Group #28356074 — subgroups

Subgroups — #28356074

Order 13851.

716 conjugacy classes of subgroups; only the 292 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 292 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	#17987468	C₃	1	9	yes	maximal, complemented
2	#17987461	C₃	1	18	yes	maximal, complemented
3	#17987445	C₃	1	6	yes	maximal, complemented
4	#17987421	C₃	1	3	yes	maximal
5	#17987405	C₃	1	3	yes	maximal, complemented
6	C₃²×C₁₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
7	C₃×#10587025	C₃²	1	9	yes	—
8	#13095307	C₃²	1	9	yes	complemented
9	#13095306	C₃²	1	9	yes	complemented
10	#13095305	C₃²	1	54	yes	complemented
11	C₃²⋊#4294	C₃²	1	9	yes	—
12	C₃²⋊#4294	C₃²	1	18	yes	—
13	C₉×#4294	C₃²	1	9	yes	complemented
14	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
15	C₃×C₁₉×3₋¹⁺²	—	3	9	no	—
16	C₃×C₁₉×3₋¹⁺²	C₃²	1	6	yes	—
17	C₃×C₁₉×3₋¹⁺²	C₃²	1	3	yes	—
18	C₃²×C₉×C₁₉	C₃²	1	2	yes	—
19	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
20	#10624884	—	19	1	no	maximal, Sylow
21	C₃³×C₁₉	—	3	3	no	—
22	C₃³×C₁₉	—	9	3	no	—
23	C₃³×C₁₉	C₃³	1	2	yes	—
24	C₃³×C₁₉	C₃³	1	1	yes	characteristic
25	C₃³×C₁₉	C₃³	1	1	yes	characteristic
26	C₁₉×3₋¹⁺²	—	3	18	no	—
27	C₁₉×3₋¹⁺²	—	3	9	no	—
28	C₃×C₉×C₁₉	—	3	6	no	—
29	C₃×C₉×C₁₉	3₋¹⁺²	1	9	yes	complemented
30	C₃×C₉×C₁₉	C₃³	1	6	yes	—
31	C₃×C₉×C₁₉	C₃³	1	3	yes	—
32	#10587025	—	9	9	no	—
33	C₃×#4294	C₃³	1	27	yes	—
34	C₃²×3₋¹⁺²	C₁₉⋊C₃	1	1	yes	characteristic
35	#6900	—	19	9	no	—
36	#6899	—	19	18	no	—
37	#6895	—	19	6	no	—
38	#6891	—	19	3	no	—
39	#6887	—	19	3	no	—
40	C₃²×C₁₉	—	3	3	no	—
41	C₃²×C₁₉	—	3	2	no	—
42	C₃²×C₁₉	—	3	1	no	—
43	C₃²×C₁₉	—	3	3	no	—
44	C₃²×C₁₉	—	3	6	no	—
45	C₃²×C₁₉	—	9	9	no	—
46	C₃²×C₁₉	C₃×3₋¹⁺²	1	3	yes	—
47	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
48	C₉×C₁₉	—	3	6	no	—
49	C₉×C₁₉	C₃×3₋¹⁺²	1	9	yes	—
50	#4294	—	3	27	no	—
51	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
52	C₃×3₋¹⁺²	—	3	9	no	—
53	C₃×3₋¹⁺²	—	19	9	no	—
54	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	6	yes	—
55	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	3	yes	—
56	C₃²×C₉	C₃×C₁₉⋊C₃	1	2	yes	—
57	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
58	C₉⋊C₉	—	19	54	no	—
59	C₉⋊C₉	—	19	9	no	—
60	C₉⋊C₉	—	19	9	no	—
61	C₃²⋊C₉	—	19	9	no	—
62	C₃²⋊C₉	—	19	18	no	—
63	C₉²	—	19	9	no	—
64	C₃×C₁₉	—	3	3	no	—
65	C₃×C₁₉	—	9	1	no	—
66	C₃×C₁₉	—	9	2	no	—
67	C₃×C₁₉	C₃²×3₋¹⁺²	1	1	yes	characteristic
68	C₃×C₁₉	3₋¹⁺⁴	1	3	yes	—
69	C₃³	—	3	3	no	—
70	C₃³	—	9	3	no	—
71	C₃³	C₃²×C₁₉⋊C₃	1	2	yes	—
72	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
73	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
74	3₋¹⁺²	—	3	18	no	—
75	3₋¹⁺²	—	3	9	no	—
76	3₋¹⁺²	—	171	9	no	—
77	C₃×C₉	—	3	6	no	—
78	C₃×C₉	—	19	27	no	—
79	C₃×C₉	#10587025	1	9	yes	complemented
80	C₃×C₉	C₃²×C₁₉⋊C₃	1	6	yes	—
81	C₃×C₉	C₃²×C₁₉⋊C₃	1	3	yes	—
82	C₁₉	#10624884	1	1	yes	characteristic, complemented, Sylow
83	C₃²	—	3	3	no	—
84	C₃²	—	3	2	no	—
85	C₃²	—	3	1	no	—
86	C₃²	—	3	3	no	—
87	C₃²	—	3	6	no	—
88	C₃²	—	9	9	no	—
89	C₃²	C₃×#10587025	1	3	yes	—
90	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
91	C₉	—	3	6	no	—
92	C₉	—	57	27	no	—
93	C₉	C₃×#10587025	1	9	yes	—
94	C₃	—	3	3	no	—
95	C₃	—	9	1	no	—
96	C₃	—	9	2	no	—
97	C₃	C₃²×#10587025	1	1	yes	characteristic
98	C₃	#17987546	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17987468

