Group #30767708 — subgroups

Subgroups — #30767708

Order 17152.

214 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 77 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	#22562078	C₂	1	1	yes	maximal, characteristic
2	#22559028	C₂	1	1	yes	maximal, characteristic
3	#22559027	C₂	1	1	yes	maximal, characteristic
4	#22558274	C₂	1	1	yes	maximal, characteristic
5	#22558180	C₂	1	1	yes	maximal, characteristic
6	#22558131	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1334	C₂	1	1	yes	maximal, characteristic
8	C₂×#14523989	C₄	1	1	yes	characteristic
9	Q₈×C₆₇⋊C₈	—	2	1	no	—
10	#16863520	—	2	1	no	—
11	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
12	#16863223	C₄	1	1	yes	characteristic
13	#16863200	C₄	1	1	yes	characteristic
14	#16863199	C₄	1	1	yes	characteristic
15	#16863172	C₂²	1	1	yes	characteristic
16	#16863147	C₂²	1	1	yes	characteristic
17	Q₈⋊(C₆₇⋊C₈)	—	2	1	no	—
18	#16863126	—	2	1	no	—
19	#16863117	C₂²	1	1	yes	characteristic
20	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
21	#346×C₆₇	C₂²	1	1	yes	characteristic
22	#334×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₁₃₄	—	2	1	no	—
25	#14523989	C₈	1	4	yes	complemented
26	C₂×C₆₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₆₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
28	#14523977	C₂×C₄	1	1	yes	characteristic
29	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
30	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
31	Dic₆₇⋊C₈	—	4	1	no	—
32	#14523968	—	2	1	no	—
33	C₄×C₆₇⋊C₈	—	4	1	no	—
34	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
35	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
36	C₈×Dic₆₇	—	4	1	no	—
37	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₄×Q₈×C₆₇	—	2	1	no	—
39	C₂×C₁₆×C₆₇	—	2	1	no	—
40	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₄⋊C₈×C₆₇	—	2	1	no	—
42	#103×C₆₇	—	2	1	no	—
43	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
44	C₂×Dic₁₃₄	—	2	1	no	—
45	Dic₂₆₈	—	2	1	no	—
46	Dic₂₆₈	C₂×C₈	1	2	yes	—
47	C₂×C₆₇⋊C₈	—	2	1	no	—
48	C₂×C₆₇⋊C₈	—	2	1	no	—
49	C₂×C₆₇⋊C₈	—	4	1	no	—
50	C₄⋊Dic₆₇	—	2	1	no	—
51	Dic₆₇⋊C₄	—	4	1	no	—
52	C₄×Dic₆₇	—	4	1	no	—
53	C₆₇⋊C₁₆	—	2	1	no	—
54	C₆₇⋊C₁₆	C₂×C₈	1	2	yes	—
55	C₂×Q₈×C₆₇	—	2	1	no	—
56	Q₁₆×C₆₇	—	2	1	no	—
57	Q₁₆×C₆₇	C₂×C₈	1	2	yes	—
58	C₂×C₈×C₆₇	—	4	1	no	—
59	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
61	C₄⋊C₄×C₆₇	—	2	1	no	—
62	C₄⋊C₄×C₆₇	—	4	1	no	—
63	C₄²×C₆₇	—	4	1	no	—
64	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₁₆×C₆₇	—	4	1	no	—
66	Q₈×C₆₇	—	2	2	no	—
67	Q₈×C₆₇	—	4	1	no	—
68	C₂×C₄×C₆₇	—	4	1	no	—
69	C₂×C₄×C₆₇	—	4	1	no	—
70	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₂×Dic₆₇	—	4	1	no	—
74	C₂×Dic₆₇	—	4	1	no	—
75	Dic₁₃₄	—	2	2	no	—
76	Dic₁₃₄	—	4	1	no	—
77	C₈×C₆₇	—	2	1	no	—
78	C₈×C₆₇	—	8	1	no	—
79	C₈×C₆₇	C₂²×C₈	1	1	yes	characteristic
80	C₈×C₆₇	#131	1	1	yes	characteristic
81	C₆₇⋊C₈	—	4	1	no	—
82	C₆₇⋊C₈	—	4	1	no	—
83	C₆₇⋊C₈	—	8	1	no	—
84	C₂²×C₆₇	#442	1	1	yes	characteristic
85	C₄×C₆₇	—	2	1	no	—
86	C₄×C₆₇	—	2	1	no	—
