Group #10619415 — subgroups

Subgroups — #10619415

Order 640.

1424 conjugacy classes of subgroups; only the 750 normal subgroups are shown.

normal subgroups only

This group has 1424 conjugacy classes of subgroups — too many to draw. Only its 750 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	#66287	C₂	1	1	yes	maximal, characteristic
2	#66264	C₂	1	10	yes	maximal, complemented
3	D₅⋊#131	C₂	1	10	yes	maximal, complemented
4	#66260	C₂	1	5	yes	maximal, complemented
5	C₂×#4133	C₂	1	5	yes	maximal, complemented
6	C₅×2₋¹⁺⁴	C₄	1	1	yes	characteristic
7	#4149	C₄	1	10	yes	—
8	D₅×#41	C₂²	1	10	yes	—
9	#4147	C₄	1	5	yes	—
10	C₂×#713	C₂²	1	5	yes	—
11	#4133	C₂²	1	10	yes	complemented
12	#4133	C₂²	1	20	yes	complemented
13	#4133	C₂²	1	10	yes	complemented
14	#4131	C₂²	1	10	yes	complemented
15	#4131	C₂²	1	30	yes	complemented
16	D₅⋊M₁₆	C₂²	1	30	yes	complemented
17	C₂×#700	C₂²	1	15	yes	complemented
18	C₂×D₅⋊C₈	C₂²	1	15	yes	complemented
19	#3573	—	5	1	no	maximal, Sylow
20	C₅×#41	C₂×C₄	1	10	yes	—
21	C₂×C₅×Q₈	C₂×C₄	1	5	yes	—
22	#713	C₂×C₄	1	20	yes	—
23	#713	C₂³	1	10	yes	—
24	Q₈×D₅	C₂×C₄	1	10	yes	—
25	#711	C₂³	1	10	yes	—
26	D₄×D₅	C₂×C₄	1	30	yes	—
27	#709	C₂×C₄	1	30	yes	—
28	C₂×D₂₀	C₂×C₄	1	15	yes	—
29	C₂×C₄×D₅	C₂³	1	15	yes	—
30	#704	C₂³	1	20	yes	—
31	C₂×#170	C₂³	1	5	yes	—
32	C₂×#170	C₂³	1	15	yes	—
33	#700	C₂³	1	30	yes	—
34	#700	C₂³	1	10	yes	—
35	D₅⋊C₈	C₂³	1	10	yes	—
36	D₅⋊C₈	C₂³	1	30	yes	—
37	#585	—	5	1	no	—
38	#568	—	5	5	no	—
39	#568	—	5	10	no	—
40	C₂×#131	—	5	5	no	—
41	C₂×#131	—	5	10	no	—
42	C₂²×D₅	C₂²×C₄	1	15	yes	—
43	C₅×Q₈	C₂²×C₄	1	10	yes	—
44	C₅×D₄	C₂²×C₄	1	10	yes	—
45	C₂×C₄×C₅	C₂²×C₄	1	15	yes	—
46	C₂²⋊D₅	C₂²×C₄	1	20	yes	—
47	C₂×Dic₅	C₂⁴	1	5	yes	—
48	D₂₀	C₂²×C₄	1	30	yes	—
