Group #16860524 — subgroups

Subgroups — #16860524

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	#14521885	C₂	1	15	yes	maximal, complemented
2	#14521884	C₂	1	20	yes	maximal, complemented
3	C₂×#11758567	C₂	1	15	yes	maximal, complemented
4	2₋¹⁺⁴×D₃₁	C₂	1	12	yes	maximal, complemented
5	C₃₁×#585	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×2₋¹⁺⁴	C₂²	1	6	yes	complemented
7	C₃₁×H(5,𝔽₂)	C₂²	1	10	yes	complemented
8	C₂×#41×C₃₁	C₂²	1	15	yes	complemented
9	#11758567	C₂²	1	120	yes	complemented
10	#11758567	C₂²	1	60	yes	complemented
11	#11758566	C₂²	1	20	yes	complemented
12	#41×D₃₁	C₂²	1	120	yes	complemented
13	#11758564	C₂²	1	30	yes	complemented
14	C₂×Q₈×D₃₁	C₂²	1	30	yes	complemented
15	#11758561	C₂²	1	90	yes	complemented
16	C₂×#92497	C₂²	1	90	yes	complemented
17	C₂×#92495	C₂²	1	45	yes	complemented
18	C₂²×Dic₆₂	C₂²	1	15	yes	complemented
19	#41×C₃₁	C₂³	1	60	yes	—
20	#41×C₃₁	C₂³	1	20	yes	—
21	C₂×Q₈×C₃₁	C₂³	1	15	yes	—
22	C₂×D₄×C₃₁	C₂³	1	45	yes	—
23	C₂²×C₄×C₃₁	C₂³	1	15	yes	—
24	C₂×C₂²⋊D₃₁	C₂³	1	90	yes	—
25	C₂²×Dic₃₁	C₂³	1	30	yes	—
26	#92499	C₂³	1	40	yes	—
27	Q₈×D₃₁	C₂³	1	120	yes	—
28	#92497	C₂³	1	360	yes	—
29	D₄×D₃₁	C₂³	1	120	yes	—
30	#92495	C₂³	1	60	yes	—
31	#92495	C₂³	1	180	yes	—
32	C₂×D₁₂₄	C₂³	1	15	yes	—
33	C₂×C₄×D₃₁	C₂³	1	90	yes	—
34	C₂×Dic₆₂	C₂³	1	45	yes	—
35	C₂×Dic₆₂	C₂³	1	90	yes	—
36	C₂³×C₃₁	C₂⁴	1	15	yes	—
37	C₂²×D₃₁	C₂⁴	1	30	yes	—
38	Q₈×C₃₁	C₂⁴	1	20	yes	—
39	D₄×C₃₁	C₂⁴	1	60	yes	—
40	C₂×C₄×C₃₁	C₂⁴	1	15	yes	—
41	C₂×C₄×C₃₁	C₂⁴	1	45	yes	—
42	C₂²⋊D₃₁	C₂⁴	1	120	yes	—
43	C₂×Dic₃₁	C₂⁴	1	90	yes	—
44	D₁₂₄	C₂⁴	1	6	yes	—
45	D₁₂₄	C₂⁴	1	30	yes	—
46	C₄×D₃₁	C₂⁴	1	120	yes	—
47	Dic₆₂	C₂⁴	1	10	yes	—
48	Dic₆₂	C₂⁴	1	90	yes	—
49	2₋¹⁺⁶	—	31	1	no	maximal, Sylow
50	C₂²×C₃₁	—	4	15	no	—
51	C₂²×C₃₁	C₂⁵	1	15	yes	—