87	C₄×C₆₇	—	4	2	no	—
88	C₄×C₆₇	—	8	1	no	—
89	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
90	C₄×C₆₇	#444	1	1	yes	characteristic
91	Dic₆₇	—	4	2	no	—
92	Dic₆₇	—	8	1	no	—
93	#12066	—	67	1	no	maximal, Sylow
94	C₂×C₆₇	#1580	1	1	yes	characteristic
95	C₂×C₆₇	#2180	1	1	yes	characteristic
96	C₂×C₆₇	#2180	1	1	yes	characteristic
97	#2176	—	67	1	no	—
98	#1579	—	67	1	no	—
99	#1579	—	67	1	no	—
100	#1368	—	67	1	no	—
101	#1334	—	67	1	no	—
102	#1334	D₆₇	1	1	yes	characteristic
103	#1313	—	67	1	no	—
104	C₆₇	#12066	1	1	yes	characteristic, complemented, Sylow
105	C₂×Q₃₂	—	67	1	no	—
106	C₈×Q₈	—	134	1	no	—
107	C₈×Q₈	—	134	1	no	—
108	C₄×Q₁₆	—	67	1	no	—
109	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
110	#367	—	67	1	no	—
111	#358	—	67	1	no	—
112	#358	—	67	1	no	—
113	#346	—	67	1	no	—
114	#346	C₂×D₆₇	1	1	yes	characteristic
115	#334	—	67	1	no	—
116	#334	C₂×D₆₇	1	1	yes	characteristic
117	Q₈⋊C₈	—	134	1	no	—
118	Q₈⋊C₈	—	134	1	no	—
119	#322	—	67	1	no	—
120	C₂×Q₁₆	—	67	1	no	—
121	C₂×Q₁₆	C₄×D₆₇	1	1	yes	characteristic
122	C₄×Q₈	—	2	1	no	—
123	C₄×Q₈	—	134	1	no	—
124	Q₃₂	—	67	4	no	—
125	C₂×C₁₆	—	2	1	no	—
126	C₂×C₁₆	—	67	1	no	—
127	C₂×C₁₆	—	67	1	no	—
128	#107	—	67	1	no	—
129	#107	C₄×D₆₇	1	1	yes	characteristic
130	C₄⋊C₈	—	2	1	no	—
131	C₄⋊C₈	—	268	1	no	—
132	C₄⋊C₈	—	268	1	no	—
133	C₄⋊C₈	—	134	1	no	—
134	#103	—	2	1	no	—
135	#103	—	134	1	no	—
136	C₄×C₈	—	268	1	no	—
137	C₄×C₈	—	268	1	no	—
138	C₄×C₈	—	67	1	no	—
139	C₄×C₈	—	67	1	no	—
140	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
141	C₂×Q₈	—	2	1	no	—
142	C₂×Q₈	—	134	1	no	—
143	Q₁₆	—	2	1	no	—
144	Q₁₆	—	67	2	no	—
145	Q₁₆	—	134	1	no	—
146	Q₁₆	C₈×D₆₇	1	2	yes	—
147	C₂×C₈	—	4	1	no	—
148	C₂×C₈	—	268	1	no	—
149	C₂×C₈	—	134	1	no	—
150	C₂×C₈	—	134	1	no	—
151	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
152	C₂×C₈	#11761627	1	1	yes	characteristic
153	C₄⋊C₄	—	2	1	no	—
154	C₄⋊C₄	—	4	1	no	—
155	C₄⋊C₄	—	268	1	no	—
156	C₄⋊C₄	—	134	1	no	—
157	C₄²	—	4	1	no	—
158	C₄²	—	268	1	no	—
159	C₄²	D₄×D₆₇	1	1	yes	characteristic
160	C₁₆	—	4	1	no	—
161	C₁₆	—	67	2	no	—
162	C₁₆	—	134	1	no	—
163	Q₈	—	2	2	no	—
164	Q₈	—	4	1	no	—
165	Q₈	—	268	1	no	—
166	Q₈	—	134	2	no	—
167	C₂×C₄	—	4	1	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	268	1	no	—
170	C₂×C₄	—	268	1	no	—
171	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
172	C₂×C₄	#14524067	1	1	yes	characteristic
173	C₂×C₄	#14524070	1	1	yes	characteristic
174	C₈	—	2	1	no	—
175	C₈	—	8	1	no	—
176	C₈	—	268	1	no	—
177	C₈	—	268	1	no	—
178	C₈	—	536	1	no	—
179	C₈	C₂×C₈×D₆₇	1	1	yes	characteristic
180	C₈	#14524051	1	1	yes	characteristic
181	C₂²	#16863505	1	1	yes	characteristic
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	4	2	no	—
185	C₄	—	8	1	no	—
186	C₄	—	268	2	no	—
187	C₄	—	536	1	no	—
188	C₄	(C₂²⋊D₆₇)⋊C₈	1	1	yes	characteristic
189	C₄	#16863515	1	1	yes	characteristic
190	C₂	#22559030	1	1	yes	characteristic
191	C₂	#22562097	1	1	yes	characteristic
192	C₂	#22562103	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22562078

