Group #21466531 — subgroups

Subgroups — C₃₁×#9772 #21466531

Order 7936.

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

normal subgroups only

This group has 478 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	C₂×C₃₁×#494	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1848	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1546	C₂	1	4	yes	maximal
4	C₃₁×#1450	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆×C₃₁	—	2	2	no	—
6	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₃₁×#494	—	2	1	no	—
8	C₃₁×#494	—	2	1	no	—
9	C₃₁×#494	C₄	1	4	yes	—
10	C₃₁×#431	C₂²	1	2	yes	—
11	C₂×C₃₁×#108	—	2	2	no	—
12	C₃₁×#423	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#332	—	2	8	no	—
15	C₃₁×#322	—	2	1	no	—
16	C₈²×C₃₁	—	2	1	no	—
17	C₂²×Q₈×C₃₁	—	2	2	no	—
18	C₂×Q₁₆×C₃₁	—	2	4	no	—
19	C₂×Q₁₆×C₃₁	—	2	4	no	—
20	C₂×Q₁₆×C₃₁	—	4	4	no	—
21	C₂×Q₁₆×C₃₁	—	4	4	no	—
22	C₂×M₁₆×C₃₁	—	2	2	no	—
23	C₂²×C₈×C₃₁	D₄	1	2	yes	—
24	C₃₁×C₄⋊Q₈	—	2	4	no	—
25	C₃₁×C₄⋊Q₈	—	2	2	no	—
26	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
28	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
29	C₃₁×#108	—	4	4	no	—
30	C₃₁×C₄⋊C₈	—	2	4	no	—
31	C₃₁×C₄⋊C₈	D₄	1	4	yes	—
32	C₃₁×C₂²⋊C₈	—	2	2	no	—
33	C₄×C₈×C₃₁	—	2	2	no	—
34	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
36	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
37	C₂×Q₈×C₃₁	—	2	4	no	—
38	C₂×Q₈×C₃₁	—	4	2	no	—
39	C₂×Q₈×C₃₁	—	4	4	no	—
40	C₂×Q₈×C₃₁	—	4	4	no	—
41	C₂²×C₄×C₃₁	—	4	2	no	—
42	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₃₁	#41	1	2	yes	—
44	Q₁₆×C₃₁	—	4	8	no	—
45	Q₁₆×C₃₁	—	4	8	no	—
46	M₁₆×C₃₁	—	4	4	no	—
47	C₂×C₈×C₃₁	—	2	4	no	—
48	C₂×C₈×C₃₁	—	4	2	no	—
49	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
50	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₃₁	D₈	1	4	yes	—
52	C₂×C₈×C₃₁	SD₁₆	1	4	yes	—
53	C₄⋊C₄×C₃₁	—	2	8	no	—
54	C₄⋊C₄×C₃₁	—	4	4	no	—
55	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₃₁	C₂×D₄	1	2	yes	—
58	#9772	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂³×C₃₁	#124	1	1	yes	characteristic
60	Q₈×C₃₁	—	4	8	no	—
61	Q₈×C₃₁	—	8	2	no	—
62	Q₈×C₃₁	—	8	2	no	—
63	C₂×C₄×C₃₁	—	2	4	no	—
64	C₂×C₄×C₃₁	—	4	4	no	—
65	C₂×C₄×C₃₁	—	8	4	no	—
66	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
68	C₂×C₄×C₃₁	#121	1	2	yes	—
69	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
70	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
71	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
72	C₈×C₃₁	—	4	4	no	—
73	C₈×C₃₁	—	8	2	no	—
74	C₈×C₃₁	#102	1	4	yes	—
75	C₈×C₃₁	#102	1	4	yes	—
76	C₂×#494	C₂×C₃₁	1	1	yes	characteristic
77	#1848	C₂×C₃₁	1	1	yes	characteristic
78	#1546	C₂×C₃₁	1	4	yes	—
79	#1450	C₂×C₃₁	1	1	yes	characteristic
80	C₂²×C₃₁	—	2	2	no	—
81	C₂²×C₃₁	#390	1	1	yes	characteristic
82	C₂²×C₃₁	#486	1	2	yes	—
83	C₄×C₃₁	—	2	2	no	—
