Group #21466970 — subgroups

Subgroups — C₃₁×#10211 #21466970

Order 7936.

406 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 94 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×#3237	C₂	1	3	yes	maximal
2	C₃₁×#1515	C₂	1	3	yes	maximal
3	C₃₁×#1308	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#533	C₂²	1	3	yes	—
5	C₃₁×#501	—	2	3	no	—
6	C₃₁×#491	—	2	3	no	—
7	C₃₁×#485	—	2	6	no	—
8	C₃₁×#477	—	2	3	no	—
9	C₃₁×#475	—	2	3	no	—
10	C₃₁×#432	C₂²	1	3	yes	—
11	C₃₁×#383	C₂²	1	1	yes	characteristic
12	C₃₁×#328	—	2	6	no	—
13	C₃₁×C₄⋊Q₈	D₄	1	3	yes	—
14	C₃₁×C₄⋊Q₈	D₄	1	3	yes	—
15	C₃₁×#125	—	2	3	no	—
16	C₃₁×C₂²⋊Q₈	—	2	3	no	—
17	C₃₁×C₂²⋊Q₈	—	2	3	no	—
18	C₄×Q₈×C₃₁	—	2	3	no	—
19	C₄×Q₈×C₃₁	—	2	3	no	—
20	C₃₁×#117	—	2	3	no	—
21	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
22	C₃₁×#107	—	2	6	no	—
23	C₃₁×#106	—	2	6	no	—
24	C₃₁×C₄⋊C₈	—	2	3	no	—
25	C₃₁×#103	—	2	6	no	—
26	C₃₁×#103	—	2	6	no	—
27	C₃₁×C₂²⋊C₈	D₄	1	6	yes	—
28	C₃₁×#97	—	2	3	no	—
29	C₃₁×#95	—	2	3	no	—
30	C₂×Q₈×C₃₁	—	2	3	no	—
31	C₂×Q₈×C₃₁	—	2	3	no	—
32	C₂²×C₄×C₃₁	—	4	1	no	—
33	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
34	C₂×C₈×C₃₁	—	2	6	no	—
35	C₄⋊C₄×C₃₁	—	2	3	no	—
36	C₄⋊C₄×C₃₁	—	2	3	no	—
37	C₄⋊C₄×C₃₁	—	2	3	no	—
38	C₄⋊C₄×C₃₁	—	2	3	no	—
39	C₄⋊C₄×C₃₁	—	4	3	no	—
40	C₄⋊C₄×C₃₁	—	4	3	no	—
41	C₂²⋊C₄×C₃₁	—	4	3	no	—
42	C₄²×C₃₁	—	4	3	no	—
43	C₄²×C₃₁	C₂×D₄	1	3	yes	—
44	C₄²×C₃₁	#41	1	1	yes	characteristic
45	#10211	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂³×C₃₁	#127	1	1	yes	characteristic
47	Q₈×C₃₁	—	4	6	no	—
48	Q₈×C₃₁	—	8	3	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	—	8	3	no	—
53	C₂×C₄×C₃₁	#120	1	3	yes	—
54	C₂×C₄×C₃₁	#121	1	3	yes	—
55	C₈×C₃₁	—	4	6	no	—
56	#3237	C₂×C₃₁	1	3	yes	—
57	#1515	C₂×C₃₁	1	3	yes	—
58	#1308	C₂×C₃₁	1	1	yes	characteristic
59	C₂²×C₃₁	—	2	3	no	—
60	C₂²×C₃₁	#392	1	1	yes	characteristic
61	C₄×C₃₁	—	2	3	no	—
62	C₄×C₃₁	—	2	3	no	—
63	C₄×C₃₁	—	8	3	no	—
64	C₄×C₃₁	—	8	3	no	—
65	C₄×C₃₁	—	16	1	no	—
66	#533	C₂²×C₃₁	1	3	yes	—
67	#501	—	2	3	no	—
68	#491	—	2	3	no	—
69	#485	—	2	6	no	—
70	#477	—	2	3	no	—
71	#475	—	2	3	no	—
72	#432	C₂²×C₃₁	1	3	yes	—
73	#383	C₂²×C₃₁	1	1	yes	characteristic
74	#328	—	2	6	no	—
75	C₂×C₃₁	—	4	1	no	—
76	C₂×C₃₁	#2009	1	3	yes	—
77	C₄⋊Q₈	D₄×C₃₁	1	3	yes	—
78	C₄⋊Q₈	D₄×C₃₁	1	3	yes	—
79	#125	—	2	3	no	—
80	C₂²⋊Q₈	—	2	3	no	—
81	C₂²⋊Q₈	—	2	3	no	—
82	C₄×Q₈	—	2	3	no	—
83	C₄×Q₈	—	2	3	no	—
84	#117	—	2	3	no	—
85	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
86	#107	—	2	6	no	—
87	#106	—	2	6	no	—
88	C₄⋊C₈	—	2	3	no	—
89	#103	—	2	6	no	—
90	#103	—	2	6	no	—
91	C₂²⋊C₈	D₄×C₃₁	1	6	yes	—
92	#97	—	2	3	no	—
93	#95	—	2	3	no	—
94	C₃₁	#10211	1	1	yes	characteristic, direct product factor, Sylow
95	C₂×Q₈	—	2	3	no	—
96	C₂×Q₈	—	2	3	no	—
97	C₂²×C₄	—	4	1	no	—
98	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
99	C₂×C₈	—	2	6	no	—
100	C₄⋊C₄	—	2	3	no	—
101	C₄⋊C₄	—	2	3	no	—
102	C₄⋊C₄	—	2	3	no	—
103	C₄⋊C₄	—	2	3	no	—
104	C₄⋊C₄	—	4	3	no	—
105	C₄⋊C₄	—	4	3	no	—
106	C₂²⋊C₄	—	4	3	no	—
107	C₄²	—	4	3	no	—
108	C₄²	C₂×D₄×C₃₁	1	3	yes	—
109	C₄²	#41×C₃₁	1	1	yes	characteristic
110	C₂³	C₃₁×#127	1	1	yes	characteristic
111	Q₈	—	4	6	no	—
112	Q₈	—	8	3	no	—
113	C₂×C₄	—	4	3	no	—
114	C₂×C₄	—	4	3	no	—
115	C₂×C₄	—	4	3	no	—
116	C₂×C₄	—	8	3	no	—
117	C₂×C₄	C₃₁×#120	1	3	yes	—
118	C₂×C₄	C₃₁×#121	1	3	yes	—
119	C₈	—	4	6	no	—
120	C₂²	—	2	3	no	—
121	C₂²	C₃₁×#392	1	1	yes	characteristic
122	C₄	—	2	3	no	—
123	C₄	—	2	3	no	—
124	C₄	—	8	3	no	—
125	C₄	—	8	3	no	—
126	C₄	—	16	1	no	—
127	C₂	—	4	1	no	—
128	C₂	C₃₁×#2009	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₁×#3237

