Group #72229565 — subgroups

Subgroups — C₃×#11716763 #72229565

Order 2400.

220 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 58 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	C₃×#87862	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×#1043	C₂	1	1	yes	maximal, characteristic
3	C₃×#87767	C₂	1	1	yes	maximal, characteristic
4	#11716763	C₃	1	1	yes	maximal, characteristic, direct product factor, 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	C₃×C₅×#168	C₄	1	2	yes	complemented
9	C₃×#3992	—	5	2	no	maximal
10	#87862	C₂×C₃	1	1	yes	characteristic
11	C₄×#1043	C₂×C₃	1	1	yes	characteristic
12	#87767	C₂×C₃	1	1	yes	characteristic
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	C₃×C₄×AGL(1,𝔽₅)	—	5	2	no	—
18	C₃×C₄×AGL(1,𝔽₅)	—	10	2	no	—
19	C₃×D₅⋊C₈	—	5	2	no	—
20	C₃×C₈×D₅	—	5	2	no	—
21	C₂×#1043	C₃×C₄	1	2	yes	—
22	C₄×C₅⋊D₅	C₂²×C₃	1	1	yes	characteristic
23	#6034	C₃×C₄	1	2	yes	complemented
24	C₅×#168	C₃×C₄	1	2	yes	complemented
25	#3992	—	5	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×C₅×C₈	—	5	2	no	—
34	C₃×#168	AGL(1,𝔽₅)	1	2	yes	complemented
35	C₂×C₅⋊D₅	C₂×C₃×C₄	1	1	yes	characteristic
36	#1043	—	2	2	no	—
37	C₄×C₅²	C₂×C₃×C₄	1	1	yes	characteristic
38	#1039	C₂×C₃×C₄	1	1	yes	characteristic
39	C₃×#97	—	25	1	no	Hall
40	C₄×AGL(1,𝔽₅)	—	5	2	no	—
41	C₄×AGL(1,𝔽₅)	—	10	2	no	—
42	D₅⋊C₈	—	5	2	no	—
43	C₈×D₅	—	5	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	C₃×C₄²	1	1	yes	characteristic
54	C₅⋊D₅	C₃×M₁₆	1	2	yes	—
55	C₂×C₃×C₈	—	25	1	no	—
56	C₂×C₃×C₈	—	25	1	no	—
57	C₃×C₄²	—	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	—	5	4	no	—
63	C₅×C₈	—	5	2	no	—
64	#168	C₃×AGL(1,𝔽₅)	1	2	yes	complemented
65	#97	—	25	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	C₃×#97	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	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
81	Dic₅	—	5	2	no	—
82	Dic₅	—	10	2	no	—
83	C₂×C₈	—	25	1	no	—
84	C₂×C₈	—	25	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	C₃×C₄×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	—	5	2	no	—
99	C₈	—	25	2	no	—
100	C₂×C₃	—	25	2	no	—
101	C₂×C₃	#87830	1	1	yes	characteristic
102	C₅	—	2	2	no	—
103	C₅	C₃×#3992	1	2	yes	complemented
104	C₂²	—	25	1	no	—
105	C₄	—	25	1	no	—
106	C₄	—	50	2	no	—
107	C₄	C₃×D₅⋊AGL(1,𝔽₅)	1	1	yes	characteristic
108	C₃	#11716763	1	1	yes	characteristic, direct product factor, Sylow
109	C₂	—	25	2	no	—
110	C₂	C₃×#87830	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#87862

C₂

yes

maximal, characteristic

C₃×C₄×#1043

C₂

yes

maximal, characteristic

C₃×#87767

C₂

yes

maximal, characteristic

#11716763

C₃

yes

maximal, characteristic, direct product factor, Hall

C₂×C₃×#1043

C₄

yes

—

C₃×C₄×C₅⋊D₅

C₂²

yes

characteristic

C₃×#6034

C₄

yes

complemented

C₃×C₅×#168

C₄

yes

—

C₂×C₃

yes

characteristic

C₄×#1043

C₂×C₃

yes

characteristic

#87767

C₂×C₃

yes

characteristic

C₂×C₃×C₅⋊D₅

C₂×C₄

yes

characteristic

C₃×#1043

—

C₃×C₄×C₅²

C₂×C₄

yes

characteristic

C₃×#1039

C₂×C₄

yes

characteristic

C₃×C₄×AGL(1,𝔽₅)

—

C₃×C₄×AGL(1,𝔽₅)

