Group #16854063 — subgroups

Subgroups — #16854063

Order 3968.

644 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 237 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×#11758446	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11758501	C₂	1	2	yes	maximal
3	#14520955	C₂	1	2	yes	maximal, complemented
4	#14520947	C₂	1	4	yes	maximal, complemented
5	#14520919	C₂	1	1	yes	maximal, characteristic, complemented
6	#14520909	C₂	1	4	yes	maximal
7	C₂²×C₄²×C₃₁	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
9	C₂³×C₄×C₃₁	C₂²	1	2	yes	—
10	C₂²×Dic₆₂	C₂²	1	2	yes	—
11	C₂×C₄²×C₃₁	—	2	4	no	—
12	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Dic₃₁	C₂²	1	4	yes	—
16	C₂×C₄⋊Dic₃₁	C₂²	1	4	yes	—
17	C₂×C₄⋊Dic₃₁	C₂²	1	4	yes	—
18	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
19	#11758501	—	2	8	no	—
20	C₂×Dic₃₁⋊C₄	C₂²	1	4	yes	—
21	#11758446	—	2	4	no	—
22	C₃₁⋊#95	C₂²	1	8	yes	—
23	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₃₁	—	2	2	no	—
25	C₂²×C₄×C₃₁	—	2	4	no	—
26	C₂²×C₄×C₃₁	—	2	2	no	—
27	C₂²×C₄×C₃₁	—	2	4	no	—
28	C₂²×C₄×C₃₁	D₄	1	4	yes	—
29	C₂²×C₄×C₃₁	Q₈	1	8	yes	complemented
30	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
32	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
34	C₂²×Dic₃₁	C₂³	1	4	yes	—
35	C₂²×Dic₃₁	C₂³	1	4	yes	—
36	C₂×Dic₆₂	—	2	4	no	—
37	C₂×Dic₆₂	—	2	4	no	—
38	C₂×Dic₆₂	—	2	4	no	—
39	C₄²×C₃₁	—	2	2	no	—
40	C₄²×C₃₁	—	2	4	no	—
41	C₄²×C₃₁	D₄	1	4	yes	—
42	C₂²⋊Dic₃₁	—	2	8	no	—
43	C₄⋊Dic₃₁	—	2	8	no	—
44	C₄⋊Dic₃₁	—	2	4	no	—
45	C₄⋊Dic₃₁	—	2	8	no	—
46	Dic₃₁⋊C₄	—	2	8	no	—
47	C₂³×C₃₁	—	2	1	no	—
48	C₂³×C₃₁	—	2	1	no	—
49	C₂³×C₃₁	—	2	2	no	—
50	C₂³×C₃₁	C₂×D₄	1	2	yes	—
51	C₂³×C₃₁	C₂×Q₈	1	4	yes	—
52	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
53	C₂×C₄×C₃₁	—	2	1	no	—
54	C₂×C₄×C₃₁	—	2	2	no	—
55	C₂×C₄×C₃₁	—	2	1	no	—
56	C₂×C₄×C₃₁	—	2	2	no	—
57	C₂×C₄×C₃₁	—	2	8	no	—
58	C₂×C₄×C₃₁	—	2	16	no	—
59	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₃₁	C₂×D₄	1	4	yes	—
63	C₂×C₄×C₃₁	C₂×Q₈	1	4	yes	—
64	C₂×C₄×C₃₁	C₂×Q₈	1	4	yes	—
65	C₂×C₄×C₃₁	#41	1	2	yes	—
66	C₂×C₄×C₃₁	#41	1	4	yes	—
67	C₂×Dic₃₁	—	2	4	no	—
68	C₂×Dic₃₁	—	2	4	no	—
69	C₂×Dic₃₁	—	2	4	no	—
70	C₂×Dic₃₁	—	2	8	no	—
71	C₂×Dic₃₁	—	2	4	no	—
72	Dic₆₂	—	4	8	no	—
73	#2857	—	31	1	no	maximal, Sylow
74	C₂²×C₃₁	—	2	2	no	—
75	C₂²×C₃₁	—	2	4	no	—
76	C₂²×C₃₁	—	2	2	no	—
77	C₂²×C₃₁	—	2	4	no	—
78	C₂²×C₃₁	C₄⋊Q₈	1	4	yes	complemented
79	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
80	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
81	C₂²×C₃₁	C₂²×Q₈	1	2	yes	—
82	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
83	C₂²×C₃₁	C₂×#41	1	2	yes	—
84	C₄×C₃₁	—	2	2	no	—
85	C₄×C₃₁	—	2	4	no	—
86	C₄×C₃₁	#121	1	4	yes	—
87	C₄×C₃₁	C₂²⋊Q₈	1	8	yes	complemented
88	Dic₃₁	—	4	4	no	—
89	Dic₃₁	—	4	4	no	—
90	C₂×C₄⋊Q₈	—	31	1	no	—
91	C₂×C₂²⋊Q₈	—	31	2	no	—
92	C₂²×C₄²	D₃₁	1	1	yes	characteristic
93	#391	—	31	2	no	—
94	#389	—	31	4	no	—
95	#384	—	31	1	no	—
