Group #21470278 — subgroups

Subgroups — C₃₁×#13519 #21470278

Order 7936.

252 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 106 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	C₃₁×#2161	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2150	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1945	C₂	1	2	yes	maximal
4	C₃₁×#1389	C₂	1	2	yes	maximal
5	C₃₁×#1384	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
7	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
8	C₂×C₃₁×#108	C₂²	1	2	yes	—
9	C₃₁×#427	C₂²	1	2	yes	—
10	C₃₁×#424	—	2	2	no	—
11	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#368	—	2	2	no	—
13	C₃₁×C₄⋊C₁₆	—	2	2	no	—
14	C₃₁×#339	—	2	2	no	—
15	C₂×M₁₆×C₃₁	—	2	2	no	—
16	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
18	C₃₁×#117	—	2	2	no	—
19	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
20	C₃₁×M₃₂	—	2	2	no	—
21	C₃₁×M₃₂	—	2	2	no	—
22	C₂×C₁₆×C₃₁	—	2	1	no	—
23	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
24	C₃₁×#108	—	2	2	no	—
25	C₃₁×#108	—	2	2	no	—
26	C₃₁×#107	—	2	2	no	—
27	C₃₁×#106	Q₈	1	4	yes	—
28	C₃₁×C₄⋊C₈	—	4	4	no	—
29	C₄×C₈×C₃₁	—	2	1	no	—
30	C₄×C₈×C₃₁	—	2	1	no	—
31	C₂²×C₄×C₃₁	—	2	1	no	—
32	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
33	M₁₆×C₃₁	—	4	2	no	—
34	M₁₆×C₃₁	—	4	2	no	—
35	C₂×C₈×C₃₁	—	2	1	no	—
36	C₂×C₈×C₃₁	—	2	2	no	—
37	C₂×C₈×C₃₁	—	4	2	no	—
38	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
40	C₂×C₈×C₃₁	#41	1	2	yes	—
41	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
42	C₄⋊C₄×C₃₁	—	2	4	no	—
43	C₂²⋊C₄×C₃₁	—	4	2	no	—
44	C₄²×C₃₁	—	2	1	no	—
45	C₄²×C₃₁	—	4	2	no	—
46	C₄²×C₃₁	Q₁₆	1	1	yes	characteristic
47	C₄²×C₃₁	Q₁₆	1	1	yes	characteristic
48	C₁₆×C₃₁	—	2	2	no	—
49	C₁₆×C₃₁	—	4	1	no	—
50	#13519	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂³×C₃₁	#135	1	1	yes	characteristic
52	C₂×C₄×C₃₁	—	2	1	no	—
53	C₂×C₄×C₃₁	—	2	1	no	—
54	C₂×C₄×C₃₁	—	4	2	no	—
55	C₂×C₄×C₃₁	—	4	4	no	—
56	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
57	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
58	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
59	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
60	C₈×C₃₁	—	4	1	no	—
61	C₈×C₃₁	—	8	2	no	—
62	C₈×C₃₁	#123	1	1	yes	characteristic
63	C₈×C₃₁	#125	1	2	yes	—
64	C₈×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
65	#2161	C₂×C₃₁	1	1	yes	characteristic
66	#2150	C₂×C₃₁	1	1	yes	characteristic
67	#1945	C₂×C₃₁	1	2	yes	—
68	#1389	C₂×C₃₁	1	2	yes	—
69	#1384	C₂×C₃₁	1	1	yes	characteristic
70	C₂²×C₃₁	—	4	1	no	—
71	C₂²×C₃₁	#467	1	1	yes	characteristic
72	C₂²×C₃₁	#478	1	2	yes	—
73	C₄×C₃₁	—	2	2	no	—
74	C₄×C₃₁	—	4	1	no	—
75	C₄×C₃₁	—	8	4	no	—
76	C₄×C₃₁	#398	1	1	yes	characteristic
77	C₄×C₃₁	#469	1	1	yes	characteristic
78	C₄×C₃₁	#477	1	2	yes	—
79	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
80	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
81	C₂×#108	C₂²×C₃₁	1	2	yes	—
82	#427	C₂²×C₃₁	1	2	yes	—
83	#424	—	2	2	no	—
84	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
85	#368	—	2	2	no	—
86	C₄⋊C₁₆	—	2	2	no	—
87	#339	—	2	2	no	—
88	C₂×C₃₁	—	2	1	no	—
89	C₂×C₃₁	—	2	2	no	—
90	C₂×C₃₁	#2087	1	1	yes	characteristic
91	C₂×M₁₆	—	2	2	no	—
92	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
93	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
94	#117	—	2	2	no	—
95	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
96	M₃₂	—	2	2	no	—
97	M₃₂	—	2	2	no	—
98	C₂×C₁₆	—	2	1	no	—
99	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
100	#108	—	2	2	no	—
101	#108	—	2	2	no	—
102	#107	—	2	2	no	—
103	#106	Q₈×C₃₁	1	4	yes	—
104	C₄⋊C₈	—	4	4	no	—
105	C₄×C₈	—	2	1	no	—
106	C₄×C₈	—	2	1	no	—
107	C₃₁	#13519	1	1	yes	characteristic, direct product factor, Sylow
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
110	M₁₆	—	4	2	no	—
111	M₁₆	—	4	2	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
116	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
117	C₂×C₈	#41×C₃₁	1	2	yes	—
118	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
119	C₄⋊C₄	—	2	4	no	—
120	C₂²⋊C₄	—	4	2	no	—
121	C₄²	—	2	1	no	—
122	C₄²	—	4	2	no	—
123	C₄²	Q₁₆×C₃₁	1	1	yes	characteristic
124	C₄²	Q₁₆×C₃₁	1	1	yes	characteristic
125	C₁₆	—	2	2	no	—
126	C₁₆	—	4	1	no	—
127	C₂³	C₃₁×#135	1	1	yes	characteristic
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	4	no	—
132	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
133	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
134	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
135	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
136	C₈	—	4	1	no	—
137	C₈	—	8	2	no	—
138	C₈	C₃₁×#123	1	1	yes	characteristic
139	C₈	C₃₁×#125	1	2	yes	—
140	C₈	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
141	C₂²	—	4	1	no	—
142	C₂²	C₃₁×#467	1	1	yes	characteristic
143	C₂²	C₃₁×#478	1	2	yes	—
144	C₄	—	2	2	no	—
145	C₄	—	4	1	no	—
146	C₄	—	8	4	no	—
147	C₄	C₃₁×#398	1	1	yes	characteristic
148	C₄	C₃₁×#469	1	1	yes	characteristic
149	C₄	C₃₁×#477	1	2	yes	—
150	C₂	—	2	1	no	—
151	C₂	—	2	2	no	—
152	C₂	C₃₁×#2087	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₁×#2161