52	C₂×D₃₁	—	4	30	no	—
53	C₂×D₃₁	C₂⁵	1	12	yes	—
54	C₄×C₃₁	C₂⁵	1	1	yes	characteristic
55	C₄×C₃₁	C₂⁵	1	15	yes	—
56	Dic₃₁	C₂⁵	1	20	yes	—
57	#585	—	31	20	no	—
58	#585	—	31	15	no	—
59	#585	D₃₁	1	1	yes	characteristic, complemented
60	C₂×2₋¹⁺⁴	—	31	12	no	—
61	C₂×2₋¹⁺⁴	—	31	15	no	—
62	C₂×C₃₁	—	2	15	no	—
63	C₂×C₃₁	C₂⁶	1	1	yes	characteristic
64	D₃₁	—	2	12	no	—
65	2₋¹⁺⁴	—	31	120	no	—
66	2₋¹⁺⁴	—	31	60	no	—
67	2₋¹⁺⁴	—	31	30	no	—
68	2₋¹⁺⁴	C₂×D₃₁	1	6	yes	complemented
69	H(5,𝔽₂)	—	31	20	no	—
70	H(5,𝔽₂)	—	31	90	no	—
71	H(5,𝔽₂)	C₂×D₃₁	1	10	yes	complemented
72	C₂×#41	—	31	120	no	—
73	C₂×#41	—	31	90	no	—
74	C₂×#41	—	31	45	no	—
75	C₂×#41	C₂×D₃₁	1	15	yes	complemented
76	C₂²×Q₈	—	31	30	no	—
77	C₂²×Q₈	—	31	15	no	—
78	C₃₁	2₋¹⁺⁶	1	1	yes	characteristic, complemented, Sylow
79	#41	—	31	40	no	—
80	#41	—	31	360	no	—
81	#41	—	31	180	no	—
82	#41	—	31	60	no	—
83	#41	C₂²×D₃₁	1	60	yes	—
84	#41	C₂²×D₃₁	1	20	yes	—
85	C₂×Q₈	—	31	120	no	—
86	C₂×Q₈	—	31	90	no	—
87	C₂×Q₈	—	31	45	no	—
88	C₂×Q₈	C₂²×D₃₁	1	15	yes	—
89	C₂×D₄	—	31	90	no	—
90	C₂×D₄	—	31	120	no	—
91	C₂×D₄	—	31	15	no	—
92	C₂×D₄	C₂²×D₃₁	1	45	yes	—
93	C₂²×C₄	—	31	30	no	—
94	C₂²×C₄	—	31	90	no	—
95	C₂²×C₄	C₂²×D₃₁	1	15	yes	—
96	C₂³	—	31	30	no	—
97	C₂³	C₂³×D₃₁	1	15	yes	—
98	Q₈	—	31	90	no	—
99	Q₈	—	31	10	no	—
100	Q₈	C₂³×D₃₁	1	20	yes	—
101	D₄	—	31	120	no	—
102	D₄	—	31	30	no	—
103	D₄	—	31	6	no	—
104	D₄	C₂³×D₃₁	1	60	yes	—
105	C₂×C₄	—	31	90	no	—
106	C₂×C₄	—	31	120	no	—
107	C₂×C₄	C₂³×D₃₁	1	45	yes	—
108	C₂×C₄	C₂³×D₃₁	1	15	yes	—
109	C₂²	—	4	15	no	—
110	C₂²	—	31	12	no	—
111	C₂²	—	124	30	no	—
112	C₂²	C₂⁴×D₃₁	1	15	yes	—
113	C₄	—	31	20	no	—
114	C₄	C₂⁴×D₃₁	1	15	yes	—
115	C₄	C₂⁴×D₃₁	1	1	yes	characteristic
116	C₂	—	2	15	no	—
117	C₂	—	62	12	no	—
118	C₂	C₂⁵×D₃₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521885