C₂

yes

maximal, characteristic

#22559028

C₂

yes

maximal, characteristic

#22559027

C₂

yes

maximal, characteristic

#22558274

C₂

yes

maximal, characteristic

#22558180

C₂

yes

maximal, characteristic

#22558131

C₂

yes

maximal, characteristic

C₆₇×#1334

C₂

yes

maximal, characteristic

C₂×#14523989

C₄

yes

characteristic

Q₈×C₆₇⋊C₈

—

#16863520

—

C₄×Dic₂₆₈

C₂²

yes

characteristic

#16863223

C₄

yes

characteristic

#16863200

C₄

yes

characteristic

#16863199

C₄

yes

characteristic

#16863172

C₂²

yes

characteristic

#16863147

C₂²

yes

characteristic

Q₈⋊(C₆₇⋊C₈)

—

#16863126

—

#16863117

C₂²

yes

characteristic

C₄×Q₁₆×C₆₇

C₂²

yes

characteristic

#346×C₆₇

C₂²

yes

characteristic

#334×C₆₇

C₂²

yes

characteristic

C₂×Dic₂₆₈

C₂×C₄

yes

characteristic

C₄×Dic₁₃₄

—

#14523989

C₈

yes

complemented

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

characteristic

#14523977

C₂×C₄

yes

characteristic

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

Dic₆₇⋊C₈

—

#14523968

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

D₄

yes

characteristic

C₄×C₆₇⋊C₈

D₄

yes

characteristic

C₈×Dic₆₇

—

C₂×Q₁₆×C₆₇

C₂×C₄

yes

characteristic

C₄×Q₈×C₆₇

—

C₂×C₁₆×C₆₇

—

#107×C₆₇

C₂×C₄

yes

characteristic

C₄⋊C₈×C₆₇

—

#103×C₆₇

—

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×Dic₁₃₄

—

Dic₂₆₈

—

Dic₂₆₈

C₂×C₈

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

—

C₆₇⋊C₁₆

C₂×C₈

yes

—

C₂×Q₈×C₆₇

—

Q₁₆×C₆₇

—

Q₁₆×C₆₇

C₂×C₈

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

#41

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₁₆×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄×D₄

