Group #10606047 — subgroups

Subgroups — #10606047

Order 640.

630 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 133 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×#4022	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#4020	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#383	C₂	1	1	yes	maximal, characteristic, complemented
4	#64961	C₂	1	6	yes	maximal, complemented
5	#64960	C₂	1	6	yes	maximal, complemented
6	C₂²×D₂₀	C₂²	1	6	yes	—
7	C₂²×D₂₀	C₂²	1	4	yes	—
8	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×#685	C₂²	1	3	yes	—
10	C₂×#685	C₂²	1	12	yes	—
11	C₂×#683	C₂²	1	3	yes	—
12	#4022	—	2	4	no	—
13	#4020	—	2	4	no	—
14	C₅×#95	C₂²	1	6	yes	—
15	#2818	—	5	1	no	maximal, Sylow
16	C₂³×D₅	C₂³	1	1	yes	characteristic
17	C₂³×D₅	C₂³	1	4	yes	—
18	C₂²×C₄×C₅	C₂³	1	4	yes	—
19	C₂²×C₄×C₅	C₂³	1	3	yes	—
20	C₂²×Dic₅	C₂³	1	3	yes	—
21	C₂×D₂₀	—	2	12	no	—
22	C₂×D₂₀	—	4	12	no	—
23	C₂×D₂₀	—	4	12	no	—
24	C₄²×C₅	D₄	1	4	yes	complemented
25	#685	—	2	6	no	—
26	#685	—	4	12	no	—
27	#683	—	2	6	no	—
28	C₂×#127	—	5	1	no	—
29	C₂×#126	—	5	1	no	—
30	#394	—	5	6	no	—
31	#392	—	5	6	no	—
32	#383	D₅	1	1	yes	characteristic, complemented
33	C₂³×C₅	C₂⁴	1	1	yes	characteristic
34	C₂²×D₅	—	2	1	no	—
35	C₂²×D₅	—	2	4	no	—
36	C₂²×D₅	—	2	12	no	—
37	C₂²×D₅	—	2	12	no	—
38	C₂²×D₅	—	2	6	no	—
39	C₂×C₄×C₅	—	2	6	no	—
40	C₂×C₄×C₅	—	2	12	no	—
41	C₂×C₄×C₅	C₂×D₄	1	6	yes	—
42	C₂×Dic₅	—	2	3	no	—
43	C₂×Dic₅	—	2	6	no	—
44	D₂₀	—	4	24	no	—
45	D₂₀	—	4	16	no	—
46	C₂²×D₄	—	5	6	no	—
47	C₂²×D₄	—	5	4	no	—
48	#127	—	10	4	no	—
49	#126	—	10	4	no	—
50	C₂×C₄⋊C₄	—	5	3	no	—
51	C₂×C₂²⋊C₄	—	5	3	no	—
52	C₂×C₂²⋊C₄	—	5	12	no	—
53	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
54	#95	C₂×D₅	1	6	yes	—
55	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
56	C₂²×C₅	H(5,𝔽₂)	1	6	yes	—
57	C₂×D₅	—	4	12	no	—
58	C₂×D₅	—	4	3	no	—
59	C₂×D₅	—	4	4	no	—
60	C₂×D₅	—	4	4	no	—
61	C₂×D₅	—	4	12	no	—
62	C₄×C₅	—	2	6	no	—
63	C₄×C₅	—	4	4	no	—
64	Dic₅	—	4	3	no	—
65	C₂⁴	—	5	1	no	—
66	C₂⁴	—	5	4	no	—
67	C₂×D₄	—	10	12	no	—
68	C₂×D₄	—	20	12	no	—
69	C₂×D₄	—	20	12	no	—
70	C₂²×C₄	—	5	3	no	—
71	C₂²×C₄	C₂²×D₅	1	4	yes	—
72	C₂²×C₄	C₂²×D₅	1	3	yes	—
73	C₄⋊C₄	—	10	6	no	—
74	C₂²⋊C₄	—	10	6	no	—
75	C₂²⋊C₄	—	20	12	no	—
76	C₄²	D₂₀	1	4	yes	complemented
77	C₂×C₅	#535	1	3	yes	—
78	C₂×C₅	#560	1	4	yes	—
79	D₅	—	8	4	no	—
80	D₅	—	8	1	no	—
81	C₂³	—	10	1	no	—
82	C₂³	—	10	4	no	—
83	C₂³	—	10	12	no	—
84	C₂³	—	10	12	no	—
85	C₂³	—	10	6	no	—
86	C₂³	C₂³×D₅	1	1	yes	characteristic
87	D₄	—	20	24	no	—
88	D₄	—	20	16	no	—
89	C₂×C₄	—	2	6	no	—
90	C₂×C₄	—	2	12	no	—
91	C₂×C₄	—	10	3	no	—
92	C₂×C₄	—	10	6	no	—
93	C₂×C₄	C₂×D₂₀	1	6	yes	—
94	C₅	#2818	1	1	yes	characteristic, complemented, Sylow
95	C₂²	—	20	12	no	—
96	C₂²	—	20	3	no	—
97	C₂²	—	20	4	no	—
98	C₂²	—	20	4	no	—
99	C₂²	—	20	12	no	—
100	C₂²	C₂²×D₂₀	1	1	yes	characteristic
101	C₂²	#4149	1	6	yes	—
102	C₄	—	2	6	no	—
103	C₄	—	4	4	no	—
104	C₄	—	20	3	no	—
105	C₂	—	40	4	no	—
106	C₂	—	40	1	no	—
107	C₂	#65859	1	3	yes	—
108	C₂	#66045	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#4022

