Group #30760604 — subgroups

Subgroups — #30760604

Order 17152.

292 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 95 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	#22560363	C₂	1	1	yes	maximal, characteristic
2	#22558866	C₂	1	2	yes	maximal
3	#22558104	C₂	1	2	yes	maximal
4	#22558038	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1995	C₂	1	1	yes	maximal, characteristic
6	#16863449	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
8	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
10	#16863145	—	2	2	no	—
11	#16863142	—	2	2	no	—
12	#533×C₆₇	C₂²	1	1	yes	characteristic
13	#423×C₆₇	C₂²	1	1	yes	characteristic
14	#376×C₆₇	C₂²	1	1	yes	characteristic
15	#339×C₆₇	—	2	1	no	—
16	#339×C₆₇	—	2	1	no	—
17	C₂²×C₆₇⋊C₈	—	2	2	no	—
18	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
19	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
20	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
21	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
22	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
23	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₄×C₆₇⋊C₈	—	2	2	no	—
25	C₂×M₁₆×C₆₇	—	2	1	no	—
26	C₄⋊Q₈×C₆₇	D₄	1	2	yes	—
27	#125×C₆₇	—	2	1	no	—
28	C₂²⋊Q₈×C₆₇	—	2	2	no	—
29	C₄×Q₈×C₆₇	—	2	2	no	—
30	#117×C₆₇	—	2	1	no	—
31	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₆₇	—	2	2	no	—
35	#95×C₆₇	—	2	2	no	—
36	C₂×C₆₇⋊C₈	—	4	2	no	—
37	C₂×C₆₇⋊C₈	—	4	2	no	—
38	C₂×C₆₇⋊C₈	—	4	2	no	—
39	C₂×C₆₇⋊C₈	—	4	2	no	—
40	C₂×Q₈×C₆₇	—	2	2	no	—
41	C₂²×C₄×C₆₇	—	2	1	no	—
42	C₂²×C₄×C₆₇	—	2	1	no	—
43	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	#41	1	2	yes	—
45	M₁₆×C₆₇	—	4	1	no	—
46	M₁₆×C₆₇	—	4	1	no	—
47	C₂×C₈×C₆₇	—	4	1	no	—
48	C₄⋊C₄×C₆₇	—	2	2	no	—
49	C₄⋊C₄×C₆₇	—	2	2	no	—
50	C₄⋊C₄×C₆₇	—	4	2	no	—
51	C₂²⋊C₄×C₆₇	—	4	1	no	—
52	C₄²×C₆₇	—	2	1	no	—
53	C₄²×C₆₇	—	2	1	no	—
54	C₄²×C₆₇	—	4	1	no	—
55	C₄²×C₆₇	SD₁₆	1	2	yes	—
56	C₄²×C₆₇	Q₁₆	1	2	yes	—
57	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₆₇	#41	1	2	yes	—
59	C₄²×C₆₇	#41	1	1	yes	characteristic
60	C₂³×C₆₇	#123	1	1	yes	characteristic
61	Q₈×C₆₇	—	8	2	no	—
62	C₂×C₄×C₆₇	—	2	2	no	—
63	C₂×C₄×C₆₇	—	2	2	no	—
64	C₂×C₄×C₆₇	—	4	2	no	—
65	C₂×C₄×C₆₇	—	4	2	no	—
66	C₂×C₄×C₆₇	—	4	2	no	—
67	C₂×C₄×C₆₇	—	4	2	no	—
68	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
69	C₂×C₄×C₆₇	#123	1	2	yes	—
70	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
71	C₂×C₄×C₆₇	#126	1	2	yes	—
72	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
74	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
76	C₈×C₆₇	—	8	1	no	—
77	C₆₇⋊C₈	—	8	2	no	—
78	C₆₇⋊C₈	—	8	2	no	—
79	C₂²×C₆₇	—	2	2	no	—
80	C₂²×C₆₇	#397	1	1	yes	characteristic
81	C₂²×C₆₇	#483	1	1	yes	characteristic
82	C₂²×C₆₇	#484	1	1	yes	characteristic
83	C₄×C₆₇	—	2	2	no	—
84	C₄×C₆₇	—	2	2	no	—
85	C₄×C₆₇	—	4	2	no	—
86	C₄×C₆₇	—	4	2	no	—
87	C₄×C₆₇	—	8	2	no	—
88	C₄×C₆₇	#460	1	1	yes	characteristic
