Group #10603882 — subgroups

Subgroups — #10603882

Order 640.

224 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 74 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	#65689	C₂	1	1	yes	maximal, characteristic
2	#65688	C₂	1	1	yes	maximal, characteristic
3	D₅×#97	C₂	1	1	yes	maximal, characteristic
4	#64893	C₂	1	4	yes	maximal
5	C₂×C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
6	C₂×C₄⋊AGL(1,𝔽₅)	C₂²	1	2	yes	—
7	C₂×C₄⋊AGL(1,𝔽₅)	C₂²	1	2	yes	—
8	C₂×C₈×D₅	C₂²	1	2	yes	—
9	C₄²×D₅	C₂²	1	1	yes	characteristic
10	(C₂×AGL(1,𝔽₅))⋊C₄	—	2	2	no	—
11	C₅×#97	C₄	1	1	yes	characteristic, complemented
12	#3947	C₄	1	2	yes	complemented
13	#3935	C₄	1	1	yes	characteristic, complemented
14	#1836	—	5	1	no	maximal, Sylow
15	C₂²×AGL(1,𝔽₅)	—	2	2	no	—
16	C₂²×AGL(1,𝔽₅)	—	2	2	no	—
17	C₂×C₄×D₅	D₄	1	2	yes	—
18	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
19	C₄⋊AGL(1,𝔽₅)	—	2	4	no	—
20	C₄⋊AGL(1,𝔽₅)	—	2	4	no	—
21	C₄⋊AGL(1,𝔽₅)	—	4	4	no	—
22	C₂×C₅×C₈	C₂×C₄	1	2	yes	—
23	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
25	C₄×Dic₅	Q₈	1	2	yes	—
26	C₂×#168	C₂×C₄	1	2	yes	—
27	C₈×D₅	—	2	2	no	—
28	C₈×D₅	—	2	2	no	—
29	C₂×#97	—	5	1	no	—
30	#384	—	5	1	no	—
31	#382	—	5	1	no	—
32	#340	—	5	4	no	—
33	C₂²×D₅	C₂×D₄	1	1	yes	characteristic
34	C₂×AGL(1,𝔽₅)	—	4	2	no	—
35	C₂×AGL(1,𝔽₅)	—	4	2	no	—
36	C₂×AGL(1,𝔽₅)	—	4	4	no	—
37	C₂×AGL(1,𝔽₅)	—	4	2	no	—
38	C₂×AGL(1,𝔽₅)	—	4	2	no	—
39	C₂×C₄×C₅	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
41	C₂×Dic₅	C₄⋊C₄	1	2	yes	—
42	C₂×Dic₅	C₂×Q₈	1	1	yes	characteristic
43	C₄×D₅	—	2	2	no	—
44	C₄×D₅	—	2	2	no	—
45	C₄×D₅	#41	1	2	yes	—
46	C₄×D₅	#41	1	2	yes	—
47	C₅×C₈	—	2	2	no	—
48	#168	—	2	2	no	—
49	C₂²×C₈	—	5	2	no	—
50	C₂×C₄⋊C₄	—	5	2	no	—
51	C₂×C₄⋊C₄	—	5	2	no	—
52	C₂×C₄⋊C₄	—	10	2	no	—
53	C₂×C₄²	—	5	1	no	—
54	#97	—	5	2	no	—
55	#97	—	5	1	no	—
56	#97	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
57	#95	—	10	2	no	—
58	C₂²×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×D₅	#124	1	1	yes	characteristic
60	C₂×D₅	#124	1	1	yes	characteristic
61	C₂×D₅	AGL(1,ℤ/8)	1	2	yes	—
62	C₂×D₅	#137	1	2	yes	—
63	AGL(1,𝔽₅)	—	8	2	no	—
64	AGL(1,𝔽₅)	—	8	2	no	—
65	C₄×C₅	—	2	2	no	—
66	C₄×C₅	#117	1	2	yes	—
67	Dic₅	—	2	2	no	—
68	Dic₅	#125	1	2	yes	—
69	C₂²×C₄	—	5	1	no	—
70	C₂²×C₄	—	5	2	no	—
71	C₂²×C₄	—	10	2	no	—
72	C₂²×C₄	—	10	2	no	—
73	C₂×C₈	—	5	2	no	—
74	C₂×C₈	—	10	2	no	—
75	C₂×C₈	—	10	2	no	—
76	C₂×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
77	C₄⋊C₄	—	10	4	no	—
78	C₄⋊C₄	—	10	4	no	—
79	C₄⋊C₄	—	20	4	no	—
80	C₄²	—	5	2	no	—
81	C₄²	—	5	1	no	—
82	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
83	C₂×C₅	#382	1	1	yes	characteristic
84	C₂×C₅	#428	1	2	yes	—
85	D₅	#489	1	2	yes	—
86	D₅	#490	1	1	yes	characteristic
87	D₅	#491	1	1	yes	characteristic
88	C₂³	—	5	1	no	—
89	C₂×C₄	—	5	1	no	—
90	C₂×C₄	—	5	2	no	—
91	C₂×C₄	—	5	2	no	—
92	C₂×C₄	—	5	2	no	—
93	C₂×C₄	—	10	2	no	—
94	C₂×C₄	—	10	2	no	—
95	C₂×C₄	—	20	2	no	—
96	C₂×C₄	—	20	2	no	—
97	C₂×C₄	—	20	4	no	—
98	C₂×C₄	—	20	2	no	—
99	C₂×C₄	—	20	2	no	—
100	C₂×C₄	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
101	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
102	C₈	—	2	2	no	—
103	C₈	—	10	2	no	—
104	C₅	#1836	1	1	yes	characteristic, complemented, Sylow
105	C₂²	—	5	1	no	—
106	C₂²	—	5	1	no	—
107	C₂²	—	5	2	no	—
108	C₂²	—	5	2	no	—
109	C₂²	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
110	C₄	—	2	2	no	—
111	C₄	—	5	2	no	—
112	C₄	—	10	2	no	—
113	C₄	—	40	2	no	—
114	C₄	—	40	2	no	—
115	C₄	#4130	1	2	yes	—
116	C₂	—	5	1	no	—
117	C₂	—	5	2	no	—
118	C₂	—	5	1	no	—
119	C₂	#65689	1	1	yes	characteristic
120	C₂	#65727	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#65689

