Group #65859 — subgroups

Subgroups — #65859

Order 320.

334 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 111 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×#709	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	#4041	C₂	1	4	yes	maximal, complemented
5	#4028	C₂	1	4	yes	maximal, complemented
6	#4021	C₂	1	2	yes	maximal, complemented
7	#4020	C₂	1	2	yes	maximal, complemented
8	C₂³×D₅	C₂²	1	2	yes	—
9	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊D₅	C₂²	1	2	yes	—
11	#709	—	2	4	no	—
12	C₂×D₂₀	—	2	4	no	—
13	C₂×D₂₀	C₂²	1	2	yes	—
14	C₂×D₂₀	C₂²	1	2	yes	—
15	C₂×D₂₀	C₂²	1	1	yes	characteristic, complemented
16	C₂×D₂₀	C₂²	1	8	yes	complemented
17	C₂×C₄×D₅	C₂²	1	2	yes	complemented
18	C₂×Dic₁₀	C₂²	1	1	yes	characteristic, complemented
19	C₅×C₄⋊C₄	C₂²	1	2	yes	complemented
20	C₅×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₄²×C₅	C₂²	1	2	yes	complemented
22	#685	C₂²	1	8	yes	complemented
23	#535	—	5	1	no	maximal, Sylow
24	C₂³×C₅	C₂³	1	1	yes	characteristic
25	C₂²×D₅	—	2	4	no	—
26	C₂²×D₅	—	4	4	no	—
27	C₂²×D₅	C₂³	1	4	yes	—
28	C₂²×D₅	C₂³	1	2	yes	—
29	C₂×C₄×C₅	D₄	1	4	yes	complemented
30	C₂×C₄×C₅	C₂³	1	4	yes	—
31	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂²⋊D₅	—	2	4	no	—
34	C₂×Dic₅	C₂³	1	2	yes	—
35	D₂₀	—	2	4	no	—
36	D₂₀	—	2	4	no	—
37	D₂₀	—	2	2	no	—
38	D₂₀	—	4	8	no	—
39	C₄×D₅	—	2	4	no	—
40	Dic₁₀	—	2	2	no	—
41	C₂×#41	—	5	1	no	—
42	C₂²×D₄	—	5	1	no	—
43	#127	—	5	2	no	—
44	#124	—	5	2	no	—
45	#121	—	5	4	no	—
46	#120	—	5	4	no	—
47	#117	D₅	1	1	yes	characteristic, complemented
48	C₂²×C₅	—	2	2	no	—
49	C₂²×C₅	C₂×D₄	1	2	yes	—
50	C₂²×C₅	C₂⁴	1	1	yes	characteristic
51	C₂×D₅	—	2	2	no	—
52	C₂×D₅	—	2	4	no	—
53	C₂×D₅	—	2	4	no	—
54	C₂×D₅	—	2	8	no	—
55	C₂×D₅	—	4	8	no	—
56	C₄×C₅	—	2	4	no	—
57	C₄×C₅	C₂×D₄	1	2	yes	—
58	C₄×C₅	C₂×D₄	1	2	yes	—
59	Dic₅	—	2	2	no	—
60	C₂⁴	—	5	2	no	—
61	#41	—	10	4	no	—
62	C₂×Q₈	—	5	1	no	—
63	C₂×D₄	—	5	2	no	—
64	C₂×D₄	—	5	2	no	—
65	C₂×D₄	—	5	2	no	—
66	C₂×D₄	—	5	1	no	—
67	C₂×D₄	—	5	8	no	—
68	C₂×D₄	—	10	4	no	—
69	C₂²×C₄	—	5	2	no	—
70	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
71	C₄⋊C₄	C₂×D₅	1	2	yes	complemented
72	C₂²⋊C₄	—	5	8	no	—
73	C₂²⋊C₄	C₂×D₅	1	2	yes	—
74	C₄²	C₂×D₅	1	2	yes	complemented
75	C₂×C₅	—	2	2	no	—
76	C₂×C₅	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₅	H(5,𝔽₂)	1	2	yes	—
78	D₅	—	4	2	no	—
79	D₅	—	4	4	no	—
80	C₂³	—	5	4	no	—
81	C₂³	—	5	2	no	—
82	C₂³	—	10	4	no	—
83	C₂³	—	20	4	no	—
84	C₂³	C₂²×D₅	1	1	yes	characteristic
85	Q₈	—	10	2	no	—
86	D₄	—	10	4	no	—
87	D₄	—	10	4	no	—
88	D₄	—	10	4	no	—
89	D₄	—	10	2	no	—
90	D₄	—	20	8	no	—
91	C₂×C₄	—	5	2	no	—
92	C₂×C₄	—	10	4	no	—
93	C₂×C₄	D₂₀	1	4	yes	complemented
94	C₂×C₄	C₂²×D₅	1	4	yes	—
95	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
96	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
97	C₅	#535	1	1	yes	characteristic, complemented, Sylow
98	C₂²	—	2	2	no	—
99	C₂²	—	10	2	no	—
100	C₂²	—	10	4	no	—
101	C₂²	—	10	4	no	—
102	C₂²	—	10	8	no	—
103	C₂²	—	20	8	no	—
104	C₂²	C₂×D₂₀	1	2	yes	—
105	C₂²	C₂³×D₅	1	1	yes	characteristic
106	C₄	—	2	4	no	—
107	C₄	—	10	2	no	—
108	C₄	C₂×D₂₀	1	2	yes	—
109	C₄	C₂×D₂₀	1	2	yes	—
110	C₂	—	2	2	no	—
111	C₂	—	20	4	no	—
112	C₂	—	20	2	no	—
113	C₂	C₂²×D₂₀	1	1	yes	characteristic
114	C₂	#4149	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#709

