Group #28356050 — subgroups

Subgroups — #28356050

Order 13851.

1124 conjugacy classes of subgroups; only the 292 normal subgroups are shown.

normal subgroups only

This group has 1124 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×#10587025	C₃	1	2	yes	maximal, complemented
2	C₃²×C₃²⋊(C₁₉⋊C₃)	C₃	1	1	yes	maximal, characteristic
3	#17987432	C₃	1	18	yes	maximal, complemented
4	#17987431	C₃	1	9	yes	maximal, complemented
5	#17987418	C₃	1	9	yes	maximal, complemented
6	C₃²×C₁₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃³×C₁₉⋊C₃	C₃²	1	3	yes	—
8	C₃×#10587025	—	3	18	no	—
9	C₃×#10587025	C₃²	1	6	yes	—
10	C₃×#10587025	C₃²	1	6	yes	complemented
11	C₃×#10587025	C₃²	1	6	yes	complemented
12	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	9	no	—
13	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	3	yes	—
14	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	6	yes	—
15	C₃²×#4294	C₃²	1	6	yes	complemented
16	#13095303	C₃²	1	81	yes	complemented
17	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
18	C₃×C₁₉×3₋¹⁺²	—	3	9	no	—
19	C₃×C₁₉×3₋¹⁺²	C₃²	1	3	yes	—
20	C₃×C₁₉×3₋¹⁺²	C₃²	1	6	yes	complemented
21	C₃²×C₉×C₁₉	C₃²	1	3	yes	complemented
22	#10624881	—	19	1	no	maximal, Sylow
23	C₃³×C₁₉	—	9	3	no	—
24	C₃³×C₁₉	H(3,𝔽₃)	1	9	yes	—
25	C₃³×C₁₉	C₃³	1	1	yes	characteristic
26	C₃³×C₁₉	C₃³	1	1	yes	characteristic
27	C₃³×C₁₉	C₃³	1	2	yes	—
28	C₃²×C₁₉⋊C₃	—	3	9	no	—
29	C₃²×C₁₉⋊C₃	—	9	9	no	—
30	C₃²×C₁₉⋊C₃	C₃³	1	9	yes	—
31	C₁₉×3₋¹⁺²	—	3	9	no	—
32	C₁₉×3₋¹⁺²	—	3	18	no	—
33	C₃×C₉×C₁₉	—	3	9	no	—
34	C₃×C₉×C₁₉	C₃³	1	9	yes	—
35	C₃²⋊(C₁₉⋊C₃)	—	3	18	no	—
36	C₃²⋊(C₁₉⋊C₃)	—	3	9	no	—
37	#10587025	—	3	18	no	—
38	#10587025	—	3	18	no	—
39	#10587025	—	3	18	no	—
40	C₃×#4294	—	3	18	no	—
41	C₃×#4294	C₃³	1	18	yes	—
42	C₃²×3₋¹⁺²	—	19	2	no	—
43	C₃²×3₋¹⁺²	C₁₉⋊C₃	1	1	yes	characteristic, complemented
44	C₃²×H(3,𝔽₃)	—	19	1	no	—
45	#6893	—	19	18	no	—
46	#6893	—	19	9	no	—
47	#6891	—	19	9	no	—
48	C₃²×C₁₉	—	3	3	no	—
49	C₃²×C₁₉	—	3	3	no	—
50	C₃²×C₁₉	—	3	6	no	—
51	C₃²×C₁₉	—	9	9	no	—
52	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	6	yes	—
53	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	3	yes	—
