Group #10602129 — subgroups

Subgroups — #10602129

Order 640.

220 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 65 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	C₅×#498	C₂	1	1	yes	maximal, characteristic, complemented
2	#65377	C₂	1	2	yes	maximal, complemented
3	#65374	C₂	1	2	yes	maximal, complemented
4	#64984	C₂	1	1	yes	maximal, characteristic
5	#64675	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂×#688	—	2	4	no	—
8	C₂×D₄₀	—	2	2	no	—
9	#4020	C₂²	1	1	yes	characteristic
10	#4020	C₂²	1	1	yes	characteristic
11	C₅×#106	—	2	2	no	—
12	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	#3941	—	2	4	no	—
14	C₄×#168	C₂²	1	2	yes	—
15	#1715	—	5	1	no	maximal, Sylow
16	C₂×C₅×Q₈	—	2	2	no	—
17	C₂×D₂₀	—	2	2	no	—
18	C₂×D₂₀	—	2	1	no	—
19	C₂×D₂₀	—	2	1	no	—
20	C₂×D₂₀	—	2	2	no	—
21	C₂×C₅×C₈	D₄	1	2	yes	—
22	C₅×C₄⋊C₄	—	2	2	no	—
23	C₅×C₄⋊C₄	—	2	2	no	—
24	C₄²×C₅	C₂³	1	1	yes	characteristic
25	#688	—	4	8	no	—
26	C₂×#168	D₄	1	4	yes	—
27	D₄₀	—	4	4	no	—
28	#498	D₅	1	1	yes	characteristic, complemented
29	#493	—	5	1	no	—
30	#492	—	5	2	no	—
31	#486	—	5	2	no	—
32	#322	—	5	1	no	—
33	C₂²×D₅	—	4	1	no	—
34	C₂²×D₅	—	4	1	no	—
35	C₅×Q₈	—	4	4	no	—
36	C₂×C₄×C₅	—	4	2	no	—
37	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
38	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
39	D₂₀	—	4	2	no	—
40	D₂₀	—	4	2	no	—
41	D₂₀	—	4	4	no	—
42	D₂₀	—	4	4	no	—
43	C₅×C₈	SD₁₆	1	4	yes	complemented
44	#168	—	2	4	no	—
45	C₂×SD₁₆	—	10	4	no	—
46	C₂×D₈	—	10	2	no	—
47	C₄⋊Q₈	C₂×D₅	1	2	yes	—
48	#127	—	5	1	no	—
49	#127	—	5	1	no	—
50	#106	—	2	2	no	—
51	#102	—	10	4	no	—
52	C₄×C₈	—	5	2	no	—
53	C₄×C₈	C₂×D₅	1	1	yes	characteristic
54	C₂²×C₅	#127	1	1	yes	characteristic
55	C₂×D₅	—	8	1	no	—
56	C₂×D₅	—	8	1	no	—
57	C₂×D₅	—	8	2	no	—
58	C₂×D₅	—	8	2	no	—
59	C₄×C₅	—	8	2	no	—
60	C₄×C₅	C₂×SD₁₆	1	2	yes	—
61	C₄×C₅	AGL(1,ℤ/8)	1	4	yes	—
62	C₂×Q₈	—	2	2	no	—
63	C₂×D₄	—	10	2	no	—
64	C₂×D₄	—	10	2	no	—
65	C₂×D₄	—	10	1	no	—
66	C₂×D₄	—	10	1	no	—
67	SD₁₆	—	20	8	no	—
68	D₈	—	20	4	no	—
69	C₂×C₈	—	5	4	no	—
70	C₂×C₈	C₂²⋊D₅	1	2	yes	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄²	C₂²×D₅	1	1	yes	characteristic
74	C₂×C₅	#492	1	1	yes	characteristic
75	C₂×C₅	#496	1	2	yes	—
76	D₅	—	16	1	no	—
77	D₅	—	16	1	no	—
78	C₂³	—	20	1	no	—
79	C₂³	—	20	1	no	—
80	Q₈	—	4	4	no	—
81	D₄	—	20	4	no	—
82	D₄	—	20	4	no	—
83	D₄	—	20	2	no	—
84	D₄	—	20	2	no	—
85	C₂×C₄	—	4	2	no	—
86	C₂×C₄	D₄×D₅	1	2	yes	—
87	C₂×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
88	C₈	—	10	4	no	—
89	C₈	#688	1	4	yes	complemented
90	C₅	#1715	1	1	yes	characteristic, complemented, Sylow
91	C₂²	—	40	1	no	—
92	C₂²	—	40	1	no	—
93	C₂²	—	40	2	no	—
94	C₂²	—	40	2	no	—
95	C₂²	#4086	1	1	yes	characteristic
96	C₄	—	8	2	no	—
97	C₄	#4060	1	4	yes	—
98	C₄	C₂×#688	1	2	yes	—
99	C₂	—	80	1	no	—
100	C₂	—	80	1	no	—
101	C₂	#65374	1	1	yes	characteristic
102	C₂	#65465	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#498

