Group #28355941 — subgroups

Subgroups — C₃×#17987418 #28355941

Order 13851.

2636 conjugacy classes of subgroups; only the 382 normal subgroups are shown.

normal subgroups only

This group has 2636 conjugacy classes of subgroups — too many to draw. Only its 382 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	C₃⁴×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic
2	C₃²×#10587025	C₃	1	2	yes	maximal, complemented
3	#17987418	C₃	1	27	yes	maximal, direct product factor
4	C₃×#13095303	C₃	1	9	yes	maximal, complemented
5	C₃²×C₁₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃³×C₁₉⋊C₃	—	3	9	no	—
7	C₃³×C₁₉⋊C₃	—	3	27	no	—
8	C₃³×C₁₉⋊C₃	C₃²	1	9	yes	—
9	C₃³×C₁₉⋊C₃	C₃²	1	3	yes	—
10	C₃×#10587025	—	3	18	no	—
11	C₃×#10587025	C₃²	1	18	yes	complemented
12	C₃²×#4294	C₃²	1	6	yes	complemented
13	#13095303	C₃²	1	81	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	9	yes	complemented
17	C₃²×C₉×C₁₉	C₃²	1	3	yes	complemented
18	C₃×#6891	—	19	1	no	maximal, Sylow
19	C₃³×C₁₉	—	3	9	no	—
20	C₃³×C₁₉	H(3,𝔽₃)	1	9	yes	—
21	C₃³×C₁₉	C₃³	1	3	yes	—
22	C₃³×C₁₉	C₃³	1	1	yes	characteristic
23	C₃²×C₁₉⋊C₃	—	3	243	no	—
24	C₃²×C₁₉⋊C₃	—	3	27	no	—
25	C₃²×C₁₉⋊C₃	—	3	9	no	—
26	C₃²×C₁₉⋊C₃	—	3	18	no	—
27	C₃²×C₁₉⋊C₃	—	3	81	no	—
28	C₃²×C₁₉⋊C₃	3₋¹⁺²	1	27	yes	complemented
29	C₃²×C₁₉⋊C₃	C₃³	1	9	yes	—
30	C₁₉×3₋¹⁺²	—	3	27	no	—
31	C₃×C₉×C₁₉	—	3	9	no	—
32	C₃×C₉×C₁₉	C₃³	1	9	yes	complemented
33	#10587025	—	3	54	no	—
34	C₃×#4294	—	3	18	no	—
35	C₃×#4294	C₃³	1	18	yes	complemented
36	C₃⁵	—	19	1	no	—
37	C₃²×3₋¹⁺²	—	19	2	no	—
38	C₃²×3₋¹⁺²	C₁₉⋊C₃	1	1	yes	characteristic, complemented
39	#6891	—	19	27	no	—
40	C₃×C₃²⋊C₉	—	19	9	no	—
41	C₃²×C₁₉	—	3	9	no	—
42	C₃²×C₁₉	—	3	27	no	—
43	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	9	yes	—
44	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	3	yes	—
45	C₃²×C₁₉	C₃×3₋¹⁺²	1	9	yes	—
46	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
47	C₃×C₁₉⋊C₃	—	3	243	no	—
48	C₃×C₁₉⋊C₃	—	3	9	no	—
49	C₃×C₁₉⋊C₃	—	3	18	no	—
50	C₃×C₁₉⋊C₃	—	3	81	no	—
51	C₃×C₁₉⋊C₃	C₃×3₋¹⁺²	1	27	yes	complemented
52	C₉×C₁₉	—	3	9	no	—
53	#4294	—	3	18	no	—
54	C₃⁴	—	19	3	no	—
55	C₃⁴	—	19	9	no	—
56	C₃⁴	—	57	27	no	—
57	C₃⁴	—	57	9	no	—
58	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
59	C₃×3₋¹⁺²	—	3	9	no	—
60	C₃×3₋¹⁺²	—	19	18	no	—
61	C₃×3₋¹⁺²	—	57	18	no	—
62	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	9	yes	complemented
63	C₃²×C₉	—	19	6	no	—
64	C₃²×C₉	C₃×C₁₉⋊C₃	1	3	yes	complemented
65	C₃²⋊C₉	—	19	81	no	—
66	C₃×C₁₉	—	3	9	no	—
67	C₃×C₁₉	#6891	1	9	yes	complemented
68	C₃×C₁₉	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
69	C₃×C₁₉	C₃²×3₋¹⁺²	1	3	yes	—
70	C₁₉⋊C₃	—	3	27	no	—
71	C₃³	—	3	9	no	—
72	C₃³	—	19	27	no	—
73	C₃³	—	19	9	no	—
74	C₃³	—	57	81	no	—
75	C₃³	—	57	243	no	—
76	C₃³	—	57	27	no	—
77	C₃³	—	57	9	no	—
78	C₃³	—	57	18	no	—
79	C₃³	C₃²⋊(C₁₉⋊C₃)	1	9	yes	—
80	C₃³	C₃²×C₁₉⋊C₃	1	3	yes	—
81	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
82	3₋¹⁺²	—	3	27	no	—
83	3₋¹⁺²	—	57	54	no	—
84	C₃×C₉	—	3	9	no	—
85	C₃×C₉	—	19	18	no	—
86	C₃×C₉	—	57	18	no	—
87	C₃×C₉	C₃²×C₁₉⋊C₃	1	9	yes	complemented
88	C₁₉	C₃×#6891	1	1	yes	characteristic, complemented, Sylow
89	C₃²	—	3	27	no	—
90	C₃²	—	3	9	no	—
91	C₃²	—	19	27	no	—
92	C₃²	—	57	243	no	—
93	C₃²	—	57	81	no	—
94	C₃²	—	57	9	no	—
95	C₃²	—	57	18	no	—
96	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	9	yes	—
97	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	3	yes	—
98	C₃²	3₋¹⁺²×C₁₉⋊C₃	1	9	yes	—
99	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
100	C₉	—	3	9	no	—
101	C₉	—	57	18	no	—
102	C₃	—	3	9	no	—
103	C₃	—	57	27	no	—
104	C₃	#17987418	1	9	yes	direct product factor
105	C₃	C₃²×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
106	C₃	C₃×3₋¹⁺²×C₁₉⋊C₃	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃⁴×C₁₉⋊C₃

