Group #28356077 — subgroups

Subgroups — #28356077

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	C₃²×#10587023	C₃	1	1	yes	maximal, characteristic
2	#17987450	C₃	1	18	yes	maximal, complemented
3	#17987448	C₃	1	9	yes	maximal, complemented
4	#17987443	C₃	1	6	yes	maximal, complemented
5	#17987420	C₃	1	3	yes	maximal
6	#17987405	C₃	1	2	yes	maximal, complemented
7	C₁₉×#6887	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃³×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
9	C₃×#10587022	C₃²	1	9	yes	—
10	C₃×#10587023	—	3	9	no	—
11	C₃×#10587023	C₃²	1	3	yes	—
12	C₃×#10587023	C₃²	1	6	yes	—
13	C₃²×#4294	C₃²	1	2	yes	—
14	#13095307	C₃²	1	6	yes	complemented
15	#13095306	C₃²	1	6	yes	complemented
16	#13095306	C₃²	1	18	yes	complemented
17	#13095305	C₃²	1	36	yes	complemented
18	C₃²⋊#4294	C₃²	1	6	yes	—
19	#13095303	C₃²	1	18	yes	—
20	C₉×#4294	C₃²	1	6	yes	complemented
21	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
22	C₁₉×C₉⋊C₉	C₃²	1	6	yes	complemented
23	C₁₉×C₃²⋊C₉	C₃²	1	3	yes	—
24	C₉²×C₁₉	C₃²	1	3	yes	complemented
25	#10624885	—	19	1	no	maximal, Sylow
26	C₃³×C₁₉	C₃³	1	1	yes	characteristic
27	C₃²×C₁₉⋊C₃	—	3	3	no	—
28	C₃²×C₁₉⋊C₃	—	9	3	no	—
29	C₃²×C₁₉⋊C₃	C₃³	1	3	yes	—
30	C₃×C₉×C₁₉	3₋¹⁺²	1	9	yes	complemented
31	C₃×C₉×C₁₉	C₃³	1	9	yes	—
32	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
33	C₃×C₉×C₁₉	C₃³	1	2	yes	—
34	C₃×#4294	—	3	6	no	—
35	C₃×#4294	C₃³	1	18	yes	—
36	C₃×#4294	C₃³	1	6	yes	—
37	#10587023	—	3	9	no	—
38	#10587023	—	3	18	no	—
39	#10587022	—	9	9	no	—
40	C₃²×3₋¹⁺²	—	19	1	no	—
41	#6896	—	19	18	no	—
42	#6896	—	19	9	no	—
43	#6895	—	19	6	no	—
44	#6891	—	19	3	no	—
45	#6887	—	19	2	no	—
46	#6887	C₁₉⋊C₃	1	1	yes	characteristic, complemented
47	C₃²×C₁₉	—	3	3	no	—
48	C₃²×C₁₉	C₃×3₋¹⁺²	1	3	yes	—
49	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
50	C₃×C₁₉⋊C₃	—	3	3	no	—
51	C₃×C₁₉⋊C₃	—	3	3	no	—
52	C₃×C₁₉⋊C₃	—	3	6	no	—
53	C₃×C₁₉⋊C₃	—	9	9	no	—
54	C₉×C₁₉	—	3	9	no	—
55	C₉×C₁₉	C₃×3₋¹⁺²	1	3	yes	—
56	C₉×C₁₉	C₃×3₋¹⁺²	1	6	yes	—
57	#4294	—	3	6	no	—
58	#4294	—	3	18	no	—
59	C₃⁴	—	19	1	no	—
60	C₃×3₋¹⁺²	—	19	6	no	—
61	C₃×3₋¹⁺²	—	19	3	no	—
62	C₃×3₋¹⁺²	—	19	9	no	—
63	C₃×3₋¹⁺²	—	57	9	no	—
64	C₃²×C₉	—	19	2	no	—
65	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
66	C₉⋊C₉	—	19	36	no	—
67	C₉⋊C₉	—	19	6	no	—
68	C₉⋊C₉	—	19	18	no	—
69	C₉⋊C₉	—	19	6	no	—
70	C₉⋊C₉	C₃×C₁₉⋊C₃	1	6	yes	complemented
71	C₃²⋊C₉	—	19	6	no	—
72	C₃²⋊C₉	—	19	18	no	—
73	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	—
74	C₉²	—	19	6	no	—
75	C₉²	C₃×C₁₉⋊C₃	1	3	yes	complemented
76	C₃×C₁₉	—	3	3	no	—
77	C₃×C₁₉	C₃²×3₋¹⁺²	1	1	yes	characteristic
78	C₃×C₁₉	3₋¹⁺⁴	1	3	yes	—
79	C₁₉⋊C₃	—	9	3	no	—
80	C₃³	—	19	3	no	—
81	C₃³	—	57	3	no	—
82	C₃³	—	171	3	no	—
83	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
84	3₋¹⁺²	—	57	18	no	—
85	3₋¹⁺²	—	57	9	no	—
86	3₋¹⁺²	—	171	9	no	—
87	C₃×C₉	—	19	18	no	—
88	C₃×C₉	—	19	6	no	—
89	C₃×C₉	—	57	6	no	—
90	C₃×C₉	#10587022	1	9	yes	complemented
91	C₃×C₉	C₃²×C₁₉⋊C₃	1	9	yes	—
92	C₃×C₉	C₃²×C₁₉⋊C₃	1	2	yes	—
93	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
94	C₁₉	#10624885	1	1	yes	characteristic, complemented, Sylow
95	C₃²	—	3	3	no	—
96	C₃²	—	57	3	no	—
97	C₃²	—	57	3	no	—
98	C₃²	—	57	6	no	—
99	C₃²	—	171	9	no	—
100	C₃²	C₃×#10587022	1	3	yes	—
101	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
102	C₉	—	3	9	no	—
103	C₉	—	57	6	no	—
104	C₉	—	57	18	no	—
105	C₉	C₃×#10587022	1	6	yes	—
106	C₉	C₃×#10587022	1	3	yes	—
107	C₃	—	3	3	no	—
108	C₃	—	171	3	no	—
109	C₃	C₃²×#10587022	1	1	yes	characteristic
110	C₃	#17987544	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#10587023

