Group #30712414 — subgroups

Subgroups — C₆₇×#33163 #30712414

Order 17152.

1074 conjugacy classes of subgroups; only the 366 normal subgroups are shown.

normal subgroups only

This group has 1074 conjugacy classes of subgroups — too many to draw. Only its 366 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	C₂×#462×C₆₇	C₂	1	6	yes	maximal
2	C₆₇×#2868	C₂	1	1	yes	maximal, characteristic
3	C₄²×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1981	C₂	1	6	yes	maximal
5	C₆₇×#1842	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₆₇	C₂²	1	4	yes	—
7	C₂×C₄⋊Q₈×C₆₇	C₂²	1	6	yes	—
8	C₂×C₄×Q₈×C₆₇	C₂²	1	3	yes	—
9	#462×C₆₇	—	2	24	no	—
10	C₂×C₄⋊C₈×C₆₇	C₂²	1	6	yes	—
11	C₂×#103×C₆₇	C₂²	1	12	yes	—
12	#384×C₆₇	—	2	3	no	—
13	#384×C₆₇	—	2	1	no	—
14	C₄×C₄⋊C₄×C₆₇	—	2	3	no	—
15	C₄×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
16	C₄³×C₆₇	—	2	1	no	—
17	C₄³×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
19	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
20	C₂×Q₁₆×C₆₇	—	2	24	no	—
21	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
22	C₄⋊Q₈×C₆₇	—	2	12	no	—
23	C₄⋊Q₈×C₆₇	—	2	12	no	—
24	C₄×Q₈×C₆₇	—	2	6	no	—
25	C₄×Q₈×C₆₇	D₄	1	12	yes	—
26	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
27	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
28	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
29	C₂×C₄⋊C₄×C₆₇	C₂³	1	3	yes	—
30	C₂×C₄²×C₆₇	—	2	3	no	—
31	C₂×C₄²×C₆₇	—	2	3	no	—
32	C₂×C₄²×C₆₇	D₄	1	3	yes	—
33	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
35	C₄⋊C₈×C₆₇	—	2	12	no	—
36	#103×C₆₇	—	2	24	no	—
37	#95×C₆₇	—	2	3	no	—
38	C₂×Q₈×C₆₇	—	2	3	no	—
39	C₂×Q₈×C₆₇	—	2	12	no	—
40	C₂×Q₈×C₆₇	—	2	12	no	—
41	C₂×Q₈×C₆₇	C₂×D₄	1	6	yes	—
42	C₂²×C₄×C₆₇	—	2	1	no	—
43	C₂²×C₄×C₆₇	—	2	3	no	—
44	C₂²×C₄×C₆₇	—	2	4	no	—
45	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
46	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
47	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
48	Q₁₆×C₆₇	—	4	16	no	—
49	C₂×C₈×C₆₇	—	2	12	no	—
50	C₄⋊C₄×C₆₇	—	2	12	no	—
51	C₄⋊C₄×C₆₇	—	2	3	no	—
52	C₄⋊C₄×C₆₇	—	4	12	no	—
53	C₄⋊C₄×C₆₇	—	4	12	no	—
54	C₄⋊C₄×C₆₇	C₂×D₄	1	6	yes	—
55	C₄²×C₆₇	—	2	12	no	—
56	C₄²×C₆₇	—	2	6	no	—
57	C₄²×C₆₇	—	4	6	no	—
58	C₄²×C₆₇	Q₁₆	1	4	yes	complemented
59	C₄²×C₆₇	C₂×D₄	1	6	yes	—
60	C₄²×C₆₇	#41	1	6	yes	—
61	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
62	Q₈×C₆₇	—	2	6	no	—
63	Q₈×C₆₇	—	4	4	no	—
64	Q₈×C₆₇	—	4	12	no	—
65	Q₈×C₆₇	#127	1	4	yes	—
66	C₂×C₄×C₆₇	—	2	6	no	—
67	C₂×C₄×C₆₇	—	2	6	no	—
68	C₂×C₄×C₆₇	—	2	6	no	—
69	C₂×C₄×C₆₇	—	4	6	no	—
70	C₂×C₄×C₆₇	—	4	12	no	—
71	C₂×C₄×C₆₇	#121	1	6	yes	—
72	C₂×C₄×C₆₇	#121	1	6	yes	—
73	C₂×C₄×C₆₇	C₂×Q₁₆	1	6	yes	—
74	C₂×C₄×C₆₇	#137	1	6	yes	—
75	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
76	C₂×C₄×C₆₇	C₂×#41	1	3	yes	—
77	C₈×C₆₇	—	4	4	no	—
78	C₂²×C₆₇	C₂×#121	1	3	yes	—
79	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
80	C₂²×C₆₇	C₂×#137	1	3	yes	—
81	C₄×C₆₇	—	2	4	no	—
82	C₄×C₆₇	—	2	6	no	—
83	C₄×C₆₇	—	4	6	no	—
84	C₄×C₆₇	—	8	4	no	—
85	C₄×C₆₇	#462	1	12	yes	complemented
