Group #21464889 — subgroups

Subgroups — C₃₁×#8130 #21464889

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×#1309	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
4	C₃₁×#432	—	2	6	no	—
5	C₃₁×#431	—	2	6	no	—
6	C₃₁×#423	C₂²	1	6	yes	—
7	C₃₁×#407	C₄	1	6	yes	—
8	C₄×M₁₆×C₃₁	C₄	1	6	yes	—
9	C₂³×C₄×C₃₁	C₂×C₄	1	3	yes	—
10	C₂×M₁₆×C₃₁	C₂×C₄	1	12	yes	—
11	C₂×C₄²×C₃₁	C₂×C₄	1	3	yes	—
12	C₂×C₄²×C₃₁	D₄	1	6	yes	—
13	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
14	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
15	C₃₁×C₄⋊C₈	—	2	12	no	—
16	C₃₁×C₂²⋊C₈	—	2	12	no	—
17	C₃₁×#97	—	2	6	no	—
18	C₄×C₈×C₃₁	—	2	6	no	—
19	C₂⁴×C₃₁	C₄²	1	1	yes	characteristic
20	C₂²×C₄×C₃₁	—	4	6	no	—
21	C₂²×C₄×C₃₁	C₄²	1	3	yes	—
22	C₂²×C₄×C₃₁	C₂²⋊C₄	1	6	yes	—
23	C₂²×C₄×C₃₁	C₄⋊C₄	1	6	yes	—
24	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
25	M₁₆×C₃₁	—	4	12	no	—
26	C₂×C₈×C₃₁	—	2	12	no	—
27	C₄²×C₃₁	C₂²⋊C₄	1	3	yes	—
28	C₄²×C₃₁	C₂²⋊C₄	1	3	yes	—
29	C₄²×C₃₁	C₄⋊C₄	1	6	yes	—
30	C₄²×C₃₁	C₂×D₄	1	3	yes	—
31	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
32	#8130	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂³×C₃₁	—	4	3	no	—
34	C₂³×C₃₁	#95	1	2	yes	—
35	C₂³×C₃₁	C₂×C₄²	1	1	yes	characteristic
36	C₂×C₄×C₃₁	—	2	6	no	—
37	C₂×C₄×C₃₁	—	2	12	no	—
38	C₂×C₄×C₃₁	—	4	3	no	—
39	C₂×C₄×C₃₁	—	4	6	no	—
40	C₂×C₄×C₃₁	#95	1	6	yes	—
41	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	3	yes	—
42	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	3	yes	—
43	C₈×C₃₁	—	4	12	no	—
44	#2920	C₂×C₃₁	1	3	yes	—
45	#1309	C₂×C₃₁	1	4	yes	complemented
46	C₂²×C₃₁	—	2	6	no	—
47	C₂²×C₃₁	—	2	3	no	—
48	C₂²×C₃₁	—	4	4	no	—
49	C₂²×C₃₁	C₂×#95	1	1	yes	characteristic
50	C₄×C₃₁	—	2	6	no	—
51	C₄×C₃₁	—	2	3	no	—
52	C₄×C₃₁	#413	1	6	yes	—
53	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
54	#432	—	2	6	no	—
55	#431	—	2	6	no	—
56	#423	C₂²×C₃₁	1	6	yes	—
57	#407	C₄×C₃₁	1	6	yes	—
58	C₄×M₁₆	C₄×C₃₁	1	6	yes	—
59	C₂×C₃₁	—	4	2	no	—
60	C₂×C₃₁	—	4	1	no	—
61	C₂×C₃₁	#1742	1	3	yes	—
62	C₂³×C₄	C₂×C₄×C₃₁	1	3	yes	—
63	C₂×M₁₆	C₂×C₄×C₃₁	1	12	yes	—
64	C₂×C₄²	C₂×C₄×C₃₁	1	3	yes	—
65	C₂×C₄²	D₄×C₃₁	1	6	yes	—
66	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
67	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
68	C₄⋊C₈	—	2	12	no	—
69	C₂²⋊C₈	—	2	12	no	—
70	#97	—	2	6	no	—
71	C₄×C₈	—	2	6	no	—
72	C₃₁	#8130	1	1	yes	characteristic, direct product factor, Sylow
73	C₂⁴	C₄²×C₃₁	1	1	yes	characteristic
74	C₂²×C₄	—	4	6	no	—
75	C₂²×C₄	C₄²×C₃₁	1	3	yes	—
76	C₂²×C₄	C₂²⋊C₄×C₃₁	1	6	yes	—
77	C₂²×C₄	C₄⋊C₄×C₃₁	1	6	yes	—
78	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
79	M₁₆	—	4	12	no	—
80	C₂×C₈	—	2	12	no	—
81	C₄²	C₂²⋊C₄×C₃₁	1	3	yes	—
82	C₄²	C₂²⋊C₄×C₃₁	1	3	yes	—
83	C₄²	C₄⋊C₄×C₃₁	1	6	yes	—
84	C₄²	C₂×D₄×C₃₁	1	3	yes	—
85	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
86	C₂³	—	4	3	no	—
87	C₂³	C₃₁×#95	1	2	yes	—
88	C₂³	C₂×C₄²×C₃₁	1	1	yes	characteristic
89	C₂×C₄	—	2	6	no	—
90	C₂×C₄	—	2	12	no	—
91	C₂×C₄	—	4	3	no	—
92	C₂×C₄	—	4	6	no	—
93	C₂×C₄	C₃₁×#95	1	6	yes	—
94	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	3	yes	—
95	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	3	yes	—
96	C₈	—	4	12	no	—
97	C₂²	—	2	6	no	—
98	C₂²	—	2	3	no	—
99	C₂²	—	4	4	no	—
100	C₂²	C₂×C₃₁×#95	1	1	yes	characteristic
101	C₄	—	2	6	no	—
102	C₄	—	2	3	no	—
103	C₄	C₃₁×#413	1	6	yes	—
104	C₂	—	4	2	no	—
105	C₂	—	4	1	no	—
106	C₂	C₃₁×#1742	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₁×#2920

