Group #10619413 — subgroups

Subgroups — #10619413

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	#66285	C₂	1	1	yes	maximal, characteristic
2	#66264	C₂	1	6	yes	maximal, complemented
3	D₅⋊#131	C₂	1	6	yes	maximal, complemented
4	#66256	C₂	1	9	yes	maximal, complemented
5	C₂×#4131	C₂	1	9	yes	maximal, complemented
6	C₅×H(5,𝔽₂)	C₄	1	1	yes	characteristic
7	#4150	C₄	1	6	yes	—
8	D₅×#41	C₂²	1	6	yes	—
9	#4144	C₄	1	9	yes	—
10	C₂×#711	C₂²	1	9	yes	—
11	C₂×#704	C₂²	1	18	yes	complemented
12	C₂²×#170	C₂²	1	6	yes	complemented
13	#4133	C₂²	1	2	yes	complemented
14	#4133	C₂²	1	6	yes	complemented
15	#4131	C₂²	1	18	yes	complemented
16	#4131	C₂²	1	36	yes	complemented
17	#4131	C₂²	1	18	yes	complemented
18	D₅⋊M₁₆	C₂²	1	18	yes	complemented
19	C₂×#700	C₂²	1	9	yes	complemented
20	C₂×D₅⋊C₈	C₂²	1	9	yes	complemented
21	#3573	—	5	1	no	maximal, Sylow
22	C₅×#41	C₂×C₄	1	6	yes	—
23	C₂×C₅×D₄	C₂×C₄	1	9	yes	—
24	C₂×C₂²⋊D₅	C₂×C₄	1	18	yes	—
25	C₂²×Dic₅	C₂³	1	6	yes	—
26	#713	C₂³	1	2	yes	—
27	Q₈×D₅	C₂×C₄	1	6	yes	—
28	#711	C₂×C₄	1	36	yes	—
29	#711	C₂³	1	18	yes	—
30	D₄×D₅	C₂×C₄	1	18	yes	—
31	#709	C₂×C₄	1	18	yes	—
32	C₂×C₄×D₅	C₂³	1	9	yes	—
33	C₂×Dic₁₀	C₂×C₄	1	9	yes	—
34	#704	C₂³	1	36	yes	—
35	C₂×#170	C₂³	1	9	yes	—
36	C₂×#170	C₂³	1	9	yes	—
37	C₂×#170	C₂³	1	18	yes	—
38	#700	C₂³	1	18	yes	—
39	#700	C₂³	1	6	yes	—
40	D₅⋊C₈	C₂³	1	6	yes	—
41	D₅⋊C₈	C₂³	1	18	yes	—
42	#585	—	5	1	no	—
43	#568	—	5	9	no	—
44	#568	—	5	6	no	—
45	C₂×#131	—	5	9	no	—
46	C₂×#131	—	5	6	no	—
47	C₂³×C₅	C₂²×C₄	1	6	yes	—
48	C₂²×D₅	C₂²×C₄	1	9	yes	—
49	C₅×Q₈	C₂²×C₄	1	2	yes	—
50	C₅×D₄	C₂²×C₄	1	18	yes	—
51	C₂×C₄×C₅	C₂²×C₄	1	9	yes	—
52	C₂²⋊D₅	C₂²×C₄	1	36	yes	—
53	C₂×Dic₅	C₂²×C₄	1	18	yes	—
54	C₂×Dic₅	C₂⁴	1	9	yes	—
55	D₂₀	C₂²×C₄	1	6	yes	—
56	C₄×D₅	C₂²×C₄	1	18	yes	—
57	C₄×D₅	C₂⁴	1	6	yes	—
58	Dic₁₀	C₂²×C₄	1	18	yes	—
59	#170	C₂⁴	1	1	yes	characteristic