84	C₄×C₃₁	—	8	4	no	—
85	C₄×C₃₁	C₂×#102	1	2	yes	—
86	C₄×C₃₁	#473	1	4	yes	—
87	C₄×C₃₁	#492	1	1	yes	characteristic
88	C₄×C₃₁	#493	1	1	yes	characteristic
89	C₂²×Q₁₆	—	2	2	no	—
90	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
91	#494	—	2	1	no	—
92	#494	—	2	1	no	—
93	#494	C₄×C₃₁	1	4	yes	—
94	#431	C₂²×C₃₁	1	2	yes	—
95	C₂×#108	—	2	2	no	—
96	#423	C₂²×C₃₁	1	2	yes	—
97	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
98	#332	—	2	8	no	—
99	#322	—	2	1	no	—
100	C₈²	—	2	1	no	—
101	C₂×C₃₁	—	2	2	no	—
102	C₂×C₃₁	#1957	1	1	yes	characteristic
103	C₂×C₃₁	#1979	1	2	yes	—
104	C₂²×Q₈	—	2	2	no	—
105	C₂×Q₁₆	—	2	4	no	—
106	C₂×Q₁₆	—	2	4	no	—
107	C₂×Q₁₆	—	4	4	no	—
108	C₂×Q₁₆	—	4	4	no	—
109	C₂×M₁₆	—	2	2	no	—
110	C₂²×C₈	D₄×C₃₁	1	2	yes	—
111	C₄⋊Q₈	—	2	4	no	—
112	C₄⋊Q₈	—	2	2	no	—
113	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
114	C₂×C₄⋊C₄	—	2	4	no	—
115	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
116	#108	—	4	4	no	—
117	C₄⋊C₈	—	2	4	no	—
118	C₄⋊C₈	D₄×C₃₁	1	4	yes	—
119	C₂²⋊C₈	—	2	2	no	—
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
122	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
123	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
124	C₃₁	#9772	1	1	yes	characteristic, direct product factor, Sylow
125	C₂×Q₈	—	2	4	no	—
126	C₂×Q₈	—	4	2	no	—
127	C₂×Q₈	—	4	4	no	—
128	C₂×Q₈	—	4	4	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
131	C₂²×C₄	#41×C₃₁	1	2	yes	—
132	Q₁₆	—	4	8	no	—
133	Q₁₆	—	4	8	no	—
134	M₁₆	—	4	4	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
138	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
139	C₂×C₈	D₈×C₃₁	1	4	yes	—
140	C₂×C₈	SD₁₆×C₃₁	1	4	yes	—
141	C₄⋊C₄	—	2	8	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
144	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
145	C₄²	C₂×D₄×C₃₁	1	2	yes	—
146	C₂³	C₃₁×#124	1	1	yes	characteristic
147	Q₈	—	4	8	no	—
148	Q₈	—	8	2	no	—
149	Q₈	—	8	2	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	4	4	no	—
152	C₂×C₄	—	8	4	no	—
153	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
154	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
155	C₂×C₄	C₃₁×#121	1	2	yes	—
156	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
157	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
158	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
159	C₈	—	4	4	no	—
160	C₈	—	8	2	no	—
161	C₈	C₃₁×#102	1	4	yes	—
162	C₈	C₃₁×#102	1	4	yes	—
163	C₂²	—	2	2	no	—
164	C₂²	C₃₁×#390	1	1	yes	characteristic
165	C₂²	C₃₁×#486	1	2	yes	—
166	C₄	—	2	2	no	—
167	C₄	—	8	4	no	—
168	C₄	C₂×C₃₁×#102	1	2	yes	—
169	C₄	C₃₁×#473	1	4	yes	—
170	C₄	C₃₁×#492	1	1	yes	characteristic
171	C₄	C₃₁×#493	1	1	yes	characteristic
172	C₂	—	2	2	no	—
173	C₂	C₃₁×#1957	1	1	yes	characteristic
174	C₂	C₃₁×#1979	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃₁×#494

