Group #10598889 — subgroups

Subgroups — #10598889

Order 640.

164 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 61 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	#64971	C₂	1	1	yes	maximal, characteristic
2	#64970	C₂	1	1	yes	maximal, characteristic
3	#64970	C₂	1	1	yes	maximal, characteristic
4	#64970	C₂	1	1	yes	maximal, characteristic
5	#64970	C₂	1	1	yes	maximal, characteristic
6	#64969	C₂	1	1	yes	maximal, characteristic
7	C₅×#322	C₂	1	1	yes	maximal, characteristic
8	#4015	C₂²	1	1	yes	characteristic
9	#4015	C₂²	1	1	yes	characteristic
10	#4015	C₂²	1	1	yes	characteristic
11	#4015	C₂²	1	1	yes	characteristic
12	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	#3950	—	2	2	no	—
16	#3949	—	2	2	no	—
17	#3948	—	2	2	no	—
18	#3948	—	2	2	no	—
19	#3948	—	2	2	no	—
20	#3948	—	2	2	no	—
21	#1718	—	5	1	no	maximal, Sylow
22	C₂×Dic₁₀	—	2	1	no	—
23	C₂×Dic₁₀	—	2	1	no	—
24	C₂×Dic₁₀	—	2	1	no	—
25	C₂×Dic₁₀	—	2	1	no	—
26	C₂×C₅×C₈	D₄	1	2	yes	—
27	C₂×C₅×C₈	D₄	1	2	yes	—
28	C₂×C₅×C₈	D₄	1	2	yes	—
29	C₄²×C₅	C₂³	1	1	yes	characteristic
30	C₄⋊Dic₅	—	2	1	no	—
31	C₄⋊Dic₅	—	2	1	no	—
32	C₄⋊Dic₅	—	2	1	no	—
33	C₄⋊Dic₅	—	2	1	no	—
34	C₄⋊Dic₅	—	2	1	no	—
35	C₄⋊Dic₅	—	2	1	no	—
36	C₄⋊Dic₅	—	2	1	no	—
37	C₄⋊Dic₅	—	2	1	no	—
38	#500	—	5	1	no	—
39	#498	—	5	1	no	—
40	#487	—	5	1	no	—
41	#487	—	5	1	no	—
42	#487	—	5	1	no	—
43	#487	—	5	1	no	—
44	#322	D₅	1	1	yes	characteristic
45	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₅	—	4	1	no	—
49	C₂×Dic₅	—	4	1	no	—
50	C₂×Dic₅	—	4	1	no	—
51	C₂×Dic₅	—	4	1	no	—
52	Dic₁₀	—	4	2	no	—
53	Dic₁₀	—	4	2	no	—
54	Dic₁₀	—	4	2	no	—
55	Dic₁₀	—	4	2	no	—
56	C₅×C₈	—	2	2	no	—
57	C₅×C₈	—	2	2	no	—
58	C₅×C₈	—	2	2	no	—
59	C₄⋊Q₈	—	5	1	no	—
60	C₄⋊Q₈	—	5	1	no	—
61	C₄⋊Q₈	—	5	1	no	—
62	C₄⋊Q₈	—	5	1	no	—
63	#107	—	10	2	no	—
64	#106	—	10	2	no	—
65	#103	—	10	2	no	—
66	#103	—	10	2	no	—
67	#103	—	10	2	no	—
68	#103	—	10	2	no	—
69	C₄×C₈	C₂×D₅	1	1	yes	characteristic
70	C₄×C₈	C₂×D₅	1	1	yes	characteristic
71	C₄×C₈	C₂×D₅	1	1	yes	characteristic
72	C₂²×C₅	#127	1	1	yes	characteristic
73	C₄×C₅	#135	1	2	yes	—
74	C₄×C₅	#137	1	2	yes	—
75	C₄×C₅	#137	1	2	yes	—
76	Dic₅	—	8	1	no	—
77	Dic₅	—	8	1	no	—
78	Dic₅	—	8	1	no	—
79	Dic₅	—	8	1	no	—
80	C₂×Q₈	—	10	1	no	—
81	C₂×Q₈	—	10	1	no	—
82	C₂×Q₈	—	10	1	no	—
83	C₂×Q₈	—	10	1	no	—
84	C₂×C₈	D₂₀	1	2	yes	—
85	C₂×C₈	D₂₀	1	2	yes	—
86	C₂×C₈	D₂₀	1	2	yes	—
87	C₄⋊C₄	—	10	1	no	—
88	C₄⋊C₄	—	10	1	no	—
89	C₄⋊C₄	—	10	1	no	—
90	C₄⋊C₄	—	10	1	no	—
91	C₄⋊C₄	—	10	1	no	—
92	C₄⋊C₄	—	10	1	no	—
93	C₄⋊C₄	—	10	1	no	—
94	C₄⋊C₄	—	10	1	no	—
95	C₄²	C₂²×D₅	1	1	yes	characteristic
96	C₂×C₅	#495	1	1	yes	characteristic
97	C₂×C₅	#497	1	1	yes	characteristic
98	C₂×C₅	#497	1	1	yes	characteristic
99	Q₈	—	20	2	no	—
100	Q₈	—	20	2	no	—
101	Q₈	—	20	2	no	—
102	Q₈	—	20	2	no	—
103	C₂×C₄	—	20	1	no	—
104	C₂×C₄	—	20	1	no	—
105	C₂×C₄	—	20	1	no	—
106	C₂×C₄	—	20	1	no	—
107	C₂×C₄	C₂×D₂₀	1	1	yes	characteristic
108	C₂×C₄	C₂×D₂₀	1	1	yes	characteristic
109	C₂×C₄	C₂×D₂₀	1	1	yes	characteristic
110	C₈	—	2	2	no	—
111	C₈	—	2	2	no	—
112	C₈	—	2	2	no	—
113	C₅	#1718	1	1	yes	characteristic, complemented, Sylow
114	C₂²	#4020	1	1	yes	characteristic
115	C₄	—	40	1	no	—
116	C₄	—	40	1	no	—
117	C₄	—	40	1	no	—
118	C₄	—	40	1	no	—
119	C₄	#4050	1	2	yes	—
120	C₄	#4055	1	2	yes	—
121	C₄	#4055	1	2	yes	—
122	C₂	#64985	1	1	yes	characteristic
123	C₂	#65004	1	1	yes	characteristic
124	C₂	#65004	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#64971

