Group #21489922 — subgroups

Subgroups — C₃₁×#33163 #21489922

Order 7936.

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×C₃₁×#462	C₂	1	6	yes	maximal
2	C₃₁×#2868	C₂	1	1	yes	maximal, characteristic
3	C₄²×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1981	C₂	1	6	yes	maximal
5	C₃₁×#1842	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₃₁	C₂²	1	4	yes	—
7	C₂×C₃₁×C₄⋊Q₈	C₂²	1	6	yes	—
8	C₂×C₄×Q₈×C₃₁	C₂²	1	3	yes	—
9	C₃₁×#462	—	2	24	no	—
10	C₂×C₃₁×C₄⋊C₈	C₂²	1	6	yes	—
11	C₂×C₃₁×#103	C₂²	1	12	yes	—
12	C₃₁×#384	—	2	3	no	—
13	C₃₁×#384	—	2	1	no	—
14	C₄×C₄⋊C₄×C₃₁	—	2	3	no	—
15	C₄×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
16	C₄³×C₃₁	—	2	1	no	—
17	C₄³×C₃₁	C₂²	1	1	yes	characteristic
18	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
19	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
20	C₂×Q₁₆×C₃₁	—	2	24	no	—
21	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
22	C₃₁×C₄⋊Q₈	—	2	12	no	—
23	C₃₁×C₄⋊Q₈	—	2	12	no	—
24	C₄×Q₈×C₃₁	—	2	6	no	—
25	C₄×Q₈×C₃₁	D₄	1	12	yes	—
26	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
27	C₂×C₄⋊C₄×C₃₁	—	2	6	no	—
28	C₂×C₄⋊C₄×C₃₁	—	2	6	no	—
29	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
30	C₂×C₄²×C₃₁	—	2	3	no	—
31	C₂×C₄²×C₃₁	—	2	3	no	—
32	C₂×C₄²×C₃₁	D₄	1	3	yes	—
33	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
35	C₃₁×C₄⋊C₈	—	2	12	no	—
36	C₃₁×#103	—	2	24	no	—
37	C₃₁×#95	—	2	3	no	—
38	C₂×Q₈×C₃₁	—	2	3	no	—
39	C₂×Q₈×C₃₁	—	2	12	no	—
40	C₂×Q₈×C₃₁	—	2	12	no	—
41	C₂×Q₈×C₃₁	C₂×D₄	1	6	yes	—
42	C₂²×C₄×C₃₁	—	2	1	no	—
43	C₂²×C₄×C₃₁	—	2	3	no	—
44	C₂²×C₄×C₃₁	—	2	4	no	—
45	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
46	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
47	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
48	Q₁₆×C₃₁	—	4	16	no	—
49	C₂×C₈×C₃₁	—	2	12	no	—
50	C₄⋊C₄×C₃₁	—	2	12	no	—
51	C₄⋊C₄×C₃₁	—	2	3	no	—
52	C₄⋊C₄×C₃₁	—	4	12	no	—
53	C₄⋊C₄×C₃₁	—	4	12	no	—
54	C₄⋊C₄×C₃₁	C₂×D₄	1	6	yes	—
55	C₄²×C₃₁	—	2	12	no	—
56	C₄²×C₃₁	—	2	6	no	—
57	C₄²×C₃₁	—	4	6	no	—
58	C₄²×C₃₁	Q₁₆	1	4	yes	complemented
59	C₄²×C₃₁	C₂×D₄	1	6	yes	—
60	C₄²×C₃₁	#41	1	6	yes	—
61	#33163	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
63	Q₈×C₃₁	—	2	6	no	—
64	Q₈×C₃₁	—	4	4	no	—
65	Q₈×C₃₁	—	4	12	no	—
66	Q₈×C₃₁	#127	1	4	yes	—
67	C₂×C₄×C₃₁	—	2	6	no	—
68	C₂×C₄×C₃₁	—	2	6	no	—
69	C₂×C₄×C₃₁	—	2	6	no	—
70	C₂×C₄×C₃₁	—	4	6	no	—
71	C₂×C₄×C₃₁	—	4	12	no	—
72	C₂×C₄×C₃₁	#121	1	6	yes	—
73	C₂×C₄×C₃₁	#121	1	6	yes	—
74	C₂×C₄×C₃₁	C₂×Q₁₆	1	6	yes	—
75	C₂×C₄×C₃₁	#137	1	6	yes	—
76	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
77	C₂×C₄×C₃₁	C₂×#41	1	3	yes	—
78	C₈×C₃₁	—	4	4	no	—
79	C₂×#462	C₂×C₃₁	1	6	yes	—
80	#2868	C₂×C₃₁	1	1	yes	characteristic
81	C₄²×Q₈	C₂×C₃₁	1	1	yes	characteristic
82	#1981	C₂×C₃₁	1	6	yes	—
83	#1842	C₂×C₃₁	1	1	yes	characteristic
84	C₂²×C₃₁	C₂×#121	1	3	yes	—