96	#380	—	31	4	no	—
97	C₂×C₃₁	—	2	4	no	—
98	C₂×C₃₁	C₂×#121	1	1	yes	characteristic
99	C₂×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
100	C₂×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₂×C₃₁	#532	1	1	yes	characteristic
102	C₂×C₃₁	#533	1	2	yes	—
103	C₂²×Q₈	—	31	2	no	—
104	C₂³×C₄	C₂×D₃₁	1	2	yes	—
105	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
106	C₄⋊Q₈	—	62	4	no	—
107	C₂²⋊Q₈	—	62	8	no	—
108	C₂×C₄⋊C₄	—	31	2	no	—
109	C₂×C₄⋊C₄	—	31	4	no	—
110	C₂×C₄⋊C₄	—	31	4	no	—
111	C₂×C₄⋊C₄	—	31	4	no	—
112	C₂×C₂²⋊C₄	—	31	4	no	—
113	C₂×C₄²	—	2	4	no	—
114	C₂×C₄²	C₂×D₃₁	1	2	yes	—
115	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
116	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
117	#95	—	31	8	no	—
118	C₃₁	#2857	1	1	yes	characteristic, complemented, Sylow
119	C₂⁴	C₂²×D₃₁	1	1	yes	characteristic
120	C₂×Q₈	—	62	4	no	—
121	C₂×Q₈	—	62	4	no	—
122	C₂×Q₈	—	62	4	no	—
123	C₂²×C₄	—	2	4	no	—
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	—	2	4	no	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	—	31	4	no	—
128	C₂²×C₄	—	31	4	no	—
129	C₂²×C₄	Dic₆₂	1	8	yes	complemented
130	C₂²×C₄	D₁₂₄	1	4	yes	—
131	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
132	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
133	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
134	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
135	C₄⋊C₄	—	62	8	no	—
136	C₄⋊C₄	—	62	4	no	—
137	C₄⋊C₄	—	62	8	no	—
138	C₄⋊C₄	—	62	8	no	—
139	C₂²⋊C₄	—	62	8	no	—
140	C₄²	—	2	4	no	—
141	C₄²	—	2	2	no	—
142	C₄²	C₂²⋊D₃₁	1	4	yes	—
143	C₂³	—	2	1	no	—
144	C₂³	—	2	2	no	—
145	C₂³	—	2	1	no	—
146	C₂³	C₂×Dic₆₂	1	4	yes	—
147	C₂³	C₂×D₁₂₄	1	2	yes	—
148	C₂³	C₂³×D₃₁	1	1	yes	characteristic
149	Q₈	—	124	8	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	16	no	—
155	C₂×C₄	—	2	8	no	—
156	C₂×C₄	—	62	4	no	—
157	C₂×C₄	—	62	4	no	—
158	C₂×C₄	—	62	8	no	—
159	C₂×C₄	—	62	4	no	—
160	C₂×C₄	—	62	4	no	—
161	C₂×C₄	C₂×Dic₆₂	1	4	yes	—
162	C₂×C₄	C₂×Dic₆₂	1	4	yes	—
163	C₂×C₄	C₂×D₁₂₄	1	2	yes	—
164	C₂×C₄	C₂×D₁₂₄	1	2	yes	—
165	C₂×C₄	#92495	1	4	yes	—
166	C₂×C₄	#92495	1	2	yes	—
167	C₂×C₄	C₂×C₂²⋊D₃₁	1	4	yes	—
168	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
169	C₂²	—	2	4	no	—
170	C₂²	—	2	2	no	—
171	C₂²	—	2	4	no	—
172	C₂²	—	2	2	no	—
173	C₂²	#11758446	1	4	yes	complemented
174	C₂²	C₂²×Dic₆₂	1	2	yes	—
175	C₂²	C₂²×D₁₂₄	1	1	yes	characteristic
176	C₂²	C₂×#92495	1	1	yes	characteristic
177	C₂²	C₂×#92495	1	2	yes	—
178	C₂²	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic
179	C₄	—	2	2	no	—
180	C₄	—	2	4	no	—
181	C₄	—	124	4	no	—
182	C₄	—	124	4	no	—
183	C₄	#11758501	1	8	yes	complemented
184	C₄	#11758507	1	4	yes	—
185	C₂	—	2	4	no	—
186	C₂	C₂×#11758507	1	1	yes	characteristic
187	C₂	C₂×#11758501	1	2	yes	—
188	C₂	C₂×#11758446	1	1	yes	characteristic
189	C₂	#14521599	1	1	yes	characteristic
190	C₂	#14521606	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758446

