Group #16842718 — subgroups

Subgroups — C₃₁×#2785 #16842718

Order 3968.

428 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 176 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×#395	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#394	C₂	1	3	yes	maximal
3	C₃₁×#393	C₂	1	2	yes	maximal, complemented
4	C₃₁×#388	C₂	1	6	yes	maximal, complemented
5	C₃₁×#382	C₂	1	3	yes	maximal, complemented
6	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
9	C₂×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
10	C₂×C₂²⋊C₄×C₃₁	C₂²	1	6	yes	—
11	C₂×C₂²⋊C₄×C₃₁	C₂²	1	6	yes	—
12	C₂×C₄²×C₃₁	C₂²	1	3	yes	—
13	C₃₁×#95	C₂²	1	3	yes	—
14	C₃₁×#95	C₂²	1	3	yes	—
15	C₃₁×#95	C₂²	1	6	yes	—
16	C₂⁴×C₃₁	C₂³	1	2	yes	—
17	C₂×Q₈×C₃₁	—	4	3	no	—
18	C₂×D₄×C₃₁	—	4	3	no	—
19	C₂²×C₄×C₃₁	C₂³	1	6	yes	—
20	C₂²×C₄×C₃₁	C₂³	1	3	yes	—
21	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₃₁	C₂³	1	3	yes	—
23	C₄⋊C₄×C₃₁	—	4	3	no	—
24	C₄⋊C₄×C₃₁	—	4	3	no	—
25	C₂²⋊C₄×C₃₁	—	2	12	no	—
26	C₂²⋊C₄×C₃₁	—	4	6	no	—
27	C₄²×C₃₁	—	2	6	no	—
28	C₂³×C₃₁	—	2	6	no	—
29	C₂³×C₃₁	—	2	6	no	—
30	C₂³×C₃₁	—	2	2	no	—
31	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
32	Q₈×C₃₁	—	4	4	no	—
33	D₄×C₃₁	—	4	4	no	—
34	C₂×C₄×C₃₁	—	2	3	no	—
35	C₂×C₄×C₃₁	—	2	3	no	—
36	C₂×C₄×C₃₁	—	2	6	no	—
37	C₂×C₄×C₃₁	—	2	6	no	—
38	C₂×C₄×C₃₁	—	2	6	no	—
39	C₂×C₄×C₃₁	—	2	6	no	—
40	C₂×C₄×C₃₁	—	2	3	no	—
41	C₂×C₄×C₃₁	—	2	3	no	—
42	C₂×C₄×C₃₁	#41	1	6	yes	—
43	#2785	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂²×C₃₁	—	4	6	no	—
45	C₂²×C₃₁	—	4	6	no	—
46	C₂²×C₃₁	—	4	2	no	—
47	C₂²×C₃₁	C₂×#41	1	3	yes	—
48	C₂²×C₃₁	H(5,𝔽₂)	1	3	yes	—
49	C₂²×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
50	C₄×C₃₁	—	2	6	no	—
51	C₄×C₃₁	—	4	6	no	—
52	C₄×C₃₁	—	4	1	no	—
53	C₄×C₃₁	—	4	3	no	—
54	#395	C₂×C₃₁	1	1	yes	characteristic, complemented
55	#394	C₂×C₃₁	1	3	yes	—
56	#393	C₂×C₃₁	1	2	yes	complemented
57	#388	C₂×C₃₁	1	6	yes	complemented
58	#382	C₂×C₃₁	1	3	yes	complemented
59	C₂×C₃₁	—	8	2	no	—
60	C₂×C₃₁	#539	1	3	yes	—
61	C₂×C₃₁	#555	1	3	yes	—
62	C₂×C₃₁	H(3,𝔽₄)	1	1	yes	characteristic
63	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic
64	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
65	C₂×C₄⋊C₄	C₂²×C₃₁	1	3	yes	—
66	C₂×C₄⋊C₄	C₂²×C₃₁	1	3	yes	—
67	C₂×C₂²⋊C₄	C₂²×C₃₁	1	6	yes	—
68	C₂×C₂²⋊C₄	C₂²×C₃₁	1	6	yes	—
69	C₂×C₄²	C₂²×C₃₁	1	3	yes	—
70	#95	C₂²×C₃₁	1	3	yes	—
71	#95	C₂²×C₃₁	1	6	yes	—
72	#95	C₂²×C₃₁	1	3	yes	—
73	C₃₁	#2785	1	1	yes	characteristic, direct product factor, Sylow
74	C₂⁴	C₂³×C₃₁	1	2	yes	—
75	C₂×Q₈	—	4	3	no	—
76	C₂×D₄	—	4	3	no	—
77	C₂²×C₄	C₂³×C₃₁	1	6	yes	—
78	C₂²×C₄	C₂³×C₃₁	1	3	yes	—
79	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
80	C₂²×C₄	C₂³×C₃₁	1	3	yes	—
81	C₄⋊C₄	—	4	3	no	—
82	C₄⋊C₄	—	4	3	no	—
83	C₂²⋊C₄	—	2	12	no	—
84	C₂²⋊C₄	—	4	6	no	—
85	C₄²	—	2	6	no	—
86	C₂³	—	2	6	no	—
87	C₂³	—	2	6	no	—
88	C₂³	—	2	2	no	—
89	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
90	Q₈	—	4	4	no	—
91	D₄	—	4	4	no	—
92	C₂×C₄	—	2	3	no	—
93	C₂×C₄	—	2	3	no	—
94	C₂×C₄	—	2	6	no	—
95	C₂×C₄	—	2	6	no	—
96	C₂×C₄	—	2	6	no	—
97	C₂×C₄	—	2	6	no	—
98	C₂×C₄	—	2	3	no	—
99	C₂×C₄	—	2	3	no	—
100	C₂×C₄	#41×C₃₁	1	6	yes	—
101	C₂²	—	4	6	no	—
102	C₂²	—	4	6	no	—
103	C₂²	—	4	2	no	—
104	C₂²	C₂×#41×C₃₁	1	3	yes	—
105	C₂²	C₃₁×H(5,𝔽₂)	1	3	yes	—
106	C₂²	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
107	C₄	—	2	6	no	—
108	C₄	—	4	6	no	—
109	C₄	—	4	1	no	—
110	C₄	—	4	3	no	—
111	C₂	—	8	2	no	—
112	C₂	C₃₁×#539	1	3	yes	—
113	C₂	C₃₁×#555	1	3	yes	—
114	C₂	C₃₁×H(3,𝔽₄)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₁×#395