89	C₄×C₆₇	#462	1	1	yes	characteristic
90	C₄×C₆₇	#487	1	1	yes	characteristic
91	C₄×C₆₇	#489	1	1	yes	characteristic
92	#10628	—	67	1	no	maximal, Sylow
93	C₂×C₆₇	—	2	1	no	—
94	C₂×C₆₇	—	2	1	no	—
95	C₂×C₆₇	#2072	1	1	yes	characteristic
96	C₂×C₆₇	#2080	1	2	yes	—
97	#1995	D₆₇	1	1	yes	characteristic
98	#1843	—	67	1	no	—
99	#1542	—	67	2	no	—
100	#1301	—	67	2	no	—
101	#1274	—	67	1	no	—
102	C₆₇	#10628	1	1	yes	characteristic, complemented, Sylow
103	#533	C₂×D₆₇	1	1	yes	characteristic
104	#431	—	67	2	no	—
105	#423	C₂×D₆₇	1	1	yes	characteristic
106	C₂×C₄⋊C₈	—	67	2	no	—
107	C₂×C₄⋊C₈	—	134	1	no	—
108	C₂×C₄⋊C₈	—	134	1	no	—
109	#376	C₂×D₆₇	1	1	yes	characteristic
110	#339	—	2	1	no	—
111	#339	—	2	1	no	—
112	#333	—	134	2	no	—
113	#332	—	134	2	no	—
114	C₂×M₁₆	—	2	1	no	—
115	C₂²×C₈	—	134	2	no	—
116	C₄⋊Q₈	D₂₆₈	1	2	yes	—
117	#125	—	2	1	no	—
118	C₂²⋊Q₈	—	2	2	no	—
119	C₄×Q₈	—	2	2	no	—
120	#117	—	2	1	no	—
121	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
122	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
123	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	268	2	no	—
126	C₄⋊C₈	—	268	2	no	—
127	C₄⋊C₈	—	134	2	no	—
128	C₄⋊C₈	—	134	2	no	—
129	C₄⋊C₈	—	134	2	no	—
130	C₂²⋊C₈	—	134	2	no	—
131	C₄×C₈	—	134	2	no	—
132	#95	—	2	2	no	—
133	C₂×Q₈	—	2	2	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
137	C₂²×C₄	#11761627	1	2	yes	—
138	M₁₆	—	4	1	no	—
139	M₁₆	—	4	1	no	—
140	C₂×C₈	—	4	1	no	—
141	C₂×C₈	—	268	2	no	—
142	C₂×C₈	—	268	2	no	—
143	C₂×C₈	—	268	2	no	—
144	C₂×C₈	—	268	2	no	—
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₂²⋊C₄	—	4	1	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	2	1	no	—
151	C₄²	—	4	1	no	—
152	C₄²	#11761615	1	2	yes	—
153	C₄²	#11761618	1	2	yes	—
154	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
155	C₄²	#11761627	1	2	yes	—
156	C₄²	#11761628	1	1	yes	characteristic
157	C₂³	#14524024	1	1	yes	characteristic
158	Q₈	—	8	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	#14524017	1	1	yes	characteristic
166	C₂×C₄	#14524027	1	2	yes	—
167	C₂×C₄	#14524030	1	1	yes	characteristic
168	C₂×C₄	#14524036	1	2	yes	—
169	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
170	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
171	C₂×C₄	#14524068	1	1	yes	characteristic
172	C₂×C₄	#14524073	1	1	yes	characteristic
173	C₈	—	8	1	no	—
174	C₈	—	536	2	no	—
175	C₈	—	536	2	no	—
176	C₂²	—	2	2	no	—
177	C₂²	#16863321	1	1	yes	characteristic
178	C₂²	#16863724	1	1	yes	characteristic
179	C₂²	#16863729	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₄	—	8	2	no	—
185	C₄	#16863596	1	1	yes	characteristic
186	C₄	#16863606	1	1	yes	characteristic
187	C₄	#16863746	1	1	yes	characteristic
188	C₄	#16863755	1	1	yes	characteristic
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	#22561617	1	1	yes	characteristic
192	C₂	#22561659	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22560363

