Group #10608964 — subgroups

Subgroups — #10608964

Order 640.

428 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 125 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×#395	C₂	1	1	yes	maximal, characteristic, complemented
2	#65520	C₂	1	2	yes	maximal, complemented
3	#65280	C₂	1	3	yes	maximal
4	#65266	C₂	1	3	yes	maximal, complemented
5	#64959	C₂	1	6	yes	maximal, complemented
6	C₂²×C₅×Q₈	C₂²	1	1	yes	characteristic
7	C₂²×D₂₀	C₂²	1	1	yes	characteristic
8	C₂×C₅×C₄⋊C₄	C₂²	1	3	yes	—
9	C₂×#685	C₂²	1	6	yes	—
10	C₂×#685	C₂²	1	6	yes	—
11	C₂×C₄⋊Dic₅	C₂²	1	3	yes	—
12	C₂×C₄×Dic₅	C₂²	1	3	yes	—
13	C₅×#95	C₂²	1	3	yes	—
14	#3963	C₂²	1	3	yes	—
15	#3963	C₂²	1	6	yes	—
16	#2785	—	5	1	no	maximal, Sylow
17	C₂³×D₅	C₂³	1	2	yes	—
18	C₂×C₅×Q₈	—	4	3	no	—
19	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₅	C₂³	1	3	yes	—
21	C₂²×C₄×C₅	C₂³	1	3	yes	—
22	C₂²×Dic₅	C₂³	1	6	yes	—
23	C₂×D₂₀	—	4	3	no	—
24	C₅×C₄⋊C₄	—	4	3	no	—
25	#685	—	2	12	no	—
26	#685	—	4	6	no	—
27	C₄⋊Dic₅	—	4	3	no	—
28	C₄×Dic₅	—	2	6	no	—
29	#395	D₅	1	1	yes	characteristic, complemented
30	#394	—	5	3	no	—
31	#393	—	5	2	no	—
32	#388	—	5	6	no	—
33	#382	—	5	3	no	—
34	C₂³×C₅	C₂⁴	1	1	yes	characteristic
35	C₂²×D₅	—	2	6	no	—
36	C₂²×D₅	—	2	6	no	—
37	C₂²×D₅	—	2	2	no	—
38	C₅×Q₈	—	4	4	no	—
39	C₂×C₄×C₅	—	2	3	no	—
40	C₂×C₄×C₅	—	2	3	no	—
41	C₂×C₄×C₅	—	2	6	no	—
42	C₂×C₄×C₅	—	2	3	no	—
43	C₂×C₄×C₅	—	2	3	no	—
44	C₂×C₄×C₅	#41	1	6	yes	—
45	C₂×Dic₅	—	2	6	no	—
46	C₂×Dic₅	—	2	6	no	—
47	C₂×Dic₅	—	2	6	no	—
48	D₂₀	—	4	4	no	—
49	C₂²×Q₈	C₂×D₅	1	1	yes	characteristic
50	C₂²×D₄	—	5	1	no	—
51	C₂×C₄⋊C₄	—	5	3	no	—
52	C₂×C₄⋊C₄	C₂×D₅	1	3	yes	—
53	C₂×C₂²⋊C₄	—	5	6	no	—
54	C₂×C₂²⋊C₄	—	5	6	no	—
55	C₂×C₄²	—	5	3	no	—
56	#95	—	5	3	no	—
57	#95	—	5	6	no	—
58	#95	C₂×D₅	1	3	yes	—
59	C₂²×C₅	C₂×#41	1	3	yes	—
60	C₂²×C₅	H(5,𝔽₂)	1	3	yes	—
61	C₂²×C₅	2₋¹⁺⁴	1	1	yes	characteristic
62	C₂×D₅	—	4	6	no	—
63	C₂×D₅	—	4	6	no	—
64	C₂×D₅	—	4	2	no	—
65	C₄×C₅	—	2	6	no	—
66	C₄×C₅	—	4	1	no	—
67	C₄×C₅	—	4	3	no	—
68	Dic₅	—	4	6	no	—
69	C₂⁴	—	5	2	no	—
70	C₂×Q₈	—	4	3	no	—
71	C₂×D₄	—	20	3	no	—
72	C₂²×C₄	—	5	6	no	—
73	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
74	C₂²×C₄	C₂²×D₅	1	3	yes	—
75	C₂²×C₄	C₂²×D₅	1	3	yes	—
76	C₄⋊C₄	—	4	3	no	—
77	C₄⋊C₄	—	20	3	no	—
78	C₂²⋊C₄	—	10	12	no	—
79	C₂²⋊C₄	—	20	6	no	—
80	C₄²	—	10	6	no	—
81	C₂×C₅	#539	1	3	yes	—
82	C₂×C₅	#555	1	3	yes	—
83	C₂×C₅	H(3,𝔽₄)	1	1	yes	characteristic
84	D₅	—	8	2	no	—
85	C₂³	—	10	6	no	—
86	C₂³	—	10	6	no	—
87	C₂³	—	10	2	no	—
88	C₂³	C₂³×D₅	1	1	yes	characteristic
89	Q₈	—	4	4	no	—
90	D₄	—	20	4	no	—
91	C₂×C₄	—	2	3	no	—
92	C₂×C₄	—	2	3	no	—
93	C₂×C₄	—	2	6	no	—
94	C₂×C₄	—	2	3	no	—
95	C₂×C₄	—	2	3	no	—
96	C₂×C₄	—	10	6	no	—
97	C₂×C₄	—	10	6	no	—
98	C₂×C₄	—	10	6	no	—
99	C₂×C₄	#713	1	6	yes	—
100	C₅	#2785	1	1	yes	characteristic, complemented, Sylow
101	C₂²	—	20	6	no	—
102	C₂²	—	20	2	no	—
103	C₂²	—	20	6	no	—
104	C₂²	#4144	1	3	yes	—
105	C₂²	C₂×#713	1	3	yes	—
106	C₂²	#4150	1	1	yes	characteristic
107	C₄	—	2	6	no	—
108	C₄	—	4	1	no	—
109	C₄	—	4	3	no	—
110	C₄	—	20	6	no	—
111	C₂	—	40	2	no	—
112	C₂	#65928	1	1	yes	characteristic
113	C₂	#65979	1	3	yes	—
114	C₂	#66068	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#395