85	C₂²×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
86	C₂²×C₃₁	C₂×#137	1	3	yes	—
87	C₄×C₃₁	—	2	4	no	—
88	C₄×C₃₁	—	2	6	no	—
89	C₄×C₃₁	—	4	6	no	—
90	C₄×C₃₁	—	8	4	no	—
91	C₄×C₃₁	#462	1	12	yes	complemented
92	C₄×C₃₁	C₂×#127	1	1	yes	characteristic
93	C₄×C₃₁	#532	1	3	yes	—
94	C₂²×Q₁₆	C₂²×C₃₁	1	4	yes	—
95	C₂×C₄⋊Q₈	C₂²×C₃₁	1	6	yes	—
96	C₂×C₄×Q₈	C₂²×C₃₁	1	3	yes	—
97	#462	—	2	24	no	—
98	C₂×C₄⋊C₈	C₂²×C₃₁	1	6	yes	—
99	C₂×#103	C₂²×C₃₁	1	12	yes	—
100	#384	—	2	3	no	—
101	#384	—	2	1	no	—
102	C₄×C₄⋊C₄	—	2	3	no	—
103	C₄×C₄⋊C₄	C₂²×C₃₁	1	3	yes	—
104	C₄³	—	2	1	no	—
105	C₄³	C₂²×C₃₁	1	1	yes	characteristic
106	C₂×C₃₁	#2856	1	1	yes	characteristic
107	C₂×C₃₁	C₂×#462	1	3	yes	—
108	C₂×C₃₁	#3039	1	3	yes	—
109	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
110	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
111	C₂×Q₁₆	—	2	24	no	—
112	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
113	C₄⋊Q₈	—	2	12	no	—
114	C₄⋊Q₈	—	2	12	no	—
115	C₄×Q₈	—	2	6	no	—
116	C₄×Q₈	D₄×C₃₁	1	12	yes	—
117	C₂×C₄⋊C₄	—	2	3	no	—
118	C₂×C₄⋊C₄	—	2	6	no	—
119	C₂×C₄⋊C₄	—	2	6	no	—
120	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
121	C₂×C₄²	—	2	3	no	—
122	C₂×C₄²	—	2	3	no	—
123	C₂×C₄²	D₄×C₃₁	1	3	yes	—
124	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
125	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
126	C₄⋊C₈	—	2	12	no	—
127	#103	—	2	24	no	—
128	#95	—	2	3	no	—
129	C₃₁	#33163	1	1	yes	characteristic, direct product factor, Sylow
130	C₂×Q₈	—	2	3	no	—
131	C₂×Q₈	—	2	12	no	—
132	C₂×Q₈	—	2	12	no	—
133	C₂×Q₈	C₂×D₄×C₃₁	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×C₃₁	1	3	yes	—
138	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
139	C₂²×C₄	C₂⁴×C₃₁	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×C₃₁	1	6	yes	—
147	C₄²	—	2	12	no	—
148	C₄²	—	2	6	no	—
149	C₄²	—	4	6	no	—
150	C₄²	Q₁₆×C₃₁	1	4	yes	complemented
151	C₄²	C₂×D₄×C₃₁	1	6	yes	—
152	C₄²	#41×C₃₁	1	6	yes	—
153	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
154	Q₈	—	2	6	no	—
155	Q₈	—	4	4	no	—
156	Q₈	—	4	12	no	—
157	Q₈	C₃₁×#127	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	C₃₁×#121	1	6	yes	—
164	C₂×C₄	C₃₁×#121	1	6	yes	—
165	C₂×C₄	C₂×Q₁₆×C₃₁	1	6	yes	—
166	C₂×C₄	C₃₁×#137	1	6	yes	—
167	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
168	C₂×C₄	C₂×#41×C₃₁	1	3	yes	—
169	C₈	—	4	4	no	—
170	C₂²	C₂×C₃₁×#121	1	3	yes	—
171	C₂²	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
172	C₂²	C₂×C₃₁×#137	1	3	yes	—
173	C₄	—	2	4	no	—
174	C₄	—	2	6	no	—
175	C₄	—	4	6	no	—
176	C₄	—	8	4	no	—
177	C₄	C₃₁×#462	1	12	yes	complemented
178	C₄	C₂×C₃₁×#127	1	1	yes	characteristic
179	C₄	C₃₁×#532	1	3	yes	—
180	C₂	C₃₁×#2856	1	1	yes	characteristic
181	C₂	C₂×C₃₁×#462	1	3	yes	—
182	C₂	C₃₁×#3039	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃₁×#462