C₂

yes

maximal, characteristic, complemented

C₃₁×#394

C₂

yes

maximal

C₃₁×#393

C₂

yes

maximal, complemented

C₃₁×#388

C₂

yes

maximal, complemented

C₃₁×#382

C₂

yes

maximal, complemented

C₂²×Q₈×C₃₁

C₂²

yes

characteristic

C₂²×D₄×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₂²⋊C₄×C₃₁

C₂²

yes

—

C₂×C₂²⋊C₄×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

—

C₃₁×#95

C₂²

yes

—

C₃₁×#95

C₂²

yes

—

C₃₁×#95

C₂²

yes

—

C₂⁴×C₃₁

C₂³

yes

—

C₂×Q₈×C₃₁

—

C₂×D₄×C₃₁

—

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₂²⋊C₄×C₃₁

—

C₂²⋊C₄×C₃₁

—

C₄²×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

C₂⁴

yes

characteristic

Q₈×C₃₁

—

D₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

#41

yes

—

#2785

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂×#41

yes

—

C₂²×C₃₁

H(5,𝔽₂)

yes

—

C₂²×C₃₁

2₋¹⁺⁴

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

#395

C₂×C₃₁

yes

characteristic, complemented

#394

C₂×C₃₁

yes

—

#393

C₂×C₃₁

yes

complemented

#388

C₂×C₃₁

yes

complemented

#382

C₂×C₃₁

yes

complemented

C₂×C₃₁

—

C₂×C₃₁

#539

yes

—

C₂×C₃₁

#555

yes

—

C₂×C₃₁

H(3,𝔽₄)

yes

characteristic

C₂²×Q₈

C₂²×C₃₁

yes

characteristic

C₂²×D₄

C₂²×C₃₁

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₃₁

yes

—

C₂×C₄⋊C₄

C₂²×C₃₁

yes

—

C₂×C₂²⋊C₄

C₂²×C₃₁

yes

—

C₂×C₂²⋊C₄

C₂²×C₃₁

yes

—

C₂×C₄²

C₂²×C₃₁

yes

—

#95

C₂²×C₃₁

yes

—

#95

C₂²×C₃₁

yes

—

#95

C₂²×C₃₁

yes

—

C₃₁

#2785

yes

characteristic, direct product factor, Sylow

C₂⁴

C₂³×C₃₁

yes

—

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

C₂³×C₃₁

yes

—

C₂²×C₄

C₂³×C₃₁

yes

—

C₂²×C₄

C₂³×C₃₁

yes

characteristic

C₂²×C₄

C₂³×C₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂⁴×C₃₁

yes

characteristic

Q₈

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

#41×C₃₁

yes

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₂×#41×C₃₁

yes

—

105

C₂²

C₃₁×H(5,𝔽₂)

yes

—

106

C₂²

C₃₁×2₋¹⁺⁴

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₂

—

112

C₂

C₃₁×#539

yes

—

113

C₂

C₃₁×#555

yes

—

114

C₂

C₃₁×H(3,𝔽₄)

yes

characteristic

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

subgroup

G/H

center, socle

C₂³×C₃₁

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₃₁

prime radical

C₂²×D₄×C₃₁

C₂²

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₃₁ ◁ G

N	H ≅ G/N	type
1. C₃₁×#395	C₂	semidirect
3. C₃₁×#393	C₂	semidirect
4. C₃₁×#388	C₂	semidirect
5. C₃₁×#382	C₂	semidirect
43. #2785	C₃₁	direct (N × H)
54. #395	C₂×C₃₁	semidirect
56. #393	C₂×C₃₁	semidirect
57. #388	C₂×C₃₁	semidirect
58. #382	C₂×C₃₁	semidirect
73. C₃₁	#2785	direct (N × H)

H ≅ G/N

type

1. C₃₁×#395

C₂

semidirect

3. C₃₁×#393

C₂

semidirect

4. C₃₁×#388

C₂

semidirect

5. C₃₁×#382

C₂

semidirect

43. #2785

C₃₁

direct (N × H)

54. #395

C₂×C₃₁

semidirect

56. #393

C₂×C₃₁

semidirect

57. #388

C₂×C₃₁

semidirect

58. #382

C₂×C₃₁

semidirect

73. C₃₁

#2785

direct (N × H)

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

subgroup	index
C₃₁×#395	2
C₃₁×#394	2
C₃₁×#393	2
C₃₁×#388	2
C₃₁×#382	2
#2785	31

subgroup

index

C₃₁×#395

C₃₁×#394

C₃₁×#393

C₃₁×#388

C₃₁×#382

#2785

Subgroups — C₃₁×#2785 #16842718

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#2785 #16842718

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2785 #16842718