Group #10602598 — subgroups

Subgroups — #10602598

Order 640.

246 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 85 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×#532	C₂	1	1	yes	maximal, characteristic
2	#65214	C₂	1	2	yes	maximal, complemented
3	#64766	C₂	1	2	yes	maximal, complemented
4	#64764	C₂	1	2	yes	maximal, complemented
5	C₂×C₅×#41	C₄	1	2	yes	—
6	C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₅×#127	C₂²	1	1	yes	characteristic
8	C₅×#124	C₄	1	2	yes	—
9	C₅×#121	—	2	2	no	—
10	C₄×C₅×D₄	—	2	2	no	—
11	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₂²⋊#168	—	2	2	no	—
13	C₄⋊#168	C₂²	1	4	yes	—
14	C₄×#168	—	2	2	no	—
15	#1543	—	5	1	no	maximal, Sylow
16	C₅×#41	—	4	4	no	—
17	C₂×C₅×Q₈	C₂×C₄	1	2	yes	—
18	C₂×C₅×D₄	—	2	2	no	—
19	C₂×C₅×D₄	—	2	2	no	—
20	C₂×C₅×D₄	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₅	—	2	2	no	—
22	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₅	D₄	1	2	yes	—
24	C₅×C₄⋊C₄	—	2	2	no	—
25	C₅×C₂²⋊C₄	—	2	4	no	—
26	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₅	D₄	1	2	yes	—
28	C₄²×C₅	C₂³	1	1	yes	characteristic
29	C₂×#168	—	2	4	no	—
30	#532	D₅	1	1	yes	characteristic
31	#431	—	5	2	no	—
32	#333	—	5	2	no	—
33	#331	—	5	2	no	—
34	C₂³×C₅	—	2	2	no	—
35	C₂³×C₅	C₂²⋊C₄	1	1	yes	characteristic
36	C₅×Q₈	—	4	2	no	—
37	C₅×D₄	—	4	2	no	—
38	C₅×D₄	—	4	4	no	—
39	C₅×D₄	—	4	4	no	—
40	C₂×C₄×C₅	—	2	2	no	—
41	C₂×C₄×C₅	—	2	4	no	—
42	C₂×C₄×C₅	—	2	2	no	—
43	C₂×C₄×C₅	—	4	4	no	—
44	C₂×C₄×C₅	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
48	#168	—	4	4	no	—
49	C₂×#41	Dic₅	1	2	yes	—
50	C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
51	#127	C₂×D₅	1	1	yes	characteristic
52	#124	Dic₅	1	2	yes	—
53	#121	—	2	2	no	—
54	C₄×D₄	—	2	2	no	—
55	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
56	C₄⋊C₈	—	5	4	no	—
57	C₂²⋊C₈	—	10	2	no	—
58	C₄×C₈	—	10	2	no	—
59	C₂²×C₅	—	2	1	no	—
60	C₂²×C₅	—	2	2	no	—
61	C₂²×C₅	—	4	2	no	—
62	C₂²×C₅	—	4	4	no	—
63	C₂²×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₄×C₅	—	2	1	no	—
65	C₄×C₅	—	2	2	no	—
66	C₄×C₅	—	4	2	no	—
67	C₄×C₅	#100	1	2	yes	—
68	C₄×C₅	#135	1	4	yes	—
69	#41	—	4	4	no	—
70	C₂×Q₈	C₂×Dic₅	1	2	yes	—
71	C₂×D₄	—	2	2	no	—
72	C₂×D₄	—	2	2	no	—
73	C₂×D₄	C₂×Dic₅	1	2	yes	—
74	C₂²×C₄	—	2	2	no	—
75	C₂²×C₄	C₂×Dic₅	1	1	yes	characteristic
76	C₂²×C₄	C₂²⋊D₅	1	2	yes	—
77	C₂×C₈	—	10	4	no	—
78	C₄⋊C₄	—	2	2	no	—
79	C₂²⋊C₄	—	2	4	no	—
80	C₄²	C₂×Dic₅	1	1	yes	characteristic
81	C₄²	C₂²⋊D₅	1	2	yes	—
82	C₄²	C₂²×D₅	1	1	yes	characteristic
83	C₂×C₅	—	4	1	no	—
84	C₂×C₅	—	8	2	no	—
85	C₂×C₅	C₂×#100	1	1	yes	characteristic
86	C₂×C₅	#416	1	2	yes	—
87	C₂³	—	2	2	no	—
88	C₂³	C₂²⋊Dic₅	1	1	yes	characteristic
89	Q₈	—	4	2	no	—
90	D₄	—	4	2	no	—
91	D₄	—	4	4	no	—
92	D₄	—	4	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	4	4	no	—
97	C₂×C₄	C₂²⋊Dic₅	1	2	yes	—
98	C₂×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
99	C₂×C₄	C₂²×Dic₅	1	1	yes	characteristic
100	C₂×C₄	C₂×C₂²⋊D₅	1	2	yes	—
101	C₈	—	20	4	no	—
102	C₅	#1543	1	1	yes	characteristic, complemented, Sylow
103	C₂²	—	2	1	no	—
104	C₂²	—	2	2	no	—
105	C₂²	—	4	2	no	—
106	C₂²	—	4	4	no	—
107	C₂²	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	2	no	—
110	C₄	—	4	2	no	—
111	C₄	#3965	1	2	yes	—
112	C₄	#4096	1	4	yes	—
113	C₂	—	4	1	no	—
114	C₂	—	8	2	no	—
115	C₂	C₂×#3965	1	1	yes	characteristic
116	C₂	#65522	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#532