C₃

yes

maximal, characteristic

C₃²×#10587025

C₃

yes

maximal, complemented

#17987418

C₃

yes

maximal, direct product factor

C₃×#13095303

C₃

yes

maximal, complemented

C₃²×C₁₉×3₋¹⁺²

C₃

yes

maximal, characteristic, complemented

C₃³×C₁₉⋊C₃

—

C₃³×C₁₉⋊C₃

—

C₃³×C₁₉⋊C₃

C₃²

yes

—

C₃³×C₁₉⋊C₃

C₃²

yes

—

C₃×#10587025

—

C₃×#10587025

C₃²

yes

complemented

C₃²×#4294

C₃²

yes

complemented

#13095303

C₃²

yes

complemented

C₃⁴×C₁₉

C₃²

yes

characteristic

C₃×C₁₉×3₋¹⁺²

—

C₃×C₁₉×3₋¹⁺²

C₃²

yes

complemented

C₃²×C₉×C₁₉

C₃²

yes

complemented

C₃×#6891

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

H(3,𝔽₃)

yes

—

C₃³×C₁₉

C₃³

yes

—

C₃³×C₁₉

C₃³

yes

characteristic

C₃²×C₁₉⋊C₃

—

243

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

3₋¹⁺²

yes

complemented

C₃²×C₁₉⋊C₃

C₃³

yes

—

C₁₉×3₋¹⁺²

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

C₃³

yes

—

—

C₃³

yes

complemented

C₃⁵

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₁₉⋊C₃

yes

characteristic, complemented

#6891

—

C₃×C₃²⋊C₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

—

C₃²×C₁₉

C₃×3₋¹⁺²

yes

—

C₃²×C₁₉

C₃⁴

yes

characteristic

C₃×C₁₉⋊C₃

—

243

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

C₃×3₋¹⁺²

yes

complemented

C₉×C₁₉

—

#4294

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

C₃×C₁₉⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₁₉⋊C₃

yes

complemented

C₃²×C₉

—

C₃²×C₉

C₃×C₁₉⋊C₃

yes

complemented

C₃²⋊C₉

—

C₃×C₁₉

—

C₃×C₁₉

#6891

yes

complemented

C₃×C₁₉

C₃²×H(3,𝔽₃)

