Group #21465585 — subgroups

Subgroups — C₃₁×#8826 #21465585

Order 7936.

338 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 198 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×#1848	C₂	1	2	yes	maximal
2	C₃₁×#1567	C₂	1	4	yes	maximal
3	C₂×C₃₁×#322	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#108	—	2	4	no	—
5	C₃₁×#423	C₂²	1	4	yes	—
6	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₃₁×#335	—	2	8	no	—
9	C₃₁×#322	C₄	1	4	yes	—
10	C₂×M₁₆×C₃₁	—	2	4	no	—
11	C₂²×C₈×C₃₁	D₄	1	2	yes	—
12	C₂²×C₈×C₃₁	Q₈	1	4	yes	—
13	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
14	C₃₁×#108	—	4	8	no	—
15	C₃₁×C₄⋊C₈	—	2	8	no	—
16	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
17	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₃₁	D₄	1	4	yes	—
19	C₄×C₈×C₃₁	Q₈	1	2	yes	—
20	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
21	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
22	M₁₆×C₃₁	—	4	8	no	—
23	C₂×C₈×C₃₁	—	4	4	no	—
24	C₂×C₈×C₃₁	—	4	4	no	—
25	C₂×C₈×C₃₁	C₄⋊C₄	1	8	yes	—
26	C₂×C₈×C₃₁	C₄⋊C₄	1	4	yes	—
27	C₂×C₈×C₃₁	D₈	1	4	yes	—
28	C₂×C₈×C₃₁	Q₁₆	1	4	yes	—
29	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
30	C₄²×C₃₁	C₂×D₄	1	2	yes	—
31	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
32	#8826	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂³×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
34	C₂×C₄×C₃₁	—	2	4	no	—
35	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
37	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
38	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
39	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
40	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
41	C₈×C₃₁	—	2	8	no	—
42	C₈×C₃₁	—	8	4	no	—
43	C₈×C₃₁	#107	1	8	yes	—
44	#1848	C₂×C₃₁	1	2	yes	—
45	#1567	C₂×C₃₁	1	4	yes	—
46	C₂×#322	C₂×C₃₁	1	1	yes	characteristic
47	C₂²×C₃₁	—	2	2	no	—
48	C₂²×C₃₁	#384	1	1	yes	characteristic
49	C₂²×C₃₁	#500	1	2	yes	—
50	C₄×C₃₁	—	2	2	no	—
51	C₄×C₃₁	C₂×#107	1	2	yes	—
52	C₄×C₃₁	#430	1	4	yes	—
53	C₄×C₃₁	#493	1	1	yes	characteristic
54	C₄×C₃₁	#494	1	1	yes	characteristic
55	C₂×#108	—	2	4	no	—
56	#423	C₂²×C₃₁	1	4	yes	—
57	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
58	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
59	#335	—	2	8	no	—
60	#322	C₄×C₃₁	1	4	yes	—
61	C₂×C₃₁	—	2	2	no	—
62	C₂×C₃₁	#1845	1	1	yes	characteristic
63	C₂×C₃₁	#1849	1	2	yes	—
64	C₂×M₁₆	—	2	4	no	—
65	C₂²×C₈	D₄×C₃₁	1	2	yes	—
66	C₂²×C₈	Q₈×C₃₁	1	4	yes	—
67	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
68	#108	—	4	8	no	—
69	C₄⋊C₈	—	2	8	no	—
70	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
71	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
72	C₄×C₈	D₄×C₃₁	1	4	yes	—
73	C₄×C₈	Q₈×C₃₁	1	2	yes	—
74	C₃₁	#8826	1	1	yes	characteristic, direct product factor, Sylow
75	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
76	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
77	M₁₆	—	4	8	no	—
78	C₂×C₈	—	4	4	no	—
79	C₂×C₈	—	4	4	no	—
80	C₂×C₈	C₄⋊C₄×C₃₁	1	8	yes	—
81	C₂×C₈	C₄⋊C₄×C₃₁	1	4	yes	—
82	C₂×C₈	D₈×C₃₁	1	4	yes	—
83	C₂×C₈	Q₁₆×C₃₁	1	4	yes	—
84	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
85	C₄²	C₂×D₄×C₃₁	1	2	yes	—
86	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
87	C₂³	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
91	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
92	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
93	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
94	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
95	C₈	—	2	8	no	—
96	C₈	—	8	4	no	—
97	C₈	C₃₁×#107	1	8	yes	—
98	C₂²	—	2	2	no	—
99	C₂²	C₃₁×#384	1	1	yes	characteristic
100	C₂²	C₃₁×#500	1	2	yes	—
101	C₄	—	2	2	no	—
102	C₄	C₂×C₃₁×#107	1	2	yes	—
103	C₄	C₃₁×#430	1	4	yes	—
104	C₄	C₃₁×#493	1	1	yes	characteristic
105	C₄	C₃₁×#494	1	1	yes	characteristic
106	C₂	—	2	2	no	—
107	C₂	C₃₁×#1845	1	1	yes	characteristic
108	C₂	C₃₁×#1849	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₁×#1848

