Group #30864696 — subgroups

Subgroups — #30864696

Order 17152.

662 conjugacy classes of subgroups; only the 255 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 255 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	#22570733	C₂	1	1	yes	maximal, characteristic
2	#22569139	C₂	1	2	yes	maximal
3	#22561189	C₂	1	2	yes	maximal
4	#22560960	C₂	1	4	yes	maximal
5	#22560091	C₂	1	4	yes	maximal
6	#22559897	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1849	C₂	1	1	yes	maximal, characteristic
8	#16863953	C₂²	1	2	yes	—
9	#16863805	C₄	1	8	yes	complemented
10	#16863523	—	2	4	no	—
11	#16863520	—	2	4	no	—
12	#16863434	C₂²	1	4	yes	—
13	#16863424	C₂²	1	4	yes	—
14	#16863354	C₂²	1	4	yes	—
15	C₂×#14523953	C₂²	1	2	yes	—
16	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
18	#16863242	C₂²	1	1	yes	characteristic
19	#16863157	C₂²	1	8	yes	—
20	C₂×#108×C₆₇	C₂²	1	4	yes	—
21	#423×C₆₇	C₂²	1	2	yes	—
22	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
24	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
25	#14524043	C₂×C₄	1	4	yes	—
26	#14524035	C₂×C₄	1	4	yes	—
27	#14524022	—	2	4	no	—
28	C₄×Dic₁₃₄	—	2	4	no	—
29	#14524002	C₂³	1	4	yes	—
30	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
31	#14523977	C₂×C₄	1	8	yes	—
32	#14523975	C₂×C₄	1	8	yes	—
33	Dic₆₇⋊C₈	—	2	8	no	—
34	#14523953	—	2	4	no	—
35	#14523953	—	2	4	no	—
36	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
37	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
38	C₈×Dic₆₇	—	2	4	no	—
39	C₈×Dic₆₇	Q₈	1	8	yes	complemented
40	C₆₇⋊#95	—	2	2	no	—
41	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
42	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
43	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
44	#108×C₆₇	—	2	8	no	—
45	C₄⋊C₈×C₆₇	—	2	4	no	—
46	#97×C₆₇	C₂×C₄	1	4	yes	—
47	C₂×Dic₁₃₄	—	2	4	no	—
48	C₂²×Dic₆₇	—	2	2	no	—
49	C₂×C₆₇⋊C₈	—	2	2	no	—
50	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
51	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
52	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
53	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
54	C₄⋊Dic₆₇	—	2	4	no	—
55	C₄⋊Dic₆₇	C₂²×C₄	1	8	yes	—
56	Dic₆₇⋊C₄	—	2	8	no	—
57	C₂²⋊Dic₆₇	—	2	4	no	—
58	C₄×Dic₆₇	—	2	4	no	—
59	C₄×Dic₆₇	—	2	2	no	—
60	C₄×Dic₆₇	C₂×Q₈	1	4	yes	—
61	C₂²×C₄×C₆₇	#41	1	2	yes	—
62	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
63	M₁₆×C₆₇	—	2	8	no	—
64	C₂×C₈×C₆₇	—	2	4	no	—
65	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
66	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
67	C₂×C₈×C₆₇	#41	1	4	yes	—
68	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
69	C₄²×C₆₇	C₂×D₄	1	2	yes	—
70	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
71	C₂×C₄×C₆₇	—	4	2	no	—
72	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
73	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
76	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
77	C₂×Dic₆₇	—	2	8	no	—
78	C₂×Dic₆₇	—	2	4	no	—
79	C₂×Dic₆₇	—	4	4	no	—
80	Dic₁₃₄	—	4	8	no	—
81	C₈×C₆₇	—	4	4	no	—
82	C₈×C₆₇	C₄×Q₈	1	8	yes	—
83	C₆₇⋊C₈	—	2	2	no	—
84	C₆₇⋊C₈	C₄⋊Q₈	1	2	yes	—
85	C₆₇⋊C₈	C₄⋊Q₈	1	2	yes	—
86	C₂²×C₆₇	—	2	2	no	—
87	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
