Group #10616417 — subgroups

Subgroups — C₅×#2822 #10616417

Order 640.

380 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 184 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×C₅×#125	C₂	1	2	yes	maximal
2	C₅×#400	C₂	1	4	yes	maximal
3	C₅×#398	C₂	1	4	yes	maximal
4	C₅×#398	C₂	1	4	yes	maximal
5	C₅×#382	C₂	1	1	yes	maximal, characteristic
6	C₅×#125	—	2	8	no	—
7	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
8	C₂×C₅×C₄⋊C₄	C₂²	1	4	yes	—
9	C₂×C₅×C₄⋊C₄	C₂²	1	4	yes	—
10	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₅×C₄⋊C₄	C₂²	1	4	yes	—
13	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₅×#95	C₂²	1	4	yes	—
15	C₅×#95	C₂²	1	8	yes	—
16	C₅×#95	C₂²	1	4	yes	—
17	#2822	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂²×C₄×C₅	C₂³	1	4	yes	—
19	C₂²×C₄×C₅	C₂³	1	4	yes	—
20	C₂²×C₄×C₅	C₂³	1	4	yes	—
21	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₅	C₂³	1	2	yes	—
23	C₅×C₄⋊C₄	—	2	8	no	—
24	C₅×C₄⋊C₄	—	2	4	no	—
25	C₅×C₄⋊C₄	—	2	8	no	—
26	C₅×C₄⋊C₄	—	2	4	no	—
27	C₅×C₄⋊C₄	—	4	2	no	—
28	C₅×C₄⋊C₄	—	4	4	no	—
29	C₄²×C₅	D₄	1	4	yes	—
30	C₂×#125	C₂×C₅	1	2	yes	—
31	#400	C₂×C₅	1	4	yes	—
32	#398	C₂×C₅	1	4	yes	—
33	#398	C₂×C₅	1	4	yes	—
34	#382	C₂×C₅	1	1	yes	characteristic
35	C₂³×C₅	C₂⁴	1	1	yes	characteristic
36	C₂×C₄×C₅	—	2	1	no	—
37	C₂×C₄×C₅	—	2	4	no	—
38	C₂×C₄×C₅	—	2	1	no	—
39	C₂×C₄×C₅	—	2	4	no	—
40	C₂×C₄×C₅	—	2	4	no	—
41	C₂×C₄×C₅	—	2	4	no	—
42	C₂×C₄×C₅	—	2	4	no	—
43	C₂×C₄×C₅	—	2	8	no	—
44	C₂×C₄×C₅	—	2	4	no	—
45	C₂×C₄×C₅	—	2	8	no	—
46	C₂×C₄×C₅	C₂×D₄	1	4	yes	—
47	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
48	#125	—	2	8	no	—
49	C₂×C₄⋊C₄	C₂²×C₅	1	4	yes	—
50	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
51	C₂×C₄⋊C₄	C₂²×C₅	1	4	yes	—
52	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
53	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
54	C₂×C₄⋊C₄	C₂²×C₅	1	4	yes	—
55	C₂×C₄²	C₂²×C₅	1	1	yes	characteristic
56	#95	C₂²×C₅	1	4	yes	—
57	#95	C₂²×C₅	1	8	yes	—
58	#95	C₂²×C₅	1	4	yes	—
59	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
60	C₂²×C₅	H(5,𝔽₂)	1	1	yes	characteristic
61	C₂²×C₅	2₋¹⁺⁴	1	4	yes	—
62	C₂²×C₅	2₋¹⁺⁴	1	1	yes	characteristic
63	C₄×C₅	—	2	4	no	—
64	C₄×C₅	—	2	2	no	—
65	C₄×C₅	—	4	4	no	—
66	C₄×C₅	—	4	4	no	—
67	C₄×C₅	—	4	4	no	—
68	C₂²×C₄	C₂³×C₅	1	4	yes	—
69	C₂²×C₄	C₂³×C₅	1	4	yes	—
70	C₂²×C₄	C₂³×C₅	1	4	yes	—
71	C₂²×C₄	C₂³×C₅	1	2	yes	—
72	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
73	C₄⋊C₄	—	2	8	no	—
74	C₄⋊C₄	—	2	4	no	—
75	C₄⋊C₄	—	2	8	no	—
76	C₄⋊C₄	—	2	4	no	—
77	C₄⋊C₄	—	4	2	no	—
78	C₄⋊C₄	—	4	4	no	—
79	C₄²	C₅×D₄	1	4	yes	—
80	C₂×C₅	#536	1	2	yes	—
81	C₂×C₅	#537	1	1	yes	characteristic
82	C₂×C₅	#563	1	2	yes	—
83	C₂×C₅	#564	1	2	yes	—
84	C₂³	C₂⁴×C₅	1	1	yes	characteristic
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	2	4	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	2	8	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	2	8	no	—
95	C₂×C₄	C₂×C₅×D₄	1	4	yes	—
96	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
97	C₅	#2822	1	1	yes	characteristic, direct product factor, Sylow
98	C₂²	C₂²×C₅×D₄	1	1	yes	characteristic
99	C₂²	C₅×H(5,𝔽₂)	1	1	yes	characteristic
100	C₂²	C₅×2₋¹⁺⁴	1	4	yes	—
101	C₂²	C₅×2₋¹⁺⁴	1	1	yes	characteristic
102	C₄	—	2	4	no	—
103	C₄	—	2	2	no	—
104	C₄	—	4	4	no	—
105	C₄	—	4	4	no	—
106	C₄	—	4	4	no	—
107	C₂	C₅×#536	1	2	yes	—
108	C₂	C₅×#537	1	1	yes	characteristic
109	C₂	C₅×#563	1	2	yes	—
110	C₂	C₅×#564	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅×#125

