Group #28356051 — subgroups

Subgroups — #28356051

Order 13851.

1124 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 1124 conjugacy classes of subgroups — too many to draw. Only its 274 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	C₃×3₋¹⁺²×C₁₉⋊C₃	C₃	1	3	yes	maximal, complemented
2	C₃²×C₃²⋊(C₁₉⋊C₃)	C₃	1	1	yes	maximal, characteristic
3	#17987430	C₃	1	27	yes	maximal, complemented
4	#17987421	C₃	1	2	yes	maximal, complemented
5	#17987418	C₃	1	6	yes	maximal, complemented
6	C₁₉×#6891	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃³×C₁₉⋊C₃	C₃²	1	2	yes	—
8	C₃³×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
9	C₃×#10587025	C₃²	1	6	yes	—
10	C₃×#10587022	C₃²	1	9	yes	complemented
11	C₃×#10587023	C₃²	1	9	yes	complemented
12	3₋¹⁺²×C₁₉⋊C₃	—	3	27	no	—
13	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	9	no	—
14	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	9	yes	—
15	C₃×C₉×C₁₉⋊C₃	C₃²	1	9	yes	complemented
16	C₃²⋊#4294	C₃²	1	18	yes	complemented
17	#13095303	C₃²	1	54	yes	complemented
18	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
19	C₃×C₁₉×3₋¹⁺²	C₃²	1	3	yes	—
20	C₁₉×C₃²⋊C₉	C₃²	1	9	yes	complemented
21	#10624881	—	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	—	3	6	no	—
27	C₃²×C₁₉⋊C₃	—	9	3	no	—
28	C₃²×C₁₉⋊C₃	—	9	6	no	—
29	C₃²×C₁₉⋊C₃	H(3,𝔽₃)	1	9	yes	—
30	C₃²×C₁₉⋊C₃	C₃³	1	6	yes	—
31	C₃²×C₁₉⋊C₃	C₃³	1	3	yes	—
32	C₁₉×3₋¹⁺²	—	3	9	no	—
33	C₃×C₉×C₁₉	C₃³	1	9	yes	—
34	C₃²⋊(C₁₉⋊C₃)	—	3	27	no	—
35	#10587025	—	3	18	no	—
36	C₃×#4294	C₃³	1	18	yes	—
37	#10587023	—	3	27	no	—
38	#10587022	—	3	27	no	—
39	C₉×C₁₉⋊C₃	—	3	27	no	—
40	C₃²×3₋¹⁺²	—	19	3	no	—
41	C₃²×H(3,𝔽₃)	—	19	1	no	—
42	#6893	—	19	27	no	—
43	#6891	—	19	2	no	—
44	#6891	—	19	6	no	—
45	#6891	C₁₉⋊C₃	1	1	yes	characteristic, complemented
46	C₃²×C₁₉	—	3	3	no	—
47	C₃²×C₁₉	—	3	3	no	—
48	C₃²×C₁₉	—	3	9	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	6	no	—
54	C₃×C₁₉⋊C₃	—	3	18	no	—
55	C₃×C₁₉⋊C₃	—	9	9	no	—
56	C₃×C₁₉⋊C₃	—	9	18	no	—
57	C₃×C₁₉⋊C₃	C₃×H(3,𝔽₃)	1	9	yes	—
58	C₉×C₁₉	—	3	9	no	—
59	#4294	—	3	18	no	—
60	C₃⁴	—	19	1	no	—
61	C₃⁴	—	19	2	no	—
62	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
63	C₃×3₋¹⁺²	—	19	6	no	—
64	C₃×3₋¹⁺²	—	19	9	no	—
65	C₃×3₋¹⁺²	—	19	9	no	—
66	C₃×3₋¹⁺²	—	57	27	no	—
67	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	3	yes	—
68	C₃×H(3,𝔽₃)	—	19	9	no	—
69	C₃×H(3,𝔽₃)	—	57	9	no	—
70	C₃²×C₉	—	19	9	no	—
71	C₃²⋊C₉	—	19	18	no	—
72	C₃²⋊C₉	—	19	54	no	—
73	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	9	yes	complemented
74	C₃×C₁₉	—	3	3	no	—
75	C₃×C₁₉	—	9	3	no	—
76	C₃×C₁₉	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
77	C₃×C₁₉	3₋¹⁺⁴	1	3	yes	—
78	C₁₉⋊C₃	—	3	9	no	—
79	C₁₉⋊C₃	—	9	6	no	—
80	C₃³	—	3	3	no	—
81	C₃³	—	9	3	no	—
82	C₃³	—	19	3	no	—
83	C₃³	—	19	9	no	—
84	C₃³	—	19	6	no	—
85	C₃³	—	57	6	no	—
86	C₃³	—	171	3	no	—
87	C₃³	—	171	6	no	—
88	C₃³	C₃²×C₁₉⋊C₃	1	3	yes	—
89	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
90	3₋¹⁺²	—	3	9	no	—
91	3₋¹⁺²	—	57	27	no	—
92	3₋¹⁺²	—	57	27	no	—
93	3₋¹⁺²	—	57	18	no	—
94	H(3,𝔽₃)	—	57	27	no	—
95	C₃×C₉	—	19	18	no	—
96	C₃×C₉	—	57	27	no	—
97	C₃×C₉	C₃²×C₁₉⋊C₃	1	9	yes	—
98	C₁₉	#10624881	1	1	yes	characteristic, complemented, Sylow
99	C₃²	—	3	3	no	—
100	C₃²	—	3	3	no	—
101	C₃²	—	3	9	no	—
102	C₃²	—	9	9	no	—
103	C₃²	—	19	9	no	—
104	C₃²	—	57	9	no	—
105	C₃²	—	57	6	no	—
106	C₃²	—	57	18	no	—
107	C₃²	—	171	9	no	—
108	C₃²	—	171	18	no	—
109	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	3	yes	—
110	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
111	C₉	—	3	9	no	—
112	C₉	—	57	18	no	—
113	C₃	—	3	3	no	—
114	C₃	—	9	3	no	—
115	C₃	—	57	9	no	—
116	C₃	—	171	6	no	—
117	C₃	C₃×H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
118	C₃	#17987546	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×3₋¹⁺²×C₁₉⋊C₃