—

C₃×D₅⋊C₈

—

C₃×C₈×D₅

—

C₂×#1043

C₃×C₄

yes

—

C₄×C₅⋊D₅

C₂²×C₃

yes

characteristic

#6034

C₃×C₄

yes

complemented

C₅×#168

C₃×C₄

yes

complemented

#3992

—

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

—

C₃×#170

—

C₃×C₅×C₈

—

C₃×#168

AGL(1,𝔽₅)

yes

complemented

C₂×C₅⋊D₅

C₂×C₃×C₄

yes

characteristic

#1043

—

C₄×C₅²

C₂×C₃×C₄

yes

characteristic

#1039

C₂×C₃×C₄

yes

characteristic

C₃×#97

—

Hall

C₄×AGL(1,𝔽₅)

—

C₄×AGL(1,𝔽₅)

—

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

C₃×C₄²

yes

characteristic

C₅⋊D₅

C₃×M₁₆

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄²

—

C₂×AGL(1,𝔽₅)

—

C₂×AGL(1,𝔽₅)

—

C₄×D₅

—

C₄×D₅

—

#170

—

C₅×C₈

—

yes

—

C₂×C₃×C₅

—

C₂×C₃×C₅

C₄×AGL(1,𝔽₅)

yes

—

C₃×D₅

—

C₃×D₅

—

C₅²

C₃×#97

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

C₂×C₃×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

C₃×C₄×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

C₃×#3992

yes

complemented

104

C₂²

—

105

C₄

—

106

C₄

—

107

C₄

C₃×D₅⋊AGL(1,𝔽₅)

yes

characteristic

108

C₃

#11716763

yes

characteristic, direct product factor, Sylow

109

C₂

—

110

C₂

C₃×#87830

yes

characteristic

	subgroup	G/H
center, hypercenter	C₃×C₄	D₅⋊AGL(1,𝔽₅)
commutator	C₂×C₅²	C₃×C₄²
Frattini	C₄	C₃×D₅⋊AGL(1,𝔽₅)
Fitting	C₃×C₄×C₅²	C₂×C₄
socle	C₂×C₃×C₅²	C₄²
prime radical	C₂×C₃×C₅⋊D₅	C₂×C₄
cosocle	C₄×C₅⋊D₅	C₂²×C₃
hypocenter	C₅²	C₃×#97
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₃×C₄

subgroup

G/H

center, hypercenter

C₃×C₄

D₅⋊AGL(1,𝔽₅)

commutator

C₂×C₅²

C₃×C₄²

Frattini

C₄

C₃×D₅⋊AGL(1,𝔽₅)

Fitting

C₃×C₄×C₅²

C₂×C₄

socle

C₂×C₃×C₅²

C₄²

prime radical

C₂×C₃×C₅⋊D₅

C₂×C₄

cosocle

C₄×C₅⋊D₅

C₂²×C₃

hypocenter

C₅²

C₃×#97

LCS

G ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₃×C₄

N	H ≅ G/N	type
4. #11716763	C₃	direct (N × H)
7. C₃×#6034	C₄	semidirect
8. C₃×C₅×#168	C₄	semidirect
23. #6034	C₃×C₄	semidirect
24. C₅×#168	C₃×C₄	semidirect
34. C₃×#168	AGL(1,𝔽₅)	semidirect
44. C₃×C₅²	#97	semidirect
64. #168	C₃×AGL(1,𝔽₅)	semidirect
70. C₅²	C₃×#97	semidirect
87. C₃×C₅	#3992	semidirect
103. C₅	C₃×#3992	semidirect
108. C₃	#11716763	direct (N × H)

H ≅ G/N

type

4. #11716763

C₃

direct (N × H)

7. C₃×#6034

C₄

semidirect

8. C₃×C₅×#168

C₄

semidirect

23. #6034

C₃×C₄

semidirect

24. C₅×#168

C₃×C₄

semidirect

34. C₃×#168

AGL(1,𝔽₅)

semidirect

44. C₃×C₅²

#97

semidirect

64. #168

C₃×AGL(1,𝔽₅)

semidirect

70. C₅²

C₃×#97

semidirect

87. C₃×C₅

#3992

semidirect

103. C₅

C₃×#3992

semidirect

108. C₃

#11716763

direct (N × H)

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

subgroup	index
C₃×#87862	2
C₃×C₄×#1043	2
C₃×#87767	2
#11716763	3
C₃×#3992	5

subgroup

index

C₃×C₄×#1043

Subgroups — C₃×#11716763 #72229565

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#11716763 #72229565

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#11716763 #72229565