Group #10598591 — subgroups

Subgroups — #10598591

Order 640.

138 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 44 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	#65683	C₂	1	1	yes	maximal, characteristic
2	C₄×#700	C₂	1	1	yes	maximal, characteristic
3	#65124	C₂	1	1	yes	maximal, characteristic
4	C₂×#700	—	2	1	no	—
5	C₄²×D₅	C₂²	1	1	yes	characteristic
6	#4003	C₄	1	2	yes	—
7	#4002	—	2	1	no	—
8	#4002	—	2	1	no	—
9	C₄⋊#170	—	2	1	no	—
10	C₄×#170	C₄	1	2	yes	—
11	C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
12	#3952	—	2	1	no	—
13	#3947	—	2	1	no	—
14	C₄⋊#168	C₄	1	1	yes	characteristic
15	#1291	—	5	1	no	maximal, Sylow
16	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×D₅	D₄	1	1	yes	characteristic
18	C₂×C₄×D₅	Q₈	1	1	yes	characteristic
19	C₂×#170	—	2	2	no	—
20	C₂×#170	—	2	2	no	—
21	#700	—	2	2	no	—
22	#700	—	2	2	no	—
23	C₂×C₅×C₈	—	2	1	no	—
24	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
25	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
26	C₄×Dic₅	D₄	1	1	yes	characteristic
27	C₄×Dic₅	D₄	1	1	yes	characteristic
28	C₂×#168	—	2	1	no	—
29	#432	—	5	1	no	—
30	#432	—	5	1	no	—
31	C₄×M₁₆	—	5	1	no	—
32	C₂²×D₅	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₅	C₄²	1	1	yes	characteristic
34	C₂×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
36	C₂×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×Dic₅	C₄⋊C₄	1	1	yes	characteristic
39	C₄×D₅	—	2	1	no	—
40	C₄×D₅	—	2	1	no	—
41	C₄×D₅	—	4	1	no	—
42	C₄×D₅	—	4	1	no	—
43	#170	—	2	4	no	—
44	#170	—	4	2	no	—
45	C₅×C₈	—	4	1	no	—
46	#168	—	4	1	no	—
47	C₂×M₁₆	—	10	1	no	—
48	C₂×C₄²	—	5	1	no	—
49	C₄⋊C₈	—	5	1	no	—
50	C₄⋊C₈	—	10	1	no	—
51	C₄⋊C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
52	C₂²⋊C₈	—	5	2	no	—
53	C₂²⋊C₈	—	10	1	no	—
54	#97	—	10	1	no	—
55	#97	—	10	1	no	—
56	#97	—	10	1	no	—
57	C₄×C₈	—	5	2	no	—
58	C₂²×C₅	#95	1	1	yes	characteristic
59	C₂×D₅	—	2	1	no	—
60	C₂×D₅	—	2	1	no	—
61	C₂×D₅	—	2	1	no	—
62	C₄×C₅	—	2	1	no	—
63	C₄×C₅	—	2	1	no	—
64	C₄×C₅	#104	1	1	yes	characteristic
65	C₄×C₅	#104	1	1	yes	characteristic
66	Dic₅	—	2	1	no	—
67	Dic₅	—	2	1	no	—
68	Dic₅	—	2	1	no	—
69	C₂²×C₄	—	5	1	no	—
70	C₂²×C₄	—	5	1	no	—
71	C₂²×C₄	—	5	1	no	—
72	M₁₆	—	10	2	no	—
73	M₁₆	—	10	2	no	—
74	C₂×C₈	—	2	1	no	—
75	C₂×C₈	—	10	2	no	—
76	C₂×C₈	—	10	2	no	—
77	C₂×C₈	—	10	1	no	—
78	C₄²	—	5	1	no	—
79	C₄²	—	5	1	no	—
80	C₄²	—	5	1	no	—
81	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
82	C₂×C₅	#338	1	1	yes	characteristic
83	C₂×C₅	#339	1	1	yes	characteristic
84	C₂×C₅	#344	1	1	yes	characteristic
85	D₅	—	4	1	no	—
86	C₂³	—	5	1	no	—
87	C₂×C₄	—	5	1	no	—
88	C₂×C₄	—	5	1	no	—
89	C₂×C₄	—	5	1	no	—
90	C₂×C₄	—	10	1	no	—
91	C₂×C₄	—	10	1	no	—
92	C₂×C₄	—	20	1	no	—
93	C₂×C₄	—	20	1	no	—
94	C₂×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
95	C₂×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
96	C₂×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
97	C₈	—	4	1	no	—
98	C₈	—	10	4	no	—
99	C₈	—	20	2	no	—
100	C₈	—	20	1	no	—
101	C₅	#1291	1	1	yes	characteristic, complemented, Sylow
102	C₂²	—	10	1	no	—
103	C₂²	—	10	1	no	—
104	C₂²	—	10	1	no	—
105	C₂²	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	2	1	no	—
108	C₄	—	10	1	no	—
109	C₄	—	10	1	no	—
110	C₄	—	10	1	no	—
111	C₄	#4008	1	1	yes	characteristic
112	C₄	#4010	1	1	yes	characteristic
113	C₂	—	20	1	no	—
114	C₂	#64875	1	1	yes	characteristic
115	C₂	#64897	1	1	yes	characteristic
116	C₂	#64902	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#65683

