Group #16814283 — subgroups

Subgroups — C₅×#10624774 #16814283

Order 3645.

326 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 98 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	C₃²×C₅×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
2	C₅×#6895	C₃	1	3	yes	maximal, complemented
3	C₅×#6868	C₃	1	3	yes	maximal, complemented
4	C₅×#6867	C₃	1	3	yes	maximal, complemented
5	C₅×#6866	C₃	1	3	yes	maximal
6	#10624774	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₃⁴×C₅	C₃²	1	1	yes	characteristic
8	C₃×C₅×3₋¹⁺²	—	3	3	no	—
9	C₃×C₅×3₋¹⁺²	—	3	3	no	—
10	C₃×C₅×3₋¹⁺²	—	3	3	no	—
11	C₃×C₅×3₋¹⁺²	—	3	3	no	—
12	C₃²×C₅×C₉	C₃²	1	1	yes	characteristic
13	C₃²×C₅×C₉	C₃²	1	1	yes	characteristic
14	C₃²×C₅×C₉	C₃²	1	1	yes	characteristic
15	C₅×C₉⋊C₉	—	3	9	no	—
16	C₅×C₃²⋊C₉	C₃²	1	9	yes	—
17	C₃²×3₋¹⁺²	C₃×C₅	1	1	yes	characteristic
18	#6895	C₃×C₅	1	3	yes	complemented
19	#6868	C₃×C₅	1	3	yes	complemented
20	#6867	C₃×C₅	1	3	yes	complemented
21	#6866	C₃×C₅	1	3	yes	—
22	C₃³×C₅	—	3	1	no	—
23	C₃³×C₅	—	3	1	no	—
24	C₃³×C₅	—	3	1	no	—
25	C₃³×C₅	—	9	3	no	—
26	C₃³×C₅	3₋¹⁺²	1	3	yes	—
27	C₃³×C₅	C₃³	1	1	yes	characteristic
28	C₅×3₋¹⁺²	—	3	27	no	—
29	C₃×C₅×C₉	—	3	1	no	—
30	C₃×C₅×C₉	—	3	1	no	—
31	C₃×C₅×C₉	—	3	1	no	—
32	C₃×C₅×C₉	—	3	1	no	—
33	C₃×C₅×C₉	—	3	1	no	—
34	C₃×C₅×C₉	—	3	1	no	—
35	C₃×C₅×C₉	—	3	1	no	—
36	C₃×C₅×C₉	—	3	1	no	—
37	C₃×C₅×C₉	—	3	1	no	—
38	C₃×C₅×C₉	—	3	9	no	—
39	C₃×C₅×C₉	H(3,𝔽₃)	1	3	yes	—
40	C₃×C₅×C₉	3₋¹⁺²	1	3	yes	—
41	C₃×C₅×C₉	3₋¹⁺²	1	3	yes	—
42	C₃⁴	C₃²×C₅	1	1	yes	characteristic
43	C₃×3₋¹⁺²	—	3	3	no	—
44	C₃×3₋¹⁺²	—	3	3	no	—
45	C₃×3₋¹⁺²	—	3	3	no	—
46	C₃×3₋¹⁺²	—	3	3	no	—
47	C₃²×C₉	C₃²×C₅	1	1	yes	characteristic
48	C₃²×C₉	C₃²×C₅	1	1	yes	characteristic
49	C₃²×C₉	C₃²×C₅	1	1	yes	characteristic
50	C₉⋊C₉	—	3	9	no	—
51	C₃²⋊C₉	C₃²×C₅	1	9	yes	—
52	C₃²×C₅	—	3	3	no	—
53	C₃²×C₅	—	3	3	no	—
54	C₃²×C₅	—	9	3	no	—
55	C₃²×C₅	—	9	3	no	—
56	C₃²×C₅	—	9	3	no	—
57	C₃²×C₅	—	9	3	no	—
58	C₃²×C₅	C₃×H(3,𝔽₃)	1	1	yes	characteristic
59	C₃²×C₅	C₃×3₋¹⁺²	1	1	yes	characteristic
60	C₃²×C₅	C₃×3₋¹⁺²	1	1	yes	characteristic
61	C₃²×C₅	C₃×3₋¹⁺²	1	1	yes	characteristic
62	C₅×C₉	—	3	3	no	—
63	C₅×C₉	—	3	3	no	—
64	C₅×C₉	—	3	3	no	—
65	C₅×C₉	—	9	9	no	—
66	C₃³	—	3	1	no	—
67	C₃³	—	3	1	no	—
68	C₃³	—	3	1	no	—
69	C₃³	—	9	3	no	—
70	C₃³	C₅×3₋¹⁺²	1	3	yes	—
71	C₃³	C₃³×C₅	1	1	yes	characteristic
72	3₋¹⁺²	—	3	27	no	—
73	C₃×C₉	—	3	1	no	—
74	C₃×C₉	—	3	1	no	—
75	C₃×C₉	—	3	1	no	—
76	C₃×C₉	—	3	1	no	—
77	C₃×C₉	—	3	1	no	—
78	C₃×C₉	—	3	1	no	—
79	C₃×C₉	—	3	1	no	—
80	C₃×C₉	—	3	1	no	—
81	C₃×C₉	—	3	1	no	—
82	C₃×C₉	—	3	9	no	—
83	C₃×C₉	C₅×H(3,𝔽₃)	1	3	yes	—
84	C₃×C₉	C₅×3₋¹⁺²	1	3	yes	—
85	C₃×C₉	C₅×3₋¹⁺²	1	3	yes	—
86	C₃×C₅	—	3	1	no	—
87	C₃×C₅	—	3	1	no	—
88	C₃×C₅	—	3	1	no	—
89	C₃×C₅	—	9	3	no	—
90	C₃×C₅	#6891	1	1	yes	characteristic
91	C₃×C₅	#6913	1	3	yes	—
92	C₃²	—	3	3	no	—
93	C₃²	—	3	3	no	—
94	C₃²	—	9	3	no	—
95	C₃²	—	9	3	no	—
96	C₃²	—	9	3	no	—
97	C₃²	—	9	3	no	—
98	C₃²	C₃×C₅×H(3,𝔽₃)	1	1	yes	characteristic
99	C₃²	C₃×C₅×3₋¹⁺²	1	1	yes	characteristic
100	C₃²	C₃×C₅×3₋¹⁺²	1	1	yes	characteristic
101	C₃²	C₃×C₅×3₋¹⁺²	1	1	yes	characteristic
102	C₉	—	3	3	no	—
103	C₉	—	3	3	no	—
104	C₉	—	3	3	no	—
105	C₉	—	9	9	no	—
106	C₅	#10624774	1	1	yes	characteristic, direct product factor, Sylow
107	C₃	—	3	1	no	—
108	C₃	—	3	1	no	—
109	C₃	—	3	1	no	—
110	C₃	—	9	3	no	—
111	C₃	C₅×#6891	1	1	yes	characteristic
112	C₃	C₅×#6913	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₅×3₋¹⁺²