C₂

yes

maximal, characteristic

C₃₁×#2150

C₂

yes

maximal, characteristic

C₃₁×#1945

C₂

yes

maximal

C₃₁×#1389

C₂

yes

maximal

C₃₁×#1384

C₂

yes

maximal, characteristic

C₂×C₃₁×M₃₂

C₂²

yes

characteristic

C₂×C₃₁×M₃₂

C₂²

yes

characteristic

C₂×C₃₁×#108

C₂²

yes

—

C₃₁×#427

C₂²

yes

—

C₃₁×#424

—

C₂×C₄×C₈×C₃₁

C₂²

yes

characteristic

C₃₁×#368

—

C₃₁×C₄⋊C₁₆

—

C₃₁×#339

—

C₂×M₁₆×C₃₁

—

C₂²×C₈×C₃₁

D₄

yes

characteristic

C₂²×C₈×C₃₁

C₂³

yes

characteristic

C₃₁×#117

—

C₂×C₄²×C₃₁

D₄

yes

characteristic

C₃₁×M₃₂

—

C₃₁×M₃₂

—

C₂×C₁₆×C₃₁

—

C₂×C₁₆×C₃₁

D₄

yes

—

C₃₁×#108

—

C₃₁×#108

—

C₃₁×#107

—

C₃₁×#106

Q₈

yes

—

C₃₁×C₄⋊C₈

—

C₄×C₈×C₃₁

—

C₄×C₈×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×D₄

yes

characteristic

M₁₆×C₃₁

—

M₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂×D₄

yes

characteristic

C₂×C₈×C₃₁

C₂×Q₈

yes

—

C₂×C₈×C₃₁

#41

yes

—

C₂×C₈×C₃₁

#41

yes

characteristic

C₄⋊C₄×C₃₁

—

C₂²⋊C₄×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

Q₁₆

yes

characteristic

C₄²×C₃₁

Q₁₆

yes

characteristic

C₁₆×C₃₁

—

C₁₆×C₃₁

—

#13519

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

#135

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

#121

yes

characteristic

C₂×C₄×C₃₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₃₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₃₁