C₂

yes

maximal, complemented

#14521884

C₂

yes

maximal, complemented

C₂×#11758567

C₂

yes

maximal, complemented

2₋¹⁺⁴×D₃₁

C₂

yes

maximal, complemented

C₃₁×#585

C₂

yes

maximal, characteristic, complemented

C₃₁×2₋¹⁺⁴

C₂²

yes

complemented

C₃₁×H(5,𝔽₂)

C₂²

yes

complemented

C₂×#41×C₃₁

C₂²

yes

complemented

#11758567

C₂²

120

yes

complemented

#11758567

C₂²

yes

complemented

#11758566

C₂²

yes

complemented

#41×D₃₁

C₂²

120

yes

complemented

#11758564

C₂²

yes

complemented

C₂×Q₈×D₃₁

C₂²

yes

complemented

#11758561

C₂²

yes

complemented

C₂×#92497

C₂²

yes

complemented

C₂×#92495

C₂²

yes

complemented

C₂²×Dic₆₂

C₂²

yes

complemented

#41×C₃₁

C₂³

yes

—

#41×C₃₁

C₂³

yes

—

C₂×Q₈×C₃₁

C₂³

yes

—

C₂×D₄×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

C₂³

yes

—

C₂×C₂²⋊D₃₁

C₂³

yes

—

C₂²×Dic₃₁

C₂³

yes

—

#92499

C₂³

yes

—

Q₈×D₃₁

C₂³

120

yes

—

#92497

C₂³

360

yes

—

D₄×D₃₁

C₂³

120

yes

—

#92495

C₂³

yes

—

#92495

C₂³

180

yes

—

C₂×D₁₂₄

C₂³

yes

—

C₂×C₄×D₃₁

C₂³

yes

—

C₂×Dic₆₂

C₂³

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

120

yes

—

C₂×Dic₃₁

C₂⁴

yes

—

D₁₂₄

C₂⁴

yes

—

D₁₂₄

C₂⁴

yes

—

C₄×D₃₁

C₂⁴

120

yes

—

Dic₆₂

C₂⁴

yes

—

Dic₆₂

C₂⁴

yes

—

2₋¹⁺⁶

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

C₂⁵

yes

—

C₂×D₃₁

—

C₂×D₃₁

C₂⁵

yes

—

C₄×C₃₁

C₂⁵

yes

characteristic

C₄×C₃₁

C₂⁵

yes

—

Dic₃₁

C₂⁵

yes

—

#585

—

#585

—

#585

D₃₁

yes

characteristic, complemented

C₂×2₋¹⁺⁴

—

C₂×2₋¹⁺⁴

—

C₂×C₃₁

—

C₂×C₃₁

C₂⁶

yes

characteristic

D₃₁

—

2₋¹⁺⁴

—

120

—

2₋¹⁺⁴

—

2₋¹⁺⁴

—

2₋¹⁺⁴

C₂×D₃₁

yes

—

—

C₂×D₃₁

yes

complemented

C₂×#41

—

120

—

C₂×#41

—

C₂×#41

—

C₂×#41

C₂×D₃₁

yes

complemented

C₂²×Q₈

—

C₂²×Q₈

—

C₃₁

2₋¹⁺⁶

yes

characteristic, complemented, Sylow

#41

—

#41

—

360

—

#41

—

180

—

#41

—

#41

C₂²×D₃₁

yes

—

#41

C₂²×D₃₁

yes

—

C₂×Q₈

—

120

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₃₁

yes

—

C₂×D₄

—

C₂×D₄

—

120

—

C₂×D₄

—

C₂×D₄

C₂²×D₃₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₁

yes

—

C₂³

—

C₂³

C₂³×D₃₁

yes

—

Q₈

—

Q₈

—

100

Q₈

C₂³×D₃₁

yes

—

101

D₄

—

120

—

102

D₄

—

103

D₄

—

104

D₄

C₂³×D₃₁

yes

—

105

C₂×C₄

—

106

C₂×C₄

—

120

—

107

C₂×C₄

C₂³×D₃₁

yes

—

108

C₂×C₄

C₂³×D₃₁

yes

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

124

—

112

C₂²

C₂⁴×D₃₁

yes

—

113

C₄

—

114

C₄

C₂⁴×D₃₁

yes

—

115

C₄

C₂⁴×D₃₁

yes

characteristic

116

C₂

—

117

C₂

—

118

C₂

C₂⁵×D₃₁

yes

characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁵×D₃₁
commutator, socle, cosocle	C₂×C₃₁	C₂⁶
Fitting	C₃₁×#585	C₂
hypercenter	#585	D₃₁
hypocenter	C₃₁	2₋¹⁺⁶
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂ ◁ #585

