Group #28354659 — subgroups

Subgroups — #28354659

Order 13851.

362 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 130 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	#17987405	C₃	1	3	yes	maximal, complemented
2	#17987339	C₃	1	3	yes	maximal, complemented
3	#17987337	C₃	1	3	yes	maximal, complemented
4	#17987335	C₃	1	3	yes	maximal
5	C₃²×C₁₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
6	#13095307	—	3	3	no	—
7	#13095306	—	3	3	no	—
8	C₃²⋊#4294	C₃²	1	9	yes	—
9	C₉×#4294	—	3	3	no	—
10	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
11	C₃×C₁₉×3₋¹⁺²	—	3	3	no	—
12	C₃×C₁₉×3₋¹⁺²	—	3	6	no	—
13	C₃×C₁₉×3₋¹⁺²	C₉	1	9	yes	complemented
14	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
15	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
16	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
17	#10624692	—	19	1	no	maximal, Sylow
18	C₃³×C₁₉	—	3	1	no	—
19	C₃³×C₁₉	—	3	2	no	—
20	C₃³×C₁₉	—	9	3	no	—
21	C₃³×C₁₉	C₃×C₉	1	3	yes	—
22	C₃³×C₁₉	C₃³	1	1	yes	characteristic
23	C₁₉×3₋¹⁺²	—	3	27	no	—
24	C₃×C₉×C₁₉	—	3	1	no	—
25	C₃×C₉×C₁₉	—	3	1	no	—
26	C₃×C₉×C₁₉	—	3	2	no	—
27	C₃×C₉×C₁₉	—	3	2	no	—
28	C₃×C₉×C₁₉	C₃×C₉	1	3	yes	—
29	C₃×C₉×C₁₉	C₃×C₉	1	3	yes	—
30	C₃×C₉×C₁₉	C₃×C₉	1	3	yes	—
31	C₃×C₉×C₁₉	H(3,𝔽₃)	1	3	yes	—
32	C₃×C₉×C₁₉	3₋¹⁺²	1	6	yes	—
33	C₃×#4294	—	3	9	no	—
34	C₃²×3₋¹⁺²	C₁₉⋊C₃	1	1	yes	characteristic
35	#6887	—	19	3	no	—
36	#6868	—	19	3	no	—
37	#6867	—	19	3	no	—
38	#6866	—	19	3	no	—
39	C₃²×C₁₉	—	3	3	no	—
40	C₃²×C₁₉	—	3	3	no	—
41	C₃²×C₁₉	—	9	3	no	—
42	C₃²×C₁₉	—	9	6	no	—
43	C₃²×C₁₉	—	9	3	no	—
44	C₃²×C₁₉	C₃²×C₉	1	1	yes	characteristic
45	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
46	C₃²×C₁₉	C₃×3₋¹⁺²	1	2	yes	—
47	C₉×C₁₉	—	3	3	no	—
48	C₉×C₁₉	—	3	3	no	—
49	C₉×C₁₉	C₃²⋊C₉	1	9	yes	—
50	#4294	—	9	9	no	—
51	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
52	C₃×3₋¹⁺²	—	3	6	no	—
53	C₃×3₋¹⁺²	—	3	3	no	—
54	C₃×3₋¹⁺²	#4294	1	9	yes	complemented
55	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
56	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
57	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
58	C₉⋊C₉	—	57	3	no	—
59	C₉⋊C₉	—	57	3	no	—
60	C₃²⋊C₉	—	19	9	no	—
61	C₉²	—	57	3	no	—
62	C₃×C₁₉	—	3	1	no	—
63	C₃×C₁₉	—	3	2	no	—
64	C₃×C₁₉	—	9	3	no	—
65	C₃×C₁₉	C₃×C₃²⋊C₉	1	1	yes	characteristic
66	C₃×C₁₉	#6913	1	3	yes	—
67	C₃³	—	3	1	no	—
68	C₃³	—	3	2	no	—
69	C₃³	—	9	3	no	—
70	C₃³	C₃×#4294	1	3	yes	—
71	C₃³	C₃²×C₁₉⋊C₃	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	2	no	—
76	C₃×C₉	—	3	2	no	—
77	C₃×C₉	—	57	9	no	—
78	C₃×C₉	C₃×#4294	1	3	yes	—
79	C₃×C₉	C₃×#4294	1	3	yes	—
80	C₃×C₉	C₃×#4294	1	3	yes	—
81	C₃×C₉	#10587025	1	6	yes	—
82	C₃×C₉	C₃²⋊(C₁₉⋊C₃)	1	3	yes	—
83	C₁₉	#10624692	1	1	yes	characteristic, complemented, Sylow
84	C₃²	—	3	3	no	—
85	C₃²	—	3	3	no	—
86	C₃²	—	9	3	no	—
87	C₃²	—	9	6	no	—
88	C₃²	—	9	3	no	—
89	C₃²	C₃²×#4294	1	1	yes	characteristic
90	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
91	C₃²	C₃×#10587025	1	2	yes	—
92	C₉	—	3	3	no	—
93	C₉	—	3	3	no	—
94	C₉	—	171	9	no	—
95	C₉	C₃²⋊#4294	1	9	yes	—
96	C₃	—	3	1	no	—
97	C₃	—	3	2	no	—
98	C₃	—	9	3	no	—
99	C₃	C₃×C₃²⋊#4294	1	1	yes	characteristic
100	C₃	#17987528	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17987405