49	C₄×D₅	C₂²×C₄	1	30	yes	—
50	C₄×D₅	C₂⁴	1	10	yes	—
51	Dic₁₀	C₂²×C₄	1	10	yes	—
52	#170	C₂⁴	1	1	yes	characteristic
53	#170	C₂⁴	1	5	yes	—
54	#170	C₂⁴	1	10	yes	—
55	2₋¹⁺⁴	—	5	5	no	—
56	2₋¹⁺⁴	AGL(1,𝔽₅)	1	1	yes	characteristic
57	H(5,𝔽₂)	—	5	10	no	—
58	C₂×#41	—	5	10	no	—
59	C₂×#41	—	5	5	no	—
60	#131	—	5	10	no	—
61	#131	—	5	30	no	—
62	#131	—	5	10	no	—
63	#131	—	5	20	no	—
64	#131	—	5	10	no	—
65	C₂×M₁₆	—	5	15	no	—
66	C₂×M₁₆	—	5	30	no	—
67	C₂²×C₈	—	5	15	no	—
68	C₂²×C₅	C₂³×C₄	1	5	yes	—
69	C₂×D₅	—	4	15	no	—
70	C₂×D₅	C₂³×C₄	1	10	yes	—
71	C₄×C₅	C₂³×C₄	1	10	yes	—
72	Dic₅	C₂³×C₄	1	5	yes	—
73	Dic₅	C₂⁵	1	1	yes	characteristic
74	#41	—	5	10	no	—
75	#41	—	5	20	no	—
76	#41	—	5	10	no	—
77	#41	—	5	30	no	—
78	#41	C₂×AGL(1,𝔽₅)	1	10	yes	—
79	C₂×Q₈	—	5	10	no	—
80	C₂×Q₈	C₂×AGL(1,𝔽₅)	1	5	yes	—
81	C₂×D₄	—	5	30	no	—
82	C₂×D₄	—	5	15	no	—
83	C₂²×C₄	—	5	15	no	—
84	M₁₆	—	5	30	no	—
85	M₁₆	—	5	10	no	—
86	M₁₆	—	5	20	no	—
87	C₂×C₈	—	5	10	no	—
88	C₂×C₈	—	5	5	no	—
89	C₂×C₈	—	5	30	no	—
90	C₂×C₈	—	5	15	no	—
91	C₂×C₅	—	2	5	no	—
92	C₂×C₅	C₂⁴×C₄	1	1	yes	characteristic
93	D₅	—	2	10	no	—
94	C₂³	—	5	15	no	—
95	Q₈	—	5	10	no	—
96	Q₈	C₂²×AGL(1,𝔽₅)	1	10	yes	—
97	D₄	—	5	20	no	—
98	D₄	—	5	30	no	—
99	D₄	C₂²×AGL(1,𝔽₅)	1	10	yes	—
100	C₂×C₄	—	5	5	no	—
101	C₂×C₄	—	5	10	no	—
102	C₂×C₄	—	5	30	no	—
103	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	15	yes	—
104	C₈	—	5	1	no	—
105	C₈	—	5	10	no	—
106	C₈	—	5	5	no	—
107	C₅	#3573	1	1	yes	characteristic, complemented, Sylow
108	C₂²	—	5	10	no	—
109	C₂²	—	20	15	no	—
110	C₂²	C₂³×AGL(1,𝔽₅)	1	5	yes	—
111	C₄	—	5	1	no	—
112	C₄	—	5	5	no	—
113	C₄	C₂³×AGL(1,𝔽₅)	1	10	yes	—
114	C₂	—	2	5	no	—
115	C₂	—	10	10	no	—
116	C₂	C₂⁴×AGL(1,𝔽₅)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#66287

