Group #10604113 — subgroups

Subgroups — #10604113

Order 640.

184 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 96 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	#65718	C₂	1	1	yes	maximal, characteristic
2	C₂×D₅⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	D₅×M₃₂	C₂	1	1	yes	maximal, characteristic
4	#64844	C₂	1	2	yes	maximal, complemented
5	C₁₆×AGL(1,𝔽₅)	C₂	1	2	yes	maximal, complemented
6	#4130	C₄	1	1	yes	characteristic
7	D₅⋊M₁₆	C₄	1	1	yes	characteristic
8	C₂×C₈×D₅	C₂²	1	1	yes	characteristic
9	C₂×#674	C₄	1	2	yes	complemented
10	#3992	C₄	1	2	yes	—
11	C₈×AGL(1,𝔽₅)	C₂²	1	2	yes	—
12	D₅⋊C₁₆	C₄	1	2	yes	complemented
13	D₅⋊C₁₆	C₂²	1	1	yes	characteristic
14	D₅⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₅×M₃₂	C₄	1	1	yes	characteristic, complemented
16	#3944	C₄	1	1	yes	characteristic, complemented
17	#3930	C₄	1	2	yes	complemented
18	D₅×C₁₆	C₂²	1	2	yes	—
19	#2108	—	5	1	no	maximal, Sylow
20	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
21	C₂²⋊AGL(1,𝔽₅)	C₈	1	2	yes	—
22	#704	C₈	1	2	yes	—
23	C₄⋊AGL(1,𝔽₅)	C₈	1	2	yes	—
24	C₄×AGL(1,𝔽₅)	C₂×C₄	1	2	yes	—
25	#700	C₈	1	2	yes	—
26	D₅⋊C₈	C₂×C₄	1	2	yes	—
27	C₂×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×#168	C₂×C₄	1	1	yes	characteristic
29	C₈×D₅	C₂×C₄	1	1	yes	characteristic
30	C₈×D₅	C₂×C₄	1	1	yes	characteristic
31	C₈×D₅	C₂×C₄	1	1	yes	characteristic
32	C₈×D₅	C₂³	1	1	yes	characteristic
33	#674	C₂×C₄	1	2	yes	—
34	#674	C₂×C₄	1	2	yes	—
35	C₅×C₁₆	C₂×C₄	1	2	yes	—
36	#672	C₂×C₄	1	2	yes	—
37	C₂×M₃₂	—	5	1	no	—
38	C₂²×C₁₆	—	5	1	no	—
39	#405	—	5	1	no	—
40	#346	—	5	2	no	—
41	C₄×C₁₆	—	5	2	no	—
42	C₂²×D₅	C₂×C₈	1	1	yes	characteristic
43	C₂×AGL(1,𝔽₅)	C₂×C₈	1	4	yes	—
44	C₂×C₄×C₅	C₄²	1	1	yes	characteristic
45	C₂×Dic₅	C₂×C₈	1	1	yes	characteristic
46	C₄×D₅	C₄²	1	1	yes	characteristic
47	C₄×D₅	C₂×C₈	1	1	yes	characteristic
48	C₄×D₅	C₂×C₈	1	1	yes	characteristic
49	C₄×D₅	C₂²×C₄	1	1	yes	characteristic
50	#170	C₂×C₈	1	4	yes	—
51	C₅×C₈	C₄²	1	1	yes	characteristic
52	C₅×C₈	C₂²×C₄	1	1	yes	characteristic
53	#168	C₄²	1	1	yes	characteristic
54	#168	C₂²×C₄	1	1	yes	characteristic
55	C₂×M₁₆	—	5	1	no	—
56	C₂²×C₈	—	5	1	no	—
57	#117	—	5	1	no	—
58	M₃₂	—	5	1	no	—
59	M₃₂	—	5	2	no	—
60	M₃₂	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
61	C₂×C₁₆	—	5	1	no	—
62	C₂×C₁₆	—	5	1	no	—
63	C₂×C₁₆	—	5	2	no	—
64	C₂×C₁₆	—	5	2	no	—
65	C₂×C₁₆	—	5	2	no	—
66	#97	—	5	2	no	—
67	C₄×C₈	—	5	2	no	—
68	C₂²×C₅	C₄×C₈	1	1	yes	characteristic
69	C₂×D₅	—	2	2	no	—
70	C₂×D₅	C₄×C₈	1	1	yes	characteristic
71	C₂×D₅	C₂²×C₈	1	1	yes	characteristic
72	AGL(1,𝔽₅)	—	2	4	no	—
73	C₄×C₅	C₄×C₈	1	1	yes	characteristic
74	C₄×C₅	C₂×C₄²	1	1	yes	characteristic
75	Dic₅	C₄×C₈	1	1	yes	characteristic
76	Dic₅	C₂²×C₈	1	1	yes	characteristic
77	C₂²×C₄	—	5	1	no	—
78	M₁₆	—	5	2	no	—
79	M₁₆	—	5	2	no	—
80	C₂×C₈	—	5	2	no	—
81	C₂×C₈	—	5	1	no	—
82	C₂×C₈	—	5	1	no	—
83	C₂×C₈	—	5	1	no	—
84	C₂×C₈	—	5	1	no	—
85	C₂×C₈	—	5	1	no	—
86	C₂×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
87	C₄⋊C₄	—	5	2	no	—
88	C₂²⋊C₄	—	5	2	no	—
89	C₄²	—	5	2	no	—
90	C₁₆	—	5	2	no	—
91	C₁₆	—	5	2	no	—
92	C₁₆	—	5	2	no	—
93	C₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	—
94	C₂×C₅	—	2	1	no	—
95	C₂×C₅	C₂×C₄×C₈	1	1	yes	characteristic
96	D₅	—	2	1	no	—
97	D₅	#504	1	2	yes	—
98	C₂³	—	5	1	no	—
99	C₂×C₄	—	5	1	no	—
100	C₂×C₄	—	5	1	no	—
101	C₂×C₄	—	5	1	no	—
102	C₂×C₄	—	5	1	no	—
103	C₂×C₄	—	5	1	no	—
104	C₂×C₄	—	5	4	no	—
105	C₂×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
106	C₈	—	5	4	no	—
107	C₈	—	5	1	no	—
108	C₈	—	5	1	no	—
109	C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
110	C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
111	C₅	#2108	1	1	yes	characteristic, complemented, Sylow
112	C₂²	—	5	1	no	—
113	C₂²	—	5	1	no	—
114	C₂²	—	10	2	no	—
115	C₂²	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
116	C₄	—	5	1	no	—
117	C₄	—	5	1	no	—
118	C₄	—	10	4	no	—
119	C₄	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
120	C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
121	C₂	—	2	1	no	—
122	C₂	—	5	2	no	—
123	C₂	—	10	1	no	—
124	C₂	C₂×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#65718