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₂²×C₈

yes

characteristic

C₈×C₆₇

#131

yes

characteristic

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

#442

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₈×D₄

yes

characteristic

C₄×C₆₇

#444

yes

characteristic

Dic₆₇

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

—

—

100

#1368

—

101

#1334

—

102

#1334

D₆₇

yes

characteristic

103

#1313

—

104

C₆₇

#12066

yes

characteristic, complemented, Sylow

105

C₂×Q₃₂

—

106

C₈×Q₈

—

134

—

107

C₈×Q₈

—

134

—

108

C₄×Q₁₆

—

109

C₄×Q₁₆

C₂×D₆₇

yes

characteristic

110

#367

—

111

#358

—

112

#358

—

113

#346

—

114

#346

C₂×D₆₇

yes

characteristic

115

#334

—

116

#334

C₂×D₆₇

yes

characteristic

117

Q₈⋊C₈

—

134

—

118

Q₈⋊C₈

—

134

—

119

#322

—

120

C₂×Q₁₆

—

121

C₂×Q₁₆

C₄×D₆₇

yes

characteristic

122

C₄×Q₈

—

123

C₄×Q₈

—

134

—

124

Q₃₂

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

127

C₂×C₁₆

—

128

#107

—

129

#107

C₄×D₆₇

yes

characteristic

130

C₄⋊C₈

—

131

C₄⋊C₈

—

268

—

132

C₄⋊C₈

—

268

—

133

C₄⋊C₈

—

134

—

134

#103

—

135

#103

—

134

—

136

C₄×C₈

—

268

—

137

C₄×C₈

—

268

—

138

C₄×C₈

—

139

C₄×C₈

—

140

C₄×C₈

C₂²×D₆₇

yes

characteristic

141

C₂×Q₈

—

142

C₂×Q₈

—

134

—

143

Q₁₆

—

144

Q₁₆

—

145

Q₁₆

—

134

—

146

Q₁₆

C₈×D₆₇

yes

—

147

C₂×C₈

—

148

C₂×C₈

—

268

—

149

C₂×C₈

—

134

—

150

C₂×C₈

—

134

—

151

C₂×C₈

C₂×C₄×D₆₇

yes

characteristic

152

C₂×C₈

#11761627

yes

characteristic

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

268

—

156

C₄⋊C₄

—

134

—

157

C₄²

—

158

C₄²

—

268

—

159

C₄²

D₄×D₆₇

yes

characteristic

160

C₁₆

—

161

C₁₆

—

162

C₁₆

—

134

—

163

Q₈

—

164

Q₈

—

165

Q₈

—

268

—

166

Q₈

—

134

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

268

—

170

C₂×C₄

—

268

—

171

C₂×C₄

(C₂²⋊D₆₇)⋊C₄

yes

characteristic

172

C₂×C₄

#14524067

yes

characteristic

173

C₂×C₄

#14524070

yes

characteristic

174

C₈

—

175

C₈

—

176

C₈

—

268

—

177

C₈

—

268

—

178

C₈

—

536

—

179

C₈

C₂×C₈×D₆₇

yes

characteristic

180

C₈

#14524051

yes

characteristic

181

C₂²

#16863505

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

268

—

187

C₄

—

536

—

188

C₄

(C₂²⋊D₆₇)⋊C₈

yes

characteristic

189

C₄

#16863515

yes

characteristic

190

C₂

#22559030

yes

characteristic

191

C₂

#22562097

yes

characteristic

192

C₂

#22562103

yes

characteristic

	subgroup	G/H
center	C₂²	#16863505
commutator	C₈×C₆₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#1334	C₂
socle, prime radical	C₂²×C₆₇	#442
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	#1334	D₆₇
hypocenter	C₆₇	#12066
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1334

subgroup

G/H

center

C₂²

#16863505

commutator

C₈×C₆₇

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#1334

C₂

socle, prime radical

C₂²×C₆₇

#442

cosocle

C₄×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1334

N	H ≅ G/N	type
25. #14523989	C₈	semidirect
104. C₆₇	#12066	semidirect

H ≅ G/N

type

25. #14523989

C₈

semidirect

104. C₆₇

#12066

semidirect

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

subgroup	index
#22562078	2
#22559028	2
#22559027	2
#22558274	2
#22558180	2
#22558131	2
C₆₇×#1334	2
#12066	67

subgroup

index

C₆₇×#1334

Subgroups — #30767708

Conjugacy classes of subgroups

Complements

Maximal subgroups