Group #21463896 — subgroups

Subgroups — C₃₁×#7137 #21463896

Order 7936.

266 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 174 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	2	yes	maximal
2	C₂×C₃₁×#322	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₆×C₃₁	C₄	1	4	yes	—
4	C₂×C₄×C₈×C₃₁	C₄	1	2	yes	—
5	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
6	C₃₁×#346	—	2	2	no	—
7	C₃₁×#346	—	2	2	no	—
8	C₃₁×#322	—	2	2	no	—
9	C₂²×C₈×C₃₁	—	2	2	no	—
10	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
11	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
12	C₂×C₁₆×C₃₁	—	2	4	no	—
13	C₂×C₁₆×C₃₁	—	2	4	no	—
14	C₂×C₁₆×C₃₁	C₈	1	8	yes	complemented
15	C₄×C₈×C₃₁	—	2	2	no	—
16	C₄×C₈×C₃₁	C₈	1	4	yes	—
17	C₄×C₈×C₃₁	D₄	1	2	yes	—
18	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
19	C₄×C₈×C₃₁	Q₈	1	1	yes	characteristic
20	C₂²×C₄×C₃₁	C₄²	1	1	yes	characteristic
21	C₂²×C₄×C₃₁	M₁₆	1	2	yes	—
22	C₂×C₈×C₃₁	—	2	4	no	—
23	C₂×C₈×C₃₁	—	4	4	no	—
24	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
25	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
26	C₂×C₈×C₃₁	C₂×C₈	1	4	yes	—
27	C₂×C₈×C₃₁	M₁₆	1	4	yes	—
28	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
29	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
30	C₄²×C₃₁	C₂×C₈	1	2	yes	—
31	C₁₆×C₃₁	—	2	8	no	—
32	#7137	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂³×C₃₁	#97	1	1	yes	characteristic
34	C₂×C₄×C₃₁	—	2	4	no	—
35	C₂×C₄×C₃₁	#95	1	1	yes	characteristic
36	C₂×C₄×C₃₁	C₄×C₈	1	1	yes	characteristic
37	C₂×C₄×C₃₁	C₄×C₈	1	2	yes	—
38	C₂×C₄×C₃₁	#97	1	2	yes	—
39	C₂×C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
40	C₂×C₄×C₃₁	C₄⋊C₈	1	2	yes	—
41	C₈×C₃₁	—	4	4	no	—
42	C₈×C₃₁	C₂²⋊C₈	1	4	yes	—
43	C₈×C₃₁	C₄⋊C₈	1	4	yes	—
44	C₂×#346	C₂×C₃₁	1	2	yes	—
45	C₂×#322	C₂×C₃₁	1	1	yes	characteristic
46	C₂²×C₃₁	#322	1	2	yes	—
47	C₂²×C₃₁	#336	1	1	yes	characteristic
48	C₂²×C₃₁	#347	1	4	yes	—
49	C₄×C₃₁	—	2	2	no	—
50	C₄×C₃₁	—	2	2	no	—
51	C₄×C₃₁	C₈²	1	1	yes	characteristic