54	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	3	yes	—
55	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
56	C₃×C₁₉⋊C₃	—	3	9	no	—
57	C₃×C₁₉⋊C₃	—	3	27	no	—
58	C₃×C₁₉⋊C₃	—	9	27	no	—
59	C₉×C₁₉	—	3	9	no	—
60	#4294	—	3	18	no	—
61	C₃⁴	—	19	3	no	—
62	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
63	C₃×3₋¹⁺²	—	3	9	no	—
64	C₃×3₋¹⁺²	—	19	6	no	—
65	C₃×3₋¹⁺²	—	19	6	no	—
66	C₃×3₋¹⁺²	—	19	6	no	—
67	C₃×3₋¹⁺²	—	57	18	no	—
68	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	3	yes	—
69	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	6	yes	complemented
70	C₃×H(3,𝔽₃)	—	19	6	no	—
71	C₃×H(3,𝔽₃)	—	19	3	no	—
72	C₃×H(3,𝔽₃)	—	57	9	no	—
73	C₃²×C₉	—	19	6	no	—
74	C₃²×C₉	C₃×C₁₉⋊C₃	1	3	yes	complemented
75	C₃²⋊C₉	—	19	81	no	—
76	C₃×C₁₉	—	3	3	no	—
77	C₃×C₁₉	—	3	6	no	—
78	C₃×C₁₉	—	3	3	no	—
79	C₃×C₁₉	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
80	C₃×C₁₉	3₋¹⁺⁴	1	3	yes	—
81	C₁₉⋊C₃	—	9	9	no	—
82	C₃³	—	9	3	no	—
83	C₃³	—	19	9	no	—
84	C₃³	—	57	9	no	—
85	C₃³	—	171	9	no	—
86	C₃³	C₃²⋊(C₁₉⋊C₃)	1	9	yes	—
87	C₃³	C₃²×C₁₉⋊C₃	1	2	yes	—
88	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
89	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
90	3₋¹⁺²	—	3	9	no	—
91	3₋¹⁺²	—	3	18	no	—
92	3₋¹⁺²	—	57	18	no	—
93	3₋¹⁺²	—	57	18	no	—
94	3₋¹⁺²	—	57	18	no	—
95	H(3,𝔽₃)	—	57	9	no	—
96	H(3,𝔽₃)	—	57	18	no	—
97	C₃×C₉	—	3	9	no	—
98	C₃×C₉	—	19	18	no	—
99	C₃×C₉	—	57	18	no	—
100	C₃×C₉	C₃²×C₁₉⋊C₃	1	9	yes	—
101	C₁₉	#10624881	1	1	yes	characteristic, complemented, Sylow
102	C₃²	—	3	3	no	—
103	C₃²	—	3	6	no	—
104	C₃²	—	3	3	no	—
105	C₃²	—	9	9	no	—
106	C₃²	—	57	9	no	—
107	C₃²	—	57	27	no	—
108	C₃²	—	171	27	no	—
109	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	3	yes	—
110	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	6	yes	—
111	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	3	yes	—
112	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
113	C₉	—	3	9	no	—
114	C₉	—	57	18	no	—
115	C₃	—	3	3	no	—
116	C₃	—	3	3	no	—
117	C₃	—	3	6	no	—
118	C₃	—	171	9	no	—
119	C₃	C₃²×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
120	C₃	#17987546	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#10587025

