Group #16860523 — subgroups

Subgroups — #16860523

Order 3968.

5796 conjugacy classes of subgroups; only the 3201 normal subgroups are shown.

normal subgroups only

This group has 5796 conjugacy classes of subgroups — too many to draw. Only its 3201 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	#14521886	C₂	1	16	yes	maximal, complemented
2	#14521885	C₂	1	20	yes	maximal, complemented
3	C₂×#11758564	C₂	1	10	yes	maximal, complemented
4	2₋¹⁺⁴×D₃₁	C₂	1	16	yes	maximal, complemented
5	C₂×C₃₁×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×2₋¹⁺⁴	C₂²	1	16	yes	complemented
7	C₂×#41×C₃₁	C₂²	1	10	yes	complemented
8	C₂²×Q₈×C₃₁	C₂²	1	5	yes	complemented
9	#11758567	C₂²	1	80	yes	complemented
10	#11758566	C₂²	1	80	yes	complemented
11	#41×D₃₁	C₂²	1	160	yes	complemented
12	#11758564	C₂²	1	80	yes	complemented
13	#11758564	C₂²	1	40	yes	complemented
14	C₂×#92499	C₂²	1	40	yes	complemented
15	C₂×Q₈×D₃₁	C₂²	1	40	yes	complemented
16	#11758561	C₂²	1	40	yes	complemented
17	C₂×#92495	C₂²	1	60	yes	complemented
18	#41×C₃₁	C₂³	1	80	yes	—
19	C₂×Q₈×C₃₁	C₂³	1	40	yes	—
20	C₂×Q₈×C₃₁	C₂³	1	10	yes	—
21	C₂×D₄×C₃₁	C₂³	1	10	yes	—
22	C₂²×C₄×C₃₁	C₂³	1	15	yes	—
23	C₂×C₂²⋊D₃₁	C₂³	1	40	yes	—
24	#92499	C₂³	1	160	yes	—
25	Q₈×D₃₁	C₂³	1	160	yes	—
26	#92497	C₂³	1	160	yes	—
27	D₄×D₃₁	C₂³	1	160	yes	—
28	#92495	C₂³	1	80	yes	—
29	#92495	C₂³	1	240	yes	—
30	C₂×D₁₂₄	C₂³	1	60	yes	—
31	C₂×C₄×D₃₁	C₂³	1	120	yes	—
32	C₂×Dic₆₂	C₂³	1	60	yes	—
33	C₂³×C₃₁	C₂⁴	1	5	yes	—
34	C₂²×D₃₁	C₂⁴	1	40	yes	—
35	Q₈×C₃₁	C₂⁴	1	40	yes	—
36	D₄×C₃₁	C₂⁴	1	40	yes	—
37	C₂×C₄×C₃₁	C₂⁴	1	60	yes	—
38	C₂×C₄×C₃₁	C₂⁴	1	10	yes	—
39	C₂²⋊D₃₁	C₂⁴	1	16	yes	—
40	C₂²⋊D₃₁	C₂⁴	1	80	yes	—
41	C₂×Dic₃₁	C₂⁴	1	40	yes	—
42	D₁₂₄	C₂⁴	1	80	yes	—
43	C₄×D₃₁	C₂⁴	1	160	yes	—
44	Dic₆₂	C₂⁴	1	80	yes	—
45	2₋¹⁺⁶	—	31	1	no	maximal, Sylow
