Group #10603860 — subgroups

Subgroups — #10603860

Order 640.

220 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 90 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×#3996	C₂	1	2	yes	maximal
2	C₂×#3995	C₂	1	2	yes	maximal
3	#65682	C₂	1	2	yes	maximal
4	C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
5	C₂×#700	C₂²	1	4	yes	—
6	C₂×C₈×D₅	C₂²	1	1	yes	characteristic
7	C₂×C₈×D₅	C₂²	1	1	yes	characteristic
8	C₄²×D₅	C₂²	1	1	yes	characteristic
9	#4004	—	2	4	no	—
10	#3996	—	2	4	no	—
11	#3995	—	2	4	no	—
12	C₄×C₅×C₈	C₄	1	1	yes	characteristic
13	C₈×Dic₅	C₄	1	1	yes	characteristic
14	C₈×Dic₅	C₄	1	1	yes	characteristic
15	C₄×#168	C₄	1	1	yes	characteristic
16	#1848	—	5	1	no	maximal, Sylow
17	C₂×C₄×D₅	Q₈	1	2	yes	—
18	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
19	C₂×#170	—	2	4	no	—
20	#700	—	2	4	no	—
21	#700	—	2	4	no	—
22	C₂×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
25	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
26	C₄×Dic₅	D₄	1	2	yes	—
27	C₂×#168	C₂×C₄	1	1	yes	characteristic
28	C₂×#168	C₂×C₄	1	1	yes	characteristic
29	C₈×D₅	D₄	1	2	yes	—
30	C₈×D₅	D₄	1	2	yes	—
31	C₈×D₅	Q₈	1	2	yes	—
32	C₈×D₅	Q₈	1	2	yes	—
33	C₂×#108	—	5	2	no	—
34	C₂×#108	—	5	2	no	—
35	#423	—	5	2	no	—
36	C₂×C₄×C₈	—	5	1	no	—
37	C₂²×D₅	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₅	C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
40	C₂×Dic₅	C₄⋊C₄	1	2	yes	—
41	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
42	C₄×D₅	—	2	4	no	—
43	C₄×D₅	C₂×D₄	1	1	yes	characteristic
44	C₄×D₅	C₂×D₄	1	1	yes	characteristic
45	C₄×D₅	C₂×Q₈	1	1	yes	characteristic
46	C₄×D₅	C₂×Q₈	1	1	yes	characteristic
47	#170	—	4	4	no	—
48	C₅×C₈	C₄⋊C₄	1	2	yes	—
49	C₅×C₈	C₄⋊C₄	1	2	yes	—
50	#168	C₄⋊C₄	1	2	yes	—
51	#168	C₄⋊C₄	1	2	yes	—
52	C₂×M₁₆	—	5	4	no	—
53	C₂²×C₈	—	5	1	no	—
54	C₂²×C₈	—	5	1	no	—
55	C₂×C₄²	—	5	1	no	—
56	#108	—	10	4	no	—
57	#108	—	10	4	no	—
58	C₄⋊C₈	—	10	4	no	—
59	C₄×C₈	—	5	1	no	—
60	C₄×C₈	—	5	1	no	—
61	C₄×C₈	—	5	1	no	—
62	C₄×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
63	C₂²×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×D₅	—	2	2	no	—
65	C₂×D₅	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×D₅	C₄⋊Q₈	1	1	yes	characteristic
67	C₄×C₅	—	2	2	no	—
68	C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
70	Dic₅	#108	1	4	yes	complemented
71	Dic₅	#127	1	1	yes	characteristic
72	Dic₅	C₄⋊Q₈	1	1	yes	characteristic
73	C₂²×C₄	—	5	1	no	—
74	C₂²×C₄	—	5	2	no	—
75	M₁₆	—	10	4	no	—
76	M₁₆	—	10	4	no	—
77	C₂×C₈	—	5	1	no	—
78	C₂×C₈	—	5	1	no	—
79	C₂×C₈	—	5	2	no	—
80	C₂×C₈	—	5	2	no	—
81	C₂×C₈	—	5	2	no	—
82	C₂×C₈	—	5	2	no	—
83	C₂×C₈	—	10	4	no	—
84	C₂×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
85	C₂×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
86	C₄²	—	5	2	no	—
87	C₄²	—	5	1	no	—
88	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
89	C₂×C₅	#384	1	1	yes	characteristic
90	C₂×C₅	C₂×#108	1	2	yes	—
91	D₅	—	2	1	no	—
92	D₅	—	2	1	no	—
93	C₂³	—	5	1	no	—
94	C₂×C₄	—	5	1	no	—
95	C₂×C₄	—	5	2	no	—
96	C₂×C₄	—	5	1	no	—
97	C₂×C₄	—	5	1	no	—
98	C₂×C₄	—	5	1	no	—
99	C₂×C₄	—	5	1	no	—
100	C₂×C₄	—	10	4	no	—
101	C₂×C₄	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
102	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
103	C₈	—	5	2	no	—
104	C₈	—	5	2	no	—
105	C₈	—	20	4	no	—
106	C₈	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
107	C₈	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
108	C₅	#1848	1	1	yes	characteristic, complemented, Sylow
109	C₂²	—	5	1	no	—
110	C₂²	—	5	1	no	—
111	C₂²	—	10	2	no	—
112	C₂²	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	—	5	1	no	—
115	C₄	—	5	1	no	—
116	C₄	—	5	4	no	—
117	C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
118	C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
119	C₂	—	10	1	no	—
120	C₂	—	10	1	no	—
121	C₂	#65688	1	1	yes	characteristic
122	C₂	D₅⋊#108	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#3996

