Group #10610147 — subgroups

Subgroups — #10610147

Order 640.

352 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 123 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×#401	C₂	1	1	yes	maximal, characteristic
2	C₅⋊#401	C₂	1	1	yes	maximal, characteristic
3	#64949	C₂	1	3	yes	maximal
4	#64948	C₂	1	3	yes	maximal
5	#64937	C₂	1	1	yes	maximal, characteristic
6	#64936	C₂	1	3	yes	maximal
7	#64935	C₂	1	3	yes	maximal
8	C₂²×Dic₁₀	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₅	C₂²	1	3	yes	—
10	C₂×#683	C₂²	1	3	yes	—
11	C₂×#683	C₂²	1	3	yes	—
12	C₂×C₄×Dic₅	C₂²	1	3	yes	—
13	C₅×#95	C₂²	1	3	yes	—
14	C₅×#95	C₂²	1	3	yes	—
15	C₅×#95	C₂²	1	1	yes	characteristic
16	#3963	C₂²	1	3	yes	—
17	#3963	C₂²	1	3	yes	—
18	#3963	C₂²	1	3	yes	—
19	#3963	C₂²	1	3	yes	—
20	#3963	C₂²	1	3	yes	—
21	#2645	—	5	1	no	maximal, Sylow
22	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₅	C₂³	1	3	yes	—
24	C₂²×C₄×C₅	C₂³	1	3	yes	—
25	C₂²×Dic₅	C₂³	1	1	yes	characteristic
26	C₂²×Dic₅	C₂³	1	3	yes	—
27	C₂²×Dic₅	C₂³	1	3	yes	—
28	C₂²×Dic₅	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₀	—	2	6	no	—
30	C₄⋊Dic₅	—	4	3	no	—
31	#683	—	2	6	no	—
32	#683	—	4	3	no	—
33	C₄×Dic₅	—	2	6	no	—
34	#401	—	5	1	no	—
35	#401	D₅	1	1	yes	characteristic
36	#400	—	5	3	no	—
37	#400	—	5	3	no	—
38	#391	—	5	3	no	—
39	#389	—	5	3	no	—
40	#376	—	5	1	no	—
41	C₂³×C₅	C₂⁴	1	1	yes	characteristic
42	C₂×C₄×C₅	—	2	3	no	—
43	C₂×C₄×C₅	—	2	3	no	—
44	C₂×C₄×C₅	—	2	3	no	—
45	C₂×C₄×C₅	—	2	3	no	—
46	C₂×C₄×C₅	—	2	6	no	—
47	C₂×C₄×C₅	—	2	3	no	—
48	C₂×Dic₅	—	2	3	no	—
49	C₂×Dic₅	—	2	3	no	—
50	C₂×Dic₅	—	2	3	no	—
51	C₂×Dic₅	—	2	3	no	—
52	C₂×Dic₅	—	2	3	no	—
53	C₂×Dic₅	—	2	3	no	—
54	C₂×Dic₅	—	2	3	no	—
55	C₂×Dic₅	#41	1	6	yes	—
56	Dic₁₀	—	4	4	no	—
57	C₂²×Q₈	—	5	1	no	—
58	C₂×C₄⋊C₄	—	5	3	no	—
59	C₂×C₄⋊C₄	—	5	3	no	—
60	C₂×C₄⋊C₄	—	5	3	no	—
61	C₂×C₄²	—	5	3	no	—
62	#95	—	5	3	no	—
63	#95	—	5	3	no	—
64	#95	—	5	3	no	—
65	#95	—	5	3	no	—
66	#95	—	5	3	no	—
67	#95	C₂×D₅	1	1	yes	characteristic
68	#95	C₂×D₅	1	3	yes	—
69	#95	C₂×D₅	1	3	yes	—
70	C₂²×C₅	C₂×#41	1	3	yes	—
71	C₂²×C₅	H(5,𝔽₂)	1	1	yes	characteristic
72	C₂²×C₅	2₋¹⁺⁴	1	3	yes	—
73	C₄×C₅	—	4	1	no	—
74	C₄×C₅	—	4	3	no	—
75	C₄×C₅	—	4	3	no	—
76	Dic₅	—	4	1	no	—
77	Dic₅	—	4	3	no	—
78	Dic₅	—	4	3	no	—
79	Dic₅	#126	1	4	yes	—
80	C₂×Q₈	—	10	6	no	—
81	C₂²×C₄	—	5	1	no	—
82	C₂²×C₄	—	5	1	no	—
83	C₂²×C₄	—	5	3	no	—
84	C₂²×C₄	—	5	3	no	—
85	C₂²×C₄	C₂²×D₅	1	3	yes	—
86	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₅	1	3	yes	—
88	C₄⋊C₄	—	10	6	no	—
89	C₄⋊C₄	—	20	3	no	—
90	C₄⋊C₄	—	20	3	no	—
91	C₄²	—	10	6	no	—
92	C₂×C₅	C₂×#126	1	1	yes	characteristic
93	C₂×C₅	#541	1	3	yes	—
94	C₂×C₅	#542	1	3	yes	—
95	C₂³	C₂³×D₅	1	1	yes	characteristic
96	Q₈	—	20	4	no	—
97	C₂×C₄	—	2	3	no	—
98	C₂×C₄	—	2	3	no	—
99	C₂×C₄	—	2	3	no	—
100	C₂×C₄	—	2	3	no	—
101	C₂×C₄	—	2	6	no	—
102	C₂×C₄	—	2	3	no	—
103	C₂×C₄	—	5	6	no	—
104	C₂×C₄	—	10	3	no	—
105	C₂×C₄	—	10	3	no	—
106	C₂×C₄	—	10	3	no	—
107	C₂×C₄	—	10	3	no	—
108	C₂×C₄	—	10	3	no	—
109	C₂×C₄	—	10	3	no	—
110	C₂×C₄	—	10	3	no	—
111	C₅	#2645	1	1	yes	characteristic, complemented, Sylow
112	C₂²	D₅×#41	1	3	yes	—
113	C₂²	#4149	1	1	yes	characteristic
114	C₂²	#4150	1	3	yes	—
115	C₄	—	4	3	no	—
116	C₄	—	4	1	no	—
117	C₄	—	4	3	no	—
118	C₄	—	5	4	no	—
119	C₄	—	20	1	no	—
120	C₄	—	20	3	no	—
121	C₄	—	20	3	no	—
122	C₂	#66035	1	3	yes	—
123	C₂	D₅×#126	1	1	yes	characteristic
124	C₂	#66044	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#401