52	C₄×C₃₁	#322	1	1	yes	characteristic
53	C₄×C₃₁	#336	1	2	yes	—
54	C₂²×C₁₆	C₄×C₃₁	1	4	yes	—
55	C₂×C₄×C₈	C₄×C₃₁	1	2	yes	—
56	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
57	#346	—	2	2	no	—
58	#346	—	2	2	no	—
59	#322	—	2	2	no	—
60	C₂×C₃₁	#1273	1	1	yes	characteristic
61	C₂×C₃₁	#1313	1	4	yes	—
62	C₂×C₃₁	#1375	1	2	yes	—
63	C₂²×C₈	—	2	2	no	—
64	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
65	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
66	C₂×C₁₆	—	2	4	no	—
67	C₂×C₁₆	—	2	4	no	—
68	C₂×C₁₆	C₈×C₃₁	1	8	yes	complemented
69	C₄×C₈	—	2	2	no	—
70	C₄×C₈	C₈×C₃₁	1	4	yes	—
71	C₄×C₈	D₄×C₃₁	1	2	yes	—
72	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
73	C₄×C₈	Q₈×C₃₁	1	1	yes	characteristic
74	C₃₁	#7137	1	1	yes	characteristic, direct product factor, Sylow
75	C₂²×C₄	C₄²×C₃₁	1	1	yes	characteristic
76	C₂²×C₄	M₁₆×C₃₁	1	2	yes	—
77	C₂×C₈	—	2	4	no	—
78	C₂×C₈	—	4	4	no	—
79	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
80	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
81	C₂×C₈	C₂×C₈×C₃₁	1	4	yes	—
82	C₂×C₈	M₁₆×C₃₁	1	4	yes	—
83	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
84	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
85	C₄²	C₂×C₈×C₃₁	1	2	yes	—
86	C₁₆	—	2	8	no	—
87	C₂³	C₃₁×#97	1	1	yes	characteristic
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	C₃₁×#95	1	1	yes	characteristic
90	C₂×C₄	C₄×C₈×C₃₁	1	1	yes	characteristic
91	C₂×C₄	C₄×C₈×C₃₁	1	2	yes	—
92	C₂×C₄	C₃₁×#97	1	2	yes	—
93	C₂×C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
94	C₂×C₄	C₃₁×C₄⋊C₈	1	2	yes	—
95	C₈	—	4	4	no	—
96	C₈	C₃₁×C₂²⋊C₈	1	4	yes	—
97	C₈	C₃₁×C₄⋊C₈	1	4	yes	—
98	C₂²	C₃₁×#322	1	2	yes	—
99	C₂²	C₃₁×#336	1	1	yes	characteristic
100	C₂²	C₃₁×#347	1	4	yes	—
101	C₄	—	2	2	no	—
102	C₄	—	2	2	no	—
103	C₄	C₈²×C₃₁	1	1	yes	characteristic
104	C₄	C₃₁×#322	1	1	yes	characteristic
105	C₄	C₃₁×#336	1	2	yes	—
106	C₂	C₃₁×#1273	1	1	yes	characteristic
107	C₂	C₃₁×#1313	1	4	yes	—
108	C₂	C₃₁×#1375	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃₁×#346