C₂

yes

maximal, characteristic

#64970

C₂

yes

maximal, characteristic

#64970

C₂

yes

maximal, characteristic

#64970

C₂

yes

maximal, characteristic

#64970

C₂

yes

maximal, characteristic

#64969

C₂

yes

maximal, characteristic

C₅×#322

C₂

yes

maximal, characteristic

#4015

C₂²

yes

characteristic

#4015

C₂²

yes

characteristic

#4015

C₂²

yes

characteristic

#4015

C₂²

yes

characteristic

C₄×C₅×C₈

C₂²

yes

characteristic

C₄×C₅×C₈

C₂²

yes

characteristic

C₄×C₅×C₈

C₂²

yes

—

—

—

—

—

—

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

D₄

yes

—

C₂×C₅×C₈

D₄

yes

—

C₂×C₅×C₈

D₄

yes

—

C₄²×C₅

C₂³

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

—

—

—

—

—

—

D₅

yes

characteristic

C₂×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₄×C₅

C₂×D₄

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

—

—

—

—

—

—

C₄×C₈

C₂×D₅

yes

characteristic

C₄×C₈

C₂×D₅

yes

characteristic

C₄×C₈

C₂×D₅

yes

C₂²×C₅

yes

C₄×C₅

yes

—

C₄×C₅

#137

yes

—

C₄×C₅

#137

yes

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×C₈

D₂₀

yes

—

C₂×C₈

D₂₀

yes

—

C₂×C₈

D₂₀

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₅

yes

C₂×C₅

yes

C₂×C₅

yes

C₂×C₅

yes

Q₈

—

100

Q₈

—

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂×D₂₀

yes

characteristic

108

C₂×C₄

C₂×D₂₀

yes

characteristic

109

C₂×C₄

C₂×D₂₀

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

113

C₅

#1718

yes

characteristic, complemented, Sylow

114

C₂²

#4020

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

#4050

yes

—

120

C₄

#4055

yes

—

121

C₄

#4055

yes

—

122

C₂

#64985

yes

characteristic

123

C₂

#65004

yes

characteristic

124

C₂

#65004

yes

characteristic

	subgroup	G/H
center	C₂²	#4020
commutator, cosocle	C₄²×C₅	C₂³
Frattini	C₄²	C₂²×D₅
Fitting	C₅×#322	C₂
socle, prime radical	C₂²×C₅	#127
hypercenter	#322	D₅
hypocenter	C₅	#1718
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#4020

commutator, cosocle

C₄²×C₅

C₂³

Frattini

C₄²

C₂²×D₅

C₂

C₂²×C₅

D₅

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
113. C₅	#1718	semidirect

H ≅ G/N

type

113. C₅

#1718

semidirect

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

subgroup	index
#64971	2
#64970	2
#64969	2
C₅×#322	2
#1718	5

subgroup

index

Subgroups — #10598889

Conjugacy classes of subgroups

Complements

Maximal subgroups