C₂

yes

maximal, characteristic, complemented

C₂×#11758501

C₂

yes

maximal

#14520955

C₂

yes

maximal, complemented

#14520947

C₂

yes

maximal, complemented

#14520919

C₂

yes

maximal, characteristic, complemented

#14520909

C₂

yes

maximal

C₂²×C₄²×C₃₁

C₂

yes

maximal, characteristic

C₂³×C₄×C₃₁

C₂²

yes

characteristic

C₂³×C₄×C₃₁

C₂²

yes

—

C₂²×Dic₆₂

C₂²

yes

—

C₂×C₄²×C₃₁

—

C₂×C₄²×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₂²⋊Dic₃₁

C₂²

yes

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

#11758501

—

C₂×Dic₃₁⋊C₄

C₂²

yes

—

#11758446

—

C₃₁⋊#95

C₂²

yes

—

C₂⁴×C₃₁

C₂³

yes

characteristic

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

Q₈

yes

complemented

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×Dic₃₁

C₂³

yes

—

C₂²×Dic₃₁

C₂³

yes

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₄²×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

D₄

yes

—

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

Dic₃₁⋊C₄

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

C₂×D₄

yes

—

C₂³×C₃₁

C₂×Q₈

yes

—

C₂³×C₃₁

C₂⁴

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

#2857

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₄⋊Q₈

yes

complemented

C₂²×C₃₁

C₂²×D₄

yes

characteristic

C₂²×C₃₁

C₂²×D₄

yes

characteristic

C₂²×C₃₁

C₂²×Q₈

yes

—

C₂²×C₃₁

C₂×#41

yes

characteristic

C₂²×C₃₁

C₂×#41

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#121

yes

—

C₄×C₃₁

C₂²⋊Q₈

yes

complemented

Dic₃₁

—

Dic₃₁

—

C₂×C₄⋊Q₈

—

C₂×C₂²⋊Q₈

—

C₂²×C₄²

D₃₁

yes

—

—

—

—

C₂×C₃₁

—

C₂×C₃₁

C₂×#121

yes

characteristic

C₂×C₃₁

C₂×C₂²⋊Q₈

yes

—

100

C₂×C₃₁

C₂×C₄⋊Q₈

yes

characteristic

101

C₂×C₃₁

#532

yes

characteristic

102

C₂×C₃₁

#533

yes

—

103

C₂²×Q₈

—

104

C₂³×C₄

C₂×D₃₁

yes

—

105

C₂³×C₄

C₂×D₃₁

yes

characteristic

106

C₄⋊Q₈

—

107

C₂²⋊Q₈

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₂²⋊C₄

—

113

C₂×C₄²

—

114

C₂×C₄²

C₂×D₃₁

yes

—

115

C₂×C₄²

C₂×D₃₁

yes

characteristic

116

C₂×C₄²

C₂×D₃₁

yes

characteristic

117

#95

—

118

C₃₁

#2857

yes

characteristic, complemented, Sylow

119

C₂⁴

C₂²×D₃₁

yes

characteristic

120

C₂×Q₈

—

121

C₂×Q₈

—

122

C₂×Q₈

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

Dic₆₂

yes

complemented

130

C₂²×C₄

D₁₂₄

yes

—

131

C₂²×C₄

C₂²×D₃₁

yes

—

132

C₂²×C₄

C₂²×D₃₁

yes

characteristic

133

C₂²×C₄

C₂²×D₃₁

yes

—

134

C₂²×C₄

C₂²×D₃₁

yes

characteristic

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₂²⋊C₄