C₂

yes

maximal

C₂×C₃₁×#322

C₂

yes

maximal, characteristic

C₂²×C₁₆×C₃₁

C₄

yes

—

C₂×C₄×C₈×C₃₁

C₄

yes

—

C₂×C₄×C₈×C₃₁

C₂²

yes

characteristic

C₃₁×#346

—

C₃₁×#346

—

C₃₁×#322

—

C₂²×C₈×C₃₁

—

C₂²×C₈×C₃₁

C₂×C₄

yes

—

C₂×C₄²×C₃₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₃₁

—

C₂×C₁₆×C₃₁

—

C₂×C₁₆×C₃₁

C₈

yes

complemented

C₄×C₈×C₃₁

—

C₄×C₈×C₃₁

C₈

yes

—

C₄×C₈×C₃₁

D₄

yes

—

C₄×C₈×C₃₁

D₄

yes

characteristic

C₄×C₈×C₃₁

Q₈

yes

characteristic

C₂²×C₄×C₃₁

C₄²

yes

characteristic

C₂²×C₄×C₃₁

M₁₆

yes

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂²⋊C₄

yes

—

C₂×C₈×C₃₁

C₄⋊C₄

yes

—

C₂×C₈×C₃₁

C₂×C₈

yes

—

C₂×C₈×C₃₁

M₁₆

yes

—

C₄²×C₃₁

C₂²⋊C₄

yes

characteristic

C₄²×C₃₁

C₄⋊C₄

yes

characteristic

C₄²×C₃₁

C₂×C₈

yes

—

C₁₆×C₃₁

—

#7137

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

#97

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

#95

yes

characteristic

C₂×C₄×C₃₁

C₄×C₈

yes

characteristic

C₂×C₄×C₃₁

C₄×C₈

yes

—

C₂×C₄×C₃₁

#97

yes

—

C₂×C₄×C₃₁

C₂²⋊C₈

yes

—

C₂×C₄×C₃₁

C₄⋊C₈

yes

—

C₈×C₃₁

—

C₈×C₃₁

C₂²⋊C₈

yes

—

C₈×C₃₁

C₄⋊C₈

yes

—

C₂×#346

C₂×C₃₁

yes

—

C₂×#322

C₂×C₃₁

yes

characteristic

C₂²×C₃₁

#322

yes

—

C₂²×C₃₁

#336

yes

characteristic

C₂²×C₃₁

#347

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₈²

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

—

C₂²×C₁₆

C₄×C₃₁

yes

—

C₂×C₄×C₈

C₄×C₃₁

yes

—

C₂×C₄×C₈

C₂²×C₃₁

yes

—

—

—

C₂×C₃₁

#1273

yes

characteristic

C₂×C₃₁

#1313

yes

—

C₂×C₃₁

#1375

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄×C₃₁

yes

—

C₂×C₄²

C₂×C₄×C₃₁

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₈×C₃₁

yes

complemented

C₄×C₈

—

C₄×C₈

C₈×C₃₁

yes

—

C₄×C₈

D₄×C₃₁

yes

—

C₄×C₈

D₄×C₃₁

yes

characteristic

C₄×C₈

Q₈×C₃₁

yes

characteristic

C₃₁

#7137

yes

characteristic, direct product factor, Sylow

C₂²×C₄

C₄²×C₃₁

yes

characteristic

C₂²×C₄

M₁₆×C₃₁

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄×C₃₁

yes

—

C₂×C₈

C₄⋊C₄×C₃₁

yes

—

C₂×C₈

C₂×C₈×C₃₁

yes

—

C₂×C₈

M₁₆×C₃₁

yes

—

C₄²

C₂²⋊C₄×C₃₁

yes

characteristic

C₄²

C₄⋊C₄×C₃₁

yes

characteristic

C₄²

C₂×C₈×C₃₁

yes

—

C₁₆

—

C₂³

C₃₁×#97

yes

characteristic

C₂×C₄

—

C₂×C₄

C₃₁×#95

yes

characteristic

C₂×C₄

C₄×C₈×C₃₁

yes

characteristic

C₂×C₄

C₄×C₈×C₃₁

yes

—

C₂×C₄

C₃₁×#97

yes

—

C₂×C₄

C₃₁×C₂²⋊C₈

yes

—

C₂×C₄

C₃₁×C₄⋊C₈

yes

—

C₈

—

C₈

C₃₁×C₂²⋊C₈

yes

—

C₈

C₃₁×C₄⋊C₈

yes

—

C₂²

C₃₁×#322

yes

—

C₂²

C₃₁×#336

yes

characteristic

100

C₂²

C₃₁×#347

yes

—

101

C₄

—

102

C₄

—

103

C₄

C₈²×C₃₁

yes

characteristic

104

C₄

C₃₁×#322

yes

characteristic

105

C₄

C₃₁×#336

yes

—

106

C₂

C₃₁×#1273

yes

characteristic

107

C₂

C₃₁×#1313

yes

—

108

C₂

C₃₁×#1375

yes

—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₄²
commutator	C₄	C₈²×C₃₁
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₃₁
socle, prime radical	C₂³×C₃₁	#97
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

subgroup

G/H

center

C₂²×C₄×C₃₁

C₄²

commutator

C₄

C₈²×C₃₁

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₃₁

C₂³×C₃₁

G ▷ C₄ ▷ C₁

C₁ ◁ C₂²×C₄×C₃₁ ◁ G

N	H ≅ G/N	type
14. C₂×C₁₆×C₃₁	C₈	semidirect
32. #7137	C₃₁	direct (N × H)
68. C₂×C₁₆	C₈×C₃₁	semidirect
74. C₃₁	#7137	direct (N × H)

H ≅ G/N

type

14. C₂×C₁₆×C₃₁

C₈

semidirect

32. #7137

C₃₁

direct (N × H)

68. C₂×C₁₆

C₈×C₃₁

semidirect

74. C₃₁

#7137

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
C₂×C₃₁×#322	2
#7137	31

subgroup

index

C₂×C₃₁×#346

C₂×C₃₁×#322

#7137

Subgroups — C₃₁×#7137 #21463896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#7137 #21463896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#7137 #21463896