Group #10598887 — subgroups

Subgroups — #10598887

Order 640.

242 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 69 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	#64984	C₂	1	1	yes	maximal, characteristic, complemented
2	#64984	C₂	1	2	yes	maximal, complemented
3	#64983	C₂	1	2	yes	maximal, complemented
4	#64982	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄₀	—	2	4	no	—
7	C₂×D₄₀	—	2	2	no	—
8	C₂×#677	—	2	2	no	—
9	#4020	C₂²	1	1	yes	characteristic
10	#4020	C₂²	1	2	yes	—
11	#4015	C₂²	1	1	yes	characteristic
12	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₄×C₅×C₈	C₂²	1	2	yes	—
14	#3953	—	2	4	no	—
15	#1721	—	5	1	no	maximal, Sylow
16	C₂×D₂₀	—	2	2	no	—
17	C₂×D₂₀	—	2	1	no	—
18	C₂×D₂₀	—	2	2	no	—
19	C₂×D₂₀	—	2	2	no	—
20	C₂×D₂₀	—	2	2	no	—
21	C₂×Dic₁₀	—	2	1	no	—
22	C₂×C₅×C₈	D₄	1	4	yes	—
23	C₂×C₅×C₈	D₄	1	2	yes	—
24	C₄²×C₅	C₂³	1	1	yes	characteristic
25	C₄⋊Dic₅	—	2	2	no	—
26	D₄₀	—	4	8	no	—
27	D₄₀	—	4	4	no	—
28	#677	—	4	4	no	—
29	#493	—	5	2	no	—
30	#493	—	5	1	no	—
31	#492	—	5	1	no	—
32	#486	—	5	2	no	—
33	#322	D₅	1	1	yes	characteristic, complemented
34	C₂²×D₅	—	4	2	no	—
35	C₂²×D₅	—	4	1	no	—
36	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
38	C₂×Dic₅	—	4	1	no	—
39	D₂₀	—	4	2	no	—
40	D₂₀	—	4	4	no	—
41	D₂₀	—	4	4	no	—
42	D₂₀	—	4	4	no	—
43	D₂₀	—	4	4	no	—
44	Dic₁₀	—	4	2	no	—
45	C₅×C₈	—	2	4	no	—
46	C₅×C₈	D₈	1	4	yes	complemented
47	C₂×SD₁₆	—	10	2	no	—
48	C₂×D₈	—	10	4	no	—
49	C₂×D₈	—	10	2	no	—
50	C₄⋊Q₈	—	5	1	no	—
51	#127	—	5	2	no	—
52	#127	—	5	1	no	—
53	#102	—	10	4	no	—
54	C₄×C₈	C₂×D₅	1	2	yes	—
55	C₄×C₈	C₂×D₅	1	1	yes	characteristic
56	C₂²×C₅	#127	1	1	yes	characteristic
57	C₂×D₅	—	8	1	no	—
58	C₂×D₅	—	8	2	no	—
59	C₂×D₅	—	8	2	no	—
60	C₂×D₅	—	8	2	no	—
61	C₂×D₅	—	8	2	no	—
62	C₄×C₅	C₂×D₈	1	2	yes	—
63	C₄×C₅	AGL(1,ℤ/8)	1	4	yes	—
64	Dic₅	—	8	1	no	—
65	C₂×Q₈	—	10	1	no	—
66	C₂×D₄	—	10	2	no	—
67	C₂×D₄	—	10	2	no	—
68	C₂×D₄	—	10	2	no	—
69	C₂×D₄	—	10	2	no	—
70	C₂×D₄	—	10	1	no	—
71	SD₁₆	—	20	4	no	—
72	D₈	—	20	8	no	—
73	D₈	—	20	4	no	—
74	C₂×C₈	D₂₀	1	4	yes	—
75	C₂×C₈	D₂₀	1	2	yes	—
76	C₄⋊C₄	—	10	2	no	—
77	C₄²	C₂²×D₅	1	1	yes	characteristic
78	C₂×C₅	#493	1	1	yes	characteristic
79	C₂×C₅	#496	1	2	yes	—
80	D₅	—	16	1	no	—
81	D₅	—	16	2	no	—
82	C₂³	—	20	2	no	—
83	C₂³	—	20	1	no	—
84	Q₈	—	20	2	no	—
85	D₄	—	20	4	no	—
86	D₄	—	20	2	no	—
87	D₄	—	20	4	no	—
88	D₄	—	20	4	no	—
89	D₄	—	20	4	no	—
90	C₂×C₄	—	20	1	no	—
91	C₂×C₄	C₂×D₂₀	1	1	yes	characteristic
92	C₂×C₄	C₂×D₂₀	1	2	yes	—
93	C₈	—	2	4	no	—
94	C₈	D₄₀	1	4	yes	complemented
95	C₅	#1721	1	1	yes	characteristic, complemented, Sylow
96	C₂²	—	40	2	no	—
97	C₂²	—	40	1	no	—
98	C₂²	—	40	2	no	—
99	C₂²	—	40	2	no	—
100	C₂²	—	40	2	no	—
101	C₂²	#4020	1	1	yes	characteristic
102	C₄	—	40	1	no	—
103	C₄	C₂×D₄₀	1	2	yes	—
104	C₄	#4054	1	4	yes	—
105	C₂	—	80	2	no	—
106	C₂	—	80	1	no	—
107	C₂	#64984	1	1	yes	characteristic
108	C₂	#65001	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#64984