C₂

yes

maximal

C₃₁×#1567

C₂

yes

maximal

C₂×C₃₁×#322

C₂

yes

maximal, characteristic

C₂×C₃₁×#108

—

C₃₁×#423

C₂²

yes

—

C₂×C₄×C₈×C₃₁

C₂²

yes

—

C₂×C₄×C₈×C₃₁

C₂²

yes

characteristic

C₃₁×#335

—

C₃₁×#322

C₄

yes

—

C₂×M₁₆×C₃₁

—

C₂²×C₈×C₃₁

D₄

yes

—

C₂²×C₈×C₃₁

Q₈

yes

—

C₂×C₄²×C₃₁

C₂³

yes

characteristic

C₃₁×#108

—

C₃₁×C₄⋊C₈

—

C₄×C₈×C₃₁

C₂×C₄

yes

—

C₄×C₈×C₃₁

C₂×C₄

yes

—

C₄×C₈×C₃₁

D₄

yes

—

C₄×C₈×C₃₁

Q₈

yes

—

C₂²×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₃₁

C₂×Q₈

yes

—

M₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₄⋊C₄

yes

—

C₂×C₈×C₃₁

C₄⋊C₄

yes

—

C₂×C₈×C₃₁

D₈

yes

—

C₂×C₈×C₃₁

Q₁₆

yes

—

C₄²×C₃₁

C₂²×C₄

yes

characteristic

C₄²×C₃₁

C₂×D₄

yes

—

C₄²×C₃₁

C₂×Q₈

yes

characteristic

#8826

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₃₁

#127

yes

characteristic

C₂×C₄×C₃₁

C₄⋊Q₈

yes

—

C₂×C₄×C₃₁

C₂×D₈

yes

—

C₂×C₄×C₃₁

C₂×Q₁₆

yes

—

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

#107

yes

—

#1848

C₂×C₃₁

yes

—

#1567

C₂×C₃₁

yes

—

C₂×#322

C₂×C₃₁

yes

characteristic

C₂²×C₃₁

—

C₂²×C₃₁

#384

yes

characteristic

C₂²×C₃₁

#500

yes

—

C₄×C₃₁

—

C₄×C₃₁

C₂×#107

yes

—

C₄×C₃₁

#430

yes

—

C₄×C₃₁

yes

C₄×C₃₁

yes

—

C₂²×C₃₁

yes

—

C₂×C₄×C₈

C₂²×C₃₁

yes

—

C₂×C₄×C₈

C₂²×C₃₁

yes

characteristic

#335

—

#322

C₄×C₃₁

yes

—

C₂×C₃₁

—

C₂×C₃₁

#1845

yes

characteristic

C₂×C₃₁

#1849

yes

—

C₂×M₁₆

—

C₂²×C₈

D₄×C₃₁

yes

—

C₂²×C₈

Q₈×C₃₁

yes

—

C₂×C₄²

C₂³×C₃₁

yes

characteristic

#108

—

C₄⋊C₈

—

C₄×C₈

C₂×C₄×C₃₁

yes

—

C₄×C₈

C₂×C₄×C₃₁

yes

—

C₄×C₈

D₄×C₃₁

yes

—

C₄×C₈

Q₈×C₃₁

yes

—

C₃₁

#8826

yes

characteristic, direct product factor, Sylow

C₂²×C₄

C₂×D₄×C₃₁

yes

characteristic

C₂²×C₄

C₂×Q₈×C₃₁

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₃₁

yes

—

C₂×C₈

C₄⋊C₄×C₃₁

yes

—

C₂×C₈

D₈×C₃₁

yes

—

C₂×C₈

Q₁₆×C₃₁

yes

—

C₄²

C₂²×C₄×C₃₁

yes

characteristic

C₄²

C₂×D₄×C₃₁

yes

—

C₄²

C₂×Q₈×C₃₁

yes

characteristic

C₂³

C₃₁×C₄⋊Q₈

yes

characteristic

C₂×C₄

—

C₂×C₄

C₂×C₄⋊C₄×C₃₁

yes

characteristic

C₂×C₄

C₂×C₄⋊C₄×C₃₁

yes

—

C₂×C₄

C₃₁×#127

yes

characteristic

C₂×C₄

C₃₁×C₄⋊Q₈

yes

—

C₂×C₄

C₂×D₈×C₃₁

yes

—

C₂×C₄

C₂×Q₁₆×C₃₁

yes

—

C₈

—

C₈

—

C₈

C₃₁×#107

yes

—

C₂²

—

C₂²

C₃₁×#384

yes

characteristic

100

C₂²

C₃₁×#500

yes

—

101

C₄

—

102

C₄

C₂×C₃₁×#107

yes

—

103

C₄

C₃₁×#430

yes

—

104

C₄

C₃₁×#493

yes

characteristic

105

C₄

C₃₁×#494

yes

characteristic

106

C₂

—

107

C₂

C₃₁×#1845

yes

characteristic

108

C₂

C₃₁×#1849

yes

—

	subgroup	G/H
center	C₂×C₄×C₃₁	#127
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	#384
prime radical	C₂³×C₃₁	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

subgroup

G/H

center

C₂×C₄×C₃₁

#127

commutator

C₄²

C₂²×C₄×C₃₁

Frattini, cosocle

C₂×C₄²

C₂³×C₃₁

socle

C₂²×C₃₁

#384

prime radical

C₂³×C₃₁

C₄⋊Q₈

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

N	H ≅ G/N	type
32. #8826	C₃₁	direct (N × H)
74. C₃₁	#8826	direct (N × H)

H ≅ G/N

type

32. #8826

C₃₁

direct (N × H)

74. C₃₁

#8826

direct (N × H)

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

subgroup	index
C₃₁×#1848	2
C₃₁×#1567	2
C₂×C₃₁×#322	2
#8826	31

subgroup

index

C₃₁×#1848

C₃₁×#1567

C₂×C₃₁×#322

#8826

Subgroups — C₃₁×#8826 #21465585

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#8826 #21465585

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8826 #21465585