Group #21464885 — subgroups

Subgroups — C₃₁×#8126 #21464885

Order 7936.

520 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 230 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×#2920	C₂	1	3	yes	maximal
2	C₃₁×#1308	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
4	C₃₁×#432	—	2	3	no	—
5	C₃₁×#432	C₂²	1	6	yes	—
6	C₃₁×#431	—	2	3	no	—
7	C₃₁×#431	C₄	1	6	yes	—
8	C₃₁×#423	—	2	3	no	—
9	C₃₁×#407	C₄	1	6	yes	—
10	C₄×M₁₆×C₃₁	—	2	3	no	—
11	C₂³×C₄×C₃₁	C₂×C₄	1	3	yes	—
12	C₂×M₁₆×C₃₁	—	2	6	no	—
13	C₂×C₄²×C₃₁	C₂×C₄	1	3	yes	—
14	C₂×C₄²×C₃₁	D₄	1	3	yes	—
15	C₂×C₄²×C₃₁	D₄	1	3	yes	—
16	C₂×C₄²×C₃₁	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₃₁	Q₈	1	1	yes	characteristic
18	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
19	C₃₁×C₄⋊C₈	—	2	6	no	—
20	C₃₁×C₄⋊C₈	—	2	6	no	—
21	C₃₁×C₂²⋊C₈	—	2	6	no	—
22	C₃₁×C₂²⋊C₈	C₂×C₄	1	12	yes	—
23	C₃₁×#97	—	2	6	no	—
24	C₄×C₈×C₃₁	—	2	6	no	—
25	C₂⁴×C₃₁	C₄²	1	1	yes	characteristic
26	C₂²×C₄×C₃₁	—	4	6	no	—
27	C₂²×C₄×C₃₁	C₄²	1	3	yes	—
28	C₂²×C₄×C₃₁	C₂²⋊C₄	1	3	yes	—
29	C₂²×C₄×C₃₁	C₂²⋊C₄	1	3	yes	—
30	C₂²×C₄×C₃₁	C₄⋊C₄	1	3	yes	—
31	C₂²×C₄×C₃₁	C₄⋊C₄	1	3	yes	—
32	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
33	M₁₆×C₃₁	—	4	12	no	—
34	C₂×C₈×C₃₁	—	2	12	no	—
35	C₄²×C₃₁	C₂²⋊C₄	1	3	yes	—
36	C₄²×C₃₁	C₂²⋊C₄	1	3	yes	—
37	C₄²×C₃₁	C₄⋊C₄	1	3	yes	—
38	C₄²×C₃₁	C₄⋊C₄	1	3	yes	—
39	C₄²×C₃₁	C₂×D₄	1	3	yes	—
40	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
41	#8126	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂³×C₃₁	—	4	3	no	—
43	C₂³×C₃₁	#95	1	1	yes	characteristic
44	C₂³×C₃₁	#95	1	1	yes	characteristic
45	C₂³×C₃₁	C₂×C₄²	1	1	yes	characteristic
46	C₂×C₄×C₃₁	—	2	6	no	—
47	C₂×C₄×C₃₁	—	2	6	no	—
48	C₂×C₄×C₃₁	—	2	6	no	—
49	C₂×C₄×C₃₁	—	4	3	no	—
50	C₂×C₄×C₃₁	—	4	3	no	—
51	C₂×C₄×C₃₁	—	4	3	no	—
52	C₂×C₄×C₃₁	#95	1	3	yes	—
53	C₂×C₄×C₃₁	#95	1	3	yes	—
54	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	3	yes	—
55	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	3	yes	—
56	C₈×C₃₁	—	4	12	no	—
57	#2920	C₂×C₃₁	1	3	yes	—
58	#1308	C₂×C₃₁	1	4	yes	complemented
59	C₂²×C₃₁	—	2	3	no	—
60	C₂²×C₃₁	—	2	3	no	—
61	C₂²×C₃₁	—	2	3	no	—
62	C₂²×C₃₁	—	4	4	no	—
63	C₂²×C₃₁	C₂×#95	1	1	yes	characteristic
64	C₄×C₃₁	—	2	3	no	—
