Group #16854342 — subgroups

Subgroups — #16854342

Order 3968.

484 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 197 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×C₄²×D₃₁	C₂	1	1	yes	maximal, characteristic
2	#14521104	C₂	1	4	yes	maximal, complemented
3	#14521100	C₂	1	4	yes	maximal, complemented
4	#14521072	C₂	1	2	yes	maximal, complemented
5	#14521012	C₂	1	2	yes	maximal
6	C₄×#92474	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₃₁	C₄	1	2	yes	—
9	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
10	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
11	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₃₁	C₄	1	2	yes	—
13	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
14	C₂×#92474	C₂²	1	2	yes	—
15	C₄²×D₃₁	C₄	1	2	yes	—
16	C₄²×D₃₁	C₄	1	2	yes	—
17	C₄²×D₃₁	C₂²	1	2	yes	—
18	C₄²×D₃₁	C₂²	1	2	yes	—
19	C₃₁×#97	C₂²	1	2	yes	complemented
20	C₄×C₈×C₃₁	C₂²	1	2	yes	complemented
21	(C₂×D₃₁)⋊C₈	C₂²	1	8	yes	—
22	Dic₃₁⋊C₈	C₂²	1	8	yes	complemented
23	#11758391	C₂²	1	2	yes	complemented
24	C₄×C₃₁⋊C₈	C₂²	1	2	yes	complemented
25	C₂³×D₃₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
28	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×Dic₃₁	C₂×C₄	1	2	yes	—
30	C₂×C₄×D₃₁	—	2	4	no	—
31	C₂×C₄×D₃₁	—	2	8	no	—
32	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
33	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₃₁	C₂³	1	2	yes	—
38	M₁₆×C₃₁	—	2	4	no	—
39	C₂×C₈×C₃₁	C₂³	1	4	yes	—
40	C₄²×C₃₁	C₂×C₄	1	2	yes	—
41	C₄²×C₃₁	C₂³	1	1	yes	characteristic
42	C₄²×C₃₁	C₂³	1	1	yes	characteristic
43	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
44	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
45	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
46	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
47	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
48	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
49	C₄×Dic₃₁	C₂³	1	2	yes	—
50	#92474	—	2	4	no	—
51	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
52	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
53	C₂²×D₃₁	—	2	2	no	—
54	C₂²×D₃₁	—	2	4	no	—
55	C₂²×D₃₁	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₃₁	—	2	4	no	—
57	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₃₁	#41	1	4	yes	—
61	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
62	C₂×Dic₃₁	—	2	2	no	—
63	C₂×Dic₃₁	—	2	4	no	—
64	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
65	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
66	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
67	C₄×D₃₁	—	2	4	no	—
68	C₄×D₃₁	—	2	4	no	—
69	C₄×D₃₁	—	2	4	no	—
70	C₄×D₃₁	—	2	4	no	—
71	C₄×D₃₁	—	2	4	no	—
72	C₄×D₃₁	M₁₆	1	8	yes	complemented
73	C₈×C₃₁	—	2	4	no	—
74	C₃₁⋊C₈	—	2	4	no	—
75	#2920	—	31	1	no	maximal, Sylow
76	C₂²×C₃₁	—	2	2	no	—
77	C₂²×C₃₁	#117	1	2	yes	—
78	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
79	C₂×D₃₁	—	2	8	no	—
