Group #21490050 — subgroups

Subgroups — C₃₁×#33291 #21490050

Order 7936.

686 conjugacy classes of subgroups; only the 318 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 318 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×#2859	C₂	1	2	yes	maximal
2	C₃₁×#1991	C₂	1	8	yes	maximal
3	C₃₁×#1834	C₂	1	4	yes	maximal
4	C₄²×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₁×#125	C₂²	1	4	yes	—
6	C₂×C₄×C₈×C₃₁	C₂²	1	4	yes	—
7	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
8	C₃₁×#389	—	2	8	no	—
9	C₃₁×#384	C₂²	1	4	yes	—
10	C₃₁×#382	C₂²	1	4	yes	—
11	C₄³×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#340	C₂²	1	16	yes	—
13	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
14	C₃₁×#125	—	2	16	no	—
15	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
16	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
17	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
18	C₂×C₄²×C₃₁	D₄	1	4	yes	—
19	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
21	C₄×C₈×C₃₁	—	2	8	no	—
22	C₄×C₈×C₃₁	Q₈	1	8	yes	—
23	C₃₁×#95	—	2	8	no	—
24	C₂²×C₄×C₃₁	—	2	8	no	—
25	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
26	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
27	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
28	C₂×C₈×C₃₁	—	2	8	no	—
29	C₂×C₈×C₃₁	C₂×Q₈	1	8	yes	—
30	C₄⋊C₄×C₃₁	—	2	16	no	—
31	C₄⋊C₄×C₃₁	—	4	16	no	—
32	C₄⋊C₄×C₃₁	—	4	16	no	—
33	C₄²×C₃₁	—	2	8	no	—
34	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
35	C₄²×C₃₁	#41	1	8	yes	—
36	#33291	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
38	C₂×C₄×C₃₁	—	2	8	no	—
39	C₂×C₄×C₃₁	—	4	8	no	—
40	C₂×C₄×C₃₁	—	4	16	no	—
41	C₂×C₄×C₃₁	#124	1	8	yes	—
42	C₂×C₄×C₃₁	C₄⋊Q₈	1	4	yes	—
43	C₂×C₄×C₃₁	#135	1	8	yes	—
44	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
45	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
46	C₈×C₃₁	—	2	8	no	—
47	#2859	C₂×C₃₁	1	2	yes	—
48	#1991	C₂×C₃₁	1	8	yes	—
49	#1834	C₂×C₃₁	1	4	yes	—
50	C₄²×C₈	C₂×C₃₁	1	1	yes	characteristic
51	C₂²×C₃₁	C₂×#124	1	2	yes	—
52	C₂²×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
53	C₂²×C₃₁	C₂×#135	1	4	yes	—
54	C₄×C₃₁	—	2	8	no	—
55	C₄×C₃₁	—	8	8	no	—
56	C₄×C₃₁	#499	1	8	yes	—
57	C₄×C₃₁	#533	1	4	yes	—
58	C₂×#125	C₂²×C₃₁	1	4	yes	—
59	C₂×C₄×C₈	C₂²×C₃₁	1	4	yes	—
60	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
61	#389	—	2	8	no	—
62	#384	C₂²×C₃₁	1	4	yes	—
63	#382	C₂²×C₃₁	1	4	yes	—
64	C₄³	C₂²×C₃₁	1	1	yes	characteristic
65	#340	C₂²×C₃₁	1	16	yes	—
66	C₂×C₃₁	#2862	1	1	yes	characteristic
67	C₂×C₃₁	#3131	1	4	yes	—
68	C₂×C₃₁	C₂×#499	1	2	yes	—
69	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
70	#125	—	2	16	no	—
71	C₂×C₄⋊C₄	—	2	8	no	—
72	C₂×C₄⋊C₄	—	2	8	no	—
73	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
74	C₂×C₄²	D₄×C₃₁	1	4	yes	—
75	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
76	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
77	C₄×C₈	—	2	8	no	—
78	C₄×C₈	Q₈×C₃₁	1	8	yes	—
79	#95	—	2	8	no	—
80	C₃₁	#33291	1	1	yes	characteristic, direct product factor, Sylow
81	C₂²×C₄	—	2	8	no	—
82	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
83	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
84	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
85	C₂×C₈	—	2	8	no	—
86	C₂×C₈	C₂×Q₈×C₃₁	1	8	yes	—
87	C₄⋊C₄	—	2	16	no	—
88	C₄⋊C₄	—	4	16	no	—
89	C₄⋊C₄	—	4	16	no	—
90	C₄²	—	2	8	no	—
91	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
92	C₄²	#41×C₃₁	1	8	yes	—
93	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
94	C₂×C₄	—	2	8	no	—
95	C₂×C₄	—	4	8	no	—
96	C₂×C₄	—	4	16	no	—
97	C₂×C₄	C₃₁×#124	1	8	yes	—
98	C₂×C₄	C₃₁×C₄⋊Q₈	1	4	yes	—
99	C₂×C₄	C₃₁×#135	1	8	yes	—
100	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
101	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
102	C₈	—	2	8	no	—
103	C₂²	C₂×C₃₁×#124	1	2	yes	—
104	C₂²	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
105	C₂²	C₂×C₃₁×#135	1	4	yes	—
106	C₄	—	2	8	no	—
107	C₄	—	8	8	no	—
108	C₄	C₃₁×#499	1	8	yes	—
109	C₄	C₃₁×#533	1	4	yes	—
110	C₂	C₃₁×#2862	1	1	yes	characteristic
111	C₂	C₃₁×#3131	1	4	yes	—
112	C₂	C₂×C₃₁×#499	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₁×#2859

