	—
21	C₂×C₆₇⋊C₁₆	—	2	2	no	—
22	#14523977	—	2	2	no	—
23	#14523975	—	2	1	no	—
24	#14523975	D₄	1	2	yes	—
25	C₆₇⋊#95	—	4	1	no	—
26	C₂×M₁₆×C₆₇	—	2	2	no	—
27	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
30	#108×C₆₇	—	2	2	no	—
31	#108×C₆₇	—	2	2	no	—
32	#108×C₆₇	—	4	2	no	—
33	C₄⋊C₈×C₆₇	—	4	2	no	—
34	#97×C₆₇	—	2	2	no	—
35	C₂²×Dic₆₇	—	4	1	no	—
36	C₂²×Dic₆₇	—	4	1	no	—
37	C₄⋊Dic₆₇	—	2	2	no	—
38	C₄⋊Dic₆₇	—	4	1	no	—
39	C₄⋊Dic₆₇	—	4	1	no	—
40	C₄⋊Dic₆₇	—	4	1	no	—
41	Dic₆₇⋊C₄	—	8	2	no	—
42	C₆₇⋊C₁₆	—	2	4	no	—
43	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
46	M₁₆×C₆₇	—	4	2	no	—
47	M₁₆×C₆₇	—	4	2	no	—
48	C₂×C₈×C₆₇	—	2	1	no	—
49	C₂×C₈×C₆₇	—	2	1	no	—
50	C₂×C₈×C₆₇	—	4	2	no	—
51	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
52	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
54	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
55	C₂×C₈×C₆₇	#41	1	2	yes	—
56	C₄²×C₆₇	—	2	1	no	—
57	C₄²×C₆₇	—	2	1	no	—
58	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
59	C₂×C₄×C₆₇	—	2	1	no	—
60	C₂×C₄×C₆₇	—	2	1	no	—
61	C₂×C₄×C₆₇	—	4	1	no	—
62	C₂×C₄×C₆₇	—	4	1	no	—
63	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
64	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
65	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
67	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
69	C₂×Dic₆₇	—	4	2	no	—
70	C₂×Dic₆₇	—	8	1	no	—
71	C₂×Dic₆₇	—	8	1	no	—
72	C₂×Dic₆₇	—	8	1	no	—
73	C₂×Dic₆₇	—	8	1	no	—
74	C₂×Dic₆₇	—	8	1	no	—
75	C₈×C₆₇	—	2	2	no	—
76	C₈×C₆₇	—	8	2	no	—
77	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
78	C₈×C₆₇	#125	1	2	yes	—
79	C₂²×C₆₇	—	2	1	no	—
80	C₂²×C₆₇	—	2	1	no	—
81	C₂²×C₆₇	#461	1	1	yes	characteristic
82	C₂²×C₆₇	#481	1	1	yes	characteristic
83	C₂²×C₆₇	#498	1	1	yes	characteristic
84	C₄×C₆₇	—	4	1	no	—
85	C₄×C₆₇	—	4	1	no	—
86	C₄×C₆₇	#398	1	1	yes	characteristic
87	C₄×C₆₇	#460	1	1	yes	characteristic
88	C₄×C₆₇	#483	1	1	yes	characteristic
89	C₄×C₆₇	#501	1	1	yes	characteristic
90	Dic₆₇	—	8	2	no	—
91	Dic₆₇	—	16	1	no	—
92	#13510	—	67	1	no	maximal, Sylow
93	C₂×C₆₇	—	2	1	no	—
94	C₂×C₆₇	—	2	1	no	—
95	C₂×C₆₇	#2084	1	1	yes	characteristic
96	C₂×C₆₇	#2238	1	1	yes	characteristic
97	C₂×C₆₇	#2239	1	1	yes	characteristic
98	C₂×#365	—	67	2	no	—
99	#1944	—	67	1	no	—
100	#1849	D₆₇	1	1	yes	characteristic
101	#1387	—	67	2	no	—
102	#1377	—	67	1	no	—
103	C₆₇	#13510	1	1	yes	characteristic, complemented, Sylow
104	C₂×M₃₂	—	67	2	no	—
105	C₂×#108	—	2	1	no	—
106	C₂×#108	C₂×D₆₇	1	2	yes	—
107	C₂×#107	—	67	1	no	—
108	C₂×#106	—	67	1	no	—
