Group #28356059 — subgroups

Subgroups — #28356059

Order 13851.

1058 conjugacy classes of subgroups; only the 292 normal subgroups are shown.

normal subgroups only

This group has 1058 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	C₃×#13095332	C₃	1	3	yes	maximal, complemented
2	#17987431	C₃	1	6	yes	maximal, complemented
3	#17987422	C₃	1	27	yes	maximal, complemented
4	#17987421	C₃	1	2	yes	maximal, complemented
5	#17987417	C₃	1	1	yes	maximal, characteristic
6	C₃²×C₁₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
7	#13095332	—	3	27	no	—
8	C₃×#10587025	C₃²	1	6	yes	—
9	C₃×#10587022	C₃²	1	9	yes	complemented
10	C₃×#10587023	C₃²	1	9	yes	complemented
11	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	3	yes	—
12	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	9	yes	—
13	C₃×C₉×C₁₉⋊C₃	C₃²	1	9	yes	complemented
14	C₃²⋊#4294	C₃²	1	18	yes	—
15	#13095303	C₃²	1	54	yes	complemented
16	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
17	C₃×C₁₉×3₋¹⁺²	—	3	9	no	—
18	C₃×C₁₉×3₋¹⁺²	C₃²	1	9	yes	—
19	C₃²×C₉×C₁₉	C₃²	1	2	yes	—
20	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
21	#10624882	—	19	1	no	maximal, Sylow
22	C₃³×C₁₉	—	3	3	no	—
23	C₃³×C₁₉	—	9	3	no	—
24	C₃³×C₁₉	C₃³	1	3	yes	—
25	C₃³×C₁₉	C₃³	1	1	yes	characteristic
26	C₃²×C₁₉⋊C₃	C₃³	1	9	yes	—
27	C₁₉×3₋¹⁺²	—	3	27	no	—
28	C₃×C₉×C₁₉	—	3	6	no	—
29	C₃×C₉×C₁₉	H(3,𝔽₃)	1	9	yes	complemented
30	C₃×C₉×C₁₉	C₃³	1	6	yes	—
31	C₃×C₉×C₁₉	C₃³	1	3	yes	—
32	C₃²⋊(C₁₉⋊C₃)	—	3	9	no	—
33	C₃²⋊(C₁₉⋊C₃)	—	9	9	no	—
34	#10587025	—	3	18	no	—
35	C₃×#4294	C₃³	1	18	yes	—
36	#10587023	—	3	27	no	—
37	#10587022	—	3	27	no	—
38	C₉×C₁₉⋊C₃	—	3	27	no	—
39	C₃×#737	—	19	3	no	—
40	C₃²×3₋¹⁺²	C₁₉⋊C₃	1	1	yes	characteristic, complemented
41	#6893	—	19	6	no	—
42	C₉⋊H(3,𝔽₃)	—	19	27	no	—
43	#6891	—	19	2	no	—
44	C₃²⋊H(3,𝔽₃)	—	19	1	no	—
45	C₃²×C₁₉	—	3	3	no	—
46	C₃²×C₁₉	—	3	3	no	—
47	C₃²×C₁₉	—	3	3	no	—
48	C₃²×C₁₉	—	3	6	no	—
49	C₃²×C₁₉	—	9	9	no	—
50	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	3	yes	—
51	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
52	C₃×C₁₉⋊C₃	—	3	9	no	—
53	C₃×C₁₉⋊C₃	—	3	27	no	—
54	C₉×C₁₉	—	3	6	no	—
55	C₉×C₁₉	C₃×H(3,𝔽₃)	1	9	yes	—
56	#4294	—	3	18	no	—
57	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
58	#737	—	57	27	no	—
59	C₃×3₋¹⁺²	—	3	9	no	—
60	C₃×3₋¹⁺²	—	19	6	no	—
61	C₃×3₋¹⁺²	—	19	9	no	—
62	C₃×3₋¹⁺²	—	19	9	no	—
63	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	9	yes	—
64	C₃×H(3,𝔽₃)	—	19	3	no	—
65	C₃×H(3,𝔽₃)	—	19	9	no	—
66	C₃²×C₉	—	19	9	no	—
67	C₃²×C₉	C₃×C₁₉⋊C₃	1	2	yes	—
68	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
69	C₃²⋊C₉	—	19	18	no	—
70	C₃²⋊C₉	—	19	54	no	—
71	C₃×C₁₉	—	3	3	no	—
72	C₃×C₁₉	—	9	3	no	—
73	C₃×C₁₉	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
74	C₃×C₁₉	3₋¹⁺⁴	1	3	yes	—
75	C₁₉⋊C₃	—	9	9	no	—
76	C₃³	—	3	3	no	—
77	C₃³	—	9	3	no	—
78	C₃³	—	19	9	no	—
79	C₃³	C₃²×C₁₉⋊C₃	1	3	yes	—
80	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
81	3₋¹⁺²	—	3	27	no	—
82	3₋¹⁺²	—	57	18	no	—
83	3₋¹⁺²	—	57	27	no	—
84	3₋¹⁺²	—	57	27	no	—
85	H(3,𝔽₃)	—	57	9	no	—
86	H(3,𝔽₃)	—	171	9	no	—
87	C₃×C₉	—	3	6	no	—
88	C₃×C₉	—	19	18	no	—
89	C₃×C₉	—	57	27	no	—
90	C₃×C₉	C₃²⋊(C₁₉⋊C₃)	1	9	yes	complemented
91	C₃×C₉	C₃²×C₁₉⋊C₃	1	6	yes	—
92	C₃×C₉	C₃²×C₁₉⋊C₃	1	3	yes	—
93	C₁₉	#10624882	1	1	yes	characteristic, complemented, Sylow
94	C₃²	—	3	3	no	—
95	C₃²	—	3	3	no	—
96	C₃²	—	3	3	no	—
97	C₃²	—	3	6	no	—
98	C₃²	—	9	9	no	—
99	C₃²	—	57	9	no	—
100	C₃²	—	57	27	no	—
101	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	3	yes	—
102	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
103	C₉	—	3	6	no	—
104	C₉	—	57	18	no	—
105	C₉	C₃×C₃²⋊(C₁₉⋊C₃)	1	9	yes	—
106	C₃	—	3	3	no	—
107	C₃	—	9	3	no	—
108	C₃	—	171	9	no	—
109	C₃	C₃²×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
110	C₃	#17987546	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#13095332

