Group #10600608 — subgroups

Subgroups — #10600608

Order 640.

224 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂×C₅×#97	C₂	1	1	yes	maximal, characteristic
2	#65225	C₂	1	1	yes	maximal, characteristic
3	#65224	C₂	1	1	yes	maximal, characteristic
4	#64771	C₂	1	4	yes	maximal
5	C₂²×C₅×C₈	C₂²	1	2	yes	—
6	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₅	—	2	2	no	—
8	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
10	C₅×#97	C₄	1	4	yes	complemented
11	#3963	—	2	2	no	—
12	#1836	—	5	1	no	maximal, Sylow
13	C₂²×C₄×C₅	D₄	1	2	yes	—
14	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
15	C₂²×Dic₅	—	2	2	no	—
16	C₂²×Dic₅	—	2	2	no	—
17	C₂×C₅×C₈	—	2	2	no	—
18	C₂×C₅×C₈	—	2	2	no	—
19	C₂×C₅×C₈	C₂×C₄	1	4	yes	—
20	C₄²×C₅	C₂×C₄	1	2	yes	—
21	C₄²×C₅	Q₈	1	2	yes	—
22	C₄⋊Dic₅	—	2	4	no	—
23	C₄⋊Dic₅	—	2	4	no	—
24	C₄⋊Dic₅	—	4	4	no	—
25	C₂×#97	D₅	1	1	yes	characteristic
26	#384	—	5	1	no	—
27	#382	—	5	1	no	—
28	#340	—	5	4	no	—
29	C₂³×C₅	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₅	—	2	2	no	—
31	C₂×C₄×C₅	—	2	2	no	—
32	C₂×C₄×C₅	C₄⋊C₄	1	4	yes	—
33	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₅	#41	1	2	yes	—
36	C₂×C₄×C₅	#41	1	2	yes	—
37	C₂×Dic₅	—	4	2	no	—
38	C₂×Dic₅	—	4	2	no	—
39	C₂×Dic₅	—	4	4	no	—
40	C₂×Dic₅	—	4	2	no	—
41	C₂×Dic₅	—	4	2	no	—
42	C₅×C₈	—	2	4	no	—
43	C₂²×C₈	C₂×D₅	1	2	yes	—
44	C₂×C₄⋊C₄	—	5	2	no	—
45	C₂×C₄⋊C₄	—	5	2	no	—
46	C₂×C₄⋊C₄	—	10	2	no	—
47	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
48	#97	Dic₅	1	4	yes	complemented
49	#95	—	10	2	no	—
50	C₂²×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₅	#124	1	1	yes	characteristic
52	C₂²×C₅	#124	1	1	yes	characteristic
53	C₂²×C₅	AGL(1,ℤ/8)	1	2	yes	—
54	C₂²×C₅	#137	1	2	yes	—
55	C₄×C₅	—	2	4	no	—
56	C₄×C₅	#117	1	2	yes	—
57	C₄×C₅	#125	1	2	yes	—
58	Dic₅	—	8	2	no	—
59	Dic₅	—	8	2	no	—
60	C₂²×C₄	—	10	2	no	—
61	C₂²×C₄	—	10	2	no	—
62	C₂²×C₄	D₂₀	1	2	yes	—
63	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	C₂×Dic₅	1	4	yes	—
67	C₄⋊C₄	—	10	4	no	—
68	C₄⋊C₄	—	10	4	no	—
69	C₄⋊C₄	—	20	4	no	—
70	C₄²	Dic₁₀	1	2	yes	—
71	C₄²	C₂×Dic₅	1	2	yes	—
72	C₂×C₅	#382	1	1	yes	characteristic
73	C₂×C₅	#428	1	2	yes	—
74	C₂×C₅	#489	1	2	yes	—
75	C₂×C₅	#490	1	1	yes	characteristic
76	C₂×C₅	#491	1	1	yes	characteristic
77	C₂³	C₂×D₂₀	1	1	yes	characteristic
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	20	2	no	—
81	C₂×C₄	—	20	2	no	—
82	C₂×C₄	—	20	4	no	—
83	C₂×C₄	—	20	2	no	—
84	C₂×C₄	—	20	2	no	—
85	C₂×C₄	C₄⋊Dic₅	1	4	yes	—
86	C₂×C₄	C₂×Dic₁₀	1	1	yes	characteristic
87	C₂×C₄	#709	1	2	yes	—
88	C₂×C₄	#709	1	2	yes	—
89	C₂×C₄	C₂²×Dic₅	1	1	yes	characteristic
90	C₈	—	2	4	no	—
91	C₅	#1836	1	1	yes	characteristic, complemented, Sylow
92	C₂²	#4021	1	1	yes	characteristic
93	C₂²	#4021	1	1	yes	characteristic
94	C₂²	#4054	1	2	yes	—
95	C₂²	#4055	1	2	yes	—
96	C₂²	C₂×C₄⋊Dic₅	1	1	yes	characteristic
97	C₄	—	2	4	no	—
98	C₄	—	40	2	no	—
99	C₄	—	40	2	no	—
100	C₄	#4016	1	2	yes	—
101	C₄	#4072	1	2	yes	—
102	C₂	#64992	1	1	yes	characteristic
103	C₂	#65002	1	1	yes	characteristic
104	C₂	#65003	1	2	yes	—
105	C₂	#65225	1	1	yes	characteristic
106	C₂	#65410	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅×#97

