Group #10609825 — subgroups

Subgroups — #10609825

Order 640.

372 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 117 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×#399	C₂	1	1	yes	maximal, characteristic
2	#65273	C₂	1	6	yes	maximal, complemented
3	#65242	C₂	1	3	yes	maximal
4	#65235	C₂	1	3	yes	maximal
5	C₅⋊#401	C₂	1	2	yes	maximal, complemented
6	C₂³×Dic₅	C₂²	1	1	yes	characteristic
7	C₂×C₅×C₄⋊C₄	C₂²	1	3	yes	—
8	C₂×C₅×C₂²⋊C₄	C₂²	1	3	yes	—
9	C₂×C₂²⋊Dic₅	C₂²	1	3	yes	—
10	C₂×C₄⋊Dic₅	C₂²	1	3	yes	—
11	C₂×#683	C₂²	1	6	yes	—
12	C₅×#95	C₂²	1	1	yes	characteristic
13	#3963	C₂²	1	6	yes	—
14	#3963	C₂²	1	6	yes	—
15	#3963	C₂²	1	3	yes	—
16	#2834	—	5	1	no	maximal, Sylow
17	C₂⁴×C₅	C₂³	1	1	yes	characteristic
18	C₂²×C₄×C₅	C₂³	1	3	yes	—
19	C₂²×C₄×C₅	C₂³	1	3	yes	—
20	C₂²×Dic₅	—	4	3	no	—
21	C₂²×Dic₅	C₂³	1	6	yes	—
22	C₂²×Dic₅	C₂³	1	2	yes	—
23	C₅×C₄⋊C₄	—	4	3	no	—
24	C₅×C₂²⋊C₄	—	4	3	no	—
25	C₂²⋊Dic₅	—	2	6	no	—
26	C₄⋊Dic₅	—	4	3	no	—
27	#683	—	4	6	no	—
28	#401	—	5	2	no	—
29	#399	D₅	1	1	yes	characteristic
30	#398	—	5	6	no	—
31	#397	—	5	3	no	—
32	#385	—	5	3	no	—
33	C₂³×C₅	—	2	3	no	—
34	C₂³×C₅	—	2	3	no	—
35	C₂³×C₅	#41	1	2	yes	—
36	C₂³×C₅	C₂⁴	1	1	yes	characteristic
37	C₂×C₄×C₅	—	2	6	no	—
38	C₂×C₄×C₅	—	2	3	no	—
39	C₂×C₄×C₅	—	2	3	no	—
40	C₂×C₄×C₅	—	2	6	no	—
41	C₂×Dic₅	—	2	6	no	—
42	C₂×Dic₅	—	2	6	no	—
43	C₂×Dic₅	—	2	6	no	—
44	C₂×Dic₅	—	2	6	no	—
45	C₂×Dic₅	—	4	4	no	—
46	C₂³×C₄	—	5	1	no	—
47	C₂×C₄⋊C₄	—	5	6	no	—
48	C₂×C₄⋊C₄	—	5	3	no	—
49	C₂×C₄⋊C₄	C₂×D₅	1	3	yes	—
50	C₂×C₂²⋊C₄	—	5	3	no	—
51	C₂×C₂²⋊C₄	C₂×D₅	1	3	yes	—
52	#95	—	5	6	no	—
53	#95	—	5	3	no	—
54	#95	—	5	6	no	—
55	#95	C₂×D₅	1	1	yes	characteristic
56	C₂²×C₅	—	2	6	no	—
57	C₂²×C₅	—	4	1	no	—
58	C₂²×C₅	—	4	3	no	—
59	C₂²×C₅	C₂×#41	1	1	yes	characteristic
60	C₂²×C₅	H(5,𝔽₂)	1	3	yes	—
61	C₂²×C₅	2₋¹⁺⁴	1	3	yes	—
62	C₄×C₅	—	4	3	no	—
63	C₄×C₅	—	4	3	no	—
64	Dic₅	—	4	6	no	—
65	Dic₅	—	4	2	no	—
66	C₂⁴	C₂²×D₅	1	1	yes	characteristic
67	C₂²×C₄	—	5	6	no	—
68	C₂²×C₄	—	5	2	no	—
69	C₂²×C₄	—	20	3	no	—
70	C₂²×C₄	C₂²×D₅	1	3	yes	—
71	C₂²×C₄	C₂²×D₅	1	3	yes	—
72	C₄⋊C₄	—	4	3	no	—
73	C₄⋊C₄	—	20	6	no	—
74	C₄⋊C₄	—	20	3	no	—
75	C₂²⋊C₄	—	4	3	no	—
76	C₂²⋊C₄	—	10	6	no	—
77	C₂×C₅	—	4	2	no	—
78	C₂×C₅	#539	1	3	yes	—
79	C₂×C₅	#560	1	1	yes	characteristic
80	C₂×C₅	#563	1	3	yes	—
81	C₂³	—	2	3	no	—
82	C₂³	—	2	3	no	—
83	C₂³	#713	1	2	yes	—
84	C₂³	C₂³×D₅	1	1	yes	characteristic
85	C₂×C₄	—	2	3	no	—
86	C₂×C₄	—	2	3	no	—
87	C₂×C₄	—	2	6	no	—
88	C₂×C₄	—	2	6	no	—
89	C₂×C₄	—	10	6	no	—
90	C₂×C₄	—	10	6	no	—
91	C₂×C₄	—	10	6	no	—
92	C₂×C₄	—	10	6	no	—
93	C₂×C₄	—	20	4	no	—
94	C₅	#2834	1	1	yes	characteristic, complemented, Sylow
95	C₂²	—	2	6	no	—
96	C₂²	—	4	1	no	—
97	C₂²	—	4	3	no	—
98	C₂²	#4144	1	3	yes	—
99	C₂²	C₂×#713	1	1	yes	characteristic
100	C₂²	#4150	1	3	yes	—
101	C₄	—	4	3	no	—
102	C₄	—	4	3	no	—
103	C₄	—	20	2	no	—
104	C₄	—	20	6	no	—
105	C₂	—	4	2	no	—
106	C₂	#65979	1	3	yes	—
107	C₂	#65985	1	3	yes	—
108	C₂	#66054	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#399