60	#170	C₂⁴	1	6	yes	—
61	#170	C₂⁴	1	9	yes	—
62	2₋¹⁺⁴	—	5	6	no	—
63	H(5,𝔽₂)	—	5	9	no	—
64	H(5,𝔽₂)	AGL(1,𝔽₅)	1	1	yes	characteristic
65	C₂×#41	—	5	6	no	—
66	C₂×#41	—	5	9	no	—
67	#131	—	5	18	no	—
68	#131	—	5	36	no	—
69	#131	—	5	18	no	—
70	#131	—	5	2	no	—
71	#131	—	5	6	no	—
72	C₂×M₁₆	—	5	9	no	—
73	C₂×M₁₆	—	5	18	no	—
74	C₂×M₁₆	—	5	18	no	—
75	C₂²×C₈	—	5	9	no	—
76	C₂²×C₈	—	5	6	no	—
77	C₂²×C₅	—	4	6	no	—
78	C₂²×C₅	C₂³×C₄	1	9	yes	—
79	C₂×D₅	—	4	9	no	—
80	C₂×D₅	C₂³×C₄	1	6	yes	—
81	C₄×C₅	C₂³×C₄	1	6	yes	—
82	Dic₅	C₂³×C₄	1	9	yes	—
83	Dic₅	C₂⁵	1	1	yes	characteristic
84	#41	—	5	2	no	—
85	#41	—	5	18	no	—
86	#41	—	5	36	no	—
87	#41	—	5	18	no	—
88	#41	C₂×AGL(1,𝔽₅)	1	6	yes	—
89	C₂×Q₈	—	5	6	no	—
90	C₂×Q₈	—	5	9	no	—
91	C₂×D₄	—	5	18	no	—
92	C₂×D₄	—	5	18	no	—
93	C₂×D₄	C₂×AGL(1,𝔽₅)	1	9	yes	—
94	C₂²×C₄	—	5	6	no	—
95	C₂²×C₄	—	5	9	no	—
96	M₁₆	—	5	18	no	—
97	M₁₆	—	5	6	no	—
98	M₁₆	—	5	36	no	—
99	C₂×C₈	—	5	6	no	—
100	C₂×C₈	—	5	9	no	—
101	C₂×C₈	—	5	18	no	—
102	C₂×C₈	—	5	9	no	—
103	C₂×C₈	—	5	18	no	—
104	C₂×C₅	—	2	9	no	—
105	C₂×C₅	C₂⁴×C₄	1	1	yes	characteristic
106	D₅	—	2	6	no	—
107	C₂³	—	5	9	no	—
108	C₂³	C₂²×AGL(1,𝔽₅)	1	6	yes	—
109	Q₈	—	5	18	no	—
110	Q₈	C₂²×AGL(1,𝔽₅)	1	2	yes	—
111	D₄	—	5	36	no	—
112	D₄	—	5	6	no	—
113	D₄	C₂²×AGL(1,𝔽₅)	1	18	yes	—
114	C₂×C₄	—	5	9	no	—
115	C₂×C₄	—	5	18	no	—
116	C₂×C₄	—	5	6	no	—
117	C₂×C₄	—	5	18	no	—
118	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	9	yes	—
119	C₈	—	5	1	no	—
120	C₈	—	5	6	no	—
121	C₈	—	5	9	no	—
122	C₅	#3573	1	1	yes	characteristic, complemented, Sylow
123	C₂²	—	4	6	no	—
124	C₂²	—	5	6	no	—
125	C₂²	—	20	9	no	—
126	C₂²	C₂³×AGL(1,𝔽₅)	1	9	yes	—
127	C₄	—	5	1	no	—
128	C₄	—	5	9	no	—
129	C₄	C₂³×AGL(1,𝔽₅)	1	6	yes	—
130	C₂	—	2	9	no	—
131	C₂	—	10	6	no	—
132	C₂	C₂⁴×AGL(1,𝔽₅)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#66285

