Group #30689113 — subgroups

Subgroups — C₆₇×#9862 #30689113

Order 17152.

514 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 174 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	C₆₇×#3147	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1848	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#1492	C₂	1	4	yes	maximal, complemented
4	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#135×C₆₇	—	2	2	no	—
6	#532×C₆₇	C₂²	1	2	yes	—
7	#495×C₆₇	—	2	2	no	—
8	#494×C₆₇	C₄	1	1	yes	characteristic
9	#493×C₆₇	C₄	1	1	yes	characteristic
10	#492×C₆₇	C₄	1	2	yes	—
11	C₂×#108×C₆₇	—	2	2	no	—
12	#423×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	D₄⋊C₈×C₆₇	—	2	8	no	—
16	#322×C₆₇	—	2	2	no	—
17	C₂×#41×C₆₇	—	2	2	no	—
18	#135×C₆₇	—	4	4	no	—
19	#135×C₆₇	—	4	4	no	—
20	C₂×Q₁₆×C₆₇	—	2	2	no	—
21	C₂×SD₁₆×C₆₇	—	2	4	no	—
22	C₂×D₈×C₆₇	—	2	2	no	—
23	C₂×M₁₆×C₆₇	—	2	2	no	—
24	C₂²×C₈×C₆₇	—	2	2	no	—
25	C₂²×C₈×C₆₇	D₄	1	2	yes	—
26	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
27	#127×C₆₇	C₂×C₄	1	2	yes	—
28	#124×C₆₇	—	2	2	no	—
29	#121×C₆₇	—	2	4	no	—
30	C₄×D₄×C₆₇	—	2	4	no	—
31	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
32	#108×C₆₇	—	4	4	no	—
33	C₄⋊C₈×C₆₇	—	2	4	no	—
34	C₄×C₈×C₆₇	—	2	2	no	—
35	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₆₇	D₄	1	4	yes	—
38	C₄×C₈×C₆₇	D₄	1	2	yes	—
39	#41×C₆₇	—	4	4	no	—
40	#41×C₆₇	—	4	4	no	—
41	C₂×Q₈×C₆₇	—	2	2	no	—
42	C₂×D₄×C₆₇	—	2	2	no	—
43	C₂×D₄×C₆₇	—	2	4	no	—
44	C₂×D₄×C₆₇	—	4	2	no	—
45	C₂²×C₄×C₆₇	—	4	2	no	—
46	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₇	#41	1	2	yes	—
48	Q₁₆×C₆₇	—	4	4	no	—
49	SD₁₆×C₆₇	—	4	8	no	—
50	D₈×C₆₇	—	4	4	no	—
51	M₁₆×C₆₇	—	4	4	no	—
52	C₂×C₈×C₆₇	—	2	4	no	—
53	C₂×C₈×C₆₇	—	4	4	no	—
54	C₂×C₈×C₆₇	—	4	2	no	—
55	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₆₇	D₈	1	4	yes	—
58	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
59	C₄⋊C₄×C₆₇	—	2	4	no	—
60	C₂²⋊C₄×C₆₇	—	4	4	no	—
61	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₆₇	C₂×D₄	1	2	yes	—
64	C₂³×C₆₇	—	4	2	no	—
65	C₂³×C₆₇	#124	1	1	yes	characteristic
66	Q₈×C₆₇	—	4	4	no	—
67	D₄×C₆₇	—	4	4	no	—
68	D₄×C₆₇	—	4	8	no	—
69	D₄×C₆₇	—	8	4	no	—
70	C₂×C₄×C₆₇	—	2	4	no	—
71	C₂×C₄×C₆₇	—	4	2	no	—
72	C₂×C₄×C₆₇	—	8	2	no	—
73	C₂×C₄×C₆₇	—	8	2	no	—
74	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
76	C₂×C₄×C₆₇	#121	1	2	yes	—
77	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
78	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
79	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
80	C₈×C₆₇	—	4	4	no	—
81	C₈×C₆₇	—	8	2	no	—
82	C₈×C₆₇	#102	1	4	yes	—
83	C₈×C₆₇	#102	1	4	yes	—
84	C₂²×C₆₇	—	2	2	no	—
85	C₂²×C₆₇	—	8	2	no	—
86	C₂²×C₆₇	—	8	4	no	—
87	C₂²×C₆₇	#390	1	1	yes	characteristic
88	C₂²×C₆₇	#486	1	2	yes	—
89	C₄×C₆₇	—	2	2	no	—
90	C₄×C₆₇	—	8	2	no	—
91	C₄×C₆₇	C₂×#102	1	2	yes	—
92	C₄×C₆₇	#444	1	4	yes	—
93	C₄×C₆₇	#492	1	1	yes	characteristic
94	C₄×C₆₇	#493	1	1	yes	characteristic
95	#9862	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂×C₆₇	—	2	2	no	—
97	C₂×C₆₇	—	16	2	no	—
98	C₂×C₆₇	#1957	1	1	yes	characteristic