88	C₂²×C₆₇	#533	1	2	yes	—
89	C₄×C₆₇	—	2	4	no	—
90	C₄×C₆₇	C₂×C₄×Q₈	1	2	yes	—
91	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
92	C₄×C₆₇	#532	1	1	yes	characteristic
93	Dic₆₇	—	4	4	no	—
94	Dic₆₇	—	4	8	no	—
95	#21735	—	67	1	no	maximal, Sylow
96	C₂×C₆₇	—	2	2	no	—
97	C₂×C₆₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
98	C₂×C₆₇	#2955	1	2	yes	—
99	#3162	—	67	1	no	—
100	#2960	—	67	2	no	—
101	#1995	—	67	2	no	—
102	#1946	—	67	4	no	—
103	#1849	D₆₇	1	1	yes	characteristic
104	#1772	—	67	4	no	—
105	#1726	—	67	1	no	—
106	C₆₇	#21735	1	1	yes	characteristic, complemented, Sylow
107	#533	—	67	2	no	—
108	#501	—	67	8	no	—
109	#446	—	134	4	no	—
110	C₈×Q₈	—	134	4	no	—
111	C₂×#108	C₂×D₆₇	1	4	yes	—
112	#427	—	67	4	no	—
113	#424	—	67	4	no	—
114	#423	C₂×D₆₇	1	2	yes	—
115	#405	—	67	4	no	—
116	C₂×#97	—	67	2	no	—
117	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
118	C₂×C₄×C₈	—	67	2	no	—
119	C₂×C₄×C₈	—	67	1	no	—
120	#376	—	67	1	no	—
121	#339	—	67	8	no	—
122	C₂×M₁₆	C₂²×D₆₇	1	4	yes	—
123	C₂²×C₈	—	67	1	no	—
124	C₂²×C₈	—	67	1	no	—
125	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
126	C₄⋊Q₈	—	67	4	no	—
127	#125	—	67	4	no	—
128	C₂²⋊Q₈	—	134	4	no	—
129	C₄×Q₈	—	134	4	no	—
130	#117	—	67	4	no	—
131	C₂×C₄²	—	67	2	no	—
132	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
133	#108	—	2	8	no	—
134	#107	—	67	8	no	—
135	#106	—	67	8	no	—
136	C₄⋊C₈	—	2	4	no	—
137	C₄⋊C₈	—	134	8	no	—
138	#97	—	134	4	no	—
139	#97	—	134	4	no	—
140	#97	C₄×D₆₇	1	4	yes	—
141	C₄×C₈	—	67	8	no	—
142	C₄×C₈	—	67	2	no	—
143	C₄×C₈	—	67	2	no	—
144	C₄×C₈	—	134	4	no	—
145	#95	—	134	2	no	—
146	C₂×Q₈	—	134	4	no	—
147	C₂²×C₄	—	134	2	no	—
148	C₂²×C₄	#11761629	1	2	yes	—
149	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
150	M₁₆	—	2	8	no	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	67	1	no	—
153	C₂×C₈	—	67	1	no	—
154	C₂×C₈	—	67	4	no	—
155	C₂×C₈	—	67	2	no	—
156	C₂×C₈	—	134	2	no	—
157	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
158	C₂×C₈	Q₈×D₆₇	1	4	yes	—
159	C₂×C₈	#11761627	1	4	yes	—
160	C₄⋊C₄	—	67	8	no	—
161	C₄⋊C₄	—	134	8	no	—
162	C₄⋊C₄	—	134	4	no	—
163	C₂²⋊C₄	—	134	4	no	—
164	C₄²	—	67	4	no	—
165	C₄²	—	134	4	no	—
166	C₄²	—	134	2	no	—
167	C₄²	C₂×C₄×D₆₇	1	2	yes	—
168	C₄²	D₄×D₆₇	1	2	yes	—
169	C₂³	C₂×#11761629	1	1	yes	characteristic
170	Q₈	—	268	8	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	268	4	no	—
173	C₂×C₄	—	134	8	no	—
174	C₂×C₄	—	134	4	no	—
175	C₂×C₄	#14524005	1	4	yes	—
176	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
177	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
178	C₂×C₄	C₂×Q₈×D₆₇	1	2	yes	—
179	C₂×C₄	C₂×#11761627	1	2	yes	—
180	C₈	—	4	4	no	—
181	C₈	—	67	2	no	—
182	C₈	—	67	2	no	—
183	C₈	—	134	2	no	—
184	C₈	#14524008	1	8	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	C₂×#14524005	1	1	yes	characteristic
187	C₂²	#16863951	1	2	yes	—
188	C₄	—	2	4	no	—
189	C₄	—	268	4	no	—
190	C₄	—	268	8	no	—
191	C₄	C₂×#14524008	1	2	yes	—
192	C₄	C₄⋊Q₈×D₆₇	1	1	yes	characteristic
193	C₄	#16863944	1	1	yes	characteristic
194	C₂	—	2	2	no	—
195	C₂	#22562776	1	1	yes	characteristic
196	C₂	#22569111	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22570733

