Group #72229663 — subgroups

Subgroups — #72229663

Order 2400.

220 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 49 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	#11925534	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×#1043	C₂	1	1	yes	maximal, characteristic
3	#11925150	C₂	1	1	yes	maximal, characteristic
4	#11716763	—	3	1	no	maximal, Hall
5	C₂×C₃×#1043	C₄	1	2	yes	—
6	C₃×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
7	C₃⋊#6034	C₄	1	2	yes	complemented
8	C₅×C₃⋊#168	C₄	1	2	yes	complemented
9	#91149	—	5	2	no	maximal
10	#87862	—	3	1	no	—
11	C₄×#1043	D₃	1	1	yes	characteristic
12	#87767	—	3	1	no	—
13	C₂×C₃×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
14	C₃×#1043	—	2	2	no	—
15	C₃×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
16	C₃×#1039	C₂×C₄	1	1	yes	characteristic
17	D₅⋊(C₃⋊C₈)	—	5	2	no	—
18	C₃×C₄×AGL(1,𝔽₅)	—	5	2	no	—
19	C₃×C₄×AGL(1,𝔽₅)	—	10	2	no	—
20	D₅×C₃⋊C₈	—	5	2	no	—
21	C₂×#1043	Dic₃	1	2	yes	—
22	C₄×C₅⋊D₅	C₂×D₃	1	1	yes	characteristic
23	#6034	—	3	2	no	—
24	C₅×#168	—	3	2	no	—
25	#3992	—	15	2	no	—
26	C₂×C₃×C₅²	C₄²	1	1	yes	characteristic
27	C₃×C₅⋊D₅	M₁₆	1	2	yes	—
28	C₂×C₃×AGL(1,𝔽₅)	—	5	4	no	—
29	C₂×C₃×AGL(1,𝔽₅)	—	10	4	no	—
30	C₃×C₄×D₅	—	5	2	no	—
31	C₃×C₄×D₅	—	10	2	no	—
32	C₃⋊#170	—	5	4	no	—
33	C₃⋊#168	AGL(1,𝔽₅)	1	2	yes	complemented
34	C₅×C₃⋊C₈	—	5	2	no	—
35	C₂×C₅⋊D₅	C₂×Dic₃	1	1	yes	characteristic
36	#1043	—	2	2	no	—
37	C₄×C₅²	C₄×D₃	1	1	yes	characteristic
38	#1039	C₄×D₃	1	1	yes	characteristic
39	#803	—	25	1	no	Hall
40	C₄×AGL(1,𝔽₅)	—	5	2	no	—
41	C₄×AGL(1,𝔽₅)	—	10	2	no	—
42	D₅⋊C₈	—	15	2	no	—
43	C₈×D₅	—	15	2	no	—
44	C₃×C₅²	#97	1	1	yes	characteristic, complemented, Hall
45	C₂×C₃×D₅	—	5	2	no	—
46	C₂×C₃×D₅	—	10	2	no	—
47	C₃×AGL(1,𝔽₅)	—	10	4	no	—
48	C₃×AGL(1,𝔽₅)	—	10	8	no	—
49	C₃×C₄×C₅	—	2	2	no	—
50	C₃×C₄×C₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
51	C₃×Dic₅	—	5	2	no	—
52	C₃×Dic₅	—	10	2	no	—
53	C₂×C₅²	C₄×Dic₃	1	1	yes	characteristic
54	C₅⋊D₅	#208	1	2	yes	—
55	C₃×C₄²	—	25	1	no	—
56	C₂×C₃⋊C₈	—	25	1	no	—
57	C₂×C₃⋊C₈	—	25	1	no	—
58	C₂×AGL(1,𝔽₅)	—	5	4	no	—
59	C₂×AGL(1,𝔽₅)	—	10	4	no	—
60	C₄×D₅	—	5	2	no	—
61	C₄×D₅	—	10	2	no	—
62	#170	—	15	4	no	—
63	C₅×C₈	—	15	2	no	—
64	#168	—	3	2	no	—
65	#97	—	75	1	no	Sylow
66	C₂×C₃×C₅	—	2	2	no	—
67	C₂×C₃×C₅	C₄×AGL(1,𝔽₅)	1	2	yes	—
68	C₃×D₅	—	5	4	no	—
69	C₃×D₅	—	10	4	no	—
70	C₅²	#803	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₃×C₄	—	25	1	no	—
72	C₂×C₃×C₄	—	25	2	no	—
73	C₃⋊C₈	—	5	2	no	—
74	C₃⋊C₈	—	25	2	no	—
75	C₂×D₅	—	5	2	no	—
76	C₂×D₅	—	10	2	no	—
77	AGL(1,𝔽₅)	—	10	4	no	—
78	AGL(1,𝔽₅)	—	10	8	no	—
79	C₄×C₅	—	2	2	no	—
80	C₄×C₅	D₃×AGL(1,𝔽₅)	1	2	yes	—
81	Dic₅	—	5	2	no	—
82	Dic₅	—	10	2	no	—
83	C₂×C₈	—	75	1	no	—
84	C₂×C₈	—	75	1	no	—
85	C₄²	—	25	1	no	—
86	C₃×C₅	—	2	2	no	—
87	C₃×C₅	#3992	1	2	yes	complemented
88	C₂²×C₃	—	25	1	no	—
89	C₃×C₄	—	25	1	no	—
90	C₃×C₄	—	50	2	no	—
91	C₃×C₄	D₅⋊AGL(1,𝔽₅)	1	1	yes	characteristic
92	C₂×C₅	—	2	2	no	—
93	C₂×C₅	Dic₃×AGL(1,𝔽₅)	1	2	yes	—
94	D₅	—	5	4	no	—
95	D₅	—	10	4	no	—
96	C₂×C₄	—	25	1	no	—
97	C₂×C₄	—	25	2	no	—
98	C₈	—	15	2	no	—
99	C₈	—	75	2	no	—
100	C₂×C₃	—	25	2	no	—
101	C₂×C₃	#87830	1	1	yes	characteristic
102	C₅	—	2	2	no	—
103	C₅	#91149	1	2	yes	complemented
104	C₂²	—	25	1	no	—
105	C₄	—	25	1	no	—
106	C₄	—	50	2	no	—
107	C₄	#10597182	1	1	yes	characteristic
108	C₃	#11716763	1	1	yes	characteristic, complemented, Sylow
109	C₂	—	25	2	no	—
110	C₂	#11925354	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11925534