C₂

yes

maximal

C₃₁×#1991

C₂

yes

maximal

C₃₁×#1834

C₂

yes

maximal

C₄²×C₈×C₃₁

C₂

yes

maximal, characteristic

C₂×C₃₁×#125

C₂²

yes

—

C₂×C₄×C₈×C₃₁

C₂²

yes

—

C₂×C₄×C₈×C₃₁

C₂²

yes

—

C₃₁×#389

—

C₃₁×#384

C₂²

yes

—

C₃₁×#382

C₂²

yes

—

C₄³×C₃₁

C₂²

yes

characteristic

C₃₁×#340

C₂²

yes

—

C₂²×C₈×C₃₁

C₂³

yes

—

C₃₁×#125

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

C₂³

yes

—

C₂×C₄²×C₃₁

D₄

yes

—

C₂×C₄²×C₃₁

C₂³

yes

characteristic

C₂×C₄²×C₃₁

C₂³

yes

—

C₄×C₈×C₃₁

—

C₄×C₈×C₃₁

Q₈

yes

—

C₃₁×#95

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×D₄

yes

—

C₂²×C₄×C₃₁

C₂×D₄

yes

—

C₂²×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂×Q₈

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

C₂×Q₈

yes

—

C₄²×C₃₁

#41

yes

—

#33291

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

C₂²×D₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

#124

yes

—

C₂×C₄×C₃₁

C₄⋊Q₈

yes

—

C₂×C₄×C₃₁

#135

yes

—

C₂×C₄×C₃₁

C₂²×Q₈

yes

—

C₂×C₄×C₃₁

C₂×#41

yes

—

C₈×C₃₁

—

#2859

C₂×C₃₁

yes

—

#1991

C₂×C₃₁

yes

—

#1834

C₂×C₃₁

yes

—

C₄²×C₈

C₂×C₃₁

yes

characteristic

C₂²×C₃₁

C₂×#124

yes

—

C₂²×C₃₁

C₂×C₄⋊Q₈

yes

characteristic

C₂²×C₃₁

C₂×#135

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#499

yes

—

C₄×C₃₁

#533

yes

—

C₂×#125

C₂²×C₃₁

yes

—

C₂×C₄×C₈

C₂²×C₃₁

yes

—

C₂×C₄×C₈

C₂²×C₃₁

yes

—

#389

—

#384

C₂²×C₃₁

yes

—

#382

C₂²×C₃₁

yes

—

C₄³

C₂²×C₃₁

yes

characteristic

#340

C₂²×C₃₁

yes

—

C₂×C₃₁

#2862

yes

characteristic

C₂×C₃₁

#3131

yes

—

C₂×C₃₁

C₂×#499

yes

—

C₂²×C₈

C₂³×C₃₁

yes

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂³×C₃₁

yes

—

C₂×C₄²

D₄×C₃₁

yes

—

C₂×C₄²

C₂³×C₃₁

yes

characteristic

C₂×C₄²

C₂³×C₃₁

yes

—

C₄×C₈

—

C₄×C₈

Q₈×C₃₁

yes

—

#95

—

C₃₁

#33291

yes

characteristic, direct product factor, Sylow

C₂²×C₄

—

C₂²×C₄

C₂×D₄×C₃₁

yes

—

C₂²×C₄

C₂×D₄×C₃₁

yes

—

C₂²×C₄

C₂⁴×C₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

C₂×Q₈×C₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₂×Q₈×C₃₁

yes

—

C₄²

#41×C₃₁

yes

—

C₂³

C₂²×D₄×C₃₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₃₁×#124

yes

—

C₂×C₄

C₃₁×C₄⋊Q₈

yes

—

C₂×C₄

C₃₁×#135

yes

—

100

C₂×C₄

C₂²×Q₈×C₃₁

yes

—

101

C₂×C₄

C₂×#41×C₃₁

yes

—

102

C₈

—

103

C₂²

C₂×C₃₁×#124

yes

—

104

C₂²

C₂×C₃₁×C₄⋊Q₈

yes

characteristic

105

C₂²

C₂×C₃₁×#135

yes

—

106

C₄

—

107

C₄

—

108

C₄

C₃₁×#499

yes

—

109

C₄

C₃₁×#533

yes

—

110

C₂

C₃₁×#2862

yes

characteristic

111

C₂

C₃₁×#3131

yes

—

112

C₂

C₂×C₃₁×#499

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₃₁

C₂²×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂⁴×C₃₁

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

N	H ≅ G/N	type
36. #33291	C₃₁	direct (N × H)
80. C₃₁	#33291	direct (N × H)

H ≅ G/N

type

36. #33291

C₃₁

direct (N × H)

80. C₃₁

#33291

direct (N × H)

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

subgroup	index
C₃₁×#2859	2
C₃₁×#1991	2
C₃₁×#1834	2
C₄²×C₈×C₃₁	2
#33291	31

subgroup

index

C₃₁×#2859

C₃₁×#1991

C₃₁×#1834

C₄²×C₈×C₃₁

#33291

Subgroups — C₃₁×#33291 #21490050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#33291 #21490050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33291 #21490050