C₂

yes

maximal

C₅×#400

C₂

yes

maximal

C₅×#398

C₂

yes

maximal

C₅×#398

C₂

yes

maximal

C₅×#382

C₂

yes

maximal, characteristic

C₅×#125

—

C₂×C₅×C₄⋊C₄

C₂²

yes

—

C₂×C₅×C₄⋊C₄

C₂²

yes

—

C₂×C₅×C₄⋊C₄

C₂²

yes

—

C₂×C₅×C₄⋊C₄

C₂²

yes

—

C₂×C₅×C₄⋊C₄

C₂²

yes

—

C₂×C₅×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₅

C₂²

yes

characteristic

C₅×#95

C₂²

yes

—

C₅×#95

C₂²

yes

—

C₅×#95

C₂²

yes

—

#2822

C₅

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₄×C₅

C₂³

yes

—

C₂²×C₄×C₅

C₂³

yes

—

C₂²×C₄×C₅

C₂³

yes

—

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂²×C₄×C₅

C₂³

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

D₄

yes

—

C₂×#125

C₂×C₅

yes

—

#400

C₂×C₅

yes

—

#398

C₂×C₅

yes

—

#398

C₂×C₅

yes

—

#382

C₂×C₅

yes

characteristic

C₂³×C₅

C₂⁴

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×D₄

yes

—

C₂×C₄×C₅

C₂×D₄

yes

—

#125

—

C₂×C₄⋊C₄

C₂²×C₅

yes

—

C₂×C₄⋊C₄

C₂²×C₅

yes

—

C₂×C₄⋊C₄

C₂²×C₅

yes

—

C₂×C₄⋊C₄

C₂²×C₅

yes

—

C₂×C₄⋊C₄

C₂²×C₅

yes

—

C₂×C₄⋊C₄

C₂²×C₅

yes

—

C₂×C₄²

C₂²×C₅

yes

characteristic

#95

C₂²×C₅

yes

—

#95

C₂²×C₅

yes

—

#95

C₂²×C₅

yes

—

C₂²×C₅

C₂²×D₄

yes

characteristic

C₂²×C₅

H(5,𝔽₂)

yes

characteristic

C₂²×C₅

2₋¹⁺⁴

yes

—

C₂²×C₅

2₋¹⁺⁴

yes

characteristic

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₂²×C₄

C₂³×C₅

yes

—

C₂²×C₄

C₂³×C₅

yes

—

C₂²×C₄

C₂³×C₅

yes

—

C₂²×C₄

C₂³×C₅

yes

—

C₂²×C₄

C₂³×C₅

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₅×D₄

yes

—

C₂×C₅

#536

yes

—

C₂×C₅

#537

yes

characteristic

C₂×C₅

#563

yes

—

C₂×C₅

#564

yes

—

C₂³

C₂⁴×C₅

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₅×D₄

yes

—

C₂×C₄

C₂×C₅×D₄

yes

—

C₅

#2822

yes

characteristic, direct product factor, Sylow

C₂²

C₂²×C₅×D₄

yes

characteristic

C₂²

C₅×H(5,𝔽₂)

yes

characteristic

100

C₂²

C₅×2₋¹⁺⁴

yes

—

101

C₂²

C₅×2₋¹⁺⁴

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₂

C₅×#536

yes

—

108

C₂

C₅×#537

yes

characteristic

109

C₂

C₅×#563

yes

—

110

C₂

C₅×#564

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₅

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₅

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₅ ◁ G

N	H ≅ G/N	type
17. #2822	C₅	direct (N × H)
97. C₅	#2822	direct (N × H)

H ≅ G/N

type

17. #2822

C₅

direct (N × H)

97. C₅

#2822

direct (N × H)

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

subgroup	index
C₂×C₅×#125	2
C₅×#400	2
C₅×#398	2
C₅×#382	2
#2822	5

subgroup

index

C₂×C₅×#125

Subgroups — C₅×#2822 #10616417

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#2822 #10616417

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#2822 #10616417