C₂

yes

maximal, characteristic

#65214

C₂

yes

maximal, complemented

#64766

C₂

yes

maximal, complemented

#64764

C₂

yes

maximal, complemented

C₂×C₅×#41

C₄

yes

—

C₅×C₄⋊Q₈

C₂²

yes

characteristic

C₅×#127

C₂²

yes

characteristic

C₅×#124

C₄

yes

—

C₅×#121

—

C₄×C₅×D₄

—

C₂×C₄²×C₅

C₂²

yes

characteristic

C₂²⋊#168

—

C₄⋊#168

C₂²

yes

—

C₄×#168

—

#1543

—

maximal, Sylow

C₅×#41

—

C₂×C₅×Q₈

C₂×C₄

yes

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

C₂×C₄

yes

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×C₄

yes

characteristic

C₂²×C₄×C₅

D₄

yes

—

C₅×C₄⋊C₄

—

C₅×C₂²⋊C₄

—

C₄²×C₅

C₂×C₄

yes

characteristic

C₄²×C₅

D₄

yes

—

C₄²×C₅

C₂³

yes

characteristic

C₂×#168

—

#532

D₅

yes

—

—

—

C₂³×C₅

—

C₂³×C₅

C₂²⋊C₄

yes

characteristic

C₅×Q₈

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

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

C₂²×C₄

yes

characteristic

C₂×C₄×C₅

C₂×D₄

yes

—

#168

—

C₂×#41

Dic₅

yes

—

C₄⋊Q₈

C₂×D₅

yes

characteristic

#127

C₂×D₅

yes

characteristic

#124

Dic₅

yes

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×D₅

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#100

yes

—

C₄×C₅

#135

yes

—

#41

—

C₂×Q₈

C₂×Dic₅

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₅

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₅

yes

characteristic

C₂²×C₄

C₂²⋊D₅

yes

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×Dic₅

yes

characteristic

C₄²

C₂²⋊D₅

yes

—

C₄²

C₂²×D₅

yes

characteristic

C₂×C₅

—

C₂×C₅

—

C₂×C₅

C₂×#100

yes

characteristic

C₂×C₅

#416

yes

—

C₂³

—

C₂³

C₂²⋊Dic₅

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₅

yes

—

C₂×C₄

C₂²⋊Dic₅

yes

characteristic

C₂×C₄

C₂²×Dic₅

yes

characteristic

100

C₂×C₄

C₂×C₂²⋊D₅

yes

—

101

C₈

—

102

C₅

#1543

yes

characteristic, complemented, Sylow

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₂×C₂²⋊Dic₅

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#3965

yes

—

112

C₄

#4096

yes

—

113

C₂

—

114

C₂

—

115

C₂

C₂×#3965

yes

characteristic

116

C₂

#65522

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₅
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂²×D₅
Fitting, prime radical	C₅×#532	C₂
socle	C₂²×C₅	C₂×C₂²⋊C₄
cosocle	C₄²×C₅	C₂³
hypercenter	#532	D₅
hypocenter	C₅	#1543
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂×C₄

C₂²⋊Dic₅

commutator

C₂×C₄×C₅

C₂²×C₄

Frattini

C₄²

C₂²×D₅

Fitting, prime radical

C₅×#532

C₂

socle

C₂²×C₅

C₂×C₂²⋊C₄

cosocle

C₄²×C₅

C₂³

D₅

C₅

G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
2. #65214	C₂	semidirect
3. #64766	C₂	semidirect
4. #64764	C₂	semidirect
102. C₅	#1543	semidirect

H ≅ G/N

type

2. #65214

C₂

semidirect

3. #64766

C₂

semidirect

4. #64764

C₂

semidirect

102. C₅

#1543

semidirect

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

subgroup	index
C₅×#532	2
#65214	2
#64766	2
#64764	2
#1543	5

subgroup

index

Subgroups — #10602598

Conjugacy classes of subgroups

Complements

Maximal subgroups