C₂

yes

maximal, characteristic

#66264

C₂

yes

maximal, complemented

D₅⋊#131

C₂

yes

maximal, complemented

#66260

C₂

yes

maximal, complemented

C₂×#4133

C₂

yes

maximal, complemented

C₅×2₋¹⁺⁴

C₄

yes

characteristic

#4149

C₄

yes

—

D₅×#41

C₂²

yes

—

#4147

C₄

yes

—

C₂×#713

C₂²

yes

—

#4133

C₂²

yes

complemented

#4133

C₂²

yes

complemented

#4133

C₂²

yes

complemented

#4131

C₂²

yes

complemented

#4131

C₂²

yes

complemented

D₅⋊M₁₆

C₂²

yes

complemented

C₂×#700

C₂²

yes

complemented

C₂×D₅⋊C₈

C₂²

yes

complemented

#3573

—

maximal, Sylow

C₅×#41

C₂×C₄

yes

—

C₂×C₅×Q₈

C₂×C₄

yes

—

#713

C₂×C₄

yes

—

#713

C₂³

yes

—

Q₈×D₅

C₂×C₄

yes

—

#711

C₂³

yes

—

D₄×D₅

C₂×C₄

yes

—

#709

C₂×C₄

yes

—

C₂×D₂₀

C₂×C₄

yes

—

C₂×C₄×D₅

C₂³

yes

—

#704

C₂³

yes

—

C₂×#170

C₂³

yes

—

C₂×#170

C₂³

yes

—

#700

C₂³

yes

—

#700

C₂³

yes

—

D₅⋊C₈

C₂³

yes

—

D₅⋊C₈

C₂³

yes

—

—

—

—

—

—

C₂²×D₅

C₂²×C₄

yes

—

C₅×Q₈

C₂²×C₄

yes

—

C₅×D₄

C₂²×C₄

yes

—

C₂×C₄×C₅

C₂²×C₄

yes

—

C₂²⋊D₅

C₂²×C₄

yes

—

C₂×Dic₅

C₂⁴

yes

—

D₂₀

C₂²×C₄

yes

—

C₄×D₅

C₂²×C₄

yes

—

C₄×D₅

C₂⁴

yes

—

Dic₁₀

C₂²×C₄

yes

—

#170

C₂⁴

yes

characteristic

#170

C₂⁴

yes

—

#170

C₂⁴

yes

—

2₋¹⁺⁴

—

2₋¹⁺⁴

AGL(1,𝔽₅)

yes

characteristic

H(5,𝔽₂)

—

C₂×#41

—

C₂×#41

—

—

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₅

C₂³×C₄

yes

—

C₂×D₅

—

C₂×D₅

C₂³×C₄

yes

—

C₄×C₅

C₂³×C₄

yes

—

Dic₅

C₂³×C₄

yes

—

Dic₅

C₂⁵

yes

—

—

—

—

yes

—

C₂×Q₈

—

C₂×Q₈

C₂×AGL(1,𝔽₅)

yes

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₅

—

C₂×C₅

C₂⁴×C₄

yes

characteristic

D₅

—

C₂³

—

Q₈

—

Q₈

C₂²×AGL(1,𝔽₅)

yes

—

D₄

—

D₄

—

D₄

C₂²×AGL(1,𝔽₅)

yes

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

—

104

C₈

—

105

C₈

—

106

C₈

—

107

C₅

#3573

yes

characteristic, complemented, Sylow

108

C₂²

—

109

C₂²

—

110

C₂²

C₂³×AGL(1,𝔽₅)

yes

—

111

C₄

—

112

C₄

—

113

C₄

C₂³×AGL(1,𝔽₅)

yes

—

114

C₂

—

115

C₂

—

116

C₂

C₂⁴×AGL(1,𝔽₅)

yes

characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×AGL(1,𝔽₅)
commutator, socle	C₂×C₅	C₂⁴×C₄
Fitting	C₅×2₋¹⁺⁴	C₄
prime radical	#66287	C₂
cosocle	Dic₅	C₂⁵
hypercenter	2₋¹⁺⁴	AGL(1,𝔽₅)
hypocenter	C₅	#3573
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ 2₋¹⁺⁴

subgroup

G/H

center, Frattini

C₂

C₂⁴×AGL(1,𝔽₅)

commutator, socle

C₂×C₅

C₂⁴×C₄

Fitting

C₅×2₋¹⁺⁴

C₄

prime radical

#66287

C₂

cosocle

Dic₅

C₂⁵

2₋¹⁺⁴

C₅

G ▷ C₂×C₅ ▷ C₅

C₁ ◁ C₂ ◁ 2₋¹⁺⁴

N	H ≅ G/N	type
2. #66264	C₂	semidirect
3. D₅⋊#131	C₂	semidirect
4. #66260	C₂	semidirect
5. C₂×#4133	C₂	semidirect
11. #4133	C₂²	semidirect
12. #4133	C₂²	semidirect
13. #4133	C₂²	semidirect
14. #4131	C₂²	semidirect
15. #4131	C₂²	semidirect
16. D₅⋊M₁₆	C₂²	semidirect
17. C₂×#700	C₂²	semidirect
18. C₂×D₅⋊C₈	C₂²	semidirect
107. C₅	#3573	semidirect

H ≅ G/N

type

2. #66264

C₂

semidirect

3. D₅⋊#131

C₂

semidirect

4. #66260

C₂

semidirect

5. C₂×#4133

C₂

semidirect

11. #4133

C₂²

semidirect

12. #4133

C₂²

semidirect

13. #4133

C₂²

semidirect

14. #4131

C₂²

semidirect

15. #4131

C₂²

semidirect

16. D₅⋊M₁₆

C₂²

semidirect

17. C₂×#700

C₂²

semidirect

18. C₂×D₅⋊C₈

C₂²

semidirect

107. C₅

#3573

semidirect

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

subgroup	index
#66287	2
#66264	2
D₅⋊#131	2
#66260	2
C₂×#4133	2
#3573	5

subgroup

index

Subgroups — #10619415

Conjugacy classes of subgroups

Complements

Maximal subgroups