AGL(1,ℤ/8)

yes

—

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

#123

yes

characteristic

C₈×C₃₁

#125

yes

—

C₈×C₃₁

C₄⋊Q₈

yes

characteristic

#2161

C₂×C₃₁

yes

characteristic

#2150

C₂×C₃₁

yes

characteristic

#1945

C₂×C₃₁

yes

—

#1389

C₂×C₃₁

yes

—

#1384

C₂×C₃₁

yes

characteristic

C₂²×C₃₁

—

C₂²×C₃₁

#467

yes

characteristic

C₂²×C₃₁

#478

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

—

C₂×M₃₂

C₂²×C₃₁

yes

characteristic

C₂×M₃₂

C₂²×C₃₁

yes

characteristic

C₂×#108

C₂²×C₃₁

yes

—

#427

C₂²×C₃₁

yes

—

#424

—

C₂×C₄×C₈

C₂²×C₃₁

yes

characteristic

#368

—

C₄⋊C₁₆

—

#339

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

#2087

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

D₄×C₃₁

yes

characteristic

C₂²×C₈

C₂³×C₃₁

yes

characteristic

#117

—

C₂×C₄²

D₄×C₃₁

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

D₄×C₃₁

yes

—

100

#108

—

101

#108

—

102

#107

—

103

#106

Q₈×C₃₁

yes

—

104

C₄⋊C₈

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₃₁

#13519

yes

characteristic, direct product factor, Sylow

108

C₂²×C₄

—

109

C₂²×C₄

C₂×D₄×C₃₁

yes

characteristic

110

M₁₆

—

111

M₁₆

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

C₂×D₄×C₃₁

yes

characteristic

116

C₂×C₈

C₂×Q₈×C₃₁

yes

—

117

C₂×C₈

#41×C₃₁

yes

—

118

C₂×C₈

#41×C₃₁

yes

characteristic

119

C₄⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

—

123

C₄²

Q₁₆×C₃₁

yes

characteristic

124

C₄²

Q₁₆×C₃₁

yes

characteristic

125

C₁₆

—

126

C₁₆

—

127

C₂³

C₃₁×#135

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

C₃₁×#121

yes

characteristic

133

C₂×C₄

C₃₁×C₂²⋊Q₈

yes

—

134

C₂×C₄

C₂×Q₁₆×C₃₁

yes

characteristic

135

C₂×C₄

C₃₁×AGL(1,ℤ/8)

yes

—

136

C₈

—

137

C₈

—

138

C₈

C₃₁×#123

yes

characteristic

139

C₈

C₃₁×#125

yes

—

140

C₈

C₃₁×C₄⋊Q₈

yes

characteristic

141

C₂²

—

142

C₂²

C₃₁×#467

yes

characteristic

143

C₂²

C₃₁×#478

yes

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

C₃₁×#398

yes

characteristic

148

C₄

C₃₁×#469

yes

characteristic

149

C₄

C₃₁×#477

yes

—

150

C₂

—

151

C₂

—

152

C₂

C₃₁×#2087

yes

characteristic

	subgroup	G/H
center	C₄×C₃₁	#469
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂×C₃₁	#2087
prime radical	C₂³×C₃₁	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

subgroup

G/H

center

C₄×C₃₁

#469

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₃₁

C₂×C₃₁

C₂³×C₃₁

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

N	H ≅ G/N	type
50. #13519	C₃₁	direct (N × H)
107. C₃₁	#13519	direct (N × H)

H ≅ G/N

type

50. #13519

C₃₁

direct (N × H)

107. C₃₁

#13519

direct (N × H)

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

subgroup	index
C₃₁×#2161	2
C₃₁×#2150	2
C₃₁×#1945	2
C₃₁×#1389	2
C₃₁×#1384	2
#13519	31

subgroup

index

C₃₁×#2161

C₃₁×#2150

C₃₁×#1945

C₃₁×#1389

C₃₁×#1384

#13519

Subgroups — C₃₁×#13519 #21470278

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#13519 #21470278

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13519 #21470278