C₂

yes

maximal, characteristic

#65273

C₂

yes

maximal, complemented

#65242

C₂

yes

maximal

#65235

C₂

yes

maximal

C₅⋊#401

C₂

yes

maximal, complemented

C₂³×Dic₅

C₂²

yes

characteristic

C₂×C₅×C₄⋊C₄

C₂²

yes

—

C₂×C₅×C₂²⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₅

C₂²

yes

—

C₂×C₄⋊Dic₅

C₂²

yes

—

C₂×#683

C₂²

yes

—

C₅×#95

C₂²

yes

characteristic

#3963

C₂²

yes

—

#3963

C₂²

yes

—

#3963

C₂²

yes

—

#2834

—

maximal, Sylow

C₂⁴×C₅

C₂³

yes

characteristic

C₂²×C₄×C₅

C₂³

yes

—

C₂²×C₄×C₅

C₂³

yes

—

C₂²×Dic₅

—

C₂²×Dic₅

C₂³

yes

—

C₂²×Dic₅

C₂³

yes

—

C₅×C₄⋊C₄

—

C₅×C₂²⋊C₄

—

C₂²⋊Dic₅

—

C₄⋊Dic₅

—

#683

—

#401

—

#399

D₅

yes

—

—

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

#41

yes

—

C₂³×C₅

C₂⁴

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂³×C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₅

yes

—

—

—

—

C₂×D₅

yes

characteristic

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂×#41

yes

characteristic

C₂²×C₅

H(5,𝔽₂)

yes

—

C₂²×C₅

2₋¹⁺⁴

yes

—

C₄×C₅

—

C₄×C₅

—

Dic₅

—

Dic₅

—

C₂⁴

C₂²×D₅

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅

yes

—

C₂²×C₄

C₂²×D₅

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂×C₅

—

C₂×C₅

#539

yes

—

C₂×C₅

#560

yes

characteristic

C₂×C₅

#563

yes

—

C₂³

—

C₂³

—

C₂³

#713

yes

—

C₂³

C₂³×D₅

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₅

#2834

yes

characteristic, complemented, Sylow

C₂²

—

C₂²

—

C₂²

—

C₂²

#4144

yes

—

C₂²

C₂×#713

yes

characteristic

100

C₂²

#4150

yes

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₂

—

106

C₂

#65979

yes

—

107

C₂

#65985

yes

—

108

C₂

#66054

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅
commutator, socle, cosocle	C₂³×C₅	C₂⁴
Fitting	C₅×#399	C₂
prime radical	C₂⁴×C₅	C₂³
hypercenter	#399	D₅
hypocenter	C₅	#2834
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ #399

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅

commutator, socle, cosocle

C₂³×C₅

C₂⁴

Fitting

C₅×#399

C₂

prime radical

C₂⁴×C₅

C₂³

D₅

C₅

G ▷ C₂³×C₅ ▷ C₅

C₁ ◁ C₂³ ◁ #399

N	H ≅ G/N	type
2. #65273	C₂	semidirect
5. C₅⋊#401	C₂	semidirect
94. C₅	#2834	semidirect

H ≅ G/N

type

2. #65273

C₂

semidirect

5. C₅⋊#401

C₂

semidirect

94. C₅

#2834

semidirect

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

subgroup	index
C₅×#399	2
#65273	2
#65242	2
#65235	2
C₅⋊#401	2
#2834	5

subgroup

index

Subgroups — #10609825

Conjugacy classes of subgroups

Complements

Maximal subgroups