C₃

yes

maximal, characteristic

C₅×#6895

C₃

yes

maximal, complemented

C₅×#6868

C₃

yes

maximal, complemented

C₅×#6867

C₃

yes

maximal, complemented

C₅×#6866

C₃

yes

maximal

#10624774

C₅

yes

maximal, characteristic, direct product factor, Sylow

C₃⁴×C₅

C₃²

yes

characteristic

C₃×C₅×3₋¹⁺²

—

C₃×C₅×3₋¹⁺²

—

C₃×C₅×3₋¹⁺²

—

C₃×C₅×3₋¹⁺²

—

C₃²×C₅×C₉

C₃²

yes

characteristic

C₃²×C₅×C₉

C₃²

yes

characteristic

C₃²×C₅×C₉

C₃²

yes

characteristic

C₅×C₉⋊C₉

—

C₅×C₃²⋊C₉

C₃²

yes

—

C₃²×3₋¹⁺²

C₃×C₅

yes

characteristic

#6895

C₃×C₅

yes

complemented

#6868

C₃×C₅

yes

complemented

#6867

C₃×C₅

yes

complemented

#6866

C₃×C₅

yes

—

C₃³×C₅

—

C₃³×C₅

—

C₃³×C₅

—

C₃³×C₅

—

C₃³×C₅

3₋¹⁺²

yes

—

C₃³×C₅

C₃³

yes

characteristic

C₅×3₋¹⁺²

—

C₃×C₅×C₉

—

C₃×C₅×C₉

—

C₃×C₅×C₉

—

C₃×C₅×C₉

—

C₃×C₅×C₉

—

C₃×C₅×C₉

—

C₃×C₅×C₉

—

C₃×C₅×C₉

—

C₃×C₅×C₉

—

C₃×C₅×C₉

—

C₃×C₅×C₉

H(3,𝔽₃)