C₃

yes

maximal, complemented

C₃²×C₃²⋊(C₁₉⋊C₃)

C₃

yes

maximal, characteristic

#17987430

C₃

yes

maximal, complemented

#17987421

C₃

yes

maximal, complemented

#17987418

C₃

yes

maximal, complemented

C₁₉×#6891

C₃

yes

maximal, characteristic, complemented

C₃³×C₁₉⋊C₃

C₃²

yes

—

C₃³×C₁₉⋊C₃

C₃²

yes

characteristic

C₃×#10587025

C₃²

yes

—

C₃×#10587022

C₃²

yes

complemented

C₃×#10587023

C₃²

yes

complemented

3₋¹⁺²×C₁₉⋊C₃

—

C₃×C₃²⋊(C₁₉⋊C₃)

—

C₃×C₃²⋊(C₁₉⋊C₃)

C₃²

yes

—

C₃×C₉×C₁₉⋊C₃

C₃²

yes

complemented

C₃²⋊#4294

C₃²

yes

complemented

#13095303

C₃²

yes

complemented

C₃⁴×C₁₉

C₃²

yes

characteristic

C₃×C₁₉×3₋¹⁺²

C₃²

yes

—

C₁₉×C₃²⋊C₉

C₃²

yes

complemented

#10624881

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

C₃³

yes

—

C₃³×C₁₉

C₃³

yes

characteristic

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

H(3,𝔽₃)

yes

—

C₃²×C₁₉⋊C₃

C₃³

yes

—

C₃²×C₁₉⋊C₃

C₃³

yes

—

C₁₉×3₋¹⁺²

—

C₃×C₉×C₁₉

C₃³

yes

—

C₃²⋊(C₁₉⋊C₃)

—

#10587025

—

C₃×#4294

C₃³

yes

—

#10587023

—

#10587022

—

C₉×C₁₉⋊C₃

—

C₃²×3₋¹⁺²

—

C₃²×H(3,𝔽₃)