C₃

yes

maximal, complemented

#17987461

C₃

yes

maximal, complemented

#17987445

C₃

yes

maximal, complemented

#17987421

C₃

yes

maximal

#17987405

C₃

yes

maximal, complemented

C₃²×C₁₉×3₋¹⁺²

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

C₃×C₁₉×3₋¹⁺²

—

C₃×C₁₉×3₋¹⁺²

C₃²

yes

—

C₃×C₁₉×3₋¹⁺²

C₃²

yes

—

C₃²×C₉×C₁₉

C₃²

yes

—

C₃²×C₉×C₁₉

C₃²

yes

characteristic

#10624884

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

C₃³

yes

—

C₃³×C₁₉

C₃³

yes

characteristic

C₃³×C₁₉

C₃³

yes

characteristic

C₁₉×3₋¹⁺²

—

C₁₉×3₋¹⁺²

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

3₋¹⁺²

yes

complemented

C₃×C₉×C₁₉

C₃³

yes

—

C₃×C₉×C₁₉

C₃³

yes

—

#10587025

—

C₃×#4294

C₃³

yes

—

C₃²×3₋¹⁺²

C₁₉⋊C₃

yes

—

—

—

—

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃×3₋¹⁺²

yes

—

C₃²×C₁₉

C₃⁴

yes

characteristic

C₉×C₁₉

—

C₉×C₁₉

C₃×3₋¹⁺²

yes

—

#4294

—

C₃⁴

C₃×C₁₉⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₁₉⋊C₃

yes

—

C₃×3₋¹⁺²

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

—

C₃×C₁₉

C₃²×3₋¹⁺²

yes

characteristic

C₃×C₁₉

3₋¹⁺⁴

yes

—

C₃³

—

C₃³

—

C₃³

C₃²×C₁₉⋊C₃

yes

—

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

171

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

#10587025

yes

complemented

C₃×C₉

C₃²×C₁₉⋊C₃

yes

—

C₃×C₉

C₃²×C₁₉⋊C₃

yes

—

C₁₉

#10624884

yes

characteristic, complemented, Sylow

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃×#10587025

yes

—

C₃²

C₃³×C₁₉⋊C₃

yes

characteristic

C₉

—

C₉

—

C₉

C₃×#10587025

yes

—

C₃

—

C₃

—

C₃

—

C₃

C₃²×#10587025

yes

characteristic

C₃

#17987546

yes

—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₁₉⋊C₃
commutator, socle, cosocle	C₃²×C₁₉	C₃⁴
Fitting	C₃²×C₁₉×3₋¹⁺²	C₃
prime radical	C₃⁴×C₁₉	C₃²
hypercenter	C₃²×3₋¹⁺²	C₁₉⋊C₃
hypocenter	C₁₉	#10624884
LCS	G ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃²×3₋¹⁺²

subgroup

G/H

center, Frattini

C₃²

C₃³×C₁₉⋊C₃

commutator, socle, cosocle

C₃²×C₁₉

C₃⁴

Fitting

C₃²×C₁₉×3₋¹⁺²

C₃

prime radical

C₃⁴×C₁₉

C₃²

hypercenter

C₃²×3₋¹⁺²

C₁₉⋊C₃

C₁₉

G ▷ C₃²×C₁₉ ▷ C₁₉

C₁ ◁ C₃² ◁ C₃²×3₋¹⁺²

N	H ≅ G/N	type
1. #17987468	C₃	semidirect
2. #17987461	C₃	semidirect
3. #17987445	C₃	semidirect
5. #17987405	C₃	semidirect
8. #13095307	C₃²	semidirect
9. #13095306	C₃²	semidirect
10. #13095305	C₃²	semidirect
13. C₉×#4294	C₃²	semidirect
29. C₃×C₉×C₁₉	3₋¹⁺²	semidirect
79. C₃×C₉	#10587025	semidirect
82. C₁₉	#10624884	semidirect

H ≅ G/N

type

1. #17987468

C₃

semidirect

2. #17987461

C₃

semidirect

3. #17987445

C₃

semidirect

5. #17987405

C₃

semidirect

8. #13095307

C₃²

semidirect

9. #13095306

C₃²

semidirect

10. #13095305

C₃²

semidirect

13. C₉×#4294

C₃²

semidirect

29. C₃×C₉×C₁₉

3₋¹⁺²

semidirect

79. C₃×C₉

#10587025

semidirect

82. C₁₉

#10624884

semidirect

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

subgroup	index
#17987468	3
#17987461	3
#17987445	3
#17987421	3
#17987405	3
C₃²×C₁₉×3₋¹⁺²	3
#10624884	19

subgroup

index

C₃²×C₁₉×3₋¹⁺²

Subgroups — #28356074

Conjugacy classes of subgroups

Complements

Maximal subgroups