C₂

yes

maximal, characteristic

C₂×D₅⋊C₁₆

C₂

yes

maximal, characteristic

D₅×M₃₂

C₂

yes

maximal, characteristic

#64844

C₂

yes

maximal, complemented

C₁₆×AGL(1,𝔽₅)

C₂

yes

maximal, complemented

#4130

C₄

yes

characteristic

D₅⋊M₁₆

C₄

yes

characteristic

C₂×C₈×D₅

C₂²

yes

characteristic

C₂×#674

C₄

yes

complemented

#3992

C₄

yes

—

C₈×AGL(1,𝔽₅)

C₂²

yes

—

D₅⋊C₁₆

C₄

yes

complemented

D₅⋊C₁₆

C₂²

yes

characteristic

D₅⋊C₁₆

C₂²

yes

characteristic

C₅×M₃₂

C₄

yes

characteristic, complemented

#3944

C₄

yes

characteristic, complemented

#3930

C₄

yes

complemented

D₅×C₁₆

C₂²

yes

—

#2108

—

maximal, Sylow

C₂×C₄×D₅

C₂×C₄

yes

characteristic

C₂²⋊AGL(1,𝔽₅)

C₈

yes

—

#704

C₈

yes

—

C₄⋊AGL(1,𝔽₅)

C₈

yes

—

C₄×AGL(1,𝔽₅)

C₂×C₄

yes

—

#700

C₈

yes

—

D₅⋊C₈

C₂×C₄

yes

—

C₂×C₅×C₈

C₂×C₄

yes

characteristic

C₂×#168

C₂×C₄

yes

characteristic

C₈×D₅

C₂×C₄

yes

characteristic

C₈×D₅

C₂×C₄

yes

characteristic

C₈×D₅

C₂×C₄

yes

characteristic

C₈×D₅

C₂³

yes

characteristic

#674

C₂×C₄

yes

—

#674

C₂×C₄

yes

—

C₅×C₁₆

C₂×C₄

yes

—

#672

C₂×C₄

yes

—

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

#346

—

C₄×C₁₆

—

C₂²×D₅

C₂×C₈

yes

characteristic

C₂×AGL(1,𝔽₅)

