Group #30864718 — subgroups

Subgroups — #30864718

Order 17152.

662 conjugacy classes of subgroups; only the 295 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 295 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	#22569144	C₂	1	2	yes	maximal
2	#22561191	C₂	1	2	yes	maximal
3	#22560958	C₂	1	4	yes	maximal
4	#22560384	C₂	1	1	yes	maximal, characteristic
5	#22560095	C₂	1	4	yes	maximal
6	#22559897	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#3162	C₂	1	1	yes	maximal, characteristic
8	#16863527	—	2	4	no	—
9	Q₈×C₆₇⋊C₈	—	2	4	no	—
10	C₂×#14523979	C₂²	1	4	yes	—
11	#16863423	C₂²	1	4	yes	—
12	#16863420	C₂²	1	2	yes	—
13	#16863352	C₂²	1	4	yes	—
14	C₂×#14523953	C₂²	1	2	yes	—
15	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
17	#16863242	C₂²	1	1	yes	characteristic
18	#16863155	C₂²	1	8	yes	—
19	#533×C₆₇	C₂²	1	2	yes	—
20	#501×C₆₇	C₄	1	8	yes	complemented
21	#427×C₆₇	C₂²	1	4	yes	—
22	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#11761612	C₂³	1	4	yes	—
24	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
25	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
26	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
27	#14523979	—	2	8	no	—
28	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
29	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
30	#14523954	—	2	4	no	—
31	#14523953	Q₈	1	8	yes	complemented
32	C₄×C₆₇⋊C₈	—	2	4	no	—
33	C₄×C₆₇⋊C₈	D₄	1	4	yes	—
34	C₈×Dic₆₇	—	2	4	no	—
35	C₆₇⋊#95	—	2	2	no	—
36	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
37	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
38	#125×C₆₇	C₂×C₄	1	4	yes	—
39	C₂²⋊Q₈×C₆₇	—	2	4	no	—
40	C₄×Q₈×C₆₇	—	2	4	no	—
41	#117×C₆₇	C₂³	1	4	yes	—
42	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
43	#107×C₆₇	C₂×C₄	1	8	yes	—
44	#106×C₆₇	C₂×C₄	1	8	yes	—
45	#97×C₆₇	C₂×C₄	1	4	yes	—
46	C₂²×Dic₆₇	—	2	2	no	—
47	#11761612	—	2	8	no	—
48	C₂×C₆₇⋊C₈	—	2	2	no	—
49	C₂×C₆₇⋊C₈	—	2	4	no	—
50	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
51	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
52	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
53	C₄×Dic₆₇	—	2	2	no	—
54	C₄×Dic₆₇	C₂×Q₈	1	4	yes	—
55	C₂×Q₈×C₆₇	—	2	4	no	—
56	C₂²×C₄×C₆₇	#41	1	2	yes	—
57	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
58	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
59	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
60	C₂×C₈×C₆₇	#41	1	4	yes	—
61	C₄⋊C₄×C₆₇	—	2	8	no	—
62	C₄⋊C₄×C₆₇	—	2	4	no	—
63	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
64	C₂²⋊C₄×C₆₇	—	2	4	no	—
65	C₄²×C₆₇	—	2	4	no	—
66	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
67	C₄²×C₆₇	C₂×D₄	1	2	yes	—
68	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
69	Q₈×C₆₇	—	4	8	no	—
70	C₂×C₄×C₆₇	—	2	8	no	—
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	4	no	—
78	C₂×Dic₆₇	—	4	4	no	—
79	C₈×C₆₇	C₄×Q₈	1	8	yes	—
80	C₆₇⋊C₈	—	2	2	no	—
81	C₆₇⋊C₈	—	4	4	no	—
82	C₆₇⋊C₈	C₄⋊Q₈	1	4	yes	—
83	C₂²×C₆₇	—	2	2	no	—
84	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
85	C₂²×C₆₇	#533	1	2	yes	—
86	C₄×C₆₇	—	2	4	no	—
