Group #10605948 — subgroups

Subgroups — #10605948

Order 640.

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

normal subgroups only

This group has 380 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×#4016	C₂	1	2	yes	maximal
2	C₅×#382	C₂	1	1	yes	maximal, characteristic
3	#65272	C₂	1	4	yes	maximal
4	#64950	C₂	1	4	yes	maximal
5	#64948	C₂	1	4	yes	maximal
6	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
7	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₅	C₂²	1	4	yes	—
11	C₂×#683	C₂²	1	4	yes	—
12	C₂×#683	C₂²	1	4	yes	—
13	#4016	—	2	8	no	—
14	C₅×#95	C₂²	1	4	yes	—
15	#3963	C₂²	1	4	yes	—
16	#3963	C₂²	1	8	yes	—
17	#2822	—	5	1	no	maximal, Sylow
18	C₂²×C₄×C₅	C₂³	1	4	yes	—
19	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₅	C₂³	1	2	yes	—
21	C₂²×Dic₅	C₂³	1	4	yes	—
22	C₂²×Dic₅	C₂³	1	4	yes	—
23	C₅×C₄⋊C₄	—	4	2	no	—
24	C₄²×C₅	D₄	1	4	yes	—
25	C₄⋊Dic₅	—	2	4	no	—
26	C₄⋊Dic₅	—	2	4	no	—
27	C₄⋊Dic₅	—	4	4	no	—
28	#683	—	2	8	no	—
29	#683	—	2	8	no	—
30	C₂×#125	—	5	2	no	—
31	#400	—	5	4	no	—
32	#398	—	5	4	no	—
33	#398	—	5	4	no	—
34	#382	D₅	1	1	yes	characteristic
35	C₂³×C₅	C₂⁴	1	1	yes	characteristic
36	C₂×C₄×C₅	—	2	1	no	—
37	C₂×C₄×C₅	—	2	1	no	—
38	C₂×C₄×C₅	—	2	4	no	—
39	C₂×C₄×C₅	—	2	4	no	—
40	C₂×C₄×C₅	—	2	8	no	—
41	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
42	C₂×C₄×C₅	C₂×D₄	1	4	yes	—
43	C₂×Dic₅	—	2	4	no	—
44	C₂×Dic₅	—	2	4	no	—
45	C₂×Dic₅	—	2	4	no	—
46	C₂×Dic₅	—	2	4	no	—
47	C₂×Dic₅	—	2	8	no	—
48	#125	—	10	8	no	—
49	C₂×C₄⋊C₄	—	5	4	no	—
50	C₂×C₄⋊C₄	—	5	2	no	—
51	C₂×C₄⋊C₄	—	5	4	no	—
52	C₂×C₄⋊C₄	—	5	2	no	—
53	C₂×C₄⋊C₄	—	5	4	no	—
54	C₂×C₄⋊C₄	C₂×D₅	1	2	yes	—
55	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
56	#95	—	5	8	no	—
57	#95	—	5	4	no	—
58	#95	C₂×D₅	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	1	yes	characteristic
62	C₂²×C₅	2₋¹⁺⁴	1	4	yes	—
63	C₄×C₅	—	2	4	no	—
64	C₄×C₅	—	2	2	no	—
65	C₄×C₅	—	4	4	no	—
66	Dic₅	—	4	4	no	—
67	Dic₅	—	4	4	no	—
68	C₂²×C₄	—	5	4	no	—
69	C₂²×C₄	—	5	4	no	—
70	C₂²×C₄	C₂²×D₅	1	4	yes	—
71	C₂²×C₄	C₂²×D₅	1	2	yes	—
72	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
73	C₄⋊C₄	—	4	2	no	—
74	C₄⋊C₄	—	10	8	no	—
75	C₄⋊C₄	—	10	8	no	—
76	C₄⋊C₄	—	10	4	no	—
77	C₄⋊C₄	—	10	4	no	—
78	C₄⋊C₄	—	20	4	no	—
79	C₄²	D₂₀	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	C₂³×D₅	1	1	yes	characteristic
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	—	2	8	no	—
90	C₂×C₄	—	10	4	no	—
91	C₂×C₄	—	10	4	no	—
92	C₂×C₄	—	10	4	no	—
93	C₂×C₄	—	10	4	no	—
94	C₂×C₄	—	10	8	no	—
95	C₂×C₄	C₂×D₂₀	1	2	yes	—
96	C₂×C₄	C₂×D₂₀	1	4	yes	—
97	C₅	#2822	1	1	yes	characteristic, complemented, Sylow
98	C₂²	C₂²×D₂₀	1	1	yes	characteristic
99	C₂²	#4144	1	1	yes	characteristic
100	C₂²	#4147	1	1	yes	characteristic
101	C₂²	#4150	1	4	yes	—
102	C₄	—	2	2	no	—
103	C₄	—	2	4	no	—
104	C₄	—	4	4	no	—
105	C₄	—	20	4	no	—
106	C₄	—	20	4	no	—
107	C₂	#65844	1	1	yes	characteristic
108	C₂	#65860	1	2	yes	—
109	C₂	#66006	1	2	yes	—
110	C₂	C₅⋊#564	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#4016

