Group #10602534 — subgroups

Subgroups — #10602534

Order 640.

260 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 113 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	1	yes	maximal, characteristic
2	#65224	C₂	1	1	yes	maximal, characteristic
3	C₂×#3935	C₂	1	1	yes	maximal, characteristic
4	#64749	C₂	1	4	yes	maximal
5	C₅×#125	—	2	2	no	—
6	C₅×#125	C₄	1	4	yes	complemented
7	C₂×C₅×C₄⋊C₄	—	2	2	no	—
8	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₅	—	2	2	no	—
11	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
12	C₂²×#168	C₂²	1	2	yes	—
13	#3935	—	2	2	no	—
14	#1992	—	5	1	no	maximal, Sylow
15	C₂²×C₄×C₅	—	2	2	no	—
16	C₂²×C₄×C₅	D₄	1	2	yes	—
17	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
18	C₂²×Dic₅	—	2	2	no	—
19	C₅×C₄⋊C₄	—	2	8	no	—
20	C₅×C₄⋊C₄	—	2	2	no	—
21	C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
22	C₄²×C₅	C₂×C₄	1	2	yes	—
23	C₄²×C₅	D₄	1	2	yes	—
24	C₄⋊Dic₅	—	2	4	no	—
25	C₄⋊Dic₅	—	4	4	no	—
26	C₂×#168	—	2	2	no	—
27	C₂×#168	—	2	2	no	—
28	C₂×#168	Q₈	1	4	yes	—
29	C₂×#125	D₅	1	1	yes	characteristic
30	C₂×#97	—	5	1	no	—
31	#384	—	5	1	no	—
32	#340	—	5	4	no	—
33	C₂³×C₅	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₅	—	2	2	no	—
35	C₂×C₄×C₅	—	2	2	no	—
36	C₂×C₄×C₅	—	2	4	no	—
37	C₂×C₄×C₅	—	4	2	no	—
38	C₂×C₄×C₅	—	4	2	no	—
39	C₂×C₄×C₅	C₂²⋊C₄	1	4	yes	—
40	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₅	#41	1	2	yes	—
44	C₂×Dic₅	—	4	2	no	—
45	C₂×Dic₅	—	4	4	no	—
46	#168	—	2	4	no	—
47	C₂²×C₈	—	5	2	no	—
48	#125	—	2	2	no	—
49	#125	Dic₅	1	4	yes	complemented
50	C₂×C₄⋊C₄	—	2	2	no	—
51	C₂×C₄⋊C₄	—	5	2	no	—
52	C₂×C₄⋊C₄	—	10	2	no	—
53	C₂×C₄⋊C₄	C₂×D₅	1	2	yes	—
54	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
55	#97	—	10	2	no	—
56	C₂²×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₅	#124	1	1	yes	characteristic
58	C₂²×C₅	C₄⋊Q₈	1	1	yes	characteristic
59	C₂²×C₅	AGL(1,ℤ/8)	1	2	yes	—
60	C₂²×C₅	#137	1	2	yes	—
61	C₄×C₅	—	2	4	no	—
62	C₄×C₅	—	4	4	no	—
63	C₄×C₅	C₄×Q₈	1	2	yes	—
64	C₄×C₅	C₂²⋊Q₈	1	2	yes	—
65	Dic₅	—	8	2	no	—
66	C₂²×C₄	—	2	2	no	—
67	C₂²×C₄	—	10	2	no	—
68	C₂²×C₄	C₂²⋊D₅	1	2	yes	—
69	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
70	C₂×C₈	—	5	4	no	—
71	C₂×C₈	—	10	2	no	—
72	C₂×C₈	—	10	2	no	—
73	C₄⋊C₄	—	2	8	no	—
74	C₄⋊C₄	—	2	2	no	—
75	C₄⋊C₄	—	10	4	no	—
76	C₄⋊C₄	—	20	4	no	—
77	C₄⋊C₄	C₂×Dic₅	1	4	yes	—
78	C₄²	C₂×Dic₅	1	2	yes	—
79	C₄²	C₂²⋊D₅	1	2	yes	—
80	C₂×C₅	#391	1	1	yes	characteristic
81	C₂×C₅	#417	1	2	yes	—
82	C₂×C₅	#490	1	1	yes	characteristic
83	C₂×C₅	#491	1	1	yes	characteristic
84	C₂×C₅	#501	1	2	yes	—
85	C₂³	C₂×C₂²⋊D₅	1	1	yes	characteristic
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	20	2	no	—
92	C₂×C₄	—	20	4	no	—
93	C₂×C₄	C₂²⋊Dic₅	1	4	yes	—
94	C₂×C₄	Q₈×D₅	1	2	yes	—
95	C₂×C₄	#713	1	2	yes	—
96	C₂×C₄	C₂²×Dic₅	1	1	yes	characteristic
97	C₂×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
98	C₈	—	10	4	no	—
99	C₅	#1992	1	1	yes	characteristic, complemented, Sylow
100	C₂²	#4090	1	1	yes	characteristic
101	C₂²	#4093	1	1	yes	characteristic
102	C₂²	#4095	1	2	yes	—
103	C₂²	#4097	1	2	yes	—
104	C₂²	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
105	C₄	—	2	4	no	—
106	C₄	—	4	4	no	—
107	C₄	—	40	2	no	—
108	C₄	Q₈×Dic₅	1	2	yes	—
109	C₄	#4092	1	2	yes	—
110	C₂	#65354	1	2	yes	—
111	C₂	#65356	1	1	yes	characteristic
112	C₂	#65358	1	1	yes	characteristic
113	C₂	#65519	1	1	yes	characteristic
114	C₂	#65521	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅×#125