C₂

yes

maximal, characteristic

C₃₁×#1848

C₂

yes

maximal, characteristic

C₃₁×#1546

C₂

yes

maximal

C₃₁×#1450

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₃₁

—

C₂×C₃₁×C₄⋊Q₈

C₂²

yes

—

C₃₁×#494

—

C₃₁×#494

—

C₃₁×#494

C₄

yes

—

C₃₁×#431

C₂²

yes

—

C₂×C₃₁×#108

—

C₃₁×#423

C₂²

yes

—

C₂×C₄×C₈×C₃₁

C₂²

yes

characteristic

C₃₁×#332

—

C₃₁×#322

—

C₈²×C₃₁

—

C₂²×Q₈×C₃₁

—

C₂×Q₁₆×C₃₁

—

C₂×Q₁₆×C₃₁

—

C₂×Q₁₆×C₃₁

—

C₂×Q₁₆×C₃₁

—

C₂×M₁₆×C₃₁

—

C₂²×C₈×C₃₁

D₄

yes

—

C₃₁×C₄⋊Q₈

—

C₃₁×C₄⋊Q₈

—

C₃₁×C₄⋊Q₈

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄²×C₃₁

C₂³

yes

characteristic

C₃₁×#108

—

C₃₁×C₄⋊C₈

—

C₃₁×C₄⋊C₈

D₄

yes

—

C₃₁×C₂²⋊C₈

—

C₄×C₈×C₃₁

—

C₄×C₈×C₃₁

C₂×C₄

yes

—

C₄×C₈×C₃₁

D₄

yes

characteristic

C₄×C₈×C₃₁

D₄

yes

characteristic

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₃₁

#41

yes

—

Q₁₆×C₃₁

—

Q₁₆×C₃₁

—

M₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂²⋊C₄

yes

—

C₂×C₈×C₃₁

C₂²⋊C₄

yes

—

C₂×C₈×C₃₁

D₈

yes

—

C₂×C₈×C₃₁

SD₁₆

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

C₂²×C₄

yes

characteristic

C₄²×C₃₁

C₂×D₄

yes

characteristic

C₄²×C₃₁

C₂×D₄

yes

—

#9772

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

#124

yes

characteristic

Q₈×C₃₁

—

Q₈×C₃₁

—

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₃₁

C₄×D₄

yes

—

C₂×C₄×C₃₁

#121

yes

—

C₂×C₄×C₃₁

#127

yes

characteristic

C₂×C₄×C₃₁

C₂×D₈

yes

—

C₂×C₄×C₃₁

C₂×SD₁₆

yes

—

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

#102

yes

—

C₈×C₃₁

#102

yes

—

C₂×#494

C₂×C₃₁

yes

characteristic

#1848

C₂×C₃₁

yes

characteristic

#1546

C₂×C₃₁

yes

—

#1450

C₂×C₃₁

yes

characteristic

C₂²×C₃₁

—

C₂²×C₃₁

#390

yes

characteristic

C₂²×C₃₁

#486

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂×#102

yes

—

C₄×C₃₁

#473

yes

—

C₄×C₃₁

yes

C₄×C₃₁

yes

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂²×C₃₁

yes

—

#494

—

#494

—

#494

C₄×C₃₁

yes

—

#431

C₂²×C₃₁

yes

—

C₂×#108

—

#423

C₂²×C₃₁

yes

—

C₂×C₄×C₈

C₂²×C₃₁

yes

characteristic

#332

—

#322

—

100

C₈²

—

101

C₂×C₃₁

—

102

C₂×C₃₁

#1957

yes

characteristic

103

C₂×C₃₁

#1979

yes

—

104

C₂²×Q₈

—

105

C₂×Q₁₆

—

106

C₂×Q₁₆

—

107

C₂×Q₁₆

—

108

C₂×Q₁₆

—

109

C₂×M₁₆

—

110

C₂²×C₈

D₄×C₃₁

yes

—

111

C₄⋊Q₈

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

C₂×C₄×C₃₁