C₃

yes

maximal, complemented

#17987431

C₃

yes

maximal, complemented

#17987422

C₃

yes

maximal, complemented

#17987421

C₃

yes

maximal, complemented

#17987417

C₃

yes

maximal, characteristic

C₃²×C₁₉×3₋¹⁺²

C₃

yes

maximal, characteristic, complemented

#13095332

—

C₃×#10587025

C₃²

yes

—

C₃×#10587022

C₃²

yes

complemented

C₃×#10587023

C₃²

yes

complemented

C₃×C₃²⋊(C₁₉⋊C₃)

C₃²

yes

—

C₃×C₃²⋊(C₁₉⋊C₃)

C₃²

yes

—

C₃×C₉×C₁₉⋊C₃

C₃²

yes

complemented

C₃²⋊#4294

C₃²

yes

—

#13095303

C₃²

yes

complemented

C₃⁴×C₁₉

C₃²

yes

characteristic

C₃×C₁₉×3₋¹⁺²

—

C₃×C₁₉×3₋¹⁺²

C₃²

yes

—

C₃²×C₉×C₁₉

C₃²

yes

—

C₃²×C₉×C₁₉

C₃²

yes

characteristic

#10624882

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

C₃³

yes

—

C₃³×C₁₉

C₃³

yes

characteristic

C₃²×C₁₉⋊C₃

C₃³

yes

—

C₁₉×3₋¹⁺²

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

H(3,𝔽₃)

yes

complemented

C₃×C₉×C₁₉

C₃³

yes

—

C₃×C₉×C₁₉

C₃³

yes

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

#10587025

—

C₃×#4294

C₃³

yes

—

#10587023

—

#10587022

—

C₉×C₁₉⋊C₃

—

C₃×#737

—

C₃²×3₋¹⁺²

C₁₉⋊C₃

yes

characteristic, complemented

#6893

—

C₉⋊H(3,𝔽₃)