C₃

yes

maximal, characteristic

#17987450

C₃

yes

maximal, complemented

#17987448

C₃

yes

maximal, complemented

#17987443

C₃

yes

maximal, complemented

#17987420

C₃

yes

maximal

#17987405

C₃

yes

maximal, complemented

C₁₉×#6887

C₃

yes

maximal, characteristic, complemented

C₃³×C₁₉⋊C₃

C₃²

yes

characteristic

C₃×#10587022

C₃²

yes

—

C₃×#10587023

—

C₃×#10587023

C₃²

yes

—

C₃×#10587023

C₃²

yes

—

C₃²×#4294

C₃²

yes

—

#13095307

C₃²

yes

complemented

#13095306

C₃²

yes

complemented

#13095306

C₃²

yes

complemented

#13095305

C₃²

yes

complemented

C₃²⋊#4294

C₃²

yes

—

#13095303

C₃²

yes

—

C₉×#4294

C₃²

yes

complemented

C₃²×C₉×C₁₉

C₃²

yes

characteristic

C₁₉×C₉⋊C₉

C₃²

yes

complemented

C₁₉×C₃²⋊C₉

C₃²

yes

—

C₉²×C₁₉

C₃²

yes

complemented

#10624885

—

maximal, Sylow

C₃³×C₁₉

C₃³

yes

characteristic

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

C₃³

yes

—

C₃×C₉×C₁₉

3₋¹⁺²

yes

complemented

C₃×C₉×C₁₉

C₃³

yes

—

C₃×C₉×C₁₉

C₃³

yes

characteristic

C₃×C₉×C₁₉

C₃³

yes

—

C₃×#4294

—

C₃×#4294

C₃³

yes

—

C₃×#4294

C₃³

yes

—

#10587023

—

#10587023

—

#10587022

—

C₃²×3₋¹⁺²

—

—

—

—

—

—

C₁₉⋊C₃

yes

characteristic, complemented

C₃²×C₁₉

—

C₃²×C₁₉

C₃×3₋¹⁺²

yes

—

C₃²×C₁₉

C₃⁴

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₉×C₁₉

—

C₉×C₁₉

C₃×3₋¹⁺²

yes

—

C₉×C₁₉

C₃×3₋¹⁺²

yes

—

#4294

—

#4294

—

C₃⁴

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃²×C₉

—

C₃²×C₉

C₃×C₁₉⋊C₃

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

C₃×C₁₉⋊C₃

yes

complemented

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₁₉⋊C₃

yes

—

C₉²

—

C₉²

C₃×C₁₉⋊C₃

yes

complemented

C₃×C₁₉

—

C₃×C₁₉

C₃²×3₋¹⁺²

yes

characteristic

C₃×C₁₉

3₋¹⁺⁴

yes

—

C₁₉⋊C₃

—

C₃³

—

C₃³

—

C₃³

—

171

—

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

171

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

#10587022

yes

complemented

C₃×C₉

C₃²×C₁₉⋊C₃

yes

—

C₃×C₉

C₃²×C₁₉⋊C₃

yes

—

C₃×C₉

C₃²×C₁₉⋊C₃

yes

characteristic

C₁₉

#10624885

yes

characteristic, complemented, Sylow

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

