Group #28355987 — subgroups

Subgroups — #28355987

Order 13851.

824 conjugacy classes of subgroups; only the 310 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 310 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	#17987445	C₃	1	6	yes	maximal, complemented
2	#17987439	C₃	1	9	yes	maximal, complemented
3	#17987438	C₃	1	9	yes	maximal, complemented
4	#17987421	C₃	1	3	yes	maximal
5	C₉×#10587025	C₃	1	9	yes	maximal, complemented
6	#17987405	C₃	1	3	yes	maximal, complemented
7	C₃³×C₉×C₁₉	C₃	1	1	yes	maximal, characteristic
8	C₃×#10587025	C₃²	1	9	yes	—
9	#13095307	C₃²	1	9	yes	complemented
10	#13095306	C₃²	1	9	yes	complemented
11	#13095305	C₃²	1	54	yes	complemented
12	C₃²⋊#4294	C₃²	1	9	yes	—
13	C₃²⋊#4294	C₃²	1	18	yes	—
14	C₉×#4294	C₃²	1	9	yes	complemented
15	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
16	C₃²×C₉×C₁₉	—	3	9	no	—
17	C₃²×C₉×C₁₉	C₃²	1	3	yes	—
18	C₃²×C₉×C₁₉	C₃²	1	3	yes	—
19	C₃²×C₉×C₁₉	C₃²	1	3	yes	—
20	C₃²×C₉×C₁₉	C₃²	1	2	yes	—
21	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
22	#10624872	—	19	1	no	maximal, Sylow
23	C₃³×C₁₉	—	3	3	no	—
24	C₃³×C₁₉	—	3	9	no	—
25	C₃³×C₁₉	C₃³	1	1	yes	characteristic
26	C₃³×C₁₉	C₃³	1	3	yes	—
27	C₃×C₉×C₁₉	—	3	6	no	—
28	C₃×C₉×C₁₉	—	3	9	no	—
29	C₃×C₉×C₁₉	—	3	9	no	—
30	C₃×C₉×C₁₉	—	3	9	no	—
31	C₃×C₉×C₁₉	3₋¹⁺²	1	9	yes	complemented
32	C₃×C₉×C₁₉	C₃³	1	6	yes	—
33	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
34	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
35	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
36	#10587025	—	3	27	no	—
37	C₃×#4294	C₃³	1	27	yes	—
38	C₃³×C₉	C₁₉⋊C₃	1	1	yes	characteristic
39	#6895	—	19	6	no	—
40	#6894	—	19	9	no	—
41	#6894	—	19	9	no	—
42	#6891	—	19	3	no	—
43	C₉×3₋¹⁺²	—	19	9	no	—
44	#6887	—	19	3	no	—
45	C₃²×C₁₉	—	3	27	no	—
46	C₃²×C₁₉	—	3	3	no	—
47	C₃²×C₁₉	—	3	3	no	—
48	C₃²×C₁₉	—	3	6	no	—
49	C₃²×C₁₉	C₃×3₋¹⁺²	1	3	yes	—
50	C₃²×C₁₉	#737	1	9	yes	—
51	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
52	C₉×C₁₉	—	3	6	no	—
53	C₉×C₁₉	C₃×3₋¹⁺²	1	3	yes	—
54	C₉×C₁₉	C₃×3₋¹⁺²	1	3	yes	—
55	C₉×C₁₉	C₃×3₋¹⁺²	1	3	yes	—
56	#4294	—	3	27	no	—
57	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
58	C₃×3₋¹⁺²	—	19	9	no	—
59	C₃²×C₉	—	3	9	no	—
60	C₃²×C₉	C₃×C₁₉⋊C₃	1	3	yes	—
61	C₃²×C₉	C₃×C₁₉⋊C₃	1	3	yes	—
62	C₃²×C₉	C₃×C₁₉⋊C₃	1	2	yes	—
63	C₃²×C₉	C₃×C₁₉⋊C₃	1	3	yes	—
64	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
65	C₉⋊C₉	—	19	54	no	—
66	C₉⋊C₉	—	19	9	no	—
67	C₉⋊C₉	—	19	9	no	—
68	C₃²⋊C₉	—	19	9	no	—
69	C₃²⋊C₉	—	19	18	no	—
70	C₉²	—	19	9	no	—
71	C₃×C₁₉	—	3	3	no	—
72	C₃×C₁₉	—	3	9	no	—
73	C₃×C₁₉	C₃²×3₋¹⁺²	1	1	yes	characteristic
74	C₃×C₁₉	C₃×#737	1	3	yes	—
75	C₃³	—	3	3	no	—
76	C₃³	—	3	9	no	—
77	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
78	C₃³	C₃²×C₁₉⋊C₃	1	3	yes	—
79	3₋¹⁺²	—	57	27	no	—
80	C₃×C₉	—	3	6	no	—
81	C₃×C₉	—	3	9	no	—
82	C₃×C₉	—	3	9	no	—
83	C₃×C₉	—	3	9	no	—
84	C₃×C₉	—	19	27	no	—
85	C₃×C₉	#10587025	1	9	yes	complemented
86	C₃×C₉	C₃²×C₁₉⋊C₃	1	6	yes	—
87	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
88	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
89	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
90	C₁₉	#10624872	1	1	yes	characteristic, complemented, Sylow
91	C₃²	—	3	27	no	—
92	C₃²	—	3	3	no	—
93	C₃²	—	3	3	no	—
94	C₃²	—	3	6	no	—
95	C₃²	C₃×#10587025	1	3	yes	—
96	C₃²	#13095332	1	9	yes	—
97	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
98	C₉	—	3	6	no	—
99	C₉	—	57	27	no	—
100	C₉	C₃×#10587025	1	3	yes	—
101	C₉	C₃×#10587025	1	3	yes	—
102	C₉	C₃×#10587025	1	3	yes	—
103	C₃	—	3	3	no	—
104	C₃	—	3	9	no	—
105	C₃	C₃²×#10587025	1	1	yes	characteristic
106	C₃	C₃×#13095332	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17987445