109	#423	—	2	1	no	—
110	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
111	#389	—	134	1	no	—
112	#365	—	134	4	no	—
113	#340	—	134	1	no	—
114	C₂×M₁₆	—	2	2	no	—
115	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	268	1	no	—
118	C₂×C₄⋊C₄	—	134	1	no	—
119	C₂×C₄⋊C₄	—	134	1	no	—
120	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
121	M₃₂	—	67	4	no	—
122	M₃₂	—	134	2	no	—
123	C₂×C₁₆	—	134	2	no	—
124	#108	—	2	2	no	—
125	#108	—	2	2	no	—
126	#108	—	4	2	no	—
127	#107	—	134	2	no	—
128	#106	—	67	2	no	—
129	#106	—	134	1	no	—
130	C₄⋊C₈	—	4	2	no	—
131	#97	—	2	2	no	—
132	#95	—	268	1	no	—
133	C₂²×C₄	—	268	1	no	—
134	C₂²×C₄	—	268	1	no	—
135	C₂²×C₄	#11761615	1	1	yes	characteristic
136	C₂²×C₄	#11761615	1	1	yes	characteristic
137	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
138	M₁₆	—	4	2	no	—
139	M₁₆	—	4	2	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	#11761616	1	2	yes	—
144	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
145	C₂×C₈	Q₈×D₆₇	1	2	yes	—
146	C₂×C₈	#11761627	1	1	yes	characteristic
147	C₂×C₈	#11761628	1	2	yes	—
148	C₄⋊C₄	—	268	1	no	—
149	C₄⋊C₄	—	268	1	no	—
150	C₄⋊C₄	—	268	1	no	—
151	C₄⋊C₄	—	536	2	no	—
152	C₄⋊C₄	—	134	2	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	1	no	—
155	C₁₆	—	134	4	no	—
156	C₂³	C₂×#11761615	1	1	yes	characteristic
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	—	268	2	no	—
162	C₂×C₄	—	536	1	no	—
163	C₂×C₄	—	536	1	no	—
164	C₂×C₄	—	536	1	no	—
165	C₂×C₄	—	536	1	no	—
166	C₂×C₄	—	536	1	no	—
167	C₂×C₄	#14524014	1	1	yes	characteristic
168	C₂×C₄	#14524026	1	1	yes	characteristic
169	C₂×C₄	#14524041	1	1	yes	characteristic
170	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
171	C₂×C₄	#14524069	1	1	yes	characteristic
172	C₂×C₄	#14524074	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	8	2	no	—
175	C₈	#14524021	1	2	yes	—
176	C₈	#14524033	1	2	yes	—
177	C₂²	—	2	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	#16863598	1	1	yes	characteristic
180	C₂²	#16863714	1	1	yes	characteristic
181	C₂²	#16863792	1	1	yes	characteristic
182	C₄	—	4	1	no	—
183	C₄	—	4	1	no	—
184	C₄	—	536	2	no	—
185	C₄	—	1072	1	no	—
186	C₄	#16863327	1	1	yes	characteristic
187	C₄	#16863593	1	1	yes	characteristic
188	C₄	#16863726	1	1	yes	characteristic
189	C₄	#16863802	1	1	yes	characteristic
190	C₂	—	2	1	no	—
191	C₂	—	2	1	no	—
192	C₂	#22561686	1	1	yes	characteristic
193	C₂	#22562423	1	1	yes	characteristic
194	C₂	#22562428	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863220