80	C₂×D₃₁	—	2	2	no	—
81	C₂×D₃₁	—	2	2	no	—
82	C₂×D₃₁	C₂×M₁₆	1	4	yes	—
83	C₄×C₃₁	—	2	2	no	—
84	C₄×C₃₁	#117	1	2	yes	—
85	C₄×C₃₁	C₂×M₁₆	1	4	yes	—
86	C₄×C₃₁	C₂×#41	1	2	yes	—
87	Dic₃₁	—	2	2	no	—
88	Dic₃₁	—	2	2	no	—
89	Dic₃₁	C₂×M₁₆	1	4	yes	—
90	C₂²×C₄²	—	31	1	no	—
91	#432	—	31	4	no	—
92	#431	—	31	4	no	—
93	#423	—	31	2	no	—
94	#407	—	31	2	no	—
95	C₄×M₁₆	—	31	1	no	—
96	C₄×M₁₆	D₃₁	1	1	yes	characteristic, complemented
97	C₂×C₃₁	—	2	2	no	—
98	C₂×C₃₁	C₂×#117	1	1	yes	characteristic
99	C₂×C₃₁	C₂²×M₁₆	1	2	yes	—
100	D₃₁	—	2	4	no	—
101	C₂³×C₄	—	31	2	no	—
102	C₂³×C₄	—	31	1	no	—
103	C₂×M₁₆	—	31	2	no	—
104	C₂×M₁₆	C₂×D₃₁	1	2	yes	—
105	C₂×C₄²	—	31	2	no	—
106	C₂×C₄²	—	31	1	no	—
107	C₂×C₄²	—	31	2	no	—
108	C₂×C₄²	—	31	2	no	—
109	C₂×C₄²	—	31	2	no	—
110	C₂×C₄²	—	31	2	no	—
111	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
112	C₄⋊C₈	—	31	8	no	—
113	C₂²⋊C₈	—	31	8	no	—
114	#97	—	31	2	no	—
115	#97	C₂×D₃₁	1	2	yes	complemented
116	C₄×C₈	—	31	2	no	—
117	C₄×C₈	C₂×D₃₁	1	2	yes	complemented
118	C₃₁	#2920	1	1	yes	characteristic, complemented, Sylow
119	C₂⁴	—	31	1	no	—
120	C₂²×C₄	—	31	2	no	—
121	C₂²×C₄	—	31	2	no	—
122	C₂²×C₄	—	31	1	no	—
123	C₂²×C₄	—	31	2	no	—
124	C₂²×C₄	—	31	2	no	—
125	C₂²×C₄	—	31	2	no	—
126	C₂²×C₄	—	31	2	no	—
127	C₂²×C₄	—	31	2	no	—
128	C₂²×C₄	—	62	8	no	—
129	C₂²×C₄	—	62	4	no	—
130	C₂²×C₄	C₄×D₃₁	1	2	yes	—
131	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
132	M₁₆	—	2	4	no	—
133	M₁₆	—	62	4	no	—
134	C₂×C₈	—	31	4	no	—
135	C₂×C₈	C₂²×D₃₁	1	4	yes	—
136	C₄²	—	31	1	no	—
137	C₄²	—	31	2	no	—
138	C₄²	—	31	2	no	—
139	C₄²	—	31	2	no	—
140	C₄²	—	31	1	no	—
141	C₄²	—	31	2	no	—
142	C₄²	—	31	2	no	—
143	C₄²	C₄×D₃₁	1	2	yes	—
144	C₄²	C₂²×D₃₁	1	1	yes	characteristic
145	C₄²	C₂²×D₃₁	1	1	yes	characteristic
146	C₂³	—	31	2	no	—
147	C₂³	—	62	4	no	—
148	C₂³	—	62	2	no	—
149	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	31	2	no	—
152	C₂×C₄	—	31	8	no	—
153	C₂×C₄	—	31	2	no	—
154	C₂×C₄	—	31	2	no	—
155	C₂×C₄	—	62	4	no	—
156	C₂×C₄	—	62	4	no	—
157	C₂×C₄	—	62	4	no	—
158	C₂×C₄	—	62	2	no	—
159	C₂×C₄	—	62	4	no	—
160	C₂×C₄	—	62	4	no	—
161	C₂×C₄	—	62	4	no	—
162	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
163	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
164	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
165	C₂×C₄	#92495	1	4	yes	—
166	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
167	C₈	—	2	4	no	—
168	C₈	—	62	4	no	—
169	C₂²	—	2	2	no	—
170	C₂²	—	31	4	no	—
171	C₂²	—	62	8	no	—
172	C₂²	—	62	2	no	—
173	C₂²	—	62	2	no	—
174	C₂²	#11758449	1	2	yes	—
175	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	31	4	no	—
178	C₄	—	62	2	no	—
179	C₄	—	62	2	no	—
180	C₄	#11758449	1	2	yes	—
181	C₄	M₁₆×D₃₁	1	4	yes	—
182	C₄	C₂×#92495	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	—	62	4	no	—
185	C₂	C₂×#11758449	1	1	yes	characteristic
186	C₂	C₂×M₁₆×D₃₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄²×D₃₁