C₂

yes

maximal, characteristic

C₃×C₄×#1043

C₂

yes

maximal, characteristic

#11925150

C₂

yes

maximal, characteristic

#11716763

—

maximal, Hall

C₂×C₃×#1043

C₄

yes

—

C₃×C₄×C₅⋊D₅

C₂²

yes

characteristic

C₃⋊#6034

C₄

yes

complemented

C₅×C₃⋊#168

C₄

yes

—

—

D₃

yes

characteristic

#87767

—

C₂×C₃×C₅⋊D₅

C₂×C₄

yes

characteristic

C₃×#1043

—

C₃×C₄×C₅²

C₂×C₄

yes

characteristic

C₃×#1039

C₂×C₄

yes

characteristic

D₅⋊(C₃⋊C₈)

—

C₃×C₄×AGL(1,𝔽₅)

—

C₃×C₄×AGL(1,𝔽₅)

—

D₅×C₃⋊C₈

—

C₂×#1043

Dic₃

yes

—

C₄×C₅⋊D₅

C₂×D₃

yes

—

—

—

C₂×C₃×C₅²

C₄²

yes

characteristic

C₃×C₅⋊D₅

M₁₆

yes

—

C₂×C₃×AGL(1,𝔽₅)

—

C₂×C₃×AGL(1,𝔽₅)

—

C₃×C₄×D₅

—

C₃×C₄×D₅

—

—

yes

C₅×C₃⋊C₈

—

C₂×C₅⋊D₅

C₂×Dic₃

yes

characteristic

#1043

—

C₄×C₅²

C₄×D₃

yes

characteristic

#1039

C₄×D₃

yes

—

—

—

D₅⋊C₈

—

C₈×D₅

—

C₃×C₅²

#97

yes

characteristic, complemented, Hall

C₂×C₃×D₅

—

C₂×C₃×D₅

—

C₃×AGL(1,𝔽₅)

