Group #10603889 — subgroups

Subgroups — #10603889

Order 640.

180 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 66 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	#65682	C₂	1	1	yes	maximal, characteristic
2	#65681	C₂	1	1	yes	maximal, characteristic
3	D₅×#97	C₂	1	1	yes	maximal, characteristic
4	#64896	C₂	1	4	yes	maximal
5	C₂×#700	C₂²	1	2	yes	—
6	C₂×#700	C₂²	1	2	yes	—
7	C₂×C₈×D₅	C₂²	1	2	yes	—
8	C₄²×D₅	C₂²	1	1	yes	characteristic
9	#4004	—	2	2	no	—
10	C₄⋊#170	—	2	2	no	—
11	C₅×#97	C₄	1	1	yes	characteristic
12	#3947	C₄	1	2	yes	—
13	#3935	C₄	1	1	yes	characteristic
14	#1838	—	5	1	no	maximal, Sylow
15	C₂×C₄×D₅	Q₈	1	2	yes	—
16	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
17	C₂×#170	—	2	2	no	—
18	C₂×#170	—	2	2	no	—
19	#700	—	2	4	no	—
20	#700	—	2	4	no	—
21	C₂×C₅×C₈	C₂×C₄	1	2	yes	—
22	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
23	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₅	D₄	1	2	yes	—
25	C₂×#168	C₂×C₄	1	2	yes	—
26	C₈×D₅	—	2	2	no	—
27	C₈×D₅	—	2	2	no	—
28	#423	—	5	1	no	—
29	#423	—	5	1	no	—
30	C₂×#97	—	5	1	no	—
31	#341	—	5	4	no	—
32	C₂²×D₅	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₅	C₄⋊C₄	1	2	yes	—
34	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
35	C₂×Dic₅	C₄⋊C₄	1	2	yes	—
36	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
37	C₄×D₅	—	4	2	no	—
38	C₄×D₅	#41	1	2	yes	—
39	C₄×D₅	#41	1	2	yes	—
40	#170	—	4	2	no	—
41	#170	—	4	2	no	—
42	C₅×C₈	—	2	2	no	—
43	#168	—	2	2	no	—
44	C₂×M₁₆	—	5	2	no	—
45	C₂×M₁₆	—	5	2	no	—
46	C₂²×C₈	—	5	2	no	—
47	C₂×C₄²	—	5	1	no	—
48	C₄⋊C₈	—	10	2	no	—
49	C₄⋊C₈	—	10	2	no	—
50	#97	—	5	2	no	—
51	#97	—	5	1	no	—
52	#97	AGL(1,𝔽₅)	1	1	yes	characteristic
53	C₂²×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×D₅	—	2	2	no	—
55	C₂×D₅	#125	1	2	yes	—
56	C₄×C₅	—	2	2	no	—
57	C₄×C₅	#117	1	2	yes	—
58	Dic₅	—	2	2	no	—
59	Dic₅	#124	1	1	yes	characteristic
60	Dic₅	#124	1	1	yes	characteristic
61	C₂²×C₄	—	5	1	no	—
62	C₂²×C₄	—	5	2	no	—
63	M₁₆	—	10	4	no	—
64	M₁₆	—	10	4	no	—
65	C₂×C₈	—	5	2	no	—
66	C₂×C₈	—	10	2	no	—
67	C₂×C₈	—	10	2	no	—
68	C₂×C₈	—	10	2	no	—
69	C₂×C₈	—	10	2	no	—
70	C₂×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
71	C₄²	—	5	2	no	—
72	C₄²	—	5	1	no	—
73	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
74	C₂×C₅	#382	1	1	yes	characteristic
75	C₂×C₅	#430	1	2	yes	—
76	D₅	—	2	2	no	—
77	C₂³	—	5	1	no	—
78	C₂×C₄	—	5	1	no	—
79	C₂×C₄	—	5	2	no	—
80	C₂×C₄	—	5	2	no	—
81	C₂×C₄	—	5	2	no	—
82	C₂×C₄	—	20	2	no	—
83	C₂×C₄	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
84	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
85	C₈	—	2	2	no	—
86	C₈	—	10	2	no	—
87	C₈	—	20	2	no	—
88	C₈	—	20	2	no	—
89	C₅	#1838	1	1	yes	characteristic, complemented, Sylow
90	C₂²	—	5	2	no	—
91	C₂²	—	10	2	no	—
92	C₂²	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
93	C₄	—	2	2	no	—
94	C₄	—	5	1	no	—
95	C₄	—	5	1	no	—
96	C₄	—	10	2	no	—
97	C₄	#4130	1	2	yes	—
98	C₂	—	10	2	no	—
99	C₂	#65689	1	1	yes	characteristic
100	C₂	#65729	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#65682

C₂

yes

maximal, characteristic

#65681

C₂

yes

maximal, characteristic

D₅×#97

C₂

yes

maximal, characteristic

#64896

C₂

yes

maximal

C₂×#700

C₂²

yes

—

C₂×#700

C₂²

yes

—

C₂×C₈×D₅

C₂²

yes

—

C₄²×D₅

C₂²

yes

characteristic

#4004

—

C₄⋊#170

—

C₅×#97

C₄

yes

characteristic

#3947

C₄

yes

—

#3935

C₄

yes

characteristic

#1838

—

maximal, Sylow

C₂×C₄×D₅

Q₈

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

D₄

yes

—

C₂×#168

C₂×C₄

yes

—

C₈×D₅

—

C₈×D₅

—

#423

—

#423

—

C₂×#97

—

#341

—

C₂²×D₅

C₂×Q₈

yes

characteristic

C₂×C₄×C₅

C₄⋊C₄

yes

—

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×Dic₅

C₄⋊C₄

yes

—

C₂×Dic₅

C₂×D₄

yes

characteristic

C₄×D₅

—

C₄×D₅

#41

yes

—

C₄×D₅

#41

yes

—

#170

—

#170

—

C₅×C₈

—

#168

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

—

—

yes

C₂²×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×D₅

—

C₂×D₅

#125

yes

—

C₄×C₅

—

C₄×C₅

#117

yes

—

Dic₅

—

Dic₅

yes

Dic₅

yes

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×AGL(1,𝔽₅)

yes

—

C₄²

—

C₄²

—

C₄²

yes

C₂×C₅

yes

C₂×C₅

yes

—

D₅

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊AGL(1,𝔽₅)

yes

—

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₅

#1838

yes

characteristic, complemented, Sylow

C₂²

—

C₂²

—

C₂²

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

#4130

yes

—

C₂

—

C₂

#65689

yes

characteristic

100

C₂

#65729

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×Q₈
cosocle	C₂×C₄×D₅	C₂³
hypercenter	#97	AGL(1,𝔽₅)
hypocenter	C₅	#1838
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×Q₈

cosocle

C₂×C₄×D₅

C₂³

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
89. C₅	#1838	semidirect

H ≅ G/N

type

89. C₅

#1838

semidirect

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

subgroup	index
#65682	2
#65681	2
D₅×#97	2
#64896	2
#1838	5

subgroup

index

D₅×#97

Subgroups — #10603889

Conjugacy classes of subgroups

Complements

Maximal subgroups