C₃

yes

maximal, complemented

#17987439

C₃

yes

maximal, complemented

#17987438

C₃

yes

maximal, complemented

#17987421

C₃

yes

maximal

C₉×#10587025

C₃

yes

maximal, complemented

#17987405

C₃

yes

maximal, complemented

C₃³×C₉×C₁₉

C₃

yes

maximal, characteristic

C₃×#10587025

C₃²

yes

—

#13095307

C₃²

yes

complemented

#13095306

C₃²

yes

complemented

#13095305

C₃²

yes

complemented

C₃²⋊#4294

C₃²

yes

—

C₃²⋊#4294

C₃²

yes

—

C₉×#4294

C₃²

yes

complemented

C₃⁴×C₁₉

C₃²

yes

characteristic

C₃²×C₉×C₁₉

—

C₃²×C₉×C₁₉

C₃²

yes

—

C₃²×C₉×C₁₉

C₃²

yes

—

C₃²×C₉×C₁₉

C₃²

yes

—

C₃²×C₉×C₁₉

C₃²

yes

—

C₃²×C₉×C₁₉

C₃²

yes

characteristic

#10624872

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

C₃³

yes

characteristic

C₃³×C₁₉

C₃³

yes

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

3₋¹⁺²

yes

complemented

C₃×C₉×C₁₉

C₃³

yes

—

C₃×C₉×C₁₉

C₃³

yes

characteristic

C₃×C₉×C₁₉

C₃³

yes

characteristic

C₃×C₉×C₁₉

C₃³

yes

characteristic

#10587025

—

C₃×#4294

C₃³

yes

—

C₃³×C₉

C₁₉⋊C₃

yes

—

—

—

—

C₉×3₋¹⁺²

—

#6887

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃×3₋¹⁺²

yes

—

C₃²×C₁₉

#737

yes

—

C₃²×C₁₉

C₃⁴

yes

characteristic

C₉×C₁₉

—

C₉×C₁₉

C₃×3₋¹⁺²

yes

—

C₉×C₁₉

C₃×3₋¹⁺²

yes

—

C₉×C₁₉

C₃×3₋¹⁺²

yes

—

#4294

—

C₃⁴

C₃×C₁₉⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃²×C₉

—

C₃²×C₉

C₃×C₁₉⋊C₃

yes

—

C₃²×C₉

C₃×C₁₉⋊C₃

yes

—

C₃²×C₉

C₃×C₁₉⋊C₃

yes

—

C₃²×C₉

C₃×C₁₉⋊C₃

yes

—

C₃²×C₉

C₃×C₁₉⋊C₃

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₉²

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

C₃²×3₋¹⁺²

yes

characteristic

C₃×C₁₉

C₃×#737

yes

—

C₃³

—

C₃³

—

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

C₃³

C₃²×C₁₉⋊C₃

yes

—

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

#10587025

yes

complemented

C₃×C₉

C₃²×C₁₉⋊C₃

yes

—

C₃×C₉

C₃²×C₁₉⋊C₃

yes

characteristic

C₃×C₉

C₃²×C₁₉⋊C₃

yes

characteristic