86	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
87	C₄×C₆₇	#532	1	3	yes	—
88	#33163	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂×C₆₇	#2856	1	1	yes	characteristic
90	C₂×C₆₇	C₂×#462	1	3	yes	—
91	C₂×C₆₇	#3039	1	3	yes	—
92	C₂×#462	C₂×C₆₇	1	6	yes	—
93	#2868	C₂×C₆₇	1	1	yes	characteristic
94	C₄²×Q₈	C₂×C₆₇	1	1	yes	characteristic
95	#1981	C₂×C₆₇	1	6	yes	—
96	#1842	C₂×C₆₇	1	1	yes	characteristic
97	C₆₇	#33163	1	1	yes	characteristic, direct product factor, Sylow
98	C₂²×Q₁₆	C₂²×C₆₇	1	4	yes	—
99	C₂×C₄⋊Q₈	C₂²×C₆₇	1	6	yes	—
100	C₂×C₄×Q₈	C₂²×C₆₇	1	3	yes	—
101	#462	—	2	24	no	—
102	C₂×C₄⋊C₈	C₂²×C₆₇	1	6	yes	—
103	C₂×#103	C₂²×C₆₇	1	12	yes	—
104	#384	—	2	3	no	—
105	#384	—	2	1	no	—
106	C₄×C₄⋊C₄	—	2	3	no	—
107	C₄×C₄⋊C₄	C₂²×C₆₇	1	3	yes	—
108	C₄³	—	2	1	no	—
109	C₄³	C₂²×C₆₇	1	1	yes	characteristic
110	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
111	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
112	C₂×Q₁₆	—	2	24	no	—
113	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
114	C₄⋊Q₈	—	2	12	no	—
115	C₄⋊Q₈	—	2	12	no	—
116	C₄×Q₈	—	2	6	no	—
117	C₄×Q₈	D₄×C₆₇	1	12	yes	—
118	C₂×C₄⋊C₄	—	2	3	no	—
119	C₂×C₄⋊C₄	—	2	6	no	—
120	C₂×C₄⋊C₄	—	2	6	no	—
121	C₂×C₄⋊C₄	C₂³×C₆₇	1	3	yes	—
122	C₂×C₄²	—	2	3	no	—
123	C₂×C₄²	—	2	3	no	—
124	C₂×C₄²	D₄×C₆₇	1	3	yes	—
125	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
126	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
127	C₄⋊C₈	—	2	12	no	—
128	#103	—	2	24	no	—
129	#95	—	2	3	no	—
130	C₂×Q₈	—	2	3	no	—
131	C₂×Q₈	—	2	12	no	—
132	C₂×Q₈	—	2	12	no	—
133	C₂×Q₈	C₂×D₄×C₆₇	1	6	yes	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	3	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
138	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
139	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
140	Q₁₆	—	4	16	no	—
141	C₂×C₈	—	2	12	no	—
142	C₄⋊C₄	—	2	12	no	—
143	C₄⋊C₄	—	2	3	no	—
144	C₄⋊C₄	—	4	12	no	—
145	C₄⋊C₄	—	4	12	no	—
146	C₄⋊C₄	C₂×D₄×C₆₇	1	6	yes	—
147	C₄²	—	2	12	no	—
148	C₄²	—	2	6	no	—
149	C₄²	—	4	6	no	—
150	C₄²	Q₁₆×C₆₇	1	4	yes	complemented
151	C₄²	C₂×D₄×C₆₇	1	6	yes	—
152	C₄²	#41×C₆₇	1	6	yes	—
153	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
154	Q₈	—	2	6	no	—
155	Q₈	—	4	4	no	—
156	Q₈	—	4	12	no	—
157	Q₈	#127×C₆₇	1	4	yes	—
158	C₂×C₄	—	2	6	no	—
159	C₂×C₄	—	2	6	no	—
160	C₂×C₄	—	2	6	no	—
161	C₂×C₄	—	4	6	no	—
162	C₂×C₄	—	4	12	no	—
163	C₂×C₄	#121×C₆₇	1	6	yes	—
164	C₂×C₄	#121×C₆₇	1	6	yes	—
165	C₂×C₄	C₂×Q₁₆×C₆₇	1	6	yes	—
166	C₂×C₄	#137×C₆₇	1	6	yes	—
167	C₂×C₄	C₂²×D₄×C₆₇	1	3	yes	—
168	C₂×C₄	C₂×#41×C₆₇	1	3	yes	—
169	C₈	—	4	4	no	—
170	C₂²	C₂×#121×C₆₇	1	3	yes	—
171	C₂²	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
172	C₂²	C₂×#137×C₆₇	1	3	yes	—
173	C₄	—	2	6	no	—
174	C₄	—	2	4	no	—
175	C₄	—	4	6	no	—
176	C₄	—	8	4	no	—
177	C₄	#462×C₆₇	1	12	yes	complemented
178	C₄	C₂×#127×C₆₇	1	1	yes	characteristic
179	C₄	#532×C₆₇	1	3	yes	—
180	C₂	C₆₇×#2856	1	1	yes	characteristic
181	C₂	C₂×#462×C₆₇	1	3	yes	—
182	C₂	C₆₇×#3039	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#462×C₆₇