99	C₂×C₆₇	#1975	1	2	yes	—
100	#3147	C₂×C₆₇	1	1	yes	characteristic
101	#1848	C₂×C₆₇	1	1	yes	characteristic, complemented
102	#1492	C₂×C₆₇	1	4	yes	complemented
103	C₂×#322	C₂×C₆₇	1	1	yes	characteristic, complemented
104	C₆₇	#9862	1	1	yes	characteristic, direct product factor, Sylow
105	C₂×#135	—	2	2	no	—
106	#532	C₂²×C₆₇	1	2	yes	—
107	#495	—	2	2	no	—
108	#494	C₄×C₆₇	1	1	yes	characteristic
109	#493	C₄×C₆₇	1	1	yes	characteristic
110	#492	C₄×C₆₇	1	2	yes	—
111	C₂×#108	—	2	2	no	—
112	#423	C₂²×C₆₇	1	2	yes	—
113	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
114	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
115	D₄⋊C₈	—	2	8	no	—
116	#322	—	2	2	no	—
117	C₂×#41	—	2	2	no	—
118	#135	—	4	4	no	—
119	#135	—	4	4	no	—
120	C₂×Q₁₆	—	2	2	no	—
121	C₂×SD₁₆	—	2	4	no	—
122	C₂×D₈	—	2	2	no	—
123	C₂×M₁₆	—	2	2	no	—
124	C₂²×C₈	—	2	2	no	—
125	C₂²×C₈	D₄×C₆₇	1	2	yes	—
126	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
127	#127	C₂×C₄×C₆₇	1	2	yes	—
128	#124	—	2	2	no	—
129	#121	—	2	4	no	—
130	C₄×D₄	—	2	4	no	—
131	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
132	#108	—	4	4	no	—
133	C₄⋊C₈	—	2	4	no	—
134	C₄×C₈	—	2	2	no	—
135	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
136	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
137	C₄×C₈	D₄×C₆₇	1	4	yes	—
138	C₄×C₈	D₄×C₆₇	1	2	yes	—
139	#41	—	4	4	no	—
140	#41	—	4	4	no	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×D₄	—	2	2	no	—
143	C₂×D₄	—	2	4	no	—
144	C₂×D₄	—	4	2	no	—
145	C₂²×C₄	—	4	2	no	—
146	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
147	C₂²×C₄	#41×C₆₇	1	2	yes	—
148	Q₁₆	—	4	4	no	—
149	SD₁₆	—	4	8	no	—
150	D₈	—	4	4	no	—
151	M₁₆	—	4	4	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	4	4	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	C₂²⋊C₄×C₆₇	1	2	yes	—
156	C₂×C₈	C₂²⋊C₄×C₆₇	1	2	yes	—
157	C₂×C₈	D₈×C₆₇	1	4	yes	—
158	C₂×C₈	SD₁₆×C₆₇	1	4	yes	—
159	C₄⋊C₄	—	2	4	no	—
160	C₂²⋊C₄	—	4	4	no	—
161	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
162	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
163	C₄²	C₂×D₄×C₆₇	1	2	yes	—
164	C₂³	—	4	2	no	—
165	C₂³	#124×C₆₇	1	1	yes	characteristic
166	Q₈	—	4	4	no	—
167	D₄	—	4	4	no	—
168	D₄	—	4	8	no	—
169	D₄	—	8	4	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
175	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
176	C₂×C₄	#121×C₆₇	1	2	yes	—
177	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
178	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
179	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
180	C₈	—	4	4	no	—
181	C₈	—	8	2	no	—
182	C₈	#102×C₆₇	1	4	yes	—
183	C₈	#102×C₆₇	1	4	yes	—
184	C₂²	—	2	2	no	—
185	C₂²	—	8	2	no	—
186	C₂²	—	8	4	no	—
187	C₂²	#390×C₆₇	1	1	yes	characteristic
188	C₂²	#486×C₆₇	1	2	yes	—
189	C₄	—	2	2	no	—
190	C₄	—	8	2	no	—
191	C₄	C₂×#102×C₆₇	1	2	yes	—
192	C₄	#444×C₆₇	1	4	yes	—
193	C₄	#492×C₆₇	1	1	yes	characteristic
194	C₄	#493×C₆₇	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	16	2	no	—
197	C₂	C₆₇×#1957	1	1	yes	characteristic
198	C₂	C₆₇×#1975	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₇×#3147