C₃×C₉

C₃²×C₁₉⋊C₃

yes

characteristic

C₁₉

#10624872

yes

characteristic, complemented, Sylow

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃×#10587025

yes

—

C₃²

#13095332

yes

—

C₃²

C₃³×C₁₉⋊C₃

yes

characteristic

C₉

—

C₉

—

100

C₉

C₃×#10587025

yes

—

101

C₉

C₃×#10587025

yes

—

102

C₉

C₃×#10587025

yes

—

103

C₃

—

104

C₃

—

105

C₃

C₃²×#10587025

yes

characteristic

106

C₃

C₃×#13095332

yes

—

	subgroup	G/H
center	C₃×C₉	C₃²×C₁₉⋊C₃
commutator, socle, cosocle	C₃²×C₁₉	C₃⁴
Frattini	C₃²	C₃³×C₁₉⋊C₃
Fitting	C₃³×C₉×C₁₉	C₃
prime radical	C₃⁴×C₁₉	C₃²
hypercenter	C₃³×C₉	C₁₉⋊C₃
hypocenter	C₁₉	#10624872
LCS	G ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃×C₉ ◁ C₃³×C₉

subgroup

G/H

center

C₃×C₉

C₃²×C₁₉⋊C₃

commutator, socle, cosocle

C₃²×C₁₉

C₃⁴

Frattini

C₃²

C₃³×C₁₉⋊C₃

Fitting

C₃³×C₉×C₁₉

C₃

prime radical

C₃⁴×C₁₉

C₃²

hypercenter

C₃³×C₉

C₁₉⋊C₃

C₁₉

G ▷ C₃²×C₁₉ ▷ C₁₉

C₁ ◁ C₃×C₉ ◁ C₃³×C₉

N	H ≅ G/N	type
1. #17987445	C₃	semidirect
2. #17987439	C₃	semidirect
3. #17987438	C₃	semidirect
5. C₉×#10587025	C₃	semidirect
6. #17987405	C₃	semidirect
9. #13095307	C₃²	semidirect
10. #13095306	C₃²	semidirect
11. #13095305	C₃²	semidirect
14. C₉×#4294	C₃²	semidirect
31. C₃×C₉×C₁₉	3₋¹⁺²	semidirect
85. C₃×C₉	#10587025	semidirect
90. C₁₉	#10624872	semidirect

H ≅ G/N

type

1. #17987445

C₃

semidirect

2. #17987439

C₃

semidirect

3. #17987438

C₃

semidirect

5. C₉×#10587025

C₃

semidirect

6. #17987405

C₃

semidirect

9. #13095307

C₃²

semidirect

10. #13095306

C₃²

semidirect

11. #13095305

C₃²

semidirect

14. C₉×#4294

C₃²

semidirect

31. C₃×C₉×C₁₉

3₋¹⁺²

semidirect

85. C₃×C₉

#10587025

semidirect

90. C₁₉

#10624872

semidirect

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

subgroup	index
#17987445	3
#17987439	3
#17987438	3
#17987421	3
C₉×#10587025	3
#17987405	3
C₃³×C₉×C₁₉	3
#10624872	19

subgroup

index

C₃³×C₉×C₁₉

Subgroups — #28355987

Conjugacy classes of subgroups

Complements

Maximal subgroups