C₂

yes

maximal, characteristic

#14521104

C₂

yes

maximal, complemented

#14521100

C₂

yes

maximal, complemented

#14521072

C₂

yes

maximal, complemented

#14521012

C₂

yes

maximal

C₄×#92474

C₂

yes

maximal, characteristic, complemented

C₄×M₁₆×C₃₁

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₃₁

C₄

yes

—

C₂²×C₄×D₃₁

C₂²

yes

characteristic

C₂×M₁₆×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄×Dic₃₁

C₄

yes

—

C₂×C₄×Dic₃₁

C₂²

yes

characteristic

C₂×#92474

C₂²

yes

—

C₄²×D₃₁

C₄

yes

—

C₄²×D₃₁

C₄

yes

—

C₄²×D₃₁

C₂²

yes

—

C₄²×D₃₁

C₂²

yes

—

C₃₁×#97

C₂²

yes

complemented

C₄×C₈×C₃₁

C₂²

yes

complemented

(C₂×D₃₁)⋊C₈

C₂²

yes

—

Dic₃₁⋊C₈

C₂²

yes

complemented

#11758391

C₂²

yes

complemented

C₄×C₃₁⋊C₈

C₂²

yes

complemented

C₂³×D₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

C₂×C₄

yes

characteristic

C₂²×Dic₃₁

C₂×C₄

yes

—

C₂×C₄×D₃₁

—

C₂×C₄×D₃₁

—

C₂×C₄×D₃₁

C₂×C₄

yes

—

C₂×C₄×D₃₁

C₂×C₄

yes

—

C₂×C₄×D₃₁

C₂×C₄

yes

—

C₂×C₄×D₃₁

C₂×C₄

yes

—

C₂×C₄×D₃₁

C₂×C₄

yes

—

C₂×C₄×D₃₁

C₂³

yes

—

M₁₆×C₃₁

—

C₂×C₈×C₃₁

C₂³

yes

—

C₄²×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

C₂³

yes

characteristic

C₄²×C₃₁

C₂³

yes

characteristic

C₄×Dic₃₁

C₂×C₄

yes

characteristic

C₄×Dic₃₁

C₂×C₄

yes

characteristic

C₄×Dic₃₁

C₂×C₄

yes

—

C₄×Dic₃₁

C₂×C₄

yes

—

C₄×Dic₃₁

C₂×C₄

yes

—

C₄×Dic₃₁

C₂×C₄

yes

—

C₄×Dic₃₁

C₂³

yes

—

#92474

—

C₂×C₃₁⋊C₈

C₂³

yes

—

C₂³×C₃₁

C₂²×C₄

yes

characteristic

C₂²×D₃₁

—

C₂²×D₃₁

—

C₂²×D₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

C₂²×C₄

yes

—

C₂×Dic₃₁

C₂²×C₄

yes

—

C₂×Dic₃₁

C₂²×C₄

yes

—

C₄×D₃₁

—

C₄×D₃₁

—

C₄×D₃₁

—

C₄×D₃₁

—

C₄×D₃₁

—

C₄×D₃₁

M₁₆

yes

complemented

C₈×C₃₁

—

C₃₁⋊C₈

—

#2920

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

#117

yes

—

C₂²×C₃₁

C₂³×C₄

yes

characteristic

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

C₂×M₁₆

yes

—

C₄×C₃₁

—

C₄×C₃₁

#117

yes

—

C₄×C₃₁

C₂×M₁₆

yes

—

C₄×C₃₁

C₂×#41

yes

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

C₂×M₁₆

yes

—

C₂²×C₄²

—

—

—

—

—

C₄×M₁₆

—

C₄×M₁₆

D₃₁

yes

characteristic, complemented

C₂×C₃₁

—

C₂×C₃₁

C₂×#117

yes

characteristic

C₂×C₃₁

C₂²×M₁₆

yes

—

100

D₃₁

—

101

C₂³×C₄

—

102

C₂³×C₄

—

103

C₂×M₁₆

—

104

C₂×M₁₆

C₂×D₃₁

yes

—

105

C₂×C₄²

—

106

C₂×C₄²

—

107

C₂×C₄²

—

108

C₂×C₄²

—

109

C₂×C₄²

—

110

C₂×C₄²

—

111

C₂×C₄²

C₂×D₃₁

yes

characteristic

112

C₄⋊C₈

—

113

C₂²⋊C₈

—

114

#97

—

115

#97

C₂×D₃₁

yes

complemented

116

C₄×C₈

—

117

C₄×C₈

C₂×D₃₁

yes

complemented

118

C₃₁

#2920

yes

characteristic, complemented, Sylow

119