C₂

yes

maximal, characteristic, complemented

C₂×#4020

C₂

yes

maximal, characteristic, complemented

C₅×#383

C₂

yes

maximal, characteristic, complemented

#64961

C₂

yes

maximal, complemented

#64960

C₂

yes

maximal, complemented

C₂²×D₂₀

C₂²

yes

—

C₂²×D₂₀

C₂²

yes

—

C₂×C₄²×C₅

C₂²

yes

characteristic

C₂×#685

C₂²

yes

—

C₂×#685

C₂²

yes

—

C₂×#683

C₂²

yes

—

#4022

—

#4020

—

C₅×#95

C₂²

yes

—

#2818

—

maximal, Sylow

C₂³×D₅

C₂³

yes

characteristic

C₂³×D₅

C₂³

yes

—

C₂²×C₄×C₅

C₂³

yes

—

C₂²×C₄×C₅

C₂³

yes

—

C₂²×Dic₅

C₂³

yes

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₄²×C₅

D₄

yes

—

—

—

—

—

—

—

D₅

yes

characteristic, complemented

C₂³×C₅

C₂⁴

yes

characteristic

C₂²×D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×D₄

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

D₂₀

—

D₂₀

—

C₂²×D₄

—

C₂²×D₄

—

#127

—

#126

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₅

yes

characteristic

#95

C₂×D₅

yes

—

C₂²×C₅

C₂²×D₄

yes

characteristic

C₂²×C₅

H(5,𝔽₂)

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

Dic₅

—

C₂⁴

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅

yes

—

C₂²×C₄

C₂²×D₅

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

D₂₀

yes

complemented

C₂×C₅

#535

yes

—

C₂×C₅

#560

yes

—

D₅

—

D₅

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂³×D₅

yes

characteristic

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₂₀

yes

—

C₅

#2818

yes

characteristic, complemented, Sylow

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

C₂²×D₂₀

yes

characteristic

101

C₂²

#4149

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₂

—

106

C₂

—

107

C₂

#65859

yes

—

108

C₂

#66045

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅
commutator, socle, cosocle	C₂³×C₅	C₂⁴
Fitting	C₅×#383	C₂
hypercenter	#383	D₅
hypocenter	C₅	#2818
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ #383

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅

commutator, socle, cosocle

C₂³×C₅

C₂⁴

C₂

D₅

C₅

G ▷ C₂³×C₅ ▷ C₅

C₁ ◁ C₂³ ◁ #383

N	H ≅ G/N	type
1. C₂×#4022	C₂	semidirect
2. C₂×#4020	C₂	semidirect
3. C₅×#383	C₂	semidirect
4. #64961	C₂	semidirect
5. #64960	C₂	semidirect
24. C₄²×C₅	D₄	semidirect
32. #383	D₅	semidirect
76. C₄²	D₂₀	semidirect
94. C₅	#2818	semidirect

H ≅ G/N

type

1. C₂×#4022

C₂

semidirect

2. C₂×#4020

C₂

semidirect

3. C₅×#383

C₂

semidirect

4. #64961

C₂

semidirect

5. #64960

C₂

semidirect

24. C₄²×C₅

D₄

semidirect

32. #383

D₅

semidirect

76. C₄²

D₂₀

semidirect

94. C₅

#2818

semidirect

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

subgroup	index
C₂×#4022	2
C₂×#4020	2
C₅×#383	2
#64961	2
#64960	2
#2818	5

subgroup

index

Subgroups — #10606047

Conjugacy classes of subgroups

Complements

Maximal subgroups