C₂

yes

maximal, characteristic, complemented

#65377

C₂

yes

maximal, complemented

#65374

C₂

yes

maximal, complemented

#64984

C₂

yes

maximal, characteristic

#64675

C₂

yes

maximal, characteristic, complemented

C₅×C₄⋊Q₈

C₂²

yes

—

C₂×#688

—

C₂×D₄₀

—

#4020

C₂²

yes

characteristic

#4020

C₂²

yes

characteristic

C₅×#106

—

C₄×C₅×C₈

C₂²

yes

characteristic

#3941

—

C₄×#168

C₂²

yes

—

#1715

—

maximal, Sylow

C₂×C₅×Q₈

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×C₅×C₈

D₄

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂³

yes

characteristic

#688

—

C₂×#168

D₄

yes

—

D₄₀

—

#498

D₅

yes

characteristic, complemented

—

—

—

—

C₂²×D₅

—

C₂²×D₅

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×D₄

yes

—

C₂×C₄×C₅

C₂×D₄

yes

characteristic

D₂₀

—

D₂₀

—

D₂₀

—

D₂₀

—

C₅×C₈

SD₁₆

yes

complemented

#168

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

C₂×D₅

yes

—

—

—

—

—

C₄×C₈

—

C₄×C₈

C₂×D₅

yes

characteristic

C₂²×C₅

#127

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

C₂×SD₁₆

yes

—

C₄×C₅

AGL(1,ℤ/8)

yes

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

D₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₅

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₅

yes

C₂×C₅

yes

C₂×C₅

yes

—

D₅

—

D₅

—

C₂³

—

C₂³

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

D₄×D₅

yes

—

C₂×C₄

C₂×C₂²⋊D₅

yes

characteristic

C₈

—

C₈

#688

yes

complemented

C₅

#1715

yes

characteristic, complemented, Sylow

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#4086

yes

characteristic

C₄

—

C₄

#4060

yes

—

C₄

C₂×#688

yes

—

C₂

—

100

C₂

—

101

C₂

#65374

yes

characteristic

102

C₂

#65465

yes

—

	subgroup	G/H
center	C₂²	#4086
commutator, cosocle	C₄²×C₅	C₂³
Frattini	C₄²	C₂²×D₅
Fitting	C₅×#498	C₂
socle	C₂²×C₅	#127
prime radical	#64984	C₂
hypercenter	#498	D₅
hypocenter	C₅	#1715
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#4086

commutator, cosocle

C₄²×C₅

C₂³

Frattini

C₄²

C₂²×D₅

C₂

C₂²×C₅

C₂

D₅

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
1. C₅×#498	C₂	semidirect
2. #65377	C₂	semidirect
3. #65374	C₂	semidirect
5. #64675	C₂	semidirect
28. #498	D₅	semidirect
43. C₅×C₈	SD₁₆	semidirect
89. C₈	#688	semidirect
90. C₅	#1715	semidirect

H ≅ G/N

type

1. C₅×#498

C₂

semidirect

2. #65377

C₂

semidirect

3. #65374

C₂

semidirect

5. #64675

C₂

semidirect

28. #498

D₅

semidirect

43. C₅×C₈

SD₁₆

semidirect

89. C₈

#688

semidirect

90. C₅

#1715

semidirect

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

subgroup	index
C₅×#498	2
#65377	2
#65374	2
#64984	2
#64675	2
#1715	5

subgroup

index

Subgroups — #10602129

Conjugacy classes of subgroups

Complements

Maximal subgroups