C₂⁴

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

C₄×D₃₁

yes

—

131

C₂²×C₄

C₂²×D₃₁

yes

characteristic

132

M₁₆

—

133

M₁₆

—

134

C₂×C₈

—

135

C₂×C₈

C₂²×D₃₁

yes

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₄²

C₄×D₃₁

yes

—

144

C₄²

C₂²×D₃₁

yes

characteristic

145

C₄²

C₂²×D₃₁

yes

characteristic

146

C₂³

—

147

C₂³

—

148

C₂³

—

149

C₂³

C₂×C₄×D₃₁

yes

characteristic

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

—

162

C₂×C₄

C₂×C₄×D₃₁

yes

—

163

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

164

C₂×C₄

C₂×C₄×D₃₁

yes

—

165

C₂×C₄

#92495

yes

—

166

C₂×C₄

C₂³×D₃₁

yes

characteristic

167

C₈

—

168

C₈

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

#11758449

yes

—

175

C₂²

C₂²×C₄×D₃₁

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

#11758449

yes

—

181

C₄

M₁₆×D₃₁

yes

—

182

C₄

C₂×#92495

yes

—

183

C₂

—

184

C₂

—

185

C₂

C₂×#11758449

yes

characteristic

186

C₂

C₂×M₁₆×D₃₁

yes

—

	subgroup	G/H
center	C₄²	C₂²×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₄×M₁₆×C₃₁	C₂
prime radical	C₂³×D₃₁	C₂×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₄×M₁₆	D₃₁
hypocenter	C₃₁	#2920
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₄²

C₂²×D₃₁

commutator, socle

C₂²×C₃₁

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₃₁

Fitting

C₄×M₁₆×C₃₁

C₂

prime radical

C₂³×D₃₁

C₂×C₄

cosocle

C₂×C₄×C₃₁

C₂⁴

hypercenter

C₄×M₁₆

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
2. #14521104	C₂	semidirect
3. #14521100	C₂	semidirect
4. #14521072	C₂	semidirect
6. C₄×#92474	C₂	semidirect
7. C₄×M₁₆×C₃₁	C₂	semidirect
19. C₃₁×#97	C₂²	semidirect
20. C₄×C₈×C₃₁	C₂²	semidirect
22. Dic₃₁⋊C₈	C₂²	semidirect
23. #11758391	C₂²	semidirect
24. C₄×C₃₁⋊C₈	C₂²	semidirect
72. C₄×D₃₁	M₁₆	semidirect
96. C₄×M₁₆	D₃₁	semidirect
115. #97	C₂×D₃₁	semidirect
117. C₄×C₈	C₂×D₃₁	semidirect
118. C₃₁	#2920	semidirect

H ≅ G/N

type

2. #14521104

C₂

semidirect

3. #14521100

C₂

semidirect

4. #14521072

C₂

semidirect

6. C₄×#92474

C₂

semidirect

7. C₄×M₁₆×C₃₁

C₂

semidirect

19. C₃₁×#97

C₂²

semidirect

20. C₄×C₈×C₃₁

C₂²

semidirect

22. Dic₃₁⋊C₈

C₂²

semidirect

23. #11758391

C₂²

semidirect

24. C₄×C₃₁⋊C₈

C₂²

semidirect

72. C₄×D₃₁

M₁₆

semidirect

96. C₄×M₁₆

D₃₁

semidirect

115. #97

C₂×D₃₁

semidirect

117. C₄×C₈

C₂×D₃₁

semidirect

118. C₃₁

#2920

semidirect

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

subgroup	index
C₂×C₄²×D₃₁	2
#14521104	2
#14521100	2
#14521072	2
#14521012	2
C₄×#92474	2
C₄×M₁₆×C₃₁	2
#2920	31

subgroup

index

C₂×C₄²×D₃₁

C₄×M₁₆×C₃₁

Subgroups — #16854342

Conjugacy classes of subgroups

Complements

Maximal subgroups