C₂

yes

maximal, characteristic

#22569139

C₂

yes

maximal

#22561189

C₂

yes

maximal

#22560960

C₂

yes

maximal

#22560091

C₂

yes

maximal

#22559897

C₂

yes

maximal, characteristic

C₆₇×#1849

C₂

yes

maximal, characteristic

#16863953

C₂²

yes

—

#16863805

C₄

yes

—

—

C₂²

yes

—

#16863424

C₂²

yes

—

#16863354

C₂²

yes

—

C₂×#14523953

C₂²

yes

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×C₈×Dic₆₇

C₂²

yes

—

#16863242

C₂²

yes

characteristic

#16863157

C₂²

yes

—

C₂×#108×C₆₇

C₂²

yes

—

#423×C₆₇

C₂²

yes

—

C₂×#97×C₆₇

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

C₂³

yes

characteristic

C₂²×C₆₇⋊C₈

C₂³

yes

characteristic

#14524043

C₂×C₄

yes

—

#14524035

C₂×C₄

yes

—

#14524022

—

C₄×Dic₁₃₄

—

#14524002

C₂³

yes

—

C₂×C₄×Dic₆₇

C₂³

yes

—

#14523977

C₂×C₄

yes

—

#14523975

C₂×C₄

yes

—

Dic₆₇⋊C₈

—

#14523953

—

#14523953

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₈×Dic₆₇

—

C₈×Dic₆₇

Q₈

yes

complemented

C₆₇⋊#95

—

C₂×M₁₆×C₆₇

C₂³

yes

—

C₂²×C₈×C₆₇

C₂³

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#108×C₆₇

—

C₄⋊C₈×C₆₇

—

#97×C₆₇

C₂×C₄

yes

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×D₄

yes

—

C₂×C₆₇⋊C₈

C₂×D₄

yes

characteristic

C₂×C₆₇⋊C₈

C₂×D₄

yes

characteristic

C₂×C₆₇⋊C₈

C₂×Q₈

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂²×C₄

yes

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

C₂×Q₈

yes

—

C₂²×C₄×C₆₇

#41

yes

—

C₂²×C₄×C₆₇

C₂⁴

yes

characteristic

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²×C₄

yes

—

C₂×C₈×C₆₇

C₂×Q₈

yes

—

C₂×C₈×C₆₇

#41

yes

—

C₄²×C₆₇

C₂²×C₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

—

C₂³×C₆₇

C₂×#41

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

C₂³×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×Q₈

yes

—

C₂×C₄×C₆₇

C₂×#41

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

C₄×Q₈

yes

—

C₆₇⋊C₈

—

C₆₇⋊C₈

C₄⋊Q₈

yes

—

C₆₇⋊C₈

C₄⋊Q₈

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₄×D₄

yes

characteristic

C₂²×C₆₇

#533

yes

—

C₄×C₆₇

—

C₄×C₆₇

C₂×C₄×Q₈

yes

—

C₄×C₆₇

C₂×C₄⋊Q₈

yes

characteristic

C₄×C₆₇

#532

yes

characteristic

Dic₆₇

—

Dic₆₇

—

#21735

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

C₄×C₄⋊Q₈

yes

characteristic

C₂×C₆₇

#2955

yes

—

#3162

—

100

#2960

—

101

#1995

—

102

#1946

—

103

#1849

D₆₇

yes

characteristic

104

#1772

—

105

#1726

—

106

C₆₇

#21735

yes

characteristic, complemented, Sylow

107

#533

—

108

#501

—

109

#446

—

134

—

110

C₈×Q₈

—

134

—

111

C₂×#108

C₂×D₆₇

yes

—

112

#427

—

113

#424

—

114

#423

C₂×D₆₇

yes

—

115

#405

—

116

C₂×#97

—

117

C₂×#97

C₂×D₆₇

yes

characteristic

118

C₂×C₄×C₈

—

119