46	C₂²×C₃₁	—	4	5	no	—
47	C₂²×C₃₁	C₂⁵	1	10	yes	—
48	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
49	C₂×D₃₁	—	4	40	no	—
50	C₂×D₃₁	C₂⁵	1	16	yes	—
51	C₄×C₃₁	C₂⁵	1	20	yes	—
52	Dic₃₁	C₂⁵	1	16	yes	—
53	#585	—	31	16	no	—
54	#585	—	31	20	no	—
55	C₂×2₋¹⁺⁴	—	31	16	no	—
56	C₂×2₋¹⁺⁴	—	31	10	no	—
57	C₂×2₋¹⁺⁴	D₃₁	1	1	yes	characteristic, complemented
58	C₂×C₃₁	—	2	10	no	—
59	C₂×C₃₁	—	2	1	no	—
60	C₂×C₃₁	C₂⁶	1	1	yes	characteristic
61	D₃₁	—	2	16	no	—
62	2₋¹⁺⁴	—	31	80	no	—
63	2₋¹⁺⁴	—	31	80	no	—
64	2₋¹⁺⁴	—	31	40	no	—
65	2₋¹⁺⁴	C₂×D₃₁	1	16	yes	complemented
66	H(5,𝔽₂)	—	31	80	no	—
67	H(5,𝔽₂)	—	31	40	no	—
68	C₂×#41	—	31	160	no	—
69	C₂×#41	—	31	40	no	—
70	C₂×#41	—	31	60	no	—
71	C₂×#41	C₂×D₃₁	1	10	yes	complemented
72	C₂²×Q₈	—	31	40	no	—
73	C₂²×Q₈	C₂×D₃₁	1	5	yes	complemented
74	C₃₁	2₋¹⁺⁶	1	1	yes	characteristic, complemented, Sylow
75	#41	—	31	160	no	—
76	#41	—	31	160	no	—
77	#41	—	31	240	no	—
78	#41	—	31	80	no	—
79	#41	C₂²×D₃₁	1	80	yes	—
80	C₂×Q₈	—	31	160	no	—
81	C₂×Q₈	—	31	60	no	—
82	C₂×Q₈	C₂²×D₃₁	1	40	yes	—
83	C₂×Q₈	C₂²×D₃₁	1	10	yes	—
84	C₂×D₄	—	31	40	no	—
85	C₂×D₄	—	31	160	no	—
86	C₂×D₄	—	31	60	no	—
87	C₂×D₄	C₂²×D₃₁	1	10	yes	—
88	C₂²×C₄	—	31	120	no	—
89	C₂²×C₄	C₂²×D₃₁	1	15	yes	—
90	C₂³	—	31	40	no	—
91	C₂³	C₂³×D₃₁	1	5	yes	—
92	Q₈	—	31	80	no	—
93	Q₈	C₂³×D₃₁	1	40	yes	—
94	D₄	—	31	80	no	—
95	D₄	—	31	16	no	—
96	D₄	—	31	80	no	—
97	D₄	C₂³×D₃₁	1	40	yes	—
98	C₂×C₄	—	31	40	no	—
99	C₂×C₄	—	31	160	no	—
100	C₂×C₄	C₂³×D₃₁	1	60	yes	—
101	C₂×C₄	C₂³×D₃₁	1	10	yes	—
102	C₂²	—	4	5	no	—
103	C₂²	—	31	16	no	—
104	C₂²	—	124	40	no	—
105	C₂²	C₂⁴×D₃₁	1	10	yes	—
106	C₂²	C₂⁴×D₃₁	1	1	yes	characteristic
107	C₄	—	31	16	no	—
108	C₄	C₂⁴×D₃₁	1	20	yes	—
109	C₂	—	2	1	no	—
110	C₂	—	2	10	no	—
111	C₂	—	62	16	no	—
112	C₂	C₂⁵×D₃₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521886