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₀

C₂

yes

maximal, characteristic, complemented

C₅×#117

C₂

yes

maximal, characteristic, complemented

#4041

C₂

yes

maximal, complemented

#4028

C₂

yes

maximal, complemented

#4021

C₂

yes

maximal, complemented

#4020

C₂

yes

maximal, complemented

C₂³×D₅

C₂²

yes

—

C₂²×C₄×C₅

C₂²

yes

characteristic

C₂×C₂²⋊D₅

C₂²

yes

—

#709

—

C₂×D₂₀

—

C₂×D₂₀

C₂²

yes

—

C₂×D₂₀

C₂²

yes

—

C₂×D₂₀

C₂²

yes

characteristic, complemented

C₂×D₂₀

C₂²

yes

complemented

C₂×C₄×D₅

C₂²

yes

complemented

C₂×Dic₁₀

C₂²

yes

characteristic, complemented

C₅×C₄⋊C₄

C₂²

yes

complemented

C₅×C₂²⋊C₄

C₂²

yes

—

C₄²×C₅

C₂²

yes

complemented

#685

C₂²

yes

complemented

#535

—

maximal, Sylow

C₂³×C₅

C₂³

yes

characteristic

C₂²×D₅

—

C₂²×D₅

—

C₂²×D₅

C₂³

yes

—

C₂²×D₅

C₂³

yes

—

C₂×C₄×C₅

D₄

yes

complemented

C₂×C₄×C₅

C₂³

yes

—

C₂×C₄×C₅

C₂³

yes

characteristic

C₂×C₄×C₅

C₂³

yes

characteristic

C₂²⋊D₅

—

C₂×Dic₅

C₂³

yes

—

D₂₀

—

D₂₀

—

D₂₀

—

D₂₀

—

C₄×D₅

—

Dic₁₀

—

C₂×#41

—

C₂²×D₄

—

—

—

—

—

D₅

yes

characteristic, complemented

C₂²×C₅

—

C₂²×C₅

C₂×D₄

yes

—

C₂²×C₅

C₂⁴

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

C₂×D₄

yes

—

C₄×C₅

C₂×D₄

yes

—

Dic₅

—

C₂⁴

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₅

yes

characteristic

C₄⋊C₄

C₂×D₅

yes

complemented

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₅

yes

—

C₄²

C₂×D₅

yes

complemented

C₂×C₅

—

C₂×C₅

C₂²×D₄

yes

characteristic

C₂×C₅

H(5,𝔽₂)

yes

—

D₅

—

D₅

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²×D₅

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₂₀

yes

complemented

C₂×C₄

C₂²×D₅

yes

—

C₂×C₄

C₂²×D₅

yes

characteristic

C₂×C₄

C₂²×D₅

yes

characteristic

C₅

#535

yes

characteristic, complemented, Sylow

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₂×D₂₀

yes

—

105

C₂²

C₂³×D₅

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

C₂×D₂₀

yes

—

109

C₄

C₂×D₂₀

yes

—

110

C₂

—

111

C₂

—

112

C₂

—

113

C₂

C₂²×D₂₀

yes

characteristic

114

C₂

#4149

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁴
Fitting	C₅×#117	C₂
hypercenter	#117	D₅
hypocenter	C₅	#535
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅

commutator, socle, cosocle

C₂²×C₅

C₂⁴

C₂

D₅

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
1. C₂×#709	C₂	semidirect
2. C₂²×D₂₀	C₂	semidirect
3. C₅×#117	C₂	semidirect
4. #4041	C₂	semidirect
5. #4028	C₂	semidirect
6. #4021	C₂	semidirect
7. #4020	C₂	semidirect
15. C₂×D₂₀	C₂²	semidirect
16. C₂×D₂₀	C₂²	semidirect
17. C₂×C₄×D₅	C₂²	semidirect
18. C₂×Dic₁₀	C₂²	semidirect
19. C₅×C₄⋊C₄	C₂²	semidirect
21. C₄²×C₅	C₂²	semidirect
22. #685	C₂²	semidirect
29. C₂×C₄×C₅	D₄	semidirect
47. #117	D₅	semidirect
71. C₄⋊C₄	C₂×D₅	semidirect
74. C₄²	C₂×D₅	semidirect
93. C₂×C₄	D₂₀	semidirect
97. C₅	#535	semidirect

H ≅ G/N

type

1. C₂×#709

C₂

semidirect

2. C₂²×D₂₀

C₂

semidirect

3. C₅×#117

C₂

semidirect

4. #4041

C₂

semidirect

5. #4028

C₂

semidirect

6. #4021

C₂

semidirect

7. #4020

C₂

semidirect

15. C₂×D₂₀

C₂²

semidirect

16. C₂×D₂₀

C₂²

semidirect

17. C₂×C₄×D₅

C₂²

semidirect

18. C₂×Dic₁₀

C₂²

semidirect

19. C₅×C₄⋊C₄

C₂²

semidirect

21. C₄²×C₅

C₂²

semidirect

22. #685

C₂²

semidirect

29. C₂×C₄×C₅

D₄

semidirect

47. #117

D₅

semidirect

71. C₄⋊C₄

C₂×D₅

semidirect

74. C₄²

C₂×D₅

semidirect

93. C₂×C₄

D₂₀

semidirect

97. C₅

#535

semidirect

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

subgroup	index
C₂×#709	2
C₂²×D₂₀	2
C₅×#117	2
#4041	2
#4028	2
#4021	2
#4020	2
#535	5

subgroup

index

C₂²×D₂₀

Subgroups — #65859

Conjugacy classes of subgroups

Complements

Maximal subgroups