C₂

yes

maximal, characteristic

C₆₇×#1848

C₂

yes

maximal, characteristic, complemented

C₆₇×#1492

C₂

yes

maximal, complemented

C₂×#322×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#135×C₆₇

—

#532×C₆₇

C₂²

yes

—

#495×C₆₇

—

#494×C₆₇

C₄

yes

characteristic

#493×C₆₇

C₄

yes

characteristic

#492×C₆₇

C₄

yes

—

C₂×#108×C₆₇

—

#423×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

D₄⋊C₈×C₆₇

—

#322×C₆₇

—

C₂×#41×C₆₇

—

#135×C₆₇

—

#135×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₂×SD₁₆×C₆₇

—

C₂×D₈×C₆₇

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

—

C₄⋊Q₈×C₆₇

C₂×C₄

yes

—

#127×C₆₇

C₂×C₄

yes

—

#124×C₆₇

—

#121×C₆₇

—

C₄×D₄×C₆₇

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#108×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

Q₁₆×C₆₇

—

SD₁₆×C₆₇

—

D₈×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

SD₁₆

yes

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

—

C₂³×C₆₇

—

C₂³×C₆₇

#124

yes

characteristic

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

—

C₂×C₄×C₆₇

C₂×SD₁₆

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#102

yes

—

C₈×C₆₇

#102

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#390

yes

characteristic

C₂²×C₆₇

#486

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×#102

yes

—

C₄×C₆₇

#444

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#1957

yes

characteristic

C₂×C₆₇

#1975

yes

—

100

#3147

C₂×C₆₇

yes

characteristic

101

#1848

C₂×C₆₇

yes

characteristic, complemented

102

#1492

C₂×C₆₇

yes

complemented

103

C₂×#322

C₂×C₆₇

yes

characteristic, complemented

104

C₆₇

#9862

yes

characteristic, direct product factor, Sylow

105

C₂×#135

—

106

#532

C₂²×C₆₇

yes

—

107

#495

—

108

#494

C₄×C₆₇

yes

characteristic

109

#493

C₄×C₆₇

yes

characteristic

110

#492

C₄×C₆₇

yes

—

111

C₂×#108

—

112

#423

C₂²×C₆₇

yes

—

113

C₂×C₄×C₈

C₂²×C₆₇

yes

—

114

C₂×C₄×C₈

C₂²×C₆₇

yes

characteristic

115

D₄⋊C₈

—

116

#322

—

117

C₂×#41

—

118

#135

—

119

#135

—

120

C₂×Q₁₆

—

121

C₂×SD₁₆

—

122

C₂×D₈

—

123

C₂×M₁₆

—

124

C₂²×C₈

—

125

C₂²×C₈

D₄×C₆₇

yes

—

126

C₄⋊Q₈

C₂×C₄×C₆₇

yes

—

127

#127

C₂×C₄×C₆₇

yes

—

128

#124

—

129

#121

—

130

C₄×D₄

—

131

C₂×C₄²

C₂³×C₆₇

yes

characteristic

132

#108

—

133

C₄⋊C₈

—

134

C₄×C₈

—

135

C₄×C₈

C₂×C₄×C₆₇

yes

characteristic

136

C₄×C₈

C₂×C₄×C₆₇

yes

characteristic

137

C₄×C₈

D₄×C₆₇

yes

—

138

C₄×C₈

D₄×C₆₇

yes

—

139

#41

—

140

#41

—

141

C₂×Q₈