C₂×C₈

yes

—

C₂×C₄×C₅

C₄²

yes

characteristic

C₂×Dic₅

C₂×C₈

yes

characteristic

C₄×D₅

C₄²

yes

characteristic

C₄×D₅

C₂×C₈

yes

characteristic

C₄×D₅

C₂×C₈

yes

characteristic

C₄×D₅

C₂²×C₄

yes

characteristic

#170

C₂×C₈

yes

—

C₅×C₈

C₄²

yes

characteristic

C₅×C₈

C₂²×C₄

yes

characteristic

#168

C₄²

yes

characteristic

#168

C₂²×C₄

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

M₃₂

—

M₃₂

AGL(1,𝔽₅)

yes

characteristic, complemented

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

C₄×C₈

—

C₂²×C₅

C₄×C₈

yes

characteristic

C₂×D₅

—

C₂×D₅

C₄×C₈

yes

characteristic

C₂×D₅

C₂²×C₈

yes

characteristic

AGL(1,𝔽₅)

—

C₄×C₅

C₄×C₈

yes

characteristic

C₄×C₅

C₂×C₄²

yes

characteristic

Dic₅

C₄×C₈

yes

characteristic

Dic₅

C₂²×C₈

yes

characteristic

C₂²×C₄

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

C₂×AGL(1,𝔽₅)

yes

—

C₂×C₅

—

C₂×C₅

C₂×C₄×C₈

yes

characteristic

D₅

—

D₅

#504

yes

—

C₂³

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₄×AGL(1,𝔽₅)

yes

characteristic

106

C₈

—

107

C₈

—

108

C₈

—

109

C₈

C₄×AGL(1,𝔽₅)

yes

characteristic

110

C₈

C₂²×AGL(1,𝔽₅)

yes

characteristic

111

C₅

#2108

yes

characteristic, complemented, Sylow

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₈×AGL(1,𝔽₅)

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

C₈×AGL(1,𝔽₅)

yes

characteristic

120

C₄

C₂×C₄×AGL(1,𝔽₅)

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

124

C₂

C₂×C₈×AGL(1,𝔽₅)

yes

characteristic

	subgroup	G/H
center, Frattini	C₈	C₂²×AGL(1,𝔽₅)
commutator, socle	C₂×C₅	C₂×C₄×C₈
Fitting	C₅×M₃₂	C₄
prime radical	C₂²×D₅	C₂×C₈
cosocle	C₈×D₅	C₂³
hypercenter	M₃₂	AGL(1,𝔽₅)
hypocenter	C₅	#2108
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₈ ◁ M₃₂

subgroup

G/H

center, Frattini

C₈

C₂²×AGL(1,𝔽₅)

commutator, socle

C₂×C₅

C₂×C₄×C₈

Fitting

C₅×M₃₂

C₄

prime radical

C₂²×D₅

C₂×C₈

cosocle

C₈×D₅

C₂³

M₃₂

C₅

G ▷ C₂×C₅ ▷ C₅

C₁ ◁ C₈ ◁ M₃₂

N	H ≅ G/N	type
4. #64844	C₂	semidirect
5. C₁₆×AGL(1,𝔽₅)	C₂	semidirect
9. C₂×#674	C₄	semidirect
12. D₅⋊C₁₆	C₄	semidirect
15. C₅×M₃₂	C₄	semidirect
16. #3944	C₄	semidirect
17. #3930	C₄	semidirect
60. M₃₂	AGL(1,𝔽₅)	semidirect
111. C₅	#2108	semidirect

H ≅ G/N

type

4. #64844

C₂

semidirect

5. C₁₆×AGL(1,𝔽₅)

C₂

semidirect

9. C₂×#674

C₄

semidirect

12. D₅⋊C₁₆

C₄

semidirect

15. C₅×M₃₂

C₄

semidirect

16. #3944

C₄

semidirect

17. #3930

C₄

semidirect

60. M₃₂

AGL(1,𝔽₅)

semidirect

111. C₅

#2108

semidirect

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

subgroup	index
#65718	2
C₂×D₅⋊C₁₆	2
D₅×M₃₂	2
#64844	2
C₁₆×AGL(1,𝔽₅)	2
#2108	5

subgroup

index

#65718

C₂×D₅⋊C₁₆

D₅×M₃₂

#64844

C₁₆×AGL(1,𝔽₅)

#2108

Subgroups — #10604113

Conjugacy classes of subgroups

Complements

Maximal subgroups