C₂

yes

maximal

C₃₁×#2868

C₂

yes

maximal, characteristic

C₄²×Q₈×C₃₁

C₂

yes

maximal, characteristic

C₃₁×#1981

C₂

yes

maximal

C₃₁×#1842

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₃₁

C₂²

yes

—

C₂×C₃₁×C₄⋊Q₈

C₂²

yes

—

C₂×C₄×Q₈×C₃₁

C₂²

yes

—

C₃₁×#462

—

C₂×C₃₁×C₄⋊C₈

C₂²

yes

—

C₂×C₃₁×#103

C₂²

yes

—

C₃₁×#384

—

C₃₁×#384

—

C₄×C₄⋊C₄×C₃₁

—

C₄×C₄⋊C₄×C₃₁

C₂²

yes

—

C₄³×C₃₁

—

C₄³×C₃₁

C₂²

yes

characteristic

C₂²×Q₈×C₃₁

C₂³

yes

—

C₂²×Q₈×C₃₁

C₂³

yes

characteristic

C₂×Q₁₆×C₃₁

—

C₂²×C₈×C₃₁

C₂³

yes

—

C₃₁×C₄⋊Q₈

—

C₃₁×C₄⋊Q₈

—

C₄×Q₈×C₃₁

—

C₄×Q₈×C₃₁

D₄

yes

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

C₂³

yes

—

C₂×C₄²×C₃₁

—

C₂×C₄²×C₃₁

—

C₂×C₄²×C₃₁

D₄

yes

—

C₂×C₄²×C₃₁

D₄

yes

characteristic

C₂×C₄²×C₃₁

C₂³

yes

—

C₃₁×C₄⋊C₈

—

C₃₁×#103

—

C₃₁×#95

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

C₂×D₄

yes

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×D₄

yes

—

C₂²×C₄×C₃₁

C₂×D₄

yes

—

C₂²×C₄×C₃₁

C₂⁴

yes

characteristic

Q₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

C₂×D₄

yes

—

C₄²×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

Q₁₆

yes

complemented

C₄²×C₃₁

C₂×D₄

yes

—

C₄²×C₃₁

#41

yes

—

#33163

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

C₂²×D₄

yes

characteristic

Q₈×C₃₁

—

Q₈×C₃₁

—

Q₈×C₃₁

—

Q₈×C₃₁

#127

yes

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

#121

yes

—

C₂×C₄×C₃₁

#121

yes

—

C₂×C₄×C₃₁

C₂×Q₁₆

yes

—

C₂×C₄×C₃₁

#137

yes

—

C₂×C₄×C₃₁

C₂²×D₄

yes

—

C₂×C₄×C₃₁

C₂×#41

yes

—

C₈×C₃₁

—

C₂×#462

C₂×C₃₁

yes

—

#2868

C₂×C₃₁

yes

characteristic

C₄²×Q₈

C₂×C₃₁

yes

characteristic

#1981

C₂×C₃₁

yes

—

#1842

C₂×C₃₁

yes

characteristic

C₂²×C₃₁

C₂×#121

yes

—

C₂²×C₃₁

C₂²×Q₁₆

yes

characteristic

C₂²×C₃₁

C₂×#137

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

—

C₂²×Q₁₆

C₂²×C₃₁

yes

—

C₂×C₄⋊Q₈

C₂²×C₃₁

yes

—

C₂×C₄×Q₈

C₂²×C₃₁

yes

—

#462

—

C₂×C₄⋊C₈

C₂²×C₃₁

yes

—

C₂×#103

C₂²×C₃₁

yes

—

100

#384

—

101

#384

—

102

C₄×C₄⋊C₄

—

103

C₄×C₄⋊C₄

C₂²×C₃₁

yes

—

104

C₄³

—

105

C₄³

C₂²×C₃₁

yes

characteristic

106

C₂×C₃₁

#2856

yes

characteristic

107

C₂×C₃₁

C₂×#462

yes

—

108

C₂×C₃₁

#3039

yes

—

109

C₂²×Q₈

C₂³×C₃₁

yes

—

110

C₂²×Q₈

C₂³×C₃₁

yes

characteristic

111

C₂×Q₁₆

—

112

C₂²×C₈

C₂³×C₃₁

yes

—

113

C₄⋊Q₈

—

114

C₄⋊Q₈

—

115

C₄×Q₈

—

116

C₄×Q₈