subgroup

G/H

center, Frattini

C₂

C₂⁵×D₃₁

commutator, socle, cosocle

C₂×C₃₁

C₂⁶

Fitting

C₃₁×#585

C₂

hypercenter

#585

D₃₁

hypocenter

C₃₁

2₋¹⁺⁶

LCS

G ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂ ◁ #585

N	H ≅ G/N	type
1. #14521885	C₂	semidirect
2. #14521884	C₂	semidirect
3. C₂×#11758567	C₂	semidirect
4. 2₋¹⁺⁴×D₃₁	C₂	semidirect
5. C₃₁×#585	C₂	semidirect
6. C₃₁×2₋¹⁺⁴	C₂²	semidirect
7. C₃₁×H(5,𝔽₂)	C₂²	semidirect
8. C₂×#41×C₃₁	C₂²	semidirect
9. #11758567	C₂²	semidirect
10. #11758567	C₂²	semidirect
11. #11758566	C₂²	semidirect
12. #41×D₃₁	C₂²	semidirect
13. #11758564	C₂²	semidirect
14. C₂×Q₈×D₃₁	C₂²	semidirect
15. #11758561	C₂²	semidirect
16. C₂×#92497	C₂²	semidirect
17. C₂×#92495	C₂²	semidirect
18. C₂²×Dic₆₂	C₂²	semidirect
59. #585	D₃₁	semidirect
68. 2₋¹⁺⁴	C₂×D₃₁	semidirect
71. H(5,𝔽₂)	C₂×D₃₁	semidirect
75. C₂×#41	C₂×D₃₁	semidirect
78. C₃₁	2₋¹⁺⁶	semidirect

H ≅ G/N

type

1. #14521885

C₂

semidirect

2. #14521884

C₂

semidirect

3. C₂×#11758567

C₂

semidirect

4. 2₋¹⁺⁴×D₃₁

C₂

semidirect

5. C₃₁×#585

C₂

semidirect

6. C₃₁×2₋¹⁺⁴

C₂²

semidirect

7. C₃₁×H(5,𝔽₂)

C₂²

semidirect

8. C₂×#41×C₃₁

C₂²

semidirect

9. #11758567

C₂²

semidirect

10. #11758567

C₂²

semidirect

11. #11758566

C₂²

semidirect

12. #41×D₃₁

C₂²

semidirect

13. #11758564

C₂²

semidirect

14. C₂×Q₈×D₃₁

C₂²

semidirect

15. #11758561

C₂²

semidirect

16. C₂×#92497

C₂²

semidirect

17. C₂×#92495

C₂²

semidirect

18. C₂²×Dic₆₂

C₂²

semidirect

59. #585

D₃₁

semidirect

68. 2₋¹⁺⁴

C₂×D₃₁

semidirect

71. H(5,𝔽₂)

C₂×D₃₁

semidirect

75. C₂×#41

C₂×D₃₁

semidirect

78. C₃₁

2₋¹⁺⁶

semidirect

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

subgroup	index
#14521885	2
#14521884	2
C₂×#11758567	2
2₋¹⁺⁴×D₃₁	2
C₃₁×#585	2
2₋¹⁺⁶	31

subgroup

index

#14521885

#14521884

C₂×#11758567

2₋¹⁺⁴×D₃₁

C₃₁×#585

2₋¹⁺⁶

Subgroups — #16860524

Conjugacy classes of subgroups

Complements

Maximal subgroups