Group #21469651 — subgroups

Subgroups — C₃₁×#12892 #21469651

Order 7936.

274 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 158 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	C₂×C₃₁×#346	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1848	C₂	1	2	yes	maximal
3	C₃₁×#1381	C₂	1	4	yes	maximal
4	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
5	C₂×C₃₁×#108	—	2	2	no	—
6	C₂×C₃₁×#108	C₂²	1	4	yes	—
7	C₃₁×#423	—	2	2	no	—
8	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₃₁×#346	C₄	1	4	yes	—
10	C₂×M₁₆×C₃₁	—	2	4	no	—
11	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
12	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
13	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
14	C₂×C₁₆×C₃₁	—	2	4	no	—
15	C₂×C₁₆×C₃₁	C₂×C₄	1	4	yes	—
16	C₃₁×#108	—	2	8	no	—
17	C₃₁×#108	—	4	4	no	—
18	C₃₁×C₄⋊C₈	—	4	4	no	—
19	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
20	C₄×C₈×C₃₁	Q₈	1	2	yes	—
21	C₂²×C₄×C₃₁	Q₁₆	1	2	yes	—
22	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
23	M₁₆×C₃₁	—	4	4	no	—
24	M₁₆×C₃₁	—	4	4	no	—
25	C₂×C₈×C₃₁	—	4	4	no	—
26	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₃₁	SD₁₆	1	4	yes	—
28	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₈×C₃₁	#41	1	4	yes	—
31	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
32	C₄²×C₃₁	D₈	1	2	yes	—
33	C₁₆×C₃₁	—	2	4	no	—
34	#12892	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂³×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
36	C₂×C₄×C₃₁	—	4	2	no	—
37	C₂×C₄×C₃₁	#107	1	4	yes	—
38	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₃₁	#124	1	2	yes	—
40	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
41	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
42	C₈×C₃₁	—	8	4	no	—
43	C₈×C₃₁	#106	1	4	yes	—
44	C₈×C₃₁	#117	1	2	yes	—
45	C₈×C₃₁	#125	1	2	yes	—
46	C₂×#346	C₂×C₃₁	1	1	yes	characteristic
47	#1848	C₂×C₃₁	1	2	yes	—
48	#1381	C₂×C₃₁	1	4	yes	—
49	C₂²×C₃₁	—	2	2	no	—
50	C₂²×C₃₁	C₂×#107	1	1	yes	characteristic
51	C₂²×C₃₁	#487	1	2	yes	—
52	C₄×C₃₁	—	2	4	no	—
53	C₄×C₃₁	#382	1	1	yes	characteristic
54	C₄×C₃₁	C₂×#106	1	1	yes	characteristic
55	C₄×C₃₁	#486	1	2	yes	—
56	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
57	C₂×#108	—	2	2	no	—
58	C₂×#108	C₂²×C₃₁	1	4	yes	—
59	#423	—	2	2	no	—
60	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
61	#346	C₄×C₃₁	1	4	yes	—
62	C₂×C₃₁	—	2	2	no	—
63	C₂×C₃₁	#1834	1	1	yes	characteristic
64	C₂×C₃₁	#2161	1	2	yes	—
65	C₂×M₁₆	—	2	4	no	—
66	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
67	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
68	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
69	C₂×C₁₆	—	2	4	no	—
70	C₂×C₁₆	C₂×C₄×C₃₁	1	4	yes	—
71	#108	—	2	8	no	—
72	#108	—	4	4	no	—
73	C₄⋊C₈	—	4	4	no	—
74	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
75	C₄×C₈	Q₈×C₃₁	1	2	yes	—
76	C₃₁	#12892	1	1	yes	characteristic, direct product factor, Sylow
77	C₂²×C₄	Q₁₆×C₃₁	1	2	yes	—
78	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
79	M₁₆	—	4	4	no	—
80	M₁₆	—	4	4	no	—
81	C₂×C₈	—	4	4	no	—
82	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
83	C₂×C₈	SD₁₆×C₃₁	1	4	yes	—
84	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
85	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
86	C₂×C₈	#41×C₃₁	1	4	yes	—
87	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
88	C₄²	D₈×C₃₁	1	2	yes	—
89	C₁₆	—	2	4	no	—
90	C₂³	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	C₃₁×#107	1	4	yes	—
93	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
94	C₂×C₄	C₃₁×#124	1	2	yes	—
95	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
96	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
97	C₈	—	8	4	no	—
98	C₈	C₃₁×#106	1	4	yes	—
99	C₈	C₃₁×#117	1	2	yes	—
100	C₈	C₃₁×#125	1	2	yes	—
101	C₂²	—	2	2	no	—
102	C₂²	C₂×C₃₁×#107	1	1	yes	characteristic
103	C₂²	C₃₁×#487	1	2	yes	—
104	C₄	—	2	4	no	—
105	C₄	C₃₁×#382	1	1	yes	characteristic
106	C₄	C₂×C₃₁×#106	1	1	yes	characteristic
107	C₄	C₃₁×#486	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₃₁×#1834	1	1	yes	characteristic
110	C₂	C₃₁×#2161	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃₁×#346