65	C₄×C₃₁	—	2	3	no	—
66	C₄×C₃₁	—	2	3	no	—
67	C₄×C₃₁	#344	1	6	yes	—
68	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
69	#432	—	2	3	no	—
70	#432	C₂²×C₃₁	1	6	yes	—
71	#431	—	2	3	no	—
72	#431	C₄×C₃₁	1	6	yes	—
73	#423	—	2	3	no	—
74	#407	C₄×C₃₁	1	6	yes	—
75	C₄×M₁₆	—	2	3	no	—
76	C₂×C₃₁	—	4	1	no	—
77	C₂×C₃₁	—	4	1	no	—
78	C₂×C₃₁	—	4	1	no	—
79	C₂×C₃₁	C₂×#344	1	3	yes	—
80	C₂³×C₄	C₂×C₄×C₃₁	1	3	yes	—
81	C₂×M₁₆	—	2	6	no	—
82	C₂×C₄²	C₂×C₄×C₃₁	1	3	yes	—
83	C₂×C₄²	D₄×C₃₁	1	3	yes	—
84	C₂×C₄²	D₄×C₃₁	1	3	yes	—
85	C₂×C₄²	Q₈×C₃₁	1	1	yes	characteristic
86	C₂×C₄²	Q₈×C₃₁	1	1	yes	characteristic
87	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
88	C₄⋊C₈	—	2	6	no	—
89	C₄⋊C₈	—	2	6	no	—
90	C₂²⋊C₈	—	2	6	no	—
91	C₂²⋊C₈	C₂×C₄×C₃₁	1	12	yes	—
92	#97	—	2	6	no	—
93	C₄×C₈	—	2	6	no	—
94	C₃₁	#8126	1	1	yes	characteristic, direct product factor, Sylow
95	C₂⁴	C₄²×C₃₁	1	1	yes	characteristic
96	C₂²×C₄	—	4	6	no	—
97	C₂²×C₄	C₄²×C₃₁	1	3	yes	—
98	C₂²×C₄	C₂²⋊C₄×C₃₁	1	3	yes	—
99	C₂²×C₄	C₂²⋊C₄×C₃₁	1	3	yes	—
100	C₂²×C₄	C₄⋊C₄×C₃₁	1	3	yes	—
101	C₂²×C₄	C₄⋊C₄×C₃₁	1	3	yes	—
102	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
103	M₁₆	—	4	12	no	—
104	C₂×C₈	—	2	12	no	—
105	C₄²	C₂²⋊C₄×C₃₁	1	3	yes	—
106	C₄²	C₂²⋊C₄×C₃₁	1	3	yes	—
107	C₄²	C₄⋊C₄×C₃₁	1	3	yes	—
108	C₄²	C₄⋊C₄×C₃₁	1	3	yes	—
109	C₄²	C₂×D₄×C₃₁	1	3	yes	—
110	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
111	C₂³	—	4	3	no	—
112	C₂³	C₃₁×#95	1	1	yes	characteristic
113	C₂³	C₃₁×#95	1	1	yes	characteristic
114	C₂³	C₂×C₄²×C₃₁	1	1	yes	characteristic
115	C₂×C₄	—	2	6	no	—
116	C₂×C₄	—	2	6	no	—
117	C₂×C₄	—	2	6	no	—
118	C₂×C₄	—	4	3	no	—
119	C₂×C₄	—	4	3	no	—
120	C₂×C₄	—	4	3	no	—
121	C₂×C₄	C₃₁×#95	1	3	yes	—
122	C₂×C₄	C₃₁×#95	1	3	yes	—
123	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	3	yes	—
124	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	3	yes	—
125	C₈	—	4	12	no	—
126	C₂²	—	2	3	no	—
127	C₂²	—	2	3	no	—
128	C₂²	—	2	3	no	—
129	C₂²	—	4	4	no	—
130	C₂²	C₂×C₃₁×#95	1	1	yes	characteristic
131	C₄	—	2	3	no	—
132	C₄	—	2	3	no	—
133	C₄	—	2	3	no	—
134	C₄	C₃₁×#344	1	6	yes	—
135	C₂	—	4	1	no	—
136	C₂	—	4	1	no	—
137	C₂	—	4	1	no	—
138	C₂	C₂×C₃₁×#344	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₁×#2920