C₂×C₄×C₈

—

120

#376

—

121

#339

—

122

C₂×M₁₆

C₂²×D₆₇

yes

—

123

C₂²×C₈

—

124

C₂²×C₈

—

125

C₂²×C₈

C₂²×D₆₇

yes

—

126

C₄⋊Q₈

—

127

#125

—

128

C₂²⋊Q₈

—

134

—

129

C₄×Q₈

—

134

—

130

#117

—

131

C₂×C₄²

—

132

C₂×C₄²

C₂²×D₆₇

yes

characteristic

133

#108

—

134

#107

—

135

#106

—

136

C₄⋊C₈

—

137

C₄⋊C₈

—

134

—

138

#97

—

134

—

139

#97

—

134

—

140

#97

C₄×D₆₇

yes

—

141

C₄×C₈

—

142

C₄×C₈

—

143

C₄×C₈

—

144

C₄×C₈

—

134

—

145

#95

—

134

—

146

C₂×Q₈

—

134

—

147

C₂²×C₄

—

134

—

148

C₂²×C₄

#11761629

yes

—

149

C₂²×C₄

C₂³×D₆₇

yes

characteristic

150

M₁₆

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

134

—

157

C₂×C₈

C₂×C₄×D₆₇

yes

—

158

C₂×C₈

Q₈×D₆₇

yes

—

159

C₂×C₈

#11761627

yes

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

134

—

162

C₄⋊C₄

—

134

—

163

C₂²⋊C₄

—

134

—

164

C₄²

—

165

C₄²

—

134

—

166

C₄²

—

134

—

167

C₄²

C₂×C₄×D₆₇

yes

—

168

C₄²

D₄×D₆₇

yes

—

169

C₂³

C₂×#11761629

yes

characteristic

170

Q₈

—

268

—

171

C₂×C₄

—

172

C₂×C₄

—

268

—

173

C₂×C₄

—

134

—

174

C₂×C₄

—

134

—

175

C₂×C₄

#14524005

yes

—

176

C₂×C₄

C₂²×C₄×D₆₇

yes

characteristic

177

C₂×C₄

C₂×D₄×D₆₇

yes

characteristic

178

C₂×C₄

C₂×Q₈×D₆₇

yes

—

179

C₂×C₄

C₂×#11761627

yes

—

180

C₈

—

181

C₈

—

182

C₈

—

183

C₈

—

134

—

184

C₈

#14524008

yes

—

185

C₂²

—

186

C₂²

C₂×#14524005

yes

characteristic

187

C₂²

#16863951

yes

—

188

C₄

—

189

C₄

—

268

—

190

C₄

—

268

—

191

C₄

C₂×#14524008

yes

—

192

C₄

C₄⋊Q₈×D₆₇

yes

characteristic

193

C₄

#16863944

yes

characteristic

194

C₂

—

195

C₂

#22562776

yes

characteristic

196

C₂

#22569111

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1849	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₂×#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1849	D₆₇
hypocenter	C₆₇	#21735
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1849

subgroup

G/H

center

C₂×C₄

C₂×D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂³×C₄

Frattini

C₂²×C₄

C₂³×D₆₇

Fitting

C₆₇×#1849

C₂

socle

C₂²×C₆₇

C₂×C₄×D₄

prime radical

C₂³×C₆₇

C₂×#41

cosocle

C₂²×C₄×C₆₇

C₂⁴

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1849

N	H ≅ G/N	type
9. #16863805	C₄	semidirect
39. C₈×Dic₆₇	Q₈	semidirect
106. C₆₇	#21735	semidirect

H ≅ G/N

type

9. #16863805

C₄

semidirect

39. C₈×Dic₆₇

Q₈

semidirect

106. C₆₇

#21735

semidirect

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

subgroup	index
#22570733	2
#22569139	2
#22561189	2
#22560960	2
#22560091	2
#22559897	2
C₆₇×#1849	2
#21735	67

subgroup

index

C₆₇×#1849

Subgroups — #30864696

Conjugacy classes of subgroups

Complements

Maximal subgroups