—

#6891

—

C₃²⋊H(3,𝔽₃)

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

—

C₃²×C₁₉

C₃⁴

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₉×C₁₉

—

C₉×C₁₉

C₃×H(3,𝔽₃)

yes

—

#4294

—

C₃⁴

C₃×C₁₉⋊C₃

yes

characteristic

#737

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₁₉⋊C₃

yes

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

C₃×C₁₉⋊C₃

yes

—

C₃²×C₉

C₃×C₁₉⋊C₃

yes

characteristic

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

C₃²×H(3,𝔽₃)

yes

characteristic

C₃×C₁₉

3₋¹⁺⁴

yes

—

C₁₉⋊C₃

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²×C₁₉⋊C₃

yes

—

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

171

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃²⋊(C₁₉⋊C₃)

yes

complemented

C₃×C₉

C₃²×C₁₉⋊C₃

yes

—

C₃×C₉

C₃²×C₁₉⋊C₃

yes

—

C₁₉

#10624882

yes

characteristic, complemented, Sylow

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

—

102

C₃²

C₃³×C₁₉⋊C₃

yes

characteristic

103

C₉

—

104

C₉

—

105

C₉

C₃×C₃²⋊(C₁₉⋊C₃)

yes

—

106

C₃

—

107

C₃

—

108

C₃

—

171

—

109

C₃

C₃²×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

110

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	#17987417	C₃
hypercenter	C₃²×3₋¹⁺²	C₁₉⋊C₃
hypocenter	C₁₉	#10624882
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

#17987417

C₃

hypercenter

C₃²×3₋¹⁺²

C₁₉⋊C₃

C₁₉

G ▷ C₃²×C₁₉ ▷ C₁₉

C₁ ◁ C₃² ◁ C₃²×3₋¹⁺²

N	H ≅ G/N	type
1. C₃×#13095332	C₃	semidirect
2. #17987431	C₃	semidirect
3. #17987422	C₃	semidirect
4. #17987421	C₃	semidirect
6. C₃²×C₁₉×3₋¹⁺²	C₃	semidirect
9. C₃×#10587022	C₃²	semidirect
10. C₃×#10587023	C₃²	semidirect
13. C₃×C₉×C₁₉⋊C₃	C₃²	semidirect
15. #13095303	C₃²	semidirect
29. C₃×C₉×C₁₉	H(3,𝔽₃)	semidirect
40. C₃²×3₋¹⁺²	C₁₉⋊C₃	semidirect
90. C₃×C₉	C₃²⋊(C₁₉⋊C₃)	semidirect
93. C₁₉	#10624882	semidirect

H ≅ G/N

type

1. C₃×#13095332

C₃

semidirect

2. #17987431

C₃

semidirect

3. #17987422

C₃

semidirect

4. #17987421

C₃

semidirect

6. C₃²×C₁₉×3₋¹⁺²

C₃

semidirect

9. C₃×#10587022

C₃²

semidirect

10. C₃×#10587023

C₃²

semidirect

13. C₃×C₉×C₁₉⋊C₃

C₃²

semidirect

15. #13095303

C₃²

semidirect

29. C₃×C₉×C₁₉

H(3,𝔽₃)

semidirect

40. C₃²×3₋¹⁺²

C₁₉⋊C₃

semidirect

90. C₃×C₉

C₃²⋊(C₁₉⋊C₃)

semidirect

93. C₁₉

#10624882

semidirect

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

subgroup	index
C₃×#13095332	3
#17987431	3
#17987422	3
#17987421	3
#17987417	3
C₃²×C₁₉×3₋¹⁺²	3
#10624882	19

subgroup

index

C₃²×C₁₉×3₋¹⁺²

Subgroups — #28356059

Conjugacy classes of subgroups

Complements

Maximal subgroups