D₄×C₃₁

yes

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

C₂³×C₃₁

yes

—

121

C₂×C₄²

—

122

C₂×C₄²

—

123

C₂×C₄²

D₄×C₃₁

yes

—

124

C₂×C₄²

D₄×C₃₁

yes

characteristic

125

C₂×C₄²

C₂³×C₃₁

yes

—

126

C₄⋊C₈

—

127

#103

—

128

#95

—

129

C₃₁

#33163

yes

characteristic, direct product factor, Sylow

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

C₂×D₄×C₃₁

yes

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

C₂×D₄×C₃₁

yes

—

138

C₂²×C₄

C₂×D₄×C₃₁

yes

—

139

C₂²×C₄

C₂⁴×C₃₁

yes

characteristic

140

Q₁₆

—

141

C₂×C₈

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

C₂×D₄×C₃₁

yes

—

147

C₄²

—

148

C₄²

—

149

C₄²

—

150

C₄²

Q₁₆×C₃₁

yes

complemented

151

C₄²

C₂×D₄×C₃₁

yes

—

152

C₄²

#41×C₃₁

yes

—

153

C₂³

C₂²×D₄×C₃₁

yes

characteristic

154

Q₈

—

155

Q₈

—

156

Q₈

—

157

Q₈

C₃₁×#127

yes

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

C₃₁×#121

yes

—

164

C₂×C₄

C₃₁×#121

yes

—

165

C₂×C₄

C₂×Q₁₆×C₃₁

yes

—

166

C₂×C₄

C₃₁×#137

yes

—

167

C₂×C₄

C₂²×D₄×C₃₁

yes

—

168

C₂×C₄

C₂×#41×C₃₁

yes

—

169

C₈

—

170

C₂²

C₂×C₃₁×#121

yes

—

171

C₂²

C₂²×Q₁₆×C₃₁

yes

characteristic

172

C₂²

C₂×C₃₁×#137

yes

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

C₃₁×#462

yes

complemented

178

C₄

C₂×C₃₁×#127

yes

characteristic

179

C₄

C₃₁×#532

yes

—

180

C₂

C₃₁×#2856

yes

characteristic

181

C₂

C₂×C₃₁×#462

yes

—

182

C₂

C₃₁×#3039

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₃₁

C₂²×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂⁴×C₃₁

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

N	H ≅ G/N	type
58. C₄²×C₃₁	Q₁₆	semidirect
61. #33163	C₃₁	direct (N × H)
91. C₄×C₃₁	#462	semidirect
129. C₃₁	#33163	direct (N × H)
150. C₄²	Q₁₆×C₃₁	semidirect
177. C₄	C₃₁×#462	semidirect

H ≅ G/N

type

58. C₄²×C₃₁

Q₁₆

semidirect

61. #33163

C₃₁

direct (N × H)

91. C₄×C₃₁

#462

semidirect

129. C₃₁

#33163

direct (N × H)

150. C₄²

Q₁₆×C₃₁

semidirect

177. C₄

C₃₁×#462

semidirect

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

subgroup	index
C₂×C₃₁×#462	2
C₃₁×#2868	2
C₄²×Q₈×C₃₁	2
C₃₁×#1981	2
C₃₁×#1842	2
#33163	31

subgroup

index

C₂×C₃₁×#462

C₃₁×#2868

C₄²×Q₈×C₃₁

C₃₁×#1981

C₃₁×#1842

#33163

Subgroups — C₃₁×#33163 #21489922

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#33163 #21489922

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33163 #21489922