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂²×C₄

—

146

C₂²×C₄

C₂×D₄×C₆₇

yes

characteristic

147

C₂²×C₄

#41×C₆₇

yes

—

148

Q₁₆

—

149

SD₁₆

—

150

D₈

—

151

M₁₆

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

C₂²⋊C₄×C₆₇

yes

—

156

C₂×C₈

C₂²⋊C₄×C₆₇

yes

—

157

C₂×C₈

D₈×C₆₇

yes

—

158

C₂×C₈

SD₁₆×C₆₇

yes

—

159

C₄⋊C₄

—

160

C₂²⋊C₄

—

161

C₄²

C₂²×C₄×C₆₇

yes

characteristic

162

C₄²

C₂×D₄×C₆₇

yes

characteristic

163

C₄²

C₂×D₄×C₆₇

yes

—

164

C₂³

—

165

C₂³

#124×C₆₇

yes

characteristic

166

Q₈

—

167

D₄

—

168

D₄

—

169

D₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

C₂×C₂²⋊C₄×C₆₇

yes

characteristic

175

C₂×C₄

C₄×D₄×C₆₇

yes

—

176

C₂×C₄

#121×C₆₇

yes

—

177

C₂×C₄

#127×C₆₇

yes

characteristic

178

C₂×C₄

C₂×D₈×C₆₇

yes

—

179

C₂×C₄

C₂×SD₁₆×C₆₇

yes

—

180

C₈

—

181

C₈

—

182

C₈

#102×C₆₇

yes

—

183

C₈

#102×C₆₇

yes

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

#390×C₆₇

yes

characteristic

188

C₂²

#486×C₆₇

yes

—

189

C₄

—

190

C₄

—

191

C₄

C₂×#102×C₆₇

yes

—

192

C₄

#444×C₆₇

yes

—

193

C₄

#492×C₆₇

yes

characteristic

194

C₄

#493×C₆₇

yes

characteristic

195

C₂

—

196

C₂

—

197

C₂

C₆₇×#1957

yes

characteristic

198

C₂

C₆₇×#1975

yes

—

	subgroup	G/H
center	C₂×C₄×C₆₇	#127
commutator	C₄²	C₂²×C₄×C₆₇
Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
socle	C₂²×C₆₇	#390
prime radical	C₆₇×#3147	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

subgroup

G/H

center

C₂×C₄×C₆₇

#127

commutator

C₄²

C₂²×C₄×C₆₇

Frattini, cosocle

C₂×C₄²

C₂³×C₆₇

socle

C₂²×C₆₇

#390

prime radical

C₆₇×#3147

C₂

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

N	H ≅ G/N	type
2. C₆₇×#1848	C₂	semidirect
3. C₆₇×#1492	C₂	semidirect
4. C₂×#322×C₆₇	C₂	semidirect
95. #9862	C₆₇	direct (N × H)
101. #1848	C₂×C₆₇	semidirect
102. #1492	C₂×C₆₇	semidirect
103. C₂×#322	C₂×C₆₇	semidirect
104. C₆₇	#9862	direct (N × H)

H ≅ G/N

type

2. C₆₇×#1848

C₂

semidirect

3. C₆₇×#1492

C₂

semidirect

4. C₂×#322×C₆₇

C₂

semidirect

95. #9862

C₆₇

direct (N × H)

101. #1848

C₂×C₆₇

semidirect

102. #1492

C₂×C₆₇

semidirect

103. C₂×#322

C₂×C₆₇

semidirect

104. C₆₇

#9862

direct (N × H)

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

subgroup	index
C₆₇×#3147	2
C₆₇×#1848	2
C₆₇×#1492	2
C₂×#322×C₆₇	2
#9862	67

subgroup

index

C₆₇×#3147

C₆₇×#1848

C₆₇×#1492

C₂×#322×C₆₇

#9862

Subgroups — C₆₇×#9862 #30689113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#9862 #30689113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#9862 #30689113