87	C₄×C₆₇	—	4	8	no	—
88	C₄×C₆₇	C₂×C₄×Q₈	1	2	yes	—
89	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
90	C₄×C₆₇	#532	1	1	yes	characteristic
91	Dic₆₇	—	4	4	no	—
92	#21737	—	67	1	no	maximal, Sylow
93	C₂×C₆₇	—	2	2	no	—
94	C₂×C₆₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
95	C₂×C₆₇	#2955	1	2	yes	—
96	#3162	D₆₇	1	1	yes	characteristic
97	#2960	—	67	2	no	—
98	#1995	—	67	2	no	—
99	#1945	—	67	4	no	—
100	#1849	—	67	1	no	—
101	#1773	—	67	4	no	—
102	#1726	—	67	1	no	—
103	C₆₇	#21737	1	1	yes	characteristic, complemented, Sylow
104	#533	C₂×D₆₇	1	2	yes	—
105	#501	Dic₆₇	1	8	yes	complemented
106	#446	—	134	4	no	—
107	C₈×Q₈	—	134	4	no	—
108	C₂×#108	—	67	4	no	—
109	#427	C₂×D₆₇	1	4	yes	—
110	#424	—	67	4	no	—
111	#423	—	67	2	no	—
112	#405	—	67	4	no	—
113	C₂×#97	—	67	2	no	—
114	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
115	C₂×C₄×C₈	—	67	1	no	—
116	C₂×C₄×C₈	—	67	2	no	—
117	#376	—	67	1	no	—
118	#339	—	67	8	no	—
119	C₂×M₁₆	—	67	4	no	—
120	C₂²×C₈	—	67	1	no	—
121	C₂²×C₈	—	67	1	no	—
122	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
123	C₄⋊Q₈	C₂×Dic₆₇	1	4	yes	—
124	#125	C₂×Dic₆₇	1	4	yes	—
125	C₂²⋊Q₈	—	2	4	no	—
126	C₄×Q₈	—	2	4	no	—
127	#117	C₂²×D₆₇	1	4	yes	—
128	C₂×C₄²	—	67	2	no	—
129	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
130	#108	—	134	8	no	—
131	#107	C₂×Dic₆₇	1	8	yes	—
132	#106	C₂×Dic₆₇	1	8	yes	—
133	C₄⋊C₈	—	134	8	no	—
134	C₄⋊C₈	—	134	4	no	—
135	#97	—	67	8	no	—
136	#97	—	134	4	no	—
137	#97	C₂×Dic₆₇	1	4	yes	—
138	C₄×C₈	—	67	4	no	—
139	C₄×C₈	—	134	4	no	—
140	C₄×C₈	—	134	4	no	—
141	#95	—	134	2	no	—
142	C₂×Q₈	—	2	4	no	—
143	C₂²×C₄	—	134	2	no	—
144	C₂²×C₄	#11761627	1	2	yes	—
145	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
146	M₁₆	—	134	8	no	—
147	C₂×C₈	—	67	2	no	—
148	C₂×C₈	—	67	4	no	—
149	C₂×C₈	—	67	2	no	—
150	C₂×C₈	—	134	4	no	—
151	C₂×C₈	—	134	2	no	—
152	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
153	C₂×C₈	Q₈×D₆₇	1	4	yes	—
154	C₂×C₈	#11761629	1	4	yes	—
155	C₄⋊C₄	—	2	8	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	C₂²×Dic₆₇	1	8	yes	—
158	C₂²⋊C₄	—	2	4	no	—
159	C₄²	—	2	4	no	—
160	C₄²	—	67	4	no	—
161	C₄²	—	134	2	no	—
162	C₄²	C₂²×Dic₆₇	1	2	yes	—
163	C₄²	D₄×D₆₇	1	2	yes	—
164	C₂³	C₂×#11761627	1	1	yes	characteristic
165	Q₈	—	4	8	no	—
166	C₂×C₄	—	2	8	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	268	4	no	—
169	C₂×C₄	—	134	4	no	—
170	C₂×C₄	D₄×Dic₆₇	1	4	yes	—
171	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
172	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
173	C₂×C₄	C₂×Q₈×D₆₇	1	2	yes	—
174	C₂×C₄	C₂×#11761629	1	2	yes	—
175	C₈	—	268	4	no	—
176	C₈	—	67	4	no	—
177	C₈	—	134	2	no	—
178	C₈	Q₈×Dic₆₇	1	8	yes	—
179	C₂²	—	2	2	no	—
180	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
181	C₂²	#16863951	1	2	yes	—
182	C₄	—	2	4	no	—
183	C₄	—	4	8	no	—
184	C₄	—	268	4	no	—
185	C₄	C₂×Q₈×Dic₆₇	1	2	yes	—
186	C₄	C₄⋊Q₈×D₆₇	1	1	yes	characteristic
187	C₄	#16863947	1	1	yes	characteristic
188	C₂	—	2	2	no	—
189	C₂	C₄⋊Q₈×Dic₆₇	1	1	yes	characteristic
190	C₂	#22569112	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22569144

