Group #10604069 — subgroups

Subgroups — #10604069

Order 640.

148 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 56 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	#65709	C₂	1	2	yes	maximal
2	#65162	C₂	1	1	yes	maximal, characteristic
3	#64881	C₂	1	1	yes	maximal, characteristic
4	#64879	C₂	1	1	yes	maximal, characteristic
5	#64872	C₂	1	2	yes	maximal
6	C₂×Dic₂₀	C₄	1	1	yes	characteristic
7	#4033	C₂²	1	2	yes	—
8	#4002	—	2	2	no	—
9	C₄⋊#170	—	2	2	no	—
10	C₄⋊#170	C₂²	1	2	yes	—
11	C₄×#170	C₂²	1	1	yes	characteristic
12	C₄×#170	C₂²	1	1	yes	characteristic
13	C₅×#107	C₄	1	1	yes	characteristic
14	C₈×Dic₅	C₂²	1	1	yes	characteristic
15	#3942	C₄	1	2	yes	—
16	#1593	—	5	1	no	maximal, Sylow
17	C₂×Dic₁₀	C₂×C₄	1	2	yes	—
18	C₂×#170	—	2	2	no	—
19	C₂×#170	—	2	1	no	—
20	C₂×#170	D₄	1	2	yes	—
21	C₂×C₅×C₈	C₂×C₄	1	1	yes	characteristic
22	C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
23	#683	—	2	2	no	—
24	C₄×Dic₅	—	2	2	no	—
25	C₄×Dic₅	C₂³	1	1	yes	characteristic
26	C₂×#168	C₂×C₄	1	1	yes	characteristic
27	Dic₂₀	—	2	2	no	—
28	#446	—	5	2	no	—
29	C₄×Q₁₆	—	5	1	no	—
30	#334	—	5	1	no	—
31	Q₈⋊C₈	—	5	2	no	—
32	#322	—	5	1	no	—
33	C₂×C₄×C₅	—	2	2	no	—
34	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
35	C₂×Dic₅	—	2	2	no	—
36	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
37	C₂×Dic₅	#41	1	1	yes	characteristic
38	Dic₁₀	—	2	2	no	—
39	Dic₁₀	—	2	2	no	—
40	#170	—	2	2	no	—
41	#170	—	4	1	no	—
42	#170	—	4	2	no	—
43	C₅×C₈	M₁₆	1	2	yes	—
44	#168	—	2	1	no	—
45	C₂×Q₁₆	—	5	1	no	—
46	C₄×Q₈	—	5	2	no	—
47	#107	AGL(1,𝔽₅)	1	1	yes	characteristic
48	C₄⋊C₈	—	5	2	no	—
49	C₄⋊C₈	—	10	2	no	—
50	#103	—	5	2	no	—
51	#97	—	10	2	no	—
52	C₄×C₈	—	5	1	no	—
53	C₄×C₈	—	5	1	no	—
54	C₄×C₈	—	5	1	no	—
55	C₂²×C₅	C₄×D₄	1	1	yes	characteristic
56	C₄×C₅	—	4	2	no	—
57	C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
58	C₄×C₅	#131	1	1	yes	characteristic
59	Dic₅	—	2	1	no	—
60	Dic₅	—	4	2	no	—
61	Dic₅	#137	1	2	yes	—
62	C₂×Q₈	—	5	2	no	—
63	Q₁₆	—	10	2	no	—
64	C₂×C₈	—	5	2	no	—
65	C₂×C₈	—	5	1	no	—
66	C₂×C₈	—	10	1	no	—
67	C₂×C₈	—	10	2	no	—
68	C₂×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
69	C₄⋊C₄	—	10	2	no	—
70	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
71	C₄²	—	5	1	no	—
72	C₄²	—	10	2	no	—
73	C₂×C₅	#436	1	1	yes	characteristic
74	C₂×C₅	#441	1	1	yes	characteristic
75	C₂×C₅	#444	1	1	yes	characteristic
76	Q₈	—	10	2	no	—
77	Q₈	—	10	2	no	—
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	5	1	no	—
80	C₂×C₄	—	5	1	no	—
81	C₂×C₄	—	10	2	no	—
82	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
83	C₈	—	10	2	no	—
84	C₈	—	10	1	no	—
85	C₈	—	20	2	no	—
86	C₈	—	20	1	no	—
87	C₈	#700	1	2	yes	—
88	C₅	#1593	1	1	yes	characteristic, complemented, Sylow
89	C₂²	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
90	C₄	—	4	2	no	—
91	C₄	—	5	2	no	—
92	C₄	—	10	1	no	—
93	C₄	—	20	2	no	—
94	C₄	C₂×#700	1	1	yes	characteristic
95	C₄	#4133	1	1	yes	characteristic
96	C₂	#65705	1	1	yes	characteristic
97	C₂	#65733	1	1	yes	characteristic
98	C₂	#65739	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#65709