C₂

yes

maximal, characteristic, complemented

#64984

C₂

yes

maximal, complemented

#64983

C₂

yes

maximal, complemented

#64982

C₂

yes

maximal, characteristic, complemented

C₅×#322

C₂

yes

maximal, characteristic, complemented

C₂×D₄₀

—

C₂×D₄₀

—

C₂×#677

—

#4020

C₂²

yes

characteristic

#4020

C₂²

yes

—

#4015

C₂²

yes

characteristic

C₄×C₅×C₈

C₂²

yes

characteristic

C₄×C₅×C₈

C₂²

yes

—

#3953

—

#1721

—

maximal, Sylow

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

D₄

yes

—

C₂×C₅×C₈

D₄

yes

—

C₄²×C₅

C₂³

yes

characteristic

C₄⋊Dic₅

—

D₄₀

—

D₄₀

—

—

—

—

—

—

D₅

yes

characteristic, complemented

C₂²×D₅

—

C₂²×D₅

—

C₂×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₄×C₅

C₂×D₄

yes

—

C₂×Dic₅

—

D₂₀

—

D₂₀

—

D₂₀

—

D₂₀

—

D₂₀

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

D₈

yes

complemented

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

#127

—

#102

—

C₄×C₈

C₂×D₅

yes

—

C₄×C₈

C₂×D₅

yes

characteristic

C₂²×C₅

#127

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

C₂×D₈

yes

—

C₄×C₅

AGL(1,ℤ/8)

yes

—

Dic₅

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

D₈

—

D₈

—

C₂×C₈

D₂₀

yes

—

C₂×C₈

D₂₀

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₅

yes

C₂×C₅

yes

C₂×C₅

yes

—

D₅

—

D₅

—

C₂³

—

C₂³

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

C₂×D₂₀

yes

characteristic

C₂×C₄

C₂×D₂₀

yes

—

C₈

—

C₈

D₄₀

yes

complemented

C₅

#1721

yes

characteristic, complemented, Sylow

C₂²

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

#4020

yes

characteristic

102

C₄

—

103

C₄

C₂×D₄₀

yes

—

104

C₄

#4054

yes

—

105

C₂

—

106

C₂

—

107

C₂

#64984

yes

characteristic

108

C₂

#65001

yes

—

	subgroup	G/H
center	C₂²	#4020
commutator, cosocle	C₄²×C₅	C₂³
Frattini	C₄²	C₂²×D₅
Fitting	C₅×#322	C₂
socle	C₂²×C₅	#127
hypercenter	#322	D₅
hypocenter	C₅	#1721
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#4020

commutator, cosocle

C₄²×C₅

C₂³

Frattini

C₄²

C₂²×D₅

C₂

C₂²×C₅

D₅

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
1. #64984	C₂	semidirect
2. #64984	C₂	semidirect
3. #64983	C₂	semidirect
4. #64982	C₂	semidirect
5. C₅×#322	C₂	semidirect
33. #322	D₅	semidirect
46. C₅×C₈	D₈	semidirect
94. C₈	D₄₀	semidirect
95. C₅	#1721	semidirect

H ≅ G/N

type

1. #64984

C₂

semidirect

2. #64984

C₂

semidirect

3. #64983

C₂

semidirect

4. #64982

C₂

semidirect

5. C₅×#322

C₂

semidirect

33. #322

D₅

semidirect

46. C₅×C₈

D₈

semidirect

94. C₈

D₄₀

semidirect

95. C₅

#1721

semidirect

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

subgroup	index
#64984	2
#64983	2
#64982	2
C₅×#322	2
#1721	5

subgroup

index

Subgroups — #10598887

Conjugacy classes of subgroups

Complements

Maximal subgroups