—

140

C₄²

—

141

C₄²

—

142

C₄²

C₂²⋊D₃₁

yes

—

143

C₂³

—

144

C₂³

—

145

C₂³

—

146

C₂³

C₂×Dic₆₂

yes

—

147

C₂³

C₂×D₁₂₄

yes

—

148

C₂³

C₂³×D₃₁

yes

characteristic

149

Q₈

—

124

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

C₂×Dic₆₂

yes

—

162

C₂×C₄

C₂×Dic₆₂

yes

—

163

C₂×C₄

C₂×D₁₂₄

yes

—

164

C₂×C₄

C₂×D₁₂₄

yes

—

165

C₂×C₄

#92495

yes

—

166

C₂×C₄

#92495

yes

—

167

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

168

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

#11758446

yes

complemented

174

C₂²

C₂²×Dic₆₂

yes

—

175

C₂²

C₂²×D₁₂₄

yes

characteristic

176

C₂²

C₂×#92495

yes

characteristic

177

C₂²

C₂×#92495

yes

—

178

C₂²

C₂²×C₂²⋊D₃₁

yes

characteristic

179

C₄

—

180

C₄

—

181

C₄

—

124

—

182

C₄

—

124

—

183

C₄

#11758501

yes

complemented

184

C₄

#11758507

yes

—

185

C₂

—

186

C₂

C₂×#11758507

yes

characteristic

187

C₂

C₂×#11758501

yes

—

188

C₂

C₂×#11758446

yes

characteristic

189

C₂

#14521599

yes

characteristic

190

C₂

#14521606

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₁
commutator, socle, cosocle	C₂³×C₃₁	C₂⁴
Fitting	C₂²×C₄²×C₃₁	C₂
prime radical	C₂⁴×C₃₁	C₂³
hypercenter	C₂²×C₄²	D₃₁
hypocenter	C₃₁	#2857
LCS	G ▷ C₂³×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄²

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃₁

commutator, socle, cosocle

C₂³×C₃₁

C₂⁴

Fitting

C₂²×C₄²×C₃₁

C₂

prime radical

C₂⁴×C₃₁

C₂³

hypercenter

C₂²×C₄²

D₃₁

C₃₁

G ▷ C₂³×C₃₁ ▷ C₃₁

C₁ ◁ C₂³ ◁ C₂²×C₄²

N	H ≅ G/N	type
1. C₂×#11758446	C₂	semidirect
3. #14520955	C₂	semidirect
4. #14520947	C₂	semidirect
5. #14520919	C₂	semidirect
29. C₂²×C₄×C₃₁	Q₈	semidirect
78. C₂²×C₃₁	C₄⋊Q₈	semidirect
87. C₄×C₃₁	C₂²⋊Q₈	semidirect
118. C₃₁	#2857	semidirect
129. C₂²×C₄	Dic₆₂	semidirect
173. C₂²	#11758446	semidirect
183. C₄	#11758501	semidirect

H ≅ G/N

type

1. C₂×#11758446

C₂

semidirect

3. #14520955

C₂

semidirect

4. #14520947

C₂

semidirect

5. #14520919

C₂

semidirect

29. C₂²×C₄×C₃₁

Q₈

semidirect

78. C₂²×C₃₁

C₄⋊Q₈

semidirect

87. C₄×C₃₁

C₂²⋊Q₈

semidirect

118. C₃₁

#2857

semidirect

129. C₂²×C₄

Dic₆₂

semidirect

173. C₂²

#11758446

semidirect

183. C₄

#11758501

semidirect

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

subgroup	index
C₂×#11758446	2
C₂×#11758501	2
#14520955	2
#14520947	2
#14520919	2
#14520909	2
C₂²×C₄²×C₃₁	2
#2857	31

subgroup

index

C₂²×C₄²×C₃₁

Subgroups — #16854063

Conjugacy classes of subgroups

Complements

Maximal subgroups