yes

—

C₃×C₅×C₉

3₋¹⁺²

yes

—

C₃×C₅×C₉

3₋¹⁺²

yes

—

C₃⁴

C₃²×C₅

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃²×C₉

C₃²×C₅

yes

characteristic

C₃²×C₉

C₃²×C₅

yes

characteristic

C₃²×C₉

C₃²×C₅

yes

characteristic

C₉⋊C₉

—

C₃²⋊C₉

C₃²×C₅

yes

—

C₃²×C₅

—

C₃²×C₅

—

C₃²×C₅

—

C₃²×C₅

—

C₃²×C₅

—

C₃²×C₅

—

C₃²×C₅

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₅

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₅

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₅

C₃×3₋¹⁺²

yes

characteristic

C₅×C₉

—

C₅×C₉

—

C₅×C₉

—

C₅×C₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₅×3₋¹⁺²

yes

—

C₃³

C₃³×C₅

yes

characteristic

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₅×H(3,𝔽₃)

yes

—

C₃×C₉

C₅×3₋¹⁺²

yes

—

C₃×C₉

C₅×3₋¹⁺²

yes

—

C₃×C₅

—

C₃×C₅

—

C₃×C₅

—

C₃×C₅

—

C₃×C₅

#6891

yes

characteristic

C₃×C₅

#6913

yes

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃×C₅×H(3,𝔽₃)

yes

characteristic

C₃²

C₃×C₅×3₋¹⁺²

yes

characteristic

100

C₃²

C₃×C₅×3₋¹⁺²

yes

characteristic

101

C₃²

C₃×C₅×3₋¹⁺²

yes

characteristic

102

C₉

—

103

C₉

—

104

C₉

—

105

C₉

—

106

C₅

#10624774

yes

characteristic, direct product factor, Sylow

107

C₃

—

108

C₃

—

109

C₃

—

110

C₃

—

111

C₃

C₅×#6891

yes

characteristic

112

C₃

C₅×#6913

yes

—

	subgroup	G/H
center, socle	C₃²×C₅	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₅
prime radical	C₃⁴×C₅	C₃²
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₅ ◁ C₃²×C₅×C₉ ◁ G

subgroup

G/H

center, socle

C₃²×C₅

C₃×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₃³×C₅

prime radical

C₃⁴×C₅

C₃²

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃²×C₅ ◁ C₃²×C₅×C₉ ◁ G

N	H ≅ G/N	type
2. C₅×#6895	C₃	semidirect
3. C₅×#6868	C₃	semidirect
4. C₅×#6867	C₃	semidirect
6. #10624774	C₅	direct (N × H)
18. #6895	C₃×C₅	semidirect
19. #6868	C₃×C₅	semidirect
20. #6867	C₃×C₅	semidirect
106. C₅	#10624774	direct (N × H)

H ≅ G/N

type

2. C₅×#6895

C₃

semidirect

3. C₅×#6868

C₃

semidirect

4. C₅×#6867

C₃

semidirect

6. #10624774

C₅

direct (N × H)

18. #6895

C₃×C₅

semidirect

19. #6868

C₃×C₅

semidirect

20. #6867

C₃×C₅

semidirect

106. C₅

#10624774

direct (N × H)

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

subgroup	index
C₃²×C₅×3₋¹⁺²	3
C₅×#6895	3
C₅×#6868	3
C₅×#6867	3
C₅×#6866	3
#10624774	5

subgroup

index

C₃²×C₅×3₋¹⁺²

Subgroups — C₅×#10624774 #16814283

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#10624774 #16814283

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10624774 #16814283