C₂

yes

maximal, characteristic, complemented

#65520

C₂

yes

maximal, complemented

#65280

C₂

yes

maximal

#65266

C₂

yes

maximal, complemented

#64959

C₂

yes

maximal, complemented

C₂²×C₅×Q₈

C₂²

yes

characteristic

C₂²×D₂₀

C₂²

yes

characteristic

C₂×C₅×C₄⋊C₄

C₂²

yes

—

C₂×#685

C₂²

yes

—

C₂×#685

C₂²

yes

—

C₂×C₄⋊Dic₅

C₂²

yes

—

C₂×C₄×Dic₅

C₂²

yes

—

C₅×#95

C₂²

yes

—

#3963

C₂²

yes

—

#3963

C₂²

yes

—

#2785

—

maximal, Sylow

C₂³×D₅

C₂³

yes

—

C₂×C₅×Q₈

—

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂²×C₄×C₅

C₂³

yes

—

C₂²×C₄×C₅

C₂³

yes

—

C₂²×Dic₅

C₂³

yes

—

C₂×D₂₀

—

C₅×C₄⋊C₄

—

#685

—

#685

—

C₄⋊Dic₅

—

C₄×Dic₅

—

#395

D₅

yes

characteristic, complemented

—

—

—

—

C₂³×C₅

C₂⁴

yes

characteristic

C₂²×D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#41

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

D₂₀

—

C₂²×Q₈

C₂×D₅

yes

characteristic

C₂²×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#95

—

#95

—

#95

C₂×D₅

yes

—

C₂²×C₅

C₂×#41

yes

—

C₂²×C₅

H(5,𝔽₂)

yes

—

C₂²×C₅

2₋¹⁺⁴

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

Dic₅

—

C₂⁴

—

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅

yes

characteristic

C₂²×C₄

C₂²×D₅

yes

—

C₂²×C₄

C₂²×D₅

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂×C₅

#539

yes

—

C₂×C₅

#555

yes

—

C₂×C₅

H(3,𝔽₄)

yes

characteristic

D₅

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂³×D₅

yes

characteristic

Q₈

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#713

yes

—

100

C₅

#2785

yes

characteristic, complemented, Sylow

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

#4144

yes

—

105

C₂²

C₂×#713

yes

—

106

C₂²

#4150

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₂

—

112

C₂

#65928

yes

characteristic

113

C₂

#65979

yes

—

114

C₂

#66068

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅
commutator, socle, cosocle	C₂³×C₅	C₂⁴
Fitting	C₅×#395	C₂
prime radical	C₂²×D₂₀	C₂²
hypercenter	#395	D₅
hypocenter	C₅	#2785
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ #395

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅

commutator, socle, cosocle

C₂³×C₅

C₂⁴

Fitting

C₅×#395

C₂

prime radical

C₂²×D₂₀

C₂²

D₅

C₅

G ▷ C₂³×C₅ ▷ C₅

C₁ ◁ C₂³ ◁ #395

N	H ≅ G/N	type
1. C₅×#395	C₂	semidirect
2. #65520	C₂	semidirect
4. #65266	C₂	semidirect
5. #64959	C₂	semidirect
29. #395	D₅	semidirect
100. C₅	#2785	semidirect

H ≅ G/N

type

1. C₅×#395

C₂

semidirect

2. #65520

C₂

semidirect

4. #65266

C₂

semidirect

5. #64959

C₂

semidirect

29. #395

D₅

semidirect

100. C₅

#2785

semidirect

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

subgroup	index
C₅×#395	2
#65520	2
#65280	2
#65266	2
#64959	2
#2785	5

subgroup

index

Subgroups — #10608964

Conjugacy classes of subgroups

Complements

Maximal subgroups