C₂

yes

maximal

#22561191

C₂

yes

maximal

#22560958

C₂

yes

maximal

#22560384

C₂

yes

maximal, characteristic

#22560095

C₂

yes

maximal

#22559897

C₂

yes

maximal, characteristic

C₆₇×#3162

C₂

yes

maximal, characteristic

#16863527

—

Q₈×C₆₇⋊C₈

—

C₂×#14523979

C₂²

yes

—

#16863423

C₂²

yes

—

#16863420

C₂²

yes

—

#16863352

C₂²

yes

—

C₂×#14523953

C₂²

yes

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×C₈×Dic₆₇

C₂²

yes

—

#16863242

C₂²

yes

characteristic

#16863155

C₂²

yes

—

#533×C₆₇

C₂²

yes

—

#501×C₆₇

C₄

yes

complemented

#427×C₆₇

C₂²

yes

—

C₂×#97×C₆₇

C₂²

yes

characteristic

C₂×#11761612

C₂³

yes

—

C₂²×C₆₇⋊C₈

C₂³

yes

characteristic

C₂²×C₆₇⋊C₈

C₂³

yes

characteristic

C₂×C₄×Dic₆₇

C₂³

yes

—

#14523979

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

#14523954

—

#14523953

Q₈

yes

complemented

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₈×Dic₆₇

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

C₂³

yes

—

C₄⋊Q₈×C₆₇

C₂×C₄

yes

—

#125×C₆₇

C₂×C₄

yes

—

C₂²⋊Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

#117×C₆₇

C₂³

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#107×C₆₇

C₂×C₄

yes

—

#106×C₆₇

C₂×C₄

yes

—

#97×C₆₇

C₂×C₄

yes

—

C₂²×Dic₆₇

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×D₄

yes

—

C₂×C₆₇⋊C₈

C₂×D₄

yes

—

C₂×C₆₇⋊C₈

C₂×Q₈

yes

—

C₄×Dic₆₇

—

C₄×Dic₆₇

C₂×Q₈

yes

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

#41

yes

—

C₂²×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×C₈×C₆₇

C₂²×C₄

yes

—

C₂×C₈×C₆₇

C₂×Q₈

yes

—

C₂×C₈×C₆₇

#41

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂²×C₄

yes

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂²×C₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