C₂

yes

maximal

C₃₁×#1515

C₂

yes

maximal

C₃₁×#1308

C₂

yes

maximal, characteristic

C₃₁×#533

C₂²

yes

—

C₃₁×#501

—

C₃₁×#491

—

C₃₁×#485

—

C₃₁×#477

—

C₃₁×#475

—

C₃₁×#432

C₂²

yes

—

C₃₁×#383

C₂²

yes

characteristic

C₃₁×#328

—

C₃₁×C₄⋊Q₈

D₄

yes

—

C₃₁×C₄⋊Q₈

D₄

yes

—

C₃₁×#125

—

C₃₁×C₂²⋊Q₈

—

C₃₁×C₂²⋊Q₈

—

C₄×Q₈×C₃₁

—

C₄×Q₈×C₃₁

—

C₃₁×#117

—

C₂×C₄²×C₃₁

C₂³

yes

characteristic

C₃₁×#107

—

C₃₁×#106

—

C₃₁×C₄⋊C₈

—

C₃₁×#103

—

C₃₁×#103

—

C₃₁×C₂²⋊C₈

D₄

yes

—

C₃₁×#97

—

C₃₁×#95

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₈×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₂²⋊C₄×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

C₂×D₄

yes

—

C₄²×C₃₁

#41

yes

characteristic

#10211

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

#127

yes

characteristic

Q₈×C₃₁

—

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

#120

yes

—

C₂×C₄×C₃₁

#121

yes

—

C₈×C₃₁

—

#3237

C₂×C₃₁

yes

—

#1515

C₂×C₃₁

yes

—

#1308

C₂×C₃₁

yes

characteristic

C₂²×C₃₁

—

C₂²×C₃₁

#392

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

#533

C₂²×C₃₁

yes

—

—

—

—

—

—

C₂²×C₃₁

yes

—

#383

C₂²×C₃₁

yes

characteristic

#328

—

C₂×C₃₁

—

C₂×C₃₁

#2009

yes

—

C₄⋊Q₈

D₄×C₃₁

yes

—

C₄⋊Q₈

D₄×C₃₁

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂³×C₃₁

yes

characteristic

#107

—

#106

—

C₄⋊C₈

—

#103

—

#103

—

C₂²⋊C₈

D₄×C₃₁

yes

—

#97

—

#95

—

C₃₁

#10211

yes

characteristic, direct product factor, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄×C₃₁

yes

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₂²⋊C₄

—

107

C₄²

—

108

C₄²

C₂×D₄×C₃₁

yes

—

109

C₄²

#41×C₃₁

yes

characteristic

110

C₂³

C₃₁×#127

yes

characteristic

111

Q₈

—

112

Q₈

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₃₁×#120

yes

—

118

C₂×C₄

C₃₁×#121

yes

—

119

C₈

—

120

C₂²

—

121

C₂²

C₃₁×#392

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₂

—

128

C₂

C₃₁×#2009

yes

—

	subgroup	G/H
center, socle	C₂²×C₃₁	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂³×C₃₁	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₃₁

#392

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₃₁

prime radical

C₂³×C₃₁

#127

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

N	H ≅ G/N	type
45. #10211	C₃₁	direct (N × H)
94. C₃₁	#10211	direct (N × H)

H ≅ G/N

type

45. #10211

C₃₁

direct (N × H)

94. C₃₁

#10211

direct (N × H)

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

subgroup	index
C₃₁×#3237	2
C₃₁×#1515	2
C₃₁×#1308	2
#10211	31

subgroup

index

C₃₁×#3237

C₃₁×#1515

C₃₁×#1308

#10211

Subgroups — C₃₁×#10211 #21466970

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#10211 #21466970

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10211 #21466970