C₃

yes

maximal, complemented

C₃²×C₃²⋊(C₁₉⋊C₃)

C₃

yes

maximal, characteristic

#17987432

C₃

yes

maximal, complemented

#17987431

C₃

yes

maximal, complemented

#17987418

C₃

yes

maximal, complemented

C₃²×C₁₉×3₋¹⁺²

C₃

yes

maximal, characteristic, complemented

C₃³×C₁₉⋊C₃

C₃²

yes

—

C₃×#10587025

—

C₃×#10587025

C₃²

yes

—

C₃×#10587025

C₃²

yes

complemented

C₃×#10587025

C₃²

yes

complemented

C₃×C₃²⋊(C₁₉⋊C₃)

—

C₃×C₃²⋊(C₁₉⋊C₃)

C₃²

yes

—

C₃×C₃²⋊(C₁₉⋊C₃)

C₃²

yes

—

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

—

C₃×C₁₉×3₋¹⁺²

C₃²

yes

complemented

C₃²×C₉×C₁₉

C₃²

yes

complemented

#10624881

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

H(3,𝔽₃)

yes

—

C₃³×C₁₉

C₃³

yes

characteristic

C₃³×C₁₉

C₃³

yes

characteristic

C₃³×C₁₉

C₃³

yes

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

C₃³

yes

—

C₁₉×3₋¹⁺²

—

C₁₉×3₋¹⁺²

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

C₃³

yes

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

—

—

—

—

C₃³

yes

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₁₉⋊C₃

yes

characteristic, complemented

C₃²×H(3,𝔽₃)

—

#6893

—

#6893

—

#6891

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

—

C₃²×C₁₉

C₃⁴

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₉×C₁₉

—

#4294

—

C₃⁴

—

C₃⁴

C₃×C₁₉⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₁₉⋊C₃

yes

—

C₃×3₋¹⁺²

C₃×C₁₉⋊C₃

yes

complemented

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

C₃×C₁₉⋊C₃

yes

complemented

C₃²⋊C₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

C₃²×H(3,𝔽₃)

yes

characteristic

C₃×C₁₉

3₋¹⁺⁴

yes

—

C₁₉⋊C₃

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

171

—

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

—

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

—

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

100

C₃×C₉

C₃²×C₁₉⋊C₃

yes

—

101

C₁₉

#10624881

yes

characteristic, complemented, Sylow

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

171

—

109

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

—

110

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

—

111

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

—

112

C₃²

C₃³×C₁₉⋊C₃

yes

characteristic

113

C₉

—

114

C₉

—

115

C₃

—

116

C₃

—

117

C₃

—

118

C₃

—

171

—

119

C₃

C₃²×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

120

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	C₃²×C₃²⋊(C₁₉⋊C₃)	C₃
hypercenter	C₃²×3₋¹⁺²	C₁₉⋊C₃
hypocenter	C₁₉	#10624881
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

C₃²×C₃²⋊(C₁₉⋊C₃)

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. C₃²×#10587025	C₃	semidirect
3. #17987432	C₃	semidirect
4. #17987431	C₃	semidirect
5. #17987418	C₃	semidirect
6. C₃²×C₁₉×3₋¹⁺²	C₃	semidirect
10. C₃×#10587025	C₃²	semidirect
11. C₃×#10587025	C₃²	semidirect
15. C₃²×#4294	C₃²	semidirect
16. #13095303	C₃²	semidirect
20. C₃×C₁₉×3₋¹⁺²	C₃²	semidirect
21. C₃²×C₉×C₁₉	C₃²	semidirect
43. C₃²×3₋¹⁺²	C₁₉⋊C₃	semidirect
69. C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	semidirect
74. C₃²×C₉	C₃×C₁₉⋊C₃	semidirect
101. C₁₉	#10624881	semidirect

H ≅ G/N

type

1. C₃²×#10587025

C₃

semidirect

3. #17987432

C₃

semidirect

4. #17987431

C₃

semidirect

5. #17987418

C₃

semidirect

6. C₃²×C₁₉×3₋¹⁺²

C₃

semidirect

10. C₃×#10587025

C₃²

semidirect

11. C₃×#10587025

C₃²

semidirect

15. C₃²×#4294

C₃²

semidirect

16. #13095303

C₃²

semidirect

20. C₃×C₁₉×3₋¹⁺²

C₃²

semidirect

21. C₃²×C₉×C₁₉

C₃²

semidirect

43. C₃²×3₋¹⁺²

C₁₉⋊C₃

semidirect

69. C₃×3₋¹⁺²

C₃×C₁₉⋊C₃

semidirect

74. C₃²×C₉

C₃×C₁₉⋊C₃

semidirect

101. C₁₉

#10624881

semidirect

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

subgroup	index
C₃²×#10587025	3
C₃²×C₃²⋊(C₁₉⋊C₃)	3
#17987432	3
#17987431	3
#17987418	3
C₃²×C₁₉×3₋¹⁺²	3
#10624881	19

subgroup

index

C₃²×C₃²⋊(C₁₉⋊C₃)

C₃²×C₁₉×3₋¹⁺²

Subgroups — #28356050

Conjugacy classes of subgroups

Complements

Maximal subgroups