C₂

yes

maximal, characteristic

#65225

C₂

yes

maximal, characteristic

#65224

C₂

yes

maximal, characteristic

#64771

C₂

yes

maximal

C₂²×C₅×C₈

C₂²

yes

—

C₂×C₄²×C₅

C₂²

yes

characteristic

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

C₂²

yes

—

C₂×C₄⋊Dic₅

C₂²

yes

—

C₅×#97

C₄

yes

—

—

C₂²×C₄×C₅

D₄

yes

—

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂²×Dic₅

—

C₂²×Dic₅

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂×C₄

yes

—

C₄²×C₅

C₂×C₄

yes

—

C₄²×C₅

Q₈

yes

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₂×#97

D₅

yes

—

—

—

C₂³×C₅

C₂×D₄

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₄⋊C₄

yes

—

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×C₄×C₅

C₂×Q₈

yes

characteristic

C₂×C₄×C₅

#41

yes

—

C₂×C₄×C₅

#41

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₅×C₈

—

C₂²×C₈

C₂×D₅

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₅

yes

characteristic

#97

Dic₅

yes

complemented

#95

—

C₂²×C₅

C₂×C₄⋊C₄

yes

C₂²×C₅

yes

C₂²×C₅

yes

C₂²×C₅

yes

—

C₂²×C₅

#137

yes

—

C₄×C₅

—

C₄×C₅

#117

yes

—

C₄×C₅

#125

yes

—

Dic₅

—

Dic₅

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₂₀

yes

—

C₂²×C₄

C₂²×D₅

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₅

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Dic₁₀

yes

—

C₄²

C₂×Dic₅

yes

—

C₂×C₅

#382

yes

characteristic

C₂×C₅

#428

yes

—

C₂×C₅

#489

yes

—

C₂×C₅

yes

C₂×C₅

yes

C₂³

C₂×D₂₀

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₅

yes

—

C₂×C₄

C₂×Dic₁₀

yes

characteristic

C₂×C₄

#709

yes

—

C₂×C₄

#709

yes

—

C₂×C₄

C₂²×Dic₅

yes

characteristic

C₈

—

C₅

#1836

yes

characteristic, complemented, Sylow

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#4055

yes

—

C₂²

C₂×C₄⋊Dic₅

yes

characteristic

C₄

—

C₄

—

C₄

—

100

C₄

#4016

yes

—

101

C₄

#4072

yes

—

102

C₂

#64992

yes

characteristic

103

C₂

#65002

yes

characteristic

104

C₂

#65003

yes

—

105

C₂

#65225

yes

characteristic

106

C₂

#65410

yes

—

	subgroup	G/H
center	C₂³	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅
Fitting	C₂×C₅×#97	C₂
socle, prime radical	C₂³×C₅	C₂×D₄
cosocle	C₂²×C₄×C₅	C₂³
hypercenter	C₂×#97	D₅
hypocenter	C₅	#1836
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₂₀

commutator

C₂×C₄×C₅

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅

Fitting

C₂×C₅×#97

C₂

socle, prime radical

C₂³×C₅

C₂×D₄

cosocle

C₂²×C₄×C₅

C₂³

hypercenter

C₂×#97

D₅

hypocenter

C₅

#1836

LCS

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
10. C₅×#97	C₄	semidirect
48. #97	Dic₅	semidirect
91. C₅	#1836	semidirect

H ≅ G/N

type

10. C₅×#97

C₄

semidirect

48. #97

Dic₅

semidirect

91. C₅

#1836

semidirect

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

subgroup	index
C₂×C₅×#97	2
#65225	2
#65224	2
#64771	2
#1836	5

subgroup

index

C₂×C₅×#97

Subgroups — #10600608

Conjugacy classes of subgroups

Complements

Maximal subgroups