C₂

yes

maximal

#22560952

C₂

yes

maximal, characteristic

#22558319

C₂

yes

maximal

#22558293

C₂

yes

maximal, characteristic

C₆₇×#1849

C₂

yes

maximal, characteristic

C₂×#14523983

C₂²

yes

—

C₂×#14523977

C₂²

yes

characteristic

C₂×#14523975

C₂²

yes

—

—

—

C₂×#108×C₆₇

—

C₂×#108×C₆₇

C₂²

yes

—

#423×C₆₇

—

C₂×#97×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×Dic₆₇⋊C₄

—

#14523983

—

#14523983

Q₈

yes

—

C₂×C₆₇⋊C₁₆

—

#14523977

—

#14523975

—

#14523975

D₄

yes

—

C₆₇⋊#95

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

characteristic

#108×C₆₇

—

#108×C₆₇

—

#108×C₆₇

—

C₄⋊C₈×C₆₇

—

#97×C₆₇

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

SD₁₆

yes

characteristic

C₂²×C₄×C₆₇

SD₁₆

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

SD₁₆

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

—

C₂×C₈×C₆₇

#41

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₂×SD₁₆

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₂²⋊Q₈

yes

—

C₈×C₆₇

#125

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

#13510

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

—

100

#1849

D₆₇

yes

characteristic

101

#1387

—

102

#1377

—

103

C₆₇

#13510

yes

characteristic, complemented, Sylow

104

C₂×M₃₂

—

105

C₂×#108

—

106

C₂×#108

C₂×D₆₇

yes

—

107

C₂×#107

—

108

C₂×#106

—

109

#423

—

110

C₂×#97

C₂×D₆₇

yes

characteristic

111

#389

—

134

—

112

#365

—

134

—

113

#340

—

134

—

114

C₂×M₁₆

—

115

C₂²×C₈

C₂²⋊D₆₇

yes

characteristic

116

C₂²×C₈

C₂²×D₆₇

yes

characteristic

117

C₂×C₄⋊C₄

—

268

—

118

C₂×C₄⋊C₄

—

134

—

119

C₂×C₄⋊C₄

—

134

—

120

C₂×C₄²

C₂²⋊D₆₇

yes

characteristic

121

M₃₂

—

122

M₃₂

—

134

—

123

C₂×C₁₆

—

134

—

124

#108

—

125

#108

—

126

#108

—

127

#107

—

134

—

128

#106

—

129

#106

—

134

—

130

C₄⋊C₈

—

131

#97

—

132

#95

—

268

—

133

C₂²×C₄

—

268

—

134

C₂²×C₄

—

268

—

135

C₂²×C₄

#11761615

yes

characteristic

136

C₂²×C₄

#11761615

yes

characteristic

137

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

138

M₁₆

—

139

M₁₆

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

#11761616

yes

—

144

C₂×C₈

D₄×D₆₇

yes

characteristic

145

C₂×C₈

Q₈×D₆₇

yes

—

146

C₂×C₈

#11761627

yes

characteristic

147

C₂×C₈

#11761628

yes

—

148

C₄⋊C₄

—

268

—

149

C₄⋊C₄

—

268

—

150

C₄⋊C₄

—

268

—

151

C₄⋊C₄

—

536

—

152

C₄⋊C₄

—

134

—

153

C₄²

—

154

C₄²

—

155

C₁₆

—

134

—

156

C₂³

C₂×#11761615

yes

characteristic

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

268

—

162

C₂×C₄

—

536

—

163

C₂×C₄

—

536

—

164

C₂×C₄

—

536

—

165

C₂×C₄

—

536

—

166

C₂×C₄

—

536

—

167

C₂×C₄

#14524014

yes

characteristic

168

C₂×C₄

#14524026

yes

characteristic

169

C₂×C₄

#14524041

yes

characteristic

170

C₂×C₄

C₂×#11761616

yes

characteristic

171

C₂×C₄

#14524069

yes

characteristic

172

C₂×C₄

#14524074

yes

characteristic

173

C₈

—

174

C₈

—

175

C₈

#14524021

yes

—

176

C₈

#14524033

yes

—

177

C₂²

—

178

C₂²

—

179

C₂²

#16863598

yes

characteristic

180

C₂²

#16863714

yes

characteristic

181

C₂²

#16863792

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

536

—

185

C₄

—

1072

—

186

C₄

#16863327

yes

characteristic

187

C₄

#16863593

yes

characteristic

188

C₄

#16863726

yes

characteristic

189

C₄

#16863802

yes

characteristic

190

C₂

—

191

C₂

—

192

C₂

#22561686

yes

characteristic

193

C₂

#22562423

yes

characteristic

194

C₂

#22562428

yes

characteristic

	subgroup	G/H
center	C₂²	#16863714
commutator, cosocle	C₂²×C₈×C₆₇	C₂³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1849	C₂
socle	C₂²×C₆₇	#481
prime radical	C₂³×C₆₇	C₂×SD₁₆
hypercenter	#1849	D₆₇
hypocenter	C₆₇	#13510
LCS	G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1849

subgroup

G/H

center

C₂²

#16863714

commutator, cosocle

C₂²×C₈×C₆₇

C₂³

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#1849

C₂

socle

C₂²×C₆₇

#481

prime radical

C₂³×C₆₇

C₂×SD₁₆

D₆₇

C₆₇

G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1849

N	H ≅ G/N	type
103. C₆₇	#13510	semidirect

H ≅ G/N

type

103. C₆₇

#13510

semidirect

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

subgroup	index
C₂×#16863220	2
#22560952	2
#22558319	2
#22558293	2
C₆₇×#1849	2
#13510	67

subgroup

index

C₆₇×#1849

Subgroups — #30775794

Conjugacy classes of subgroups

Complements

Maximal subgroups