C₂

yes

maximal

C₃₁×#1309

C₂

yes

maximal, complemented

C₂²×C₄²×C₃₁

C₂²

yes

characteristic

C₃₁×#432

—

C₃₁×#431

—

C₃₁×#423

C₂²

yes

—

C₃₁×#407

C₄

yes

—

C₄×M₁₆×C₃₁

C₄

yes

—

C₂³×C₄×C₃₁

C₂×C₄

yes

—

C₂×M₁₆×C₃₁

C₂×C₄

yes

—

C₂×C₄²×C₃₁

C₂×C₄

yes

—

C₂×C₄²×C₃₁

D₄

yes

—

C₂×C₄²×C₃₁

Q₈

yes

—

C₂×C₄²×C₃₁

C₂³

yes

characteristic

C₃₁×C₄⋊C₈

—

C₃₁×C₂²⋊C₈

—

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

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

C₂×D₄

yes

—

C₄²×C₃₁

C₂×Q₈

yes

characteristic

#8130

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

—

C₂³×C₃₁

#95

yes

—

C₂³×C₃₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

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

—

#1309

C₂×C₃₁

yes

complemented

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂×#95

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#413

yes

—

C₂²×C₄²

C₂²×C₃₁

yes

—

—

C₂²×C₃₁

yes

—

#407

C₄×C₃₁

yes

—

C₄×M₁₆

C₄×C₃₁

yes

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

#1742

yes

—

C₂³×C₄

C₂×C₄×C₃₁

yes

—

C₂×M₁₆

C₂×C₄×C₃₁

yes

—

C₂×C₄²

C₂×C₄×C₃₁

yes

—

C₂×C₄²

D₄×C₃₁

yes

—

C₂×C₄²

Q₈×C₃₁

yes

—

C₂×C₄²

C₂³×C₃₁

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₃₁

#8130

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

C₄⋊C₄×C₃₁

yes

—

C₂²×C₄

C₂²×C₄×C₃₁

yes

—

M₁₆

—

C₂×C₈

—

C₄²

C₂²⋊C₄×C₃₁

yes

—

C₄²

C₂²⋊C₄×C₃₁

yes

—

C₄²

C₄⋊C₄×C₃₁

yes

—

C₄²

C₂×D₄×C₃₁

yes

—

C₄²

C₂×Q₈×C₃₁

yes

characteristic

C₂³

—

C₂³

C₃₁×#95

yes

—

C₂³

C₂×C₄²×C₃₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₃₁×#95

yes

—

C₂×C₄

C₂×C₂²⋊C₄×C₃₁

yes

—

C₂×C₄

C₂×C₄⋊C₄×C₃₁

yes

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

C₂×C₃₁×#95

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

C₃₁×#413

yes

—

104

C₂

—

105

C₂

—

106

C₂

C₃₁×#1742

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×#1309	C₂	semidirect
32. #8130	C₃₁	direct (N × H)
45. #1309	C₂×C₃₁	semidirect
72. C₃₁	#8130	direct (N × H)

H ≅ G/N

type

2. C₃₁×#1309

C₂

semidirect

32. #8130

C₃₁

direct (N × H)

45. #1309

C₂×C₃₁

semidirect

72. C₃₁

#8130

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×#1309	2
#8130	31

subgroup

index

C₃₁×#2920

C₃₁×#1309

#8130

Subgroups — C₃₁×#8130 #21464889

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#8130 #21464889

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8130 #21464889