C₂

yes

maximal

C₆₇×#2868

C₂

yes

maximal, characteristic

C₄²×Q₈×C₆₇

C₂

yes

maximal, characteristic

C₆₇×#1981

C₂

yes

maximal

C₆₇×#1842

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₆₇

C₂²

yes

—

C₂×C₄⋊Q₈×C₆₇

C₂²

yes

—

C₂×C₄×Q₈×C₆₇

C₂²

yes

—

#462×C₆₇

—

C₂×C₄⋊C₈×C₆₇

C₂²

yes

—

C₂×#103×C₆₇

C₂²

yes

—

#384×C₆₇

—

#384×C₆₇

—

C₄×C₄⋊C₄×C₆₇

—

C₄×C₄⋊C₄×C₆₇

C₂²

yes

—

C₄³×C₆₇

—

C₄³×C₆₇

C₂²

yes

characteristic

C₂²×Q₈×C₆₇

C₂³

yes

—

C₂²×Q₈×C₆₇

C₂³

yes

characteristic

C₂×Q₁₆×C₆₇

—

C₂²×C₈×C₆₇

C₂³

yes

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

C₄×Q₈×C₆₇

D₄

yes

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂³

yes

—

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

C₂³

yes

—

C₄⋊C₈×C₆₇

—

#103×C₆₇

—

#95×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

C₂⁴

yes

characteristic

Q₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

Q₁₆

yes

complemented

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

C₂²×D₄

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

#127

yes

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

C₂×Q₁₆

yes

—

C₂×C₄×C₆₇

#137

yes

—

C₂×C₄×C₆₇

C₂²×D₄

yes

—

C₂×C₄×C₆₇

C₂×#41

yes

—

C₈×C₆₇

—

C₂²×C₆₇

C₂×#121

yes

—

C₂²×C₆₇

C₂²×Q₁₆

yes

characteristic

C₂²×C₆₇

C₂×#137

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

#33163

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₇

#2856

yes

characteristic

C₂×C₆₇

C₂×#462

yes

—

C₂×C₆₇

#3039

yes

—

C₂×#462

C₂×C₆₇

yes

—

#2868

C₂×C₆₇

yes

characteristic

C₄²×Q₈

C₂×C₆₇

yes

characteristic

#1981

C₂×C₆₇

yes

—

#1842

C₂×C₆₇

yes

characteristic

C₆₇

#33163

yes

characteristic, direct product factor, Sylow

C₂²×Q₁₆

C₂²×C₆₇

yes

—

C₂×C₄⋊Q₈

C₂²×C₆₇

yes

—

100

C₂×C₄×Q₈

C₂²×C₆₇

yes

—

101

#462

—

102

C₂×C₄⋊C₈

C₂²×C₆₇

yes

—

103

C₂×#103

C₂²×C₆₇

yes

—

104

#384

—

105

#384

—

106

C₄×C₄⋊C₄

—

107

C₄×C₄⋊C₄

C₂²×C₆₇

yes

—

108

C₄³

—

109

C₄³

C₂²×C₆₇

yes

characteristic

110

C₂²×Q₈

C₂³×C₆₇

yes

—

111

C₂²×Q₈

C₂³×C₆₇

yes

characteristic

112

C₂×Q₁₆

—

113

C₂²×C₈

C₂³×C₆₇

yes

—

114

C₄⋊Q₈

—

115

C₄⋊Q₈

—

116

C₄×Q₈

—

117

C₄×Q₈

D₄×C₆₇

yes

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

C₂³×C₆₇