C₃

yes

maximal, complemented

#17987339

C₃

yes

maximal, complemented

#17987337

C₃

yes

maximal, complemented

#17987335

C₃

yes

maximal

C₃²×C₁₉×3₋¹⁺²

C₃

yes

maximal, characteristic

#13095307

—

#13095306

—

C₃²⋊#4294

C₃²

yes

—

C₉×#4294

—

C₃⁴×C₁₉

C₃²

yes

characteristic

C₃×C₁₉×3₋¹⁺²

—

C₃×C₁₉×3₋¹⁺²

—

C₃×C₁₉×3₋¹⁺²

C₉

yes

complemented

C₃²×C₉×C₁₉

C₃²

yes

characteristic

C₃²×C₉×C₁₉

C₃²

yes

characteristic

C₃²×C₉×C₁₉

C₃²

yes

characteristic

#10624692

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

C₃×C₉

yes

—

C₃³×C₁₉

C₃³

yes

characteristic

C₁₉×3₋¹⁺²

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

C₃×C₉

yes

—

C₃×C₉×C₁₉

C₃×C₉

yes

—

C₃×C₉×C₁₉

C₃×C₉

yes

—

C₃×C₉×C₁₉

H(3,𝔽₃)

yes

—

C₃×C₉×C₁₉

3₋¹⁺²

yes

—

C₃×#4294

—

C₃²×3₋¹⁺²

C₁₉⋊C₃

yes

—

—

—

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃²×C₉

yes

characteristic

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₉

C₃×3₋¹⁺²

yes

—

C₉×C₁₉

—

C₉×C₁₉

—

C₉×C₁₉

C₃²⋊C₉

yes

—

#4294

—

C₃⁴

C₃×C₁₉⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

#4294

yes

complemented

C₃²×C₉

C₃×C₁₉⋊C₃

yes

characteristic

C₃²×C₉

C₃×C₁₉⋊C₃

yes

characteristic

C₃²×C₉

C₃×C₁₉⋊C₃

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₉²

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

C₃×C₃²⋊C₉

yes

characteristic

C₃×C₁₉

#6913

yes

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×#4294

yes

—

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃×#4294

yes

—

C₃×C₉

C₃×#4294

yes

—

C₃×C₉

C₃×#4294

yes

—

C₃×C₉

#10587025

yes

—

C₃×C₉

C₃²⋊(C₁₉⋊C₃)

yes

—

C₁₉

#10624692

yes

characteristic, complemented, Sylow

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃²×#4294

yes

characteristic

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

C₃²

C₃×#10587025

yes

—

C₉

—

C₉

—

C₉

—

171

—

C₉

C₃²⋊#4294

yes

—

C₃

—

C₃

—

C₃

—

C₃

C₃×C₃²⋊#4294

yes

characteristic

100

C₃

#17987528

yes

—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)
commutator	C₃²×C₁₉	C₃²×C₉
Frattini	C₃³	C₃²×C₁₉⋊C₃
Fitting	C₃²×C₁₉×3₋¹⁺²	C₃
socle	C₃²×C₁₉	C₃×H(3,𝔽₃)
prime radical	C₃⁴×C₁₉	C₃²
cosocle	C₃³×C₁₉	C₃³
hypercenter	C₃²×3₋¹⁺²	C₁₉⋊C₃
hypocenter	C₁₉	#10624692
LCS	G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃²×C₉ ◁ C₃²×3₋¹⁺²

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

commutator

C₃²×C₁₉

C₃²×C₉

Frattini

C₃³

C₃²×C₁₉⋊C₃

Fitting

C₃²×C₁₉×3₋¹⁺²

C₃

socle

C₃²×C₁₉

C₃×H(3,𝔽₃)

prime radical

C₃⁴×C₁₉

C₃²

cosocle

C₃³×C₁₉

C₃³

hypercenter

C₃²×3₋¹⁺²

C₁₉⋊C₃

hypocenter

C₁₉

#10624692

LCS

G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃² ◁ C₃²×C₉ ◁ C₃²×3₋¹⁺²

N	H ≅ G/N	type
1. #17987405	C₃	semidirect
2. #17987339	C₃	semidirect
3. #17987337	C₃	semidirect
13. C₃×C₁₉×3₋¹⁺²	C₉	semidirect
54. C₃×3₋¹⁺²	#4294	semidirect
83. C₁₉	#10624692	semidirect

H ≅ G/N

type

1. #17987405

C₃

semidirect

2. #17987339

C₃

semidirect

3. #17987337

C₃

semidirect

13. C₃×C₁₉×3₋¹⁺²

C₉

semidirect

54. C₃×3₋¹⁺²

#4294

semidirect

83. C₁₉

#10624692

semidirect

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

subgroup	index
#17987405	3
#17987339	3
#17987337	3
#17987335	3
C₃²×C₁₉×3₋¹⁺²	3
#10624692	19

subgroup

index

C₃²×C₁₉×3₋¹⁺²

Subgroups — #28354659

Conjugacy classes of subgroups

Complements

Maximal subgroups