—

C₂³×C₆₇

C₂×#41

yes

characteristic

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

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

—

C₈×C₆₇

C₄×Q₈

yes

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

C₄⋊Q₈

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₄×D₄

yes

characteristic

C₂²×C₆₇

#533

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×C₄×Q₈

yes

—

C₄×C₆₇

C₂×C₄⋊Q₈

yes

characteristic

C₄×C₆₇

#532

yes

characteristic

Dic₆₇

—

#21737

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

C₄×C₄⋊Q₈

yes

characteristic

C₂×C₆₇

#2955

yes

—

#3162

D₆₇

yes

—

—

—

100

#1849

—

101

#1773

—

102

#1726

—

103

C₆₇

#21737

yes

characteristic, complemented, Sylow

104

#533

C₂×D₆₇

yes

—

105

#501

Dic₆₇

yes

complemented

106

#446

—

134

—

107

C₈×Q₈

—

134

—

108

C₂×#108

—

109

#427

C₂×D₆₇

yes

—

110

#424

—

111

#423

—

112

#405

—

113

C₂×#97

—

114

C₂×#97

C₂×D₆₇

yes

characteristic

115

C₂×C₄×C₈

—

116

C₂×C₄×C₈

—

117

#376

—

118

#339

—

119

C₂×M₁₆

—

120

C₂²×C₈

—

121

C₂²×C₈

—

122

C₂²×C₈

C₂²×D₆₇

yes

—

123

C₄⋊Q₈

C₂×Dic₆₇

yes

—

124

#125

C₂×Dic₆₇

yes

—

125

C₂²⋊Q₈

—

126

C₄×Q₈

—

127

#117

C₂²×D₆₇

yes

—

128

C₂×C₄²

—

129

C₂×C₄²

C₂²×D₆₇

yes

characteristic

130

#108

—

134

—

131

#107

C₂×Dic₆₇

yes

—

132

#106

C₂×Dic₆₇

yes

—

133

C₄⋊C₈

—

134

—

134

C₄⋊C₈

—

134

—

135

#97

—

136

#97

—

134

—

137

#97

C₂×Dic₆₇

yes

—

138

C₄×C₈

—

139

C₄×C₈

—

134

—

140

C₄×C₈

—

134

—

141

#95

—

134

—

142

C₂×Q₈

—

143

C₂²×C₄

—

134

—

144

C₂²×C₄

#11761627

yes

—

145

C₂²×C₄

C₂³×D₆₇

yes

characteristic

146

M₁₆

—

134

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

134

—

151

C₂×C₈

—

134

—

152

C₂×C₈

C₂²×Dic₆₇

yes

—

153

C₂×C₈

Q₈×D₆₇

yes

—

154

C₂×C₈

#11761629

yes

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

C₂²×Dic₆₇

yes

—

158

C₂²⋊C₄

—

159

C₄²

—

160

C₄²

—

161

C₄²

—

134

—

162

C₄²

C₂²×Dic₆₇

yes

—

163

C₄²

D₄×D₆₇

yes

—

164

C₂³

C₂×#11761627

yes

characteristic

165

Q₈

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

268

—

169

C₂×C₄

—

134

—

170

C₂×C₄

D₄×Dic₆₇

yes

—

171

C₂×C₄

C₂³×Dic₆₇

yes

characteristic

172

C₂×C₄

C₂×D₄×D₆₇

yes

characteristic

173

C₂×C₄

C₂×Q₈×D₆₇

yes

—

174

C₂×C₄

C₂×#11761629

yes

—

175

C₈

—

268

—

176

C₈

—

177

C₈

—

134

—

178

C₈

Q₈×Dic₆₇

yes

—

179

C₂²

—

180

C₂²

C₂×D₄×Dic₆₇

yes

characteristic

181

C₂²

#16863951

yes

—

182

C₄

—

183

C₄

—

184

C₄

—

268

—

185

C₄

C₂×Q₈×Dic₆₇

yes

—

186

C₄

C₄⋊Q₈×D₆₇

yes

characteristic

187

C₄

#16863947

yes

characteristic

188

C₂

—

189

C₂

C₄⋊Q₈×Dic₆₇

yes

characteristic

190

C₂

#22569112

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#3162	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₂×#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#3162	D₆₇
hypocenter	C₆₇	#21737
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3162

subgroup

G/H

center

C₂×C₄

C₂×D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂³×C₄

Frattini

C₂²×C₄

C₂³×D₆₇

Fitting

C₆₇×#3162

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 ◁ #3162

N	H ≅ G/N	type
20. #501×C₆₇	C₄	semidirect
31. #14523953	Q₈	semidirect
103. C₆₇	#21737	semidirect
105. #501	Dic₆₇	semidirect

H ≅ G/N

type

20. #501×C₆₇

C₄

semidirect

31. #14523953

Q₈

semidirect

103. C₆₇

#21737

semidirect

105. #501

Dic₆₇

semidirect

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

subgroup	index
#22569144	2
#22561191	2
#22560958	2
#22560384	2
#22560095	2
#22559897	2
C₆₇×#3162	2
#21737	67

subgroup

index

C₆₇×#3162

Subgroups — #30864718

Conjugacy classes of subgroups

Complements

Maximal subgroups