yes

characteristic

C₃×C₁₉

C₃²×3₋¹⁺²

yes

—

C₁₉⋊C₃

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

243

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

—

C₃³

C₃²×C₁₉⋊C₃

yes

—

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃²×C₁₉⋊C₃

yes

complemented

C₁₉

C₃×#6891

yes

characteristic, complemented, Sylow

C₃²

—

C₃²

—

C₃²

—

C₃²

—

243

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

—

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

—

C₃²

3₋¹⁺²×C₁₉⋊C₃

yes

—

C₃²

C₃³×C₁₉⋊C₃

yes

characteristic

100

C₉

—

101

C₉

—

102

C₃

—

103

C₃

—

104

C₃

#17987418

yes

direct product factor

105

C₃

C₃²×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

106

C₃

C₃×3₋¹⁺²×C₁₉⋊C₃

yes

—

	subgroup	G/H
center	C₃³	C₃²×C₁₉⋊C₃
commutator, cosocle	C₃²×C₁₉	C₃⁴
Frattini	C₃²	C₃³×C₁₉⋊C₃
Fitting	C₃²×C₁₉×3₋¹⁺²	C₃
socle	C₃³×C₁₉	C₃³
prime radical	C₃⁴×C₁₉⋊C₃	C₃
hypercenter	C₃²×3₋¹⁺²	C₁₉⋊C₃
hypocenter	C₁₉	C₃×#6891
LCS	G ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃³ ◁ C₃²×3₋¹⁺²

subgroup

G/H

center

C₃³

C₃²×C₁₉⋊C₃

commutator, cosocle

C₃²×C₁₉

C₃⁴

Frattini

C₃²

C₃³×C₁₉⋊C₃

Fitting

C₃²×C₁₉×3₋¹⁺²

C₃

socle

C₃³×C₁₉

C₃³

prime radical

C₃⁴×C₁₉⋊C₃

C₃

hypercenter

C₃²×3₋¹⁺²

C₁₉⋊C₃

C₁₉

G ▷ C₃²×C₁₉ ▷ C₁₉

C₁ ◁ C₃³ ◁ C₃²×3₋¹⁺²

N	H ≅ G/N	type
2. C₃²×#10587025	C₃	semidirect
3. #17987418	C₃	direct (N × H)
4. C₃×#13095303	C₃	semidirect
5. C₃²×C₁₉×3₋¹⁺²	C₃	semidirect
11. C₃×#10587025	C₃²	semidirect
12. C₃²×#4294	C₃²	semidirect
13. #13095303	C₃²	semidirect
16. C₃×C₁₉×3₋¹⁺²	C₃²	semidirect
17. C₃²×C₉×C₁₉	C₃²	semidirect
28. C₃²×C₁₉⋊C₃	3₋¹⁺²	semidirect
32. C₃×C₉×C₁₉	C₃³	semidirect
35. C₃×#4294	C₃³	semidirect
38. C₃²×3₋¹⁺²	C₁₉⋊C₃	semidirect
51. C₃×C₁₉⋊C₃	C₃×3₋¹⁺²	semidirect
62. C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	semidirect
64. C₃²×C₉	C₃×C₁₉⋊C₃	semidirect
67. C₃×C₁₉	#6891	semidirect
87. C₃×C₉	C₃²×C₁₉⋊C₃	semidirect
88. C₁₉	C₃×#6891	semidirect
104. C₃	#17987418	direct (N × H)

H ≅ G/N

type

2. C₃²×#10587025

C₃

semidirect

3. #17987418

C₃

direct (N × H)

4. C₃×#13095303

C₃

semidirect

5. C₃²×C₁₉×3₋¹⁺²

C₃

semidirect

11. C₃×#10587025

C₃²

semidirect

12. C₃²×#4294

C₃²

semidirect

13. #13095303

C₃²

semidirect

16. C₃×C₁₉×3₋¹⁺²

C₃²

semidirect

17. C₃²×C₉×C₁₉

C₃²

semidirect

28. C₃²×C₁₉⋊C₃

3₋¹⁺²

semidirect

32. C₃×C₉×C₁₉

C₃³

semidirect

35. C₃×#4294

C₃³

semidirect

38. C₃²×3₋¹⁺²

C₁₉⋊C₃

semidirect

51. C₃×C₁₉⋊C₃

C₃×3₋¹⁺²

semidirect

62. C₃×3₋¹⁺²

C₃×C₁₉⋊C₃

semidirect

64. C₃²×C₉

C₃×C₁₉⋊C₃

semidirect

67. C₃×C₁₉

#6891

semidirect

87. C₃×C₉

C₃²×C₁₉⋊C₃

semidirect

88. C₁₉

C₃×#6891

semidirect

104. C₃

#17987418

direct (N × H)

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

subgroup	index
C₃⁴×C₁₉⋊C₃	3
C₃²×#10587025	3
#17987418	3
C₃×#13095303	3
C₃²×C₁₉×3₋¹⁺²	3
C₃×#6891	19

subgroup

index

C₃⁴×C₁₉⋊C₃

C₃²×C₁₉×3₋¹⁺²

Subgroups — C₃×#17987418 #28355941

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#17987418 #28355941

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#17987418 #28355941