yes

—

122

C₂×C₄²

—

123

C₂×C₄²

—

124

C₂×C₄²

D₄×C₆₇

yes

—

125

C₂×C₄²

D₄×C₆₇

yes

characteristic

126

C₂×C₄²

C₂³×C₆₇

yes

—

127

C₄⋊C₈

—

128

#103

—

129

#95

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

C₂×D₄×C₆₇

yes

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

C₂×D₄×C₆₇

yes

—

138

C₂²×C₄

C₂×D₄×C₆₇

yes

—

139

C₂²×C₄

C₂⁴×C₆₇

yes

characteristic

140

Q₁₆

—

141

C₂×C₈

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

C₂×D₄×C₆₇

yes

—

147

C₄²

—

148

C₄²

—

149

C₄²

—

150

C₄²

Q₁₆×C₆₇

yes

complemented

151

C₄²

C₂×D₄×C₆₇

yes

—

152

C₄²

#41×C₆₇

yes

—

153

C₂³

C₂²×D₄×C₆₇

yes

characteristic

154

Q₈

—

155

Q₈

—

156

Q₈

—

157

Q₈

#127×C₆₇

yes

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

#121×C₆₇

yes

—

164

C₂×C₄

#121×C₆₇

yes

—

165

C₂×C₄

C₂×Q₁₆×C₆₇

yes

—

166

C₂×C₄

#137×C₆₇

yes

—

167

C₂×C₄

C₂²×D₄×C₆₇

yes

—

168

C₂×C₄

C₂×#41×C₆₇

yes

—

169

C₈

—

170

C₂²

C₂×#121×C₆₇

yes

—

171

C₂²

C₂²×Q₁₆×C₆₇

yes

characteristic

172

C₂²

C₂×#137×C₆₇

yes

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

#462×C₆₇

yes

complemented

178

C₄

C₂×#127×C₆₇

yes

characteristic

179

C₄

#532×C₆₇

yes

—

180

C₂

C₆₇×#2856

yes

characteristic

181

C₂

C₂×#462×C₆₇

yes

—

182

C₂

C₆₇×#3039

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₆₇

C₂²×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂⁴×C₆₇

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

N	H ≅ G/N	type
58. C₄²×C₆₇	Q₁₆	semidirect
85. C₄×C₆₇	#462	semidirect
88. #33163	C₆₇	direct (N × H)
97. C₆₇	#33163	direct (N × H)
150. C₄²	Q₁₆×C₆₇	semidirect
177. C₄	#462×C₆₇	semidirect

H ≅ G/N

type

58. C₄²×C₆₇

Q₁₆

semidirect

85. C₄×C₆₇

#462

semidirect

88. #33163

C₆₇

direct (N × H)

97. C₆₇

#33163

direct (N × H)

150. C₄²

Q₁₆×C₆₇

semidirect

177. C₄

#462×C₆₇

semidirect

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

subgroup	index
C₂×#462×C₆₇	2
C₆₇×#2868	2
C₄²×Q₈×C₆₇	2
C₆₇×#1981	2
C₆₇×#1842	2
#33163	67

subgroup

index

C₂×#462×C₆₇

C₆₇×#2868

C₄²×Q₈×C₆₇

C₆₇×#1981

C₆₇×#1842

#33163

Subgroups — C₆₇×#33163 #30712414

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#33163 #30712414

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33163 #30712414