C₂

yes

maximal, complemented

#14521885

C₂

yes

maximal, complemented

C₂×#11758564

C₂

yes

maximal, complemented

2₋¹⁺⁴×D₃₁

C₂

yes

maximal, complemented

C₂×C₃₁×2₋¹⁺⁴

C₂

yes

maximal, characteristic, complemented

C₃₁×2₋¹⁺⁴

C₂²

yes

complemented

C₂×#41×C₃₁

C₂²

yes

complemented

C₂²×Q₈×C₃₁

C₂²

yes

complemented

#11758567

C₂²

yes

complemented

#11758566

C₂²

yes

complemented

#41×D₃₁

C₂²

160

yes

complemented

#11758564

C₂²

yes

complemented

#11758564

C₂²

yes

complemented

C₂×#92499

C₂²

yes

complemented

C₂×Q₈×D₃₁

C₂²

yes

complemented

#11758561

C₂²

yes

complemented

C₂×#92495

C₂²

yes

complemented

#41×C₃₁

C₂³

yes

—

C₂×Q₈×C₃₁

C₂³

yes

—

C₂×Q₈×C₃₁

C₂³

yes

—

C₂×D₄×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

C₂³

yes

—

C₂×C₂²⋊D₃₁

C₂³

yes

—

#92499

C₂³

160

yes

—

Q₈×D₃₁

C₂³

160

yes

—

#92497

C₂³

160

yes

—

D₄×D₃₁

C₂³

160

yes

—

#92495

C₂³

yes

—

#92495

C₂³

240

yes

—

C₂×D₁₂₄

C₂³

yes

—

C₂×C₄×D₃₁

C₂³

120

yes

—

C₂×Dic₆₂

C₂³

yes

—

C₂³×C₃₁

C₂⁴

yes

—

C₂²×D₃₁

C₂⁴

yes

—

Q₈×C₃₁

C₂⁴

yes

—

D₄×C₃₁

C₂⁴

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

—

C₂²⋊D₃₁

C₂⁴

yes

—

C₂²⋊D₃₁

C₂⁴

yes

—

C₂×Dic₃₁

C₂⁴

yes

—

D₁₂₄

C₂⁴

yes

—

C₄×D₃₁

C₂⁴

160

yes

—

Dic₆₂

C₂⁴

yes

—

2₋¹⁺⁶

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

C₂⁵

yes

—

C₂²×C₃₁

C₂⁵

yes

characteristic

C₂×D₃₁

—

C₂×D₃₁

C₂⁵

yes

—

C₄×C₃₁

C₂⁵

yes

—

Dic₃₁

C₂⁵

yes

—

#585

—

#585

—

C₂×2₋¹⁺⁴

—

C₂×2₋¹⁺⁴

—

C₂×2₋¹⁺⁴

D₃₁

yes

characteristic, complemented

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

C₂⁶

yes

characteristic

D₃₁

—

2₋¹⁺⁴

—

2₋¹⁺⁴

—

2₋¹⁺⁴

—

2₋¹⁺⁴

C₂×D₃₁

yes

complemented

H(5,𝔽₂)

—

H(5,𝔽₂)

—

C₂×#41

—

160

—

C₂×#41

—

C₂×#41

—

C₂×#41

C₂×D₃₁

yes

complemented

C₂²×Q₈

—

C₂²×Q₈

C₂×D₃₁

yes

complemented

C₃₁

2₋¹⁺⁶

yes

characteristic, complemented, Sylow

#41

—

160

—

#41

—

160

—

#41

—

240

—

#41

—

#41

C₂²×D₃₁

yes

—

C₂×Q₈

—

160

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₃₁

yes

—

C₂×Q₈

C₂²×D₃₁

yes

—

C₂×D₄

—

C₂×D₄

—

160

—

C₂×D₄

—

C₂×D₄

C₂²×D₃₁

yes

—

C₂²×C₄

—

120

—

C₂²×C₄

C₂²×D₃₁

yes

—

C₂³

—

C₂³

C₂³×D₃₁

yes

—

Q₈

—

Q₈

C₂³×D₃₁

yes

—

D₄

—

D₄

—

D₄

—

D₄

C₂³×D₃₁

yes

—

C₂×C₄

—

C₂×C₄

—

160

—

100

C₂×C₄

C₂³×D₃₁

yes

—

101

C₂×C₄

C₂³×D₃₁

yes

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

124

—

105

C₂²

C₂⁴×D₃₁

yes

—

106

C₂²

C₂⁴×D₃₁

yes

characteristic

107

C₄

—

108

C₄

C₂⁴×D₃₁

yes

—

109

C₂

—

110

C₂

—

111

C₂

—

112

C₂

C₂⁵×D₃₁

yes

characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁵×D₃₁
commutator, socle, cosocle	C₂×C₃₁	C₂⁶
Fitting	C₂×C₃₁×2₋¹⁺⁴	C₂
hypercenter	C₂×2₋¹⁺⁴	D₃₁
hypocenter	C₃₁	2₋¹⁺⁶
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂ ◁ C₂×2₋¹⁺⁴