C₂

yes

maximal

C₂×#3995

C₂

yes

maximal

#65682

C₂

yes

maximal

C₄×C₈×D₅

C₂

yes

maximal, characteristic

C₂×#700

C₂²

yes

—

C₂×C₈×D₅

C₂²

yes

characteristic

C₂×C₈×D₅

C₂²

yes

characteristic

C₄²×D₅

C₂²

yes

—

—

—

C₄×C₅×C₈

C₄

yes

characteristic

C₈×Dic₅

C₄

yes

characteristic

C₈×Dic₅

C₄

yes

characteristic

C₄×#168

C₄

yes

characteristic

#1848

—

maximal, Sylow

C₂×C₄×D₅

Q₈

yes

—

C₂×C₄×D₅

C₂³

yes

—

—

—

C₂×C₅×C₈

C₂×C₄

yes

characteristic

C₂×C₅×C₈

C₂×C₄

yes

characteristic

C₄²×C₅

C₂×C₄

yes

characteristic

C₄×Dic₅

C₂×C₄

yes

characteristic

C₄×Dic₅

D₄

yes

—

C₂×#168

C₂×C₄

yes

characteristic

C₂×#168

C₂×C₄

yes

characteristic

C₈×D₅

D₄

yes

—

C₈×D₅

D₄

yes

—

C₈×D₅

Q₈

yes

—

C₈×D₅

Q₈

yes

—

C₂×#108

—

C₂×#108

—

#423

—

C₂×C₄×C₈

—

C₂²×D₅

C₂×Q₈

yes

characteristic

C₂×C₄×C₅

C₄⋊C₄

yes

—

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×Dic₅

C₄⋊C₄

yes

—

C₂×Dic₅

C₂×D₄

yes

characteristic

C₄×D₅

—

C₄×D₅

C₂×D₄

yes

characteristic

C₄×D₅

C₂×D₄

yes

characteristic

C₄×D₅

C₂×Q₈

yes

characteristic

C₄×D₅

C₂×Q₈

yes

characteristic

#170

—

C₅×C₈

C₄⋊C₄

yes

—

C₅×C₈

C₄⋊C₄

yes

—

#168

C₄⋊C₄

yes

—

#168

C₄⋊C₄

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

#108

—

#108

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

AGL(1,𝔽₅)

yes

characteristic

C₂²×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×D₅

—

C₂×D₅

C₄⋊Q₈

yes

characteristic

C₂×D₅

C₄⋊Q₈

yes

characteristic

C₄×C₅

—

C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅

C₂×C₄⋊C₄

yes

Dic₅

yes

Dic₅

yes

Dic₅

C₄⋊Q₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

C₄²

—

C₄²

yes

C₂×C₅

yes

C₂×C₅

yes

—

D₅

—

D₅

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₄⋊AGL(1,𝔽₅)

yes

—

102

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

103

C₈

—

104

C₈

—

105

C₈

—

106

C₈

C₄⋊AGL(1,𝔽₅)

yes

—

107

C₈

C₄⋊AGL(1,𝔽₅)

yes

—

108

C₅

#1848

yes

characteristic, complemented, Sylow

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

118

C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#65688

yes

characteristic

122

C₂

D₅⋊#108

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₄×C₅×C₈	C₄
socle	C₂²×C₅	C₂×C₄⋊C₄
prime radical	C₂²×D₅	C₂×Q₈
cosocle	C₂×C₄×D₅	C₂³
hypercenter	C₄×C₈	AGL(1,𝔽₅)
hypocenter	C₅	#1848
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊AGL(1,𝔽₅)

commutator

C₂×C₄×C₅

C₂²×C₄

Frattini

C₂×C₄

C₂²×AGL(1,𝔽₅)

Fitting

C₄×C₅×C₈

C₄

socle

C₂²×C₅

C₂×C₄⋊C₄

prime radical

C₂²×D₅

C₂×Q₈

cosocle

C₂×C₄×D₅

C₂³

C₄×C₈

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
70. Dic₅	#108	semidirect
108. C₅	#1848	semidirect

H ≅ G/N

type

70. Dic₅

#108

semidirect

108. C₅

#1848

semidirect

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

subgroup	index
C₂×#3996	2
C₂×#3995	2
#65682	2
C₄×C₈×D₅	2
#1848	5

subgroup

index

C₄×C₈×D₅

Subgroups — #10603860

Conjugacy classes of subgroups

Complements

Maximal subgroups