—

—

—

—

C₁₉⋊C₃

yes

characteristic, complemented

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

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

C₃×H(3,𝔽₃)

yes

—

C₉×C₁₉

—

#4294

—

C₃⁴

—

C₃⁴

—

C₃⁴

C₃×C₁₉⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₁₉⋊C₃

yes

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₁₉⋊C₃

yes

complemented

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

C₃²×H(3,𝔽₃)

yes

characteristic

C₃×C₁₉

3₋¹⁺⁴

yes

—

C₁₉⋊C₃

—

C₁₉⋊C₃

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

171

—

C₃³

—

171

—

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)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃²×C₁₉⋊C₃

yes

—

C₁₉

#10624881

yes

characteristic, complemented, Sylow

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

171

—

108

C₃²

—

171

—

109

C₃²

H(3,𝔽₃)×C₁₉⋊C₃

yes

—

110

C₃²

C₃³×C₁₉⋊C₃

yes

characteristic

111

C₉

—

112

C₉

—

113

C₃

—

114

C₃

—

115

C₃

—

116

C₃

—

171

—

117

C₃

C₃×H(3,𝔽₃)×C₁₉⋊C₃

yes

characteristic

118

C₃

#17987546

yes

—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₁₉⋊C₃
commutator, socle, cosocle	C₃²×C₁₉	C₃⁴
Fitting	C₁₉×#6891	C₃
prime radical	C₃²×C₃²⋊(C₁₉⋊C₃)	C₃
hypercenter	#6891	C₁₉⋊C₃
hypocenter	C₁₉	#10624881
LCS	G ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ #6891

subgroup

G/H

center, Frattini

C₃²

C₃³×C₁₉⋊C₃

commutator, socle, cosocle

C₃²×C₁₉

C₃⁴

Fitting

C₁₉×#6891

C₃

prime radical

C₃²×C₃²⋊(C₁₉⋊C₃)

C₃

C₁₉⋊C₃

C₁₉

G ▷ C₃²×C₁₉ ▷ C₁₉

C₁ ◁ C₃² ◁ #6891

N	H ≅ G/N	type
1. C₃×3₋¹⁺²×C₁₉⋊C₃	C₃	semidirect
3. #17987430	C₃	semidirect
4. #17987421	C₃	semidirect
5. #17987418	C₃	semidirect
6. C₁₉×#6891	C₃	semidirect
10. C₃×#10587022	C₃²	semidirect
11. C₃×#10587023	C₃²	semidirect
15. C₃×C₉×C₁₉⋊C₃	C₃²	semidirect
16. C₃²⋊#4294	C₃²	semidirect
17. #13095303	C₃²	semidirect
20. C₁₉×C₃²⋊C₉	C₃²	semidirect
45. #6891	C₁₉⋊C₃	semidirect
73. C₃²⋊C₉	C₃×C₁₉⋊C₃	semidirect
98. C₁₉	#10624881	semidirect

H ≅ G/N

type

1. C₃×3₋¹⁺²×C₁₉⋊C₃

C₃

semidirect

3. #17987430

C₃

semidirect

4. #17987421

C₃

semidirect

5. #17987418

C₃

semidirect

6. C₁₉×#6891

C₃

semidirect

10. C₃×#10587022

C₃²

semidirect

11. C₃×#10587023

C₃²

semidirect

15. C₃×C₉×C₁₉⋊C₃

C₃²

semidirect

16. C₃²⋊#4294

C₃²

semidirect

17. #13095303

C₃²

semidirect

20. C₁₉×C₃²⋊C₉

C₃²

semidirect

45. #6891

C₁₉⋊C₃

semidirect

73. C₃²⋊C₉

C₃×C₁₉⋊C₃

semidirect

98. C₁₉

#10624881

semidirect

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

subgroup	index
C₃×3₋¹⁺²×C₁₉⋊C₃	3
C₃²×C₃²⋊(C₁₉⋊C₃)	3
#17987430	3
#17987421	3
#17987418	3
C₁₉×#6891	3
#10624881	19

subgroup

index

C₃×3₋¹⁺²×C₁₉⋊C₃

C₃²×C₃²⋊(C₁₉⋊C₃)

C₁₉×#6891

Subgroups — #28356051

Conjugacy classes of subgroups

Complements

Maximal subgroups