C₂

yes

maximal, characteristic

C₅⋊#401

C₂

yes

maximal, characteristic

#64949

C₂

yes

maximal

#64948

C₂

yes

maximal

#64937

C₂

yes

maximal, characteristic

#64936

C₂

yes

maximal

#64935

C₂

yes

maximal

C₂²×Dic₁₀

C₂²

yes

characteristic

C₂×C₄⋊Dic₅

C₂²

yes

—

C₂×#683

C₂²

yes

—

C₂×#683

C₂²

yes

—

C₂×C₄×Dic₅

C₂²

yes

—

C₅×#95

C₂²

yes

—

C₅×#95

C₂²

yes

—

C₅×#95

C₂²

yes

characteristic

#3963

C₂²

yes

—

#3963

C₂²

yes

—

#3963

C₂²

yes

—

#3963

C₂²

yes

—

#3963

C₂²

yes

—

#2645

—

maximal, Sylow

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂²×C₄×C₅

C₂³

yes

—

C₂²×C₄×C₅

C₂³

yes

—

C₂²×Dic₅

C₂³

yes

characteristic

C₂²×Dic₅

C₂³

yes

—

C₂²×Dic₅

C₂³

yes

—

C₂²×Dic₅

C₂³

yes

characteristic

C₂×Dic₁₀

—

C₄⋊Dic₅

—

#683

—

#683

—

C₄×Dic₅

—

#401

—

#401

D₅

yes

—

—

—

—

—

C₂³×C₅

C₂⁴

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

#41

yes

—

Dic₁₀

—

C₂²×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

—

—

—

—

—

C₂×D₅

yes

characteristic

#95

C₂×D₅

yes

—

#95

C₂×D₅

yes

—

C₂²×C₅

C₂×#41

yes

—

C₂²×C₅

H(5,𝔽₂)

yes

characteristic

C₂²×C₅

2₋¹⁺⁴

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

#126

yes

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅

yes

—

C₂²×C₄

C₂²×D₅

yes

characteristic

C₂²×C₄

C₂²×D₅

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₂×C₅

C₂×#126

yes

characteristic

C₂×C₅

#541

yes

—

C₂×C₅

#542

yes

—

C₂³

C₂³×D₅

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₅

#2645

yes

characteristic, complemented, Sylow

112

C₂²

D₅×#41

yes

—

113

C₂²

#4149

yes

characteristic

114

C₂²

#4150

yes

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₂

#66035

yes

—

123

C₂

D₅×#126

yes

characteristic

124

C₂

#66044

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅
commutator, socle, prime radical, cosocle	C₂³×C₅	C₂⁴
Fitting	C₅×#401	C₂
hypercenter	#401	D₅
hypocenter	C₅	#2645
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ #401

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 ◁ #401

N	H ≅ G/N	type
111. C₅	#2645	semidirect

H ≅ G/N

type

111. C₅

#2645

semidirect

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

subgroup	index
C₅×#401	2
C₅⋊#401	2
#64949	2
#64948	2
#64937	2
#64936	2
#64935	2
#2645	5

subgroup

index

Subgroups — #10610147

Conjugacy classes of subgroups

Complements

Maximal subgroups