C₂

yes

maximal, characteristic

#66264

C₂

yes

maximal, complemented

D₅⋊#131

C₂

yes

maximal, complemented

#66256

C₂

yes

maximal, complemented

C₂×#4131

C₂

yes

maximal, complemented

C₅×H(5,𝔽₂)

C₄

yes

characteristic

#4150

C₄

yes

—

D₅×#41

C₂²

yes

—

#4144

C₄

yes

—

C₂×#711

C₂²

yes

—

C₂×#704

C₂²

yes

complemented

C₂²×#170

C₂²

yes

complemented

#4133

C₂²

yes

complemented

#4133

C₂²

yes

complemented

#4131

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×D₄

C₂×C₄

yes

—

C₂×C₂²⋊D₅

C₂×C₄

yes

—

C₂²×Dic₅

C₂³

yes

—

#713

C₂³

yes

—

Q₈×D₅

C₂×C₄

yes

—

#711

C₂×C₄

yes

—

#711

C₂³

yes

—

D₄×D₅

C₂×C₄

yes

—

#709

C₂×C₄

yes

—

C₂×C₄×D₅

C₂³

yes

—

C₂×Dic₁₀

C₂×C₄

yes

—

#704

C₂³

yes

—

C₂×#170

C₂³

yes

—

C₂×#170

C₂³

yes

—

C₂×#170

C₂³

yes

—

#700

C₂³

yes

—

#700

C₂³

yes

—

D₅⋊C₈

C₂³

yes

—

D₅⋊C₈

C₂³

yes

—

—

—

—

—

—

C₂³×C₅

C₂²×C₄

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

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

—

—

yes

C₂×#41

—

C₂×#41

—

—

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₅

—

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×D₄

—

C₂×D₄

—

C₂×D₄

C₂×AGL(1,𝔽₅)

yes

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₅

—

105

C₂×C₅

C₂⁴×C₄

yes

characteristic

106

D₅

—

107

C₂³

—

108

C₂³

C₂²×AGL(1,𝔽₅)

yes

—

109

Q₈

—

110

Q₈

C₂²×AGL(1,𝔽₅)

yes

—

111

D₄

—

112

D₄

—

113

D₄

C₂²×AGL(1,𝔽₅)

yes

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

—

119

C₈

—

120

C₈

—

121

C₈

—

122

C₅

#3573

yes

characteristic, complemented, Sylow

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

C₂³×AGL(1,𝔽₅)

yes

—

127

C₄

—

128

C₄

—

129

C₄

C₂³×AGL(1,𝔽₅)

yes

—

130

C₂

—

131

C₂

—

132

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×H(5,𝔽₂)	C₄
prime radical	#66285	C₂
cosocle	Dic₅	C₂⁵
hypercenter	H(5,𝔽₂)	AGL(1,𝔽₅)
hypocenter	C₅	#3573
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ H(5,𝔽₂)

subgroup

G/H

center, Frattini

C₂

C₂⁴×AGL(1,𝔽₅)

commutator, socle

C₂×C₅

C₂⁴×C₄

Fitting

C₅×H(5,𝔽₂)

C₄

prime radical

#66285

C₂

cosocle

Dic₅

C₂⁵

C₅

G ▷ C₂×C₅ ▷ C₅

C₁ ◁ C₂ ◁ H(5,𝔽₂)

N	H ≅ G/N	type
2. #66264	C₂	semidirect
3. D₅⋊#131	C₂	semidirect
4. #66256	C₂	semidirect
5. C₂×#4131	C₂	semidirect
11. C₂×#704	C₂²	semidirect
12. C₂²×#170	C₂²	semidirect
13. #4133	C₂²	semidirect
14. #4133	C₂²	semidirect
15. #4131	C₂²	semidirect
16. #4131	C₂²	semidirect
17. #4131	C₂²	semidirect
18. D₅⋊M₁₆	C₂²	semidirect
19. C₂×#700	C₂²	semidirect
20. C₂×D₅⋊C₈	C₂²	semidirect
122. C₅	#3573	semidirect

H ≅ G/N

type

2. #66264

C₂

semidirect

3. D₅⋊#131

C₂

semidirect

4. #66256

C₂

semidirect

5. C₂×#4131

C₂

semidirect

11. C₂×#704

C₂²

semidirect

12. C₂²×#170

C₂²

semidirect

13. #4133

C₂²

semidirect

14. #4133

C₂²

semidirect

15. #4131

C₂²

semidirect

16. #4131

C₂²

semidirect

17. #4131

C₂²

semidirect

18. D₅⋊M₁₆

C₂²

semidirect

19. C₂×#700

C₂²

semidirect

20. C₂×D₅⋊C₈

C₂²

semidirect

122. C₅

#3573

semidirect

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

subgroup	index
#66285	2
#66264	2
D₅⋊#131	2
#66256	2
C₂×#4131	2
#3573	5

subgroup

index

Subgroups — #10619413

Conjugacy classes of subgroups

Complements

Maximal subgroups