C₂

yes

maximal, characteristic

#22558866

C₂

yes

maximal

#22558104

C₂

yes

maximal

#22558038

C₂

yes

maximal, characteristic

C₆₇×#1995

C₂

yes

maximal, characteristic

#16863449

C₂²

yes

—

C₂×C₄⋊(C₆₇⋊C₈)

—

C₂×C₄⋊(C₆₇⋊C₈)

—

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

#16863145

—

#16863142

—

#533×C₆₇

C₂²

yes

characteristic

#423×C₆₇

C₂²

yes

characteristic

#376×C₆₇

C₂²

yes

characteristic

#339×C₆₇

—

#339×C₆₇

—

C₂²×C₆₇⋊C₈

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

—

C₂×M₁₆×C₆₇

—

C₄⋊Q₈×C₆₇

D₄

yes

—

#125×C₆₇

—

C₂²⋊Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

#117×C₆₇

—

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

#95×C₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

SD₁₆

yes

—

C₄²×C₆₇

Q₁₆

yes

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

#41

yes

—

C₄²×C₆₇

yes

C₂³×C₆₇

yes

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#121

yes

characteristic

C₂×C₄×C₆₇

#123

yes

—

C₂×C₄×C₆₇

#124

yes

characteristic

C₂×C₄×C₆₇

#126

yes

—

C₂×C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₁₆

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#2072

yes

characteristic

C₂×C₆₇

#2080

yes

—

#1995

D₆₇

yes

characteristic

#1843

—

#1542

—

100

#1301

—

101

#1274

—

102

C₆₇

#10628

yes

characteristic, complemented, Sylow

103

#533

C₂×D₆₇

yes

characteristic

104

#431

—

105

#423

C₂×D₆₇

yes

characteristic

106

C₂×C₄⋊C₈

—

107

C₂×C₄⋊C₈

—

134

—

108

C₂×C₄⋊C₈

—

134

—

109

#376

C₂×D₆₇

yes

characteristic

110

#339

—

111

#339

—

112

#333

—

134

—

113

#332

—

134

—

114

C₂×M₁₆

—

115

C₂²×C₈

—

134

—

116

C₄⋊Q₈

D₂₆₈

yes

—

117

#125

—

118

C₂²⋊Q₈

—

119

C₄×Q₈

—

120

#117

—

121

C₂×C₄²

C₂²⋊D₆₇

yes

characteristic

122

C₂×C₄²

C₂²⋊D₆₇

yes

characteristic

123

C₂×C₄²

C₂²×D₆₇

yes

characteristic

124

C₄⋊C₈

—

125

C₄⋊C₈

—

268

—

126

C₄⋊C₈

—

268

—

127

C₄⋊C₈

—

134

—

128

C₄⋊C₈

—

134

—

129

C₄⋊C₈

—

134

—

130

C₂²⋊C₈

—

134

—

131

C₄×C₈

—

134

—

132

#95

—

133

C₂×Q₈

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

137

C₂²×C₄

#11761627

yes

—

138

M₁₆

—

139

M₁₆

—

140

C₂×C₈

—

141

C₂×C₈

—

268

—

142

C₂×C₈

—

268

—

143

C₂×C₈

—

268

—

144

C₂×C₈

—

268

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₂²⋊C₄

—

149

C₄²

—

150

C₄²

—

151

C₄²

—

152

C₄²

#11761615

yes

—

153

C₄²

#11761618

yes

—

154

C₄²

C₂×D₂₆₈

yes

characteristic

155

C₄²

#11761627

yes

—

156

C₄²

#11761628

yes

characteristic

157

C₂³

#14524024

yes

characteristic

158

Q₈

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

#14524017

yes

characteristic

166

C₂×C₄

#14524027

yes

—

167

C₂×C₄

#14524030

yes

characteristic

168

C₂×C₄

#14524036

yes

—

169

C₂×C₄

C₂×#11761615

yes

characteristic

170

C₂×C₄

C₂×#11761618

yes

characteristic

171

C₂×C₄

#14524068

yes

characteristic

172

C₂×C₄

#14524073

yes

characteristic

173

C₈

—

174

C₈

—

536

—

175

C₈

—

536

—

176

C₂²

—

177

C₂²

#16863321

yes

characteristic

178

C₂²

#16863724

yes

characteristic

179

C₂²

#16863729

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

#16863596

yes

characteristic

186

C₄

#16863606

yes

characteristic

187

C₄

#16863746

yes

characteristic

188

C₄

#16863755

yes

characteristic

189

C₂

—

190

C₂

—

191

C₂

#22561617

yes

characteristic

192

C₂

#22561659

yes

—

	subgroup	G/H
center	C₂×C₄	#14524030
commutator, cosocle	C₂×C₄²×C₆₇	C₂³
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1995	C₂
socle	C₂²×C₆₇	#397
prime radical	C₂³×C₆₇	#123
hypercenter	#1995	D₆₇
hypocenter	C₆₇	#10628
LCS	G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1995

subgroup

G/H

center

C₂×C₄

#14524030

commutator, cosocle

C₂×C₄²×C₆₇

C₂³

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1995

C₂

C₂²×C₆₇

C₂³×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1995

N	H ≅ G/N	type
102. C₆₇	#10628	semidirect

H ≅ G/N

type

102. C₆₇

#10628

semidirect

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

subgroup	index
#22560363	2
#22558866	2
#22558104	2
#22558038	2
C₆₇×#1995	2
#10628	67

subgroup

index

C₆₇×#1995

Subgroups — #30760604

Conjugacy classes of subgroups

Complements

Maximal subgroups