C₂

yes

maximal, characteristic

#65224

C₂

yes

maximal, characteristic

C₂×#3935

C₂

yes

maximal, characteristic

#64749

C₂

yes

maximal

C₅×#125

—

C₅×#125

C₄

yes

complemented

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₅

C₂²

yes

characteristic

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

C₂²

yes

—

C₂²×#168

C₂²

yes

—

#3935

—

#1992

—

maximal, Sylow

C₂²×C₄×C₅

—

C₂²×C₄×C₅

D₄

yes

—

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂²×Dic₅

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

C₂×C₄

yes

—

C₄²×C₅

C₂×C₄

yes

—

C₄²×C₅

D₄

yes

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₂×#168

—

C₂×#168

—

C₂×#168

Q₈

yes

—

C₂×#125

D₅

yes

characteristic

C₂×#97

—

#384

—

#340

—

C₂³×C₅

C₂×D₄

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊C₄

yes

—

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₄×C₅

C₂×Q₈

yes

—

C₂×C₄×C₅

#41

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

#168

—

C₂²×C₈

—

#125

—

#125

Dic₅

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅

yes

—

C₂×C₄²

C₂×D₅

yes

characteristic

#97

—

C₂²×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅

#124

yes

characteristic

C₂²×C₅

C₄⋊Q₈

yes

characteristic

C₂²×C₅

AGL(1,ℤ/8)

yes

—

C₂²×C₅

#137

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₄×Q₈

yes

—

C₄×C₅

C₂²⋊Q₈

yes

—

Dic₅

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₅

yes

—

C₂²×C₄

C₂²×D₅

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₅

yes

—

C₄²

C₂×Dic₅

yes

—

C₄²

C₂²⋊D₅

yes

—

C₂×C₅

#391

yes

characteristic

C₂×C₅

#417

yes

—

C₂×C₅

yes

C₂×C₅

yes

C₂×C₅

yes

—

C₂³

C₂×C₂²⋊D₅

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₅

yes

—

C₂×C₄

Q₈×D₅

yes

—

C₂×C₄

#713

yes

—

C₂×C₄

C₂²×Dic₅

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₅

yes

characteristic

C₈

—

C₅

#1992

yes

characteristic, complemented, Sylow

100

C₂²

#4090

yes

characteristic

101

C₂²

#4093

yes

characteristic

102

C₂²

#4095

yes

—

103

C₂²

#4097

yes

—

104

C₂²

C₂×C₂²⋊Dic₅

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

Q₈×Dic₅

yes

—

109

C₄

#4092

yes

—

110

C₂

#65354

yes

—

111

C₂

#65356

yes

characteristic

112

C₂

#65358

yes

characteristic

113

C₂

#65519

yes

characteristic

114

C₂

#65521

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅
Fitting	C₂×C₅×#125	C₂
socle, prime radical	C₂³×C₅	C₂×D₄
cosocle	C₂²×C₄×C₅	C₂³
hypercenter	C₂×#125	D₅
hypocenter	C₅	#1992
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅

commutator

C₂×C₄×C₅

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅

Fitting

C₂×C₅×#125

C₂

socle, prime radical

C₂³×C₅

C₂×D₄

cosocle

C₂²×C₄×C₅

C₂³

D₅

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
6. C₅×#125	C₄	semidirect
49. #125	Dic₅	semidirect
99. C₅	#1992	semidirect

H ≅ G/N

type

6. C₅×#125

C₄

semidirect

49. #125

Dic₅

semidirect

99. C₅

#1992

semidirect

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

subgroup	index
C₂×C₅×#125	2
#65224	2
C₂×#3935	2
#64749	2
#1992	5

subgroup

index

C₂×C₅×#125

Subgroups — #10602534

Conjugacy classes of subgroups

Complements

Maximal subgroups