C₂

yes

maximal

C₅×#382

C₂

yes

maximal, characteristic

#65272

C₂

yes

maximal

#64950

C₂

yes

maximal

#64948

C₂

yes

maximal

C₂×C₅×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₅

C₂²

yes

characteristic

C₂×C₄⋊Dic₅

C₂²

yes

—

C₂×C₄⋊Dic₅

C₂²

yes

—

C₂×C₄⋊Dic₅

C₂²

yes

—

C₂×#683

C₂²

yes

—

C₂×#683

C₂²

yes

—

#4016

—

C₅×#95

C₂²

yes

—

#3963

C₂²

yes

—

#3963

C₂²

yes

—

#2822

—

maximal, Sylow

C₂²×C₄×C₅

C₂³

yes

—

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂²×C₄×C₅

C₂³

yes

—

C₂²×Dic₅

C₂³

yes

—

C₂²×Dic₅

C₂³

yes

—

C₅×C₄⋊C₄

—

C₄²×C₅

D₄

yes

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

—

—

—

—

—

—

D₅

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×D₄

yes

—

C₂×C₄×C₅

C₂×D₄

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅

yes

—

C₂×C₄²

C₂×D₅

yes

—

—

C₂×D₅

yes

—

C₂²×C₅

C₂²×D₄

yes

characteristic

C₂²×C₅

H(5,𝔽₂)

yes

characteristic

C₂²×C₅

2₋¹⁺⁴

yes

characteristic

C₂²×C₅

2₋¹⁺⁴

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

Dic₅

—

Dic₅

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅

yes

—

C₂²×C₄

C₂²×D₅

yes

—

C₂²×C₄

C₂²×D₅

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

D₂₀

yes

—

C₂×C₅

#536

yes

—

C₂×C₅

#537

yes

characteristic

C₂×C₅

#563

yes

—

C₂×C₅

#564

yes

—

C₂³

C₂³×D₅

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×D₂₀

yes

—

C₂×C₄

C₂×D₂₀

yes

—

C₅

#2822

yes

characteristic, complemented, Sylow

C₂²

C₂²×D₂₀

yes

characteristic

C₂²

#4144

yes

characteristic

100

C₂²

#4147

yes

characteristic

101

C₂²

#4150

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₂

#65844

yes

characteristic

108

C₂

#65860

yes

—

109

C₂

#66006

yes

—

110

C₂

C₅⋊#564

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅
commutator, socle, prime radical, cosocle	C₂³×C₅	C₂⁴
Fitting	C₅×#382	C₂
hypercenter	#382	D₅
hypocenter	C₅	#2822
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅

commutator, socle, prime radical, cosocle

C₂³×C₅

C₂⁴

C₂

D₅

C₅

G ▷ C₂³×C₅ ▷ C₅

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
97. C₅	#2822	semidirect

H ≅ G/N

type

97. C₅

#2822

semidirect

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

subgroup	index
C₂×#4016	2
C₅×#382	2
#65272	2
#64950	2
#64948	2
#2822	5

subgroup

index

Subgroups — #10605948

Conjugacy classes of subgroups

Complements

Maximal subgroups