yes

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄²

C₂³×C₃₁

yes

characteristic

116

#108

—

117

C₄⋊C₈

—

118

C₄⋊C₈

D₄×C₃₁

yes

—

119

C₂²⋊C₈

—

120

C₄×C₈

—

121

C₄×C₈

C₂×C₄×C₃₁

yes

—

122

C₄×C₈

D₄×C₃₁

yes

characteristic

123

C₄×C₈

D₄×C₃₁

yes

characteristic

124

C₃₁

#9772

yes

characteristic, direct product factor, Sylow

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂²×C₄

—

130

C₂²×C₄

C₂×D₄×C₃₁

yes

characteristic

131

C₂²×C₄

#41×C₃₁

yes

—

132

Q₁₆

—

133

Q₁₆

—

134

M₁₆

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

C₂²⋊C₄×C₃₁

yes

—

138

C₂×C₈

C₂²⋊C₄×C₃₁

yes

—

139

C₂×C₈

D₈×C₃₁

yes

—

140

C₂×C₈

SD₁₆×C₃₁

yes

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄²

C₂²×C₄×C₃₁

yes

characteristic

144

C₄²

C₂×D₄×C₃₁

yes

characteristic

145

C₄²

C₂×D₄×C₃₁

yes

—

146

C₂³

C₃₁×#124

yes

characteristic

147

Q₈

—

148

Q₈

—

149

Q₈

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

C₂×C₂²⋊C₄×C₃₁

yes

characteristic

154

C₂×C₄

C₄×D₄×C₃₁

yes

—

155

C₂×C₄

C₃₁×#121

yes

—

156

C₂×C₄

C₃₁×#127

yes

characteristic

157

C₂×C₄

C₂×D₈×C₃₁

yes

—

158

C₂×C₄

C₂×SD₁₆×C₃₁

yes

—

159

C₈

—

160

C₈

—

161

C₈

C₃₁×#102

yes

—

162

C₈

C₃₁×#102

yes

—

163

C₂²

—

164

C₂²

C₃₁×#390

yes

characteristic

165

C₂²

C₃₁×#486

yes

—

166

C₄

—

167

C₄

—

168

C₄

C₂×C₃₁×#102

yes

—

169

C₄

C₃₁×#473

yes

—

170

C₄

C₃₁×#492

yes

characteristic

171

C₄

C₃₁×#493

yes

characteristic

172

C₂

—

173

C₂

C₃₁×#1957

yes

characteristic

174

C₂

C₃₁×#1979

yes

—

	subgroup	G/H
center, socle	C₂²×C₃₁	#390
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂³×C₃₁	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₃₁

#390

commutator

C₄²

C₂²×C₄×C₃₁

Frattini, cosocle

C₂×C₄²

C₂³×C₃₁

C₂³×C₃₁

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

N	H ≅ G/N	type
58. #9772	C₃₁	direct (N × H)
124. C₃₁	#9772	direct (N × H)

H ≅ G/N

type

58. #9772

C₃₁

direct (N × H)

124. C₃₁

#9772

direct (N × H)

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

subgroup	index
C₂×C₃₁×#494	2
C₃₁×#1848	2
C₃₁×#1546	2
C₃₁×#1450	2
#9772	31

subgroup

index

C₂×C₃₁×#494

C₃₁×#1848

C₃₁×#1546

C₃₁×#1450

#9772

Subgroups — C₃₁×#9772 #21466531

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#9772 #21466531

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9772 #21466531