subgroup

G/H

center, Frattini

C₂

C₂⁵×D₃₁

commutator, socle, cosocle

C₂×C₃₁

C₂⁶

Fitting

C₂×C₃₁×2₋¹⁺⁴

C₂

hypercenter

C₂×2₋¹⁺⁴

D₃₁

hypocenter

C₃₁

2₋¹⁺⁶

LCS

G ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂ ◁ C₂×2₋¹⁺⁴

N	H ≅ G/N	type
1. #14521886	C₂	semidirect
2. #14521885	C₂	semidirect
3. C₂×#11758564	C₂	semidirect
4. 2₋¹⁺⁴×D₃₁	C₂	semidirect
5. C₂×C₃₁×2₋¹⁺⁴	C₂	semidirect
6. C₃₁×2₋¹⁺⁴	C₂²	semidirect
7. C₂×#41×C₃₁	C₂²	semidirect
8. C₂²×Q₈×C₃₁	C₂²	semidirect
9. #11758567	C₂²	semidirect
10. #11758566	C₂²	semidirect
11. #41×D₃₁	C₂²	semidirect
12. #11758564	C₂²	semidirect
13. #11758564	C₂²	semidirect
14. C₂×#92499	C₂²	semidirect
15. C₂×Q₈×D₃₁	C₂²	semidirect
16. #11758561	C₂²	semidirect
17. C₂×#92495	C₂²	semidirect
57. C₂×2₋¹⁺⁴	D₃₁	semidirect
65. 2₋¹⁺⁴	C₂×D₃₁	semidirect
71. C₂×#41	C₂×D₃₁	semidirect
73. C₂²×Q₈	C₂×D₃₁	semidirect
74. C₃₁	2₋¹⁺⁶	semidirect

H ≅ G/N

type

1. #14521886

C₂

semidirect

2. #14521885

C₂

semidirect

3. C₂×#11758564

C₂

semidirect

4. 2₋¹⁺⁴×D₃₁

C₂

semidirect

5. C₂×C₃₁×2₋¹⁺⁴

C₂

semidirect

6. C₃₁×2₋¹⁺⁴

C₂²

semidirect

7. C₂×#41×C₃₁

C₂²

semidirect

8. C₂²×Q₈×C₃₁

C₂²

semidirect

9. #11758567

C₂²

semidirect

10. #11758566

C₂²

semidirect

11. #41×D₃₁

C₂²

semidirect

12. #11758564

C₂²

semidirect

13. #11758564

C₂²

semidirect

14. C₂×#92499

C₂²

semidirect

15. C₂×Q₈×D₃₁

C₂²

semidirect

16. #11758561

C₂²

semidirect

17. C₂×#92495

C₂²

semidirect

57. C₂×2₋¹⁺⁴

D₃₁

semidirect

65. 2₋¹⁺⁴

C₂×D₃₁

semidirect

71. C₂×#41

C₂×D₃₁

semidirect

73. C₂²×Q₈

C₂×D₃₁

semidirect

74. C₃₁

2₋¹⁺⁶

semidirect

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

subgroup	index
#14521886	2
#14521885	2
C₂×#11758564	2
2₋¹⁺⁴×D₃₁	2
C₂×C₃₁×2₋¹⁺⁴	2
2₋¹⁺⁶	31

subgroup

index

#14521886

#14521885

C₂×#11758564

2₋¹⁺⁴×D₃₁

C₂×C₃₁×2₋¹⁺⁴

2₋¹⁺⁶

Subgroups — #16860523

Conjugacy classes of subgroups

Complements

Maximal subgroups