—

C₃×AGL(1,𝔽₅)

—

C₃×C₄×C₅

—

C₃×C₄×C₅

C₂×AGL(1,𝔽₅)

yes

—

C₃×Dic₅

—

C₃×Dic₅

—

C₂×C₅²

C₄×Dic₃

yes

characteristic

C₅⋊D₅

#208

yes

—

C₃×C₄²

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×AGL(1,𝔽₅)

—

C₂×AGL(1,𝔽₅)

—

C₄×D₅

—

C₄×D₅

—

#170

—

C₅×C₈

—

#168

—

#97

—

Sylow

C₂×C₃×C₅

—

C₂×C₃×C₅

C₄×AGL(1,𝔽₅)

yes

—

C₃×D₅

—

C₃×D₅

—

C₅²

#803

yes

characteristic, complemented, Sylow

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×D₅

—

C₂×D₅

—

AGL(1,𝔽₅)

—

AGL(1,𝔽₅)

—

C₄×C₅

—

C₄×C₅

D₃×AGL(1,𝔽₅)

yes

—

Dic₅

—

Dic₅

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₃×C₅

—

C₃×C₅

#3992

yes

complemented

C₂²×C₃

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

D₅⋊AGL(1,𝔽₅)

yes

characteristic

C₂×C₅

—

C₂×C₅

Dic₃×AGL(1,𝔽₅)

yes

—

D₅

—

D₅

—

C₂×C₄

—

C₂×C₄

—

C₈

—

C₈

—

100

C₂×C₃

—

101

C₂×C₃

#87830

yes

characteristic

102

C₅

—

103

C₅

#91149

yes

complemented

104

C₂²

—

105

C₄

—

106

C₄

—

107

C₄

#10597182

yes

characteristic

108

C₃

#11716763

yes

characteristic, complemented, Sylow

109

C₂

—

110

C₂

#11925354

yes

characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₄	#10597182
commutator, socle	C₂×C₃×C₅²	C₄²
Fitting	C₃×C₄×C₅²	C₂×C₄
prime radical	C₂×C₃×C₅⋊D₅	C₂×C₄
cosocle	C₃×C₄×C₅⋊D₅	C₂²
hypocenter	C₃×C₅²	#97
LCS	G ▷ C₂×C₃×C₅² ▷ C₃×C₅²
UCS	C₁ ◁ C₄

subgroup

G/H

center, Frattini, hypercenter

C₄

#10597182

commutator, socle

C₂×C₃×C₅²

C₄²

Fitting

C₃×C₄×C₅²

C₂×C₄

prime radical

C₂×C₃×C₅⋊D₅

C₂×C₄

cosocle

C₃×C₄×C₅⋊D₅

C₂²

hypocenter

C₃×C₅²

#97

LCS

G ▷ C₂×C₃×C₅² ▷ C₃×C₅²

UCS

C₁ ◁ C₄

N	H ≅ G/N	type
7. C₃⋊#6034	C₄	semidirect
8. C₅×C₃⋊#168	C₄	semidirect
33. C₃⋊#168	AGL(1,𝔽₅)	semidirect
44. C₃×C₅²	#97	semidirect
70. C₅²	#803	semidirect
87. C₃×C₅	#3992	semidirect
103. C₅	#91149	semidirect
108. C₃	#11716763	semidirect

H ≅ G/N

type

7. C₃⋊#6034

C₄

semidirect

8. C₅×C₃⋊#168

C₄

semidirect

33. C₃⋊#168

AGL(1,𝔽₅)

semidirect

44. C₃×C₅²

#97

semidirect

70. C₅²

#803

semidirect

87. C₃×C₅

#3992

semidirect

103. C₅

#91149

semidirect

108. C₃

#11716763

semidirect

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

subgroup	index
#11925534	2
C₃×C₄×#1043	2
#11925150	2
#11716763	3
#91149	5

subgroup

index

C₃×C₄×#1043

Subgroups — #72229663

Conjugacy classes of subgroups

Complements

Maximal subgroups