171

—

100

C₃²

C₃×#10587022

yes

—

101

C₃²

C₃³×C₁₉⋊C₃

yes

characteristic

102

C₉

—

103

C₉

—

104

C₉

—

105

C₉

C₃×#10587022

yes

—

106

C₉

C₃×#10587022

yes

—

107

C₃

—

108

C₃

—

171

—

109

C₃

C₃²×#10587022

yes

characteristic

110

C₃

#17987544

yes

—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₁₉⋊C₃
commutator, socle, cosocle	C₃²×C₁₉	C₃⁴
Fitting	C₁₉×#6887	C₃
prime radical	C₃³×C₁₉⋊C₃	C₃²
hypercenter	#6887	C₁₉⋊C₃
hypocenter	C₁₉	#10624885
LCS	G ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ #6887

subgroup

G/H

center, Frattini

C₃²

C₃³×C₁₉⋊C₃

commutator, socle, cosocle

C₃²×C₁₉

C₃⁴

Fitting

C₁₉×#6887

C₃

prime radical

C₃³×C₁₉⋊C₃

C₃²

C₁₉⋊C₃

C₁₉

G ▷ C₃²×C₁₉ ▷ C₁₉

C₁ ◁ C₃² ◁ #6887

N	H ≅ G/N	type
2. #17987450	C₃	semidirect
3. #17987448	C₃	semidirect
4. #17987443	C₃	semidirect
6. #17987405	C₃	semidirect
7. C₁₉×#6887	C₃	semidirect
14. #13095307	C₃²	semidirect
15. #13095306	C₃²	semidirect
16. #13095306	C₃²	semidirect
17. #13095305	C₃²	semidirect
20. C₉×#4294	C₃²	semidirect
22. C₁₉×C₉⋊C₉	C₃²	semidirect
24. C₉²×C₁₉	C₃²	semidirect
30. C₃×C₉×C₁₉	3₋¹⁺²	semidirect
46. #6887	C₁₉⋊C₃	semidirect
70. C₉⋊C₉	C₃×C₁₉⋊C₃	semidirect
75. C₉²	C₃×C₁₉⋊C₃	semidirect
90. C₃×C₉	#10587022	semidirect
94. C₁₉	#10624885	semidirect

H ≅ G/N

type

2. #17987450

C₃

semidirect

3. #17987448

C₃

semidirect

4. #17987443

C₃

semidirect

6. #17987405

C₃

semidirect

7. C₁₉×#6887

C₃

semidirect

14. #13095307

C₃²

semidirect

15. #13095306

C₃²

semidirect

16. #13095306

C₃²

semidirect

17. #13095305

C₃²

semidirect

20. C₉×#4294

C₃²

semidirect

22. C₁₉×C₉⋊C₉

C₃²

semidirect

24. C₉²×C₁₉

C₃²

semidirect

30. C₃×C₉×C₁₉

3₋¹⁺²

semidirect

46. #6887

C₁₉⋊C₃

semidirect

70. C₉⋊C₉

C₃×C₁₉⋊C₃

semidirect

75. C₉²

C₃×C₁₉⋊C₃

semidirect

90. C₃×C₉

#10587022

semidirect

94. C₁₉

#10624885

semidirect

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

subgroup	index
C₃²×#10587023	3
#17987450	3
#17987448	3
#17987443	3
#17987420	3
#17987405	3
C₁₉×#6887	3
#10624885	19

subgroup

index

C₁₉×#6887

Subgroups — #28356077

Conjugacy classes of subgroups

Complements

Maximal subgroups