C₂

yes

maximal

#65162

C₂

yes

maximal, characteristic

#64881

C₂

yes

maximal, characteristic

#64879

C₂

yes

maximal, characteristic

#64872

C₂

yes

maximal

C₂×Dic₂₀

C₄

yes

characteristic

#4033

C₂²

yes

—

#4002

—

C₄⋊#170

—

C₄⋊#170

C₂²

yes

—

C₄×#170

C₂²

yes

characteristic

C₄×#170

C₂²

yes

characteristic

C₅×#107

C₄

yes

characteristic

C₈×Dic₅

C₂²

yes

characteristic

#3942

C₄

yes

—

#1593

—

maximal, Sylow

C₂×Dic₁₀

C₂×C₄

yes

—

C₂×#170

—

C₂×#170

—

C₂×#170

D₄

yes

—

C₂×C₅×C₈

C₂×C₄

yes

characteristic

C₅×C₄⋊C₄

C₂×C₄

yes

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

C₂³

yes

characteristic

C₂×#168

C₂×C₄

yes

characteristic

Dic₂₀

—

#446

—

C₄×Q₁₆

—

#334

—

Q₈⋊C₈

—

#322

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

C₂×D₄

yes

characteristic

C₂×Dic₅

#41

yes

characteristic

Dic₁₀

—

Dic₁₀

—

#170

—

#170

—

#170

—

C₅×C₈

M₁₆

yes

—

#168

—

C₂×Q₁₆

—

C₄×Q₈

—

#107

AGL(1,𝔽₅)

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₅

C₄×D₄

yes

characteristic

C₄×C₅

—

C₄×C₅

C₂×M₁₆

yes

characteristic

C₄×C₅

#131

yes

characteristic

Dic₅

—

Dic₅

—

Dic₅

#137

yes

—

C₂×Q₈

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

C₂×AGL(1,𝔽₅)

yes

—

C₄²

—

C₄²

—

C₂×C₅

yes

C₂×C₅

yes

C₂×C₅

yes

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₈

#700

yes

—

C₅

#1593

yes

characteristic, complemented, Sylow

C₂²

D₄×AGL(1,𝔽₅)

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²	D₄×AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#107	C₄
socle, prime radical	C₂²×C₅	C₄×D₄
cosocle	C₄×Dic₅	C₂³
hypercenter	#107	AGL(1,𝔽₅)
hypocenter	C₅	#1593
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

subgroup

G/H

center

C₂²

D₄×AGL(1,𝔽₅)

commutator

C₂×C₄×C₅

C₂²×C₄

Frattini

C₂×C₄

C₂²×AGL(1,𝔽₅)

Fitting

C₅×#107

C₄

socle, prime radical

C₂²×C₅

C₄×D₄

cosocle

C₄×Dic₅

C₂³

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

N	H ≅ G/N	type
88. C₅	#1593	semidirect

H ≅ G/N

type

88. C₅

#1593

semidirect

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

subgroup	index
#65709	2
#65162	2
#64881	2
#64879	2
#64872	2
#1593	5

subgroup

index

Subgroups — #10604069

Conjugacy classes of subgroups

Complements

Maximal subgroups