C₂

yes

maximal, characteristic

#65688

C₂

yes

maximal, characteristic

D₅×#97

C₂

yes

maximal, characteristic

#64893

C₂

yes

maximal

C₂×C₄⋊AGL(1,𝔽₅)

—

C₂×C₄⋊AGL(1,𝔽₅)

C₂²

yes

—

C₂×C₄⋊AGL(1,𝔽₅)

C₂²

yes

—

C₂×C₈×D₅

C₂²

yes

—

C₄²×D₅

C₂²

yes

characteristic

(C₂×AGL(1,𝔽₅))⋊C₄

—

C₅×#97

C₄

yes

characteristic, complemented

#3947

C₄

yes

complemented

#3935

C₄

yes

characteristic, complemented

#1836

—

maximal, Sylow

C₂²×AGL(1,𝔽₅)

—

C₂²×AGL(1,𝔽₅)

—

C₂×C₄×D₅

D₄

yes

—

C₂×C₄×D₅

C₂³

yes

—

—

—

C₂×C₅×C₈

C₂×C₄

yes

—

C₄²×C₅

C₂×C₄

yes

characteristic

C₄×Dic₅

C₂×C₄

yes

characteristic

C₄×Dic₅

Q₈

yes

—

C₂×#168

C₂×C₄

yes

—

C₈×D₅

—

C₈×D₅

—

C₂×#97

—

#384

—

#382

—

#340

—

C₂²×D₅

C₂×D₄

yes

—

—

—

—

—

C₂×C₄×C₅

C₄⋊C₄

yes

—

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×Dic₅

C₄⋊C₄

yes

—

C₂×Dic₅

C₂×Q₈

yes

characteristic

C₄×D₅

—

C₄×D₅

—

C₄×D₅

#41

yes

—

C₄×D₅

#41

yes

—

C₅×C₈

—

#168

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

—

—

yes

characteristic, complemented

#95

—

C₂²×C₅

C₂×C₄⋊C₄

yes

C₂×D₅

yes

C₂×D₅

yes

C₂×D₅

yes

—

C₂×D₅

#137

yes

—

AGL(1,𝔽₅)

—

AGL(1,𝔽₅)

—

C₄×C₅

—

C₄×C₅

#117

yes

—

Dic₅

—

Dic₅

#125

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×AGL(1,𝔽₅)

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

yes

C₂×C₅

yes

C₂×C₅

yes

—

D₅

#489

yes

—

D₅

yes

D₅

yes

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

C₄⋊AGL(1,𝔽₅)

yes

—

101

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

102

C₈

—

103

C₈

—

104

C₅

#1836

yes

characteristic, complemented, Sylow

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

#4130

yes

—

116

C₂

—

117

C₂

—

118

C₂

—

119

C₂

#65689

yes

characteristic

120

C₂

#65727

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#97	C₄
socle	C₂²×C₅	C₂×C₄⋊C₄
prime radical	C₂²×D₅	C₂×D₄
cosocle	C₂×C₄×D₅	C₂³
hypercenter	#97	AGL(1,𝔽₅)
hypocenter	C₅	#1836
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

C₂×C₄⋊AGL(1,𝔽₅)

commutator

C₂×C₄×C₅

C₂²×C₄

Frattini

C₂×C₄

C₂²×AGL(1,𝔽₅)

Fitting

C₅×#97

C₄

socle

C₂²×C₅

C₂×C₄⋊C₄

prime radical

C₂²×D₅

C₂×D₄

cosocle

C₂×C₄×D₅

C₂³

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
11. C₅×#97	C₄	semidirect
12. #3947	C₄	semidirect
13. #3935	C₄	semidirect
56. #97	AGL(1,𝔽₅)	semidirect
104. C₅	#1836	semidirect

H ≅ G/N

type

11. C₅×#97

C₄

semidirect

12. #3947

C₄

semidirect

13. #3935

C₄

semidirect

56. #97

AGL(1,𝔽₅)

semidirect

104. C₅

#1836

semidirect

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

subgroup	index
#65689	2
#65688	2
D₅×#97	2
#64893	2
#1836	5

subgroup

index

D₅×#97

Subgroups — #10603882

Conjugacy classes of subgroups

Complements

Maximal subgroups