C₂

yes

maximal

C₃₁×#1308

C₂

yes

maximal, complemented

C₂²×C₄²×C₃₁

C₂²

yes

characteristic

C₃₁×#432

—

C₃₁×#432

C₂²

yes

—

C₃₁×#431

—

C₃₁×#431

C₄

yes

—

C₃₁×#423

—

C₃₁×#407

C₄

yes

—

C₄×M₁₆×C₃₁

—

C₂³×C₄×C₃₁

C₂×C₄

yes

—

C₂×M₁₆×C₃₁

—

C₂×C₄²×C₃₁

C₂×C₄

yes

—

C₂×C₄²×C₃₁

D₄

yes

—

C₂×C₄²×C₃₁

D₄

yes

—

C₂×C₄²×C₃₁

Q₈

yes

characteristic

C₂×C₄²×C₃₁

Q₈

yes

characteristic

C₂×C₄²×C₃₁

C₂³

yes

characteristic

C₃₁×C₄⋊C₈

—

C₃₁×C₄⋊C₈

—

C₃₁×C₂²⋊C₈

—

C₃₁×C₂²⋊C₈

C₂×C₄

yes

—

C₃₁×#97

—

C₄×C₈×C₃₁

—

C₂⁴×C₃₁

C₄²

yes

characteristic

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₄²

yes

—

C₂²×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₃₁

C₄⋊C₄

yes

—

C₂²×C₄×C₃₁

C₄⋊C₄

yes

—

C₂²×C₄×C₃₁

C₂²×C₄

yes

—

M₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₄²×C₃₁

C₂²⋊C₄

yes

—

C₄²×C₃₁

C₂²⋊C₄

yes

—

C₄²×C₃₁

C₄⋊C₄

yes

—

C₄²×C₃₁

C₄⋊C₄

yes

—

C₄²×C₃₁

C₂×D₄

yes

—

C₄²×C₃₁

C₂×Q₈

yes

characteristic

#8126

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

—

C₂³×C₃₁

yes

C₂³×C₃₁

yes

C₂³×C₃₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

#95

yes

—

C₂×C₄×C₃₁

#95

yes

—

C₂×C₄×C₃₁

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

C₂×C₄⋊C₄

yes

—

C₈×C₃₁

—

#2920

C₂×C₃₁

yes

—

#1308

C₂×C₃₁

yes

complemented

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂×#95

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#344

yes

—

C₂²×C₄²

C₂²×C₃₁

yes

characteristic

#432

—

#432

C₂²×C₃₁

yes

—

#431

—

#431

C₄×C₃₁

yes

—

#423

—

#407

C₄×C₃₁

yes

—

C₄×M₁₆

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

C₂×#344

yes

—

C₂³×C₄

C₂×C₄×C₃₁

yes

—

C₂×M₁₆

—

C₂×C₄²

C₂×C₄×C₃₁

yes

—

C₂×C₄²

D₄×C₃₁

yes

—

C₂×C₄²

D₄×C₃₁

yes

—

C₂×C₄²

Q₈×C₃₁

yes

characteristic

C₂×C₄²

Q₈×C₃₁

yes

characteristic

C₂×C₄²

C₂³×C₃₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

C₂×C₄×C₃₁

yes

—

#97

—

C₄×C₈

—

C₃₁

#8126

yes

characteristic, direct product factor, Sylow

C₂⁴

C₄²×C₃₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₄²×C₃₁

yes

—

C₂²×C₄

C₂²⋊C₄×C₃₁

yes

—

C₂²×C₄

C₂²⋊C₄×C₃₁

yes

—

100

C₂²×C₄

C₄⋊C₄×C₃₁

yes

—

101

C₂²×C₄

C₄⋊C₄×C₃₁

yes

—

102

C₂²×C₄

C₂²×C₄×C₃₁

yes

—

103

M₁₆

—

104

C₂×C₈

—

105

C₄²

C₂²⋊C₄×C₃₁

yes

—

106

C₄²

C₂²⋊C₄×C₃₁

yes

—

107

C₄²

C₄⋊C₄×C₃₁

yes

—

108

C₄²

C₄⋊C₄×C₃₁

yes

—

109

C₄²

C₂×D₄×C₃₁

yes

—

110

C₄²

C₂×Q₈×C₃₁

yes

characteristic

111

C₂³

—

112

C₂³

C₃₁×#95

yes

characteristic

113

C₂³

C₃₁×#95

yes

characteristic

114

C₂³

C₂×C₄²×C₃₁

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₃₁×#95

yes

—

122

C₂×C₄

C₃₁×#95

yes

—

123

C₂×C₄

C₂×C₂²⋊C₄×C₃₁

yes

—

124

C₂×C₄

C₂×C₄⋊C₄×C₃₁

yes

—

125

C₈

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

C₂×C₃₁×#95

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

C₃₁×#344

yes

—

135

C₂

—

136

C₂

—

137

C₂

—

138

C₂

C₂×C₃₁×#344

yes

—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#95
commutator	C₂³	C₂×C₄²×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂⁴×C₃₁	C₄²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₃₁

C₂×#95

commutator

C₂³

C₂×C₄²×C₃₁

Frattini, cosocle

C₂×C₄²

C₂³×C₃₁

prime radical

C₂⁴×C₃₁

C₄²

LCS

G ▷ C₂³ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

N	H ≅ G/N	type
2. C₃₁×#1308	C₂	semidirect
41. #8126	C₃₁	direct (N × H)
58. #1308	C₂×C₃₁	semidirect
94. C₃₁	#8126	direct (N × H)

H ≅ G/N

type

2. C₃₁×#1308

C₂

semidirect

41. #8126

C₃₁

direct (N × H)

58. #1308

C₂×C₃₁

semidirect

94. C₃₁

#8126

direct (N × H)

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

subgroup	index
C₃₁×#2920	2
C₃₁×#1308	2
#8126	31

subgroup

index

C₃₁×#2920

C₃₁×#1308

#8126

Subgroups — C₃₁×#8126 #21464885

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#8126 #21464885

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8126 #21464885