C₂

yes

maximal, characteristic

C₄×#700

C₂

yes

maximal, characteristic

#65124

C₂

yes

maximal, characteristic

C₂×#700

—

C₄²×D₅

C₂²

yes

characteristic

#4003

C₄

yes

—

—

—

—

C₄

yes

—

C₅×C₄⋊C₈

C₄

yes

—

—

C₄

yes

characteristic

#1291

—

maximal, Sylow

C₂×C₄×D₅

C₂×C₄

yes

characteristic

C₂×C₄×D₅

D₄

yes

characteristic

C₂×C₄×D₅

Q₈

yes

—

—

—

—

C₂×C₅×C₈

—

C₄²×C₅

C₂×C₄

yes

characteristic

C₄×Dic₅

C₂×C₄

yes

characteristic

C₄×Dic₅

D₄

yes

characteristic

C₄×Dic₅

D₄

yes

—

—

—

C₄×M₁₆

—

C₂²×D₅

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

C₄²

yes

characteristic

C₂×C₄×C₅

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅

C₄⋊C₄

yes

characteristic

C₂×Dic₅

C₂²⋊C₄

yes

characteristic

C₂×Dic₅

C₂²⋊C₄

yes

characteristic

C₂×Dic₅

C₄⋊C₄

yes

characteristic

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

#170

—

#170

—

C₅×C₈

—

#168

—

C₂×M₁₆

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

AGL(1,𝔽₅)

yes

characteristic

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

#97

—

C₄×C₈

—

C₂²×C₅

#95

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

yes

C₄×C₅

yes

Dic₅

—

Dic₅

—

Dic₅

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

—

C₄²

yes

C₂×C₅

yes

C₂×C₅

yes

C₂×C₅

yes

D₅

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

C₈

—

C₈

—

C₈

—

100

C₈

—

101

C₅

#1291

yes

characteristic, complemented, Sylow

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

(C₂×AGL(1,𝔽₅))⋊C₄

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#4008

yes

characteristic

112

C₄

#4010

yes

characteristic

113

C₂

—

114

C₂

#64875

yes

characteristic

115

C₂

#64897

yes

characteristic

116

C₂

#64902

yes

characteristic

	subgroup	G/H
center	C₂²	(C₂×AGL(1,𝔽₅))⋊C₄
commutator	C₂×C₄×C₅	C₄²
Frattini	C₄²	C₂×AGL(1,𝔽₅)
Fitting	C₅×C₄⋊C₈	C₄
socle	C₂²×C₅	#95
prime radical	C₂²×D₅	C₄⋊C₄
cosocle	C₄²×D₅	C₂²
hypercenter	C₄⋊C₈	AGL(1,𝔽₅)
hypocenter	C₅	#1291
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

subgroup

G/H

center

C₂²

(C₂×AGL(1,𝔽₅))⋊C₄

commutator

C₂×C₄×C₅

C₄²

Frattini

C₄²

C₂×AGL(1,𝔽₅)

Fitting

C₅×C₄⋊C₈

C₄

socle

C₂²×C₅

#95

prime radical

C₂²×D₅

C₄⋊C₄

cosocle

C₄²×D₅

C₂²

C₄⋊C₈

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

N	H ≅ G/N	type
101. C₅	#1291	semidirect

H ≅ G/N

type

101. C₅

#1291

semidirect

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

subgroup	index
#65683	2
C₄×#700	2
#65124	2
#1291	5

subgroup

index

Subgroups — #10598591

Conjugacy classes of subgroups

Complements

Maximal subgroups