C₂

yes

maximal, characteristic

C₃₁×#1848

C₂

yes

maximal

C₃₁×#1381

C₂

yes

maximal

C₂²×C₁₆×C₃₁

C₂²

yes

—

C₂×C₃₁×#108

—

C₂×C₃₁×#108

C₂²

yes

—

C₃₁×#423

—

C₂×C₄×C₈×C₃₁

C₂²

yes

characteristic

C₃₁×#346

C₄

yes

—

C₂×M₁₆×C₃₁

—

C₂²×C₈×C₃₁

D₄

yes

characteristic

C₂²×C₈×C₃₁

C₂³

yes

characteristic

C₂×C₄²×C₃₁

D₄

yes

characteristic

C₂×C₁₆×C₃₁

—

C₂×C₁₆×C₃₁

C₂×C₄

yes

—

C₃₁×#108

—

C₃₁×#108

—

C₃₁×C₄⋊C₈

—

C₄×C₈×C₃₁

C₂×C₄

yes

—

C₄×C₈×C₃₁

Q₈

yes

—

C₂²×C₄×C₃₁

Q₁₆

yes

—

C₂²×C₄×C₃₁

C₂×D₄

yes

characteristic

M₁₆×C₃₁

—

M₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₄⋊C₄

yes

—

C₂×C₈×C₃₁

SD₁₆

yes

—

C₂×C₈×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₃₁

#41

yes

—

C₄²×C₃₁

C₄⋊C₄

yes

—

C₄²×C₃₁

D₈

yes

—

C₁₆×C₃₁

—

#12892

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

#107

yes

—

C₂×C₄×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₃₁

#124

yes

—

C₂×C₄×C₃₁

C₂×D₈

yes

characteristic

C₂×C₄×C₃₁

C₂×SD₁₆

yes

—

C₈×C₃₁

—

C₈×C₃₁

#106

yes

—

C₈×C₃₁

#117

yes

—

C₈×C₃₁

#125

yes

—

C₂×#346

C₂×C₃₁

yes

characteristic

#1848

C₂×C₃₁

yes

—

#1381

C₂×C₃₁

yes

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂×#107

yes

characteristic

C₂²×C₃₁

#487

yes

—

C₄×C₃₁

—

C₄×C₃₁

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

—

C₂²×C₁₆

C₂²×C₃₁

yes

—

C₂×#108

—

C₂×#108

C₂²×C₃₁

yes

—

#423

—

C₂×C₄×C₈

C₂²×C₃₁

yes

characteristic

#346

C₄×C₃₁

yes

—

C₂×C₃₁

—

C₂×C₃₁

#1834

yes

characteristic

C₂×C₃₁

#2161

yes

—

C₂×M₁₆

—

C₂²×C₈

D₄×C₃₁

yes

characteristic

C₂²×C₈

C₂³×C₃₁

yes

characteristic

C₂×C₄²

D₄×C₃₁

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄×C₃₁

yes

—

#108

—

#108

—

C₄⋊C₈

—

C₄×C₈

C₂×C₄×C₃₁

yes

—

C₄×C₈

Q₈×C₃₁

yes

—

C₃₁

#12892

yes

characteristic, direct product factor, Sylow

C₂²×C₄

Q₁₆×C₃₁

yes

—

C₂²×C₄

C₂×D₄×C₃₁

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₃₁

yes

—

C₂×C₈

SD₁₆×C₃₁

yes

—

C₂×C₈

C₂²×C₄×C₃₁

yes

characteristic

C₂×C₈

C₂×Q₈×C₃₁

yes

characteristic

C₂×C₈

#41×C₃₁

yes

—

C₄²

C₄⋊C₄×C₃₁

yes

—

C₄²

D₈×C₃₁

yes

—

C₁₆

—

C₂³

C₂×Q₁₆×C₃₁

yes

characteristic

C₂×C₄

—

C₂×C₄

C₃₁×#107

yes

—

C₂×C₄

C₂×C₄⋊C₄×C₃₁

yes

characteristic

C₂×C₄

C₃₁×#124

yes

—

C₂×C₄

C₂×D₈×C₃₁

yes

characteristic

C₂×C₄

C₂×SD₁₆×C₃₁

yes

—

C₈

—

C₈

C₃₁×#106

yes

—

C₈

C₃₁×#117

yes

—

100

C₈

C₃₁×#125

yes

—

101

C₂²

—

102

C₂²

C₂×C₃₁×#107

yes

characteristic

103

C₂²

C₃₁×#487

yes

—

104

C₄

—

105

C₄

C₃₁×#382

yes

characteristic

106

C₄

C₂×C₃₁×#106

yes

characteristic

107

C₄

C₃₁×#486

yes

—

108

C₂

—

109

C₂

C₃₁×#1834

yes

characteristic

110

C₂

C₃₁×#2161

yes

—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×#107
prime radical	C₂³×C₃₁	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

subgroup

G/H

center

C₂×C₄×C₃₁

C₂×D₈

commutator

C₂×C₈

C₂²×C₄×C₃₁

Frattini, cosocle

C₂²×C₈

C₂³×C₃₁

socle

C₂²×C₃₁

C₂×#107

prime radical

C₂³×C₃₁

C₂×Q₁₆

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

N	H ≅ G/N	type
34. #12892	C₃₁	direct (N × H)
76. C₃₁	#12892	direct (N × H)

H ≅ G/N

type

34. #12892

C₃₁

direct (N × H)

76. C₃₁

#12892

direct (N × H)

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

subgroup	index
C₂×C₃₁×#346	2
C₃₁×#1848	2
C₃₁×#1381	2
#12892	31

subgroup

index

C₂×C₃₁×#346

C₃₁×#1848

C₃₁×#1381

#12892

Subgroups — C₃₁×#12892 #21469651

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#12892 #21469651

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12892 #21469651