Group #28356899 — subgroups

Subgroups — #28356899

Order 13851.

404 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 87 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×C₃²⋊#4296	C₃	1	3	yes	maximal
2	#17987559	C₃	1	9	yes	maximal, complemented
3	C₃×3₋¹⁺²×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic
4	C₃²×#4296	—	3	3	no	—
5	C₃²⋊#4296	—	3	6	no	—
6	C₃²⋊#4296	C₃²	1	9	yes	—
7	C₃³×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
8	C₃×#10587025	C₉	1	2	yes	complemented
9	C₃×#10587022	C₉	1	3	yes	complemented
10	C₃×#10587023	C₉	1	3	yes	complemented
11	3₋¹⁺²×C₁₉⋊C₃	—	3	3	no	—
12	3₋¹⁺²×C₁₉⋊C₃	—	3	6	no	—
13	C₃×C₉×C₁₉⋊C₃	C₃²	1	3	yes	—
14	C₃×C₁₉×3₋¹⁺²	C₉	1	1	yes	characteristic, complemented
15	#10624690	—	19	1	no	maximal, Sylow
16	C₃³×C₁₉	C₃×C₉	1	1	yes	characteristic
17	C₃²×C₁₉⋊C₃	—	9	3	no	—
18	C₃²×C₁₉⋊C₃	C₃×C₉	1	1	yes	characteristic
19	C₃²×C₁₉⋊C₃	C₃×C₉	1	1	yes	characteristic
20	C₃²×C₁₉⋊C₃	H(3,𝔽₃)	1	3	yes	—
21	C₃²×C₁₉⋊C₃	3₋¹⁺²	1	6	yes	—
22	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
23	C₃×#4296	—	3	9	no	—
24	C₃×#4296	—	9	9	no	—
25	C₁₉×3₋¹⁺²	—	3	3	no	—
26	C₃×C₉×C₁₉	C₃×C₉	1	3	yes	complemented
27	#10587025	—	3	6	no	—
28	C₃×#4294	C₃×C₉	1	6	yes	complemented
29	#10587023	—	3	9	no	—
30	#10587022	—	3	9	no	—
31	C₉×C₁₉⋊C₃	—	3	3	no	—
32	C₉×C₁₉⋊C₃	—	3	6	no	—
33	C₃²×3₋¹⁺²	—	19	1	no	—
34	C₃×C₃²⋊C₉	—	19	3	no	—
35	#6867	—	19	9	no	—
36	C₃²×C₁₉	—	9	1	no	—
37	C₃²×C₁₉	C₃²⋊C₉	1	3	yes	—
38	C₃²×C₁₉	C₃²×C₉	1	1	yes	characteristic
39	C₃×C₁₉⋊C₃	—	3	3	no	—
40	C₃×C₁₉⋊C₃	—	3	6	no	—
41	C₃×C₁₉⋊C₃	—	3	3	no	—
42	C₃×C₁₉⋊C₃	—	9	1	no	—
43	C₃×C₁₉⋊C₃	—	9	1	no	—
44	C₃×C₁₉⋊C₃	—	9	6	no	—
45	C₃×C₁₉⋊C₃	C₃²⋊C₉	1	3	yes	—
46	C₃×C₁₉⋊C₃	C₃²⋊C₉	1	3	yes	—
47	C₃×C₁₉⋊C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
48	C₃×C₁₉⋊C₃	C₃×3₋¹⁺²	1	2	yes	—
49	#4296	—	9	9	no	—
50	C₉×C₁₉	—	3	3	no	—
51	#4294	—	3	6	no	—
52	C₃⁴	—	19	1	no	—
53	C₃×3₋¹⁺²	—	19	2	no	—
54	C₃×3₋¹⁺²	—	19	3	no	—
55	C₃×3₋¹⁺²	—	19	3	no	—
56	C₃×3₋¹⁺²	—	57	3	no	—
57	C₃×3₋¹⁺²	—	57	6	no	—
58	C₃×3₋¹⁺²	#4296	1	1	yes	characteristic, complemented
59	C₃²×C₉	—	19	3	no	—
60	C₃²×C₉	—	57	3	no	—
61	C₃²⋊C₉	—	19	9	no	—
62	C₃²⋊C₉	—	57	6	no	—
63	C₃×C₁₉	—	3	3	no	—
64	C₃×C₁₉	—	3	1	no	—
65	C₃×C₁₉	C₃×C₃²⋊C₉	1	1	yes	characteristic
66	C₁₉⋊C₃	—	3	3	no	—
67	C₁₉⋊C₃	—	3	3	no	—
68	C₁₉⋊C₃	—	3	2	no	—
69	C₁₉⋊C₃	#6910	1	3	yes	—
70	C₃³	—	19	3	no	—
71	C₃³	—	19	1	no	—
72	C₃³	—	19	6	no	—
73	C₃³	—	19	1	no	—
74	C₃³	—	19	1	no	—
75	C₃³	—	171	3	no	—
76	C₃³	C₃×#4296	1	1	yes	characteristic
77	3₋¹⁺²	—	3	3	no	—
78	3₋¹⁺²	—	57	9	no	—
79	3₋¹⁺²	—	57	9	no	—
80	3₋¹⁺²	—	57	6	no	—
81	C₃×C₉	—	19	6	no	—
82	C₃×C₉	—	57	3	no	—
83	C₃×C₉	—	57	6	no	—
84	C₃×C₉	—	57	9	no	—
85	C₃×C₉	—	171	9	no	—
86	C₃×C₉	C₃×#4296	1	3	yes	complemented
87	C₁₉	#10624690	1	1	yes	characteristic, complemented, Sylow
88	C₃²	—	9	1	no	—
89	C₃²	—	19	1	no	—
90	C₃²	—	19	2	no	—
91	C₃²	—	19	3	no	—
92	C₃²	—	19	3	no	—
93	C₃²	—	57	3	no	—
94	C₃²	—	57	6	no	—
95	C₃²	—	57	3	no	—
96	C₃²	—	171	6	no	—
97	C₃²	—	171	1	no	—
98	C₃²	—	171	1	no	—
99	C₃²	C₃²⋊#4296	1	3	yes	—
100	C₃²	C₃²×#4296	1	1	yes	characteristic
101	C₉	—	3	3	no	—
102	C₉	—	57	6	no	—
103	C₉	—	171	9	no	—
104	C₃	—	3	3	no	—
105	C₃	—	3	1	no	—
106	C₃	—	19	3	no	—
107	C₃	—	57	2	no	—
108	C₃	—	57	3	no	—
109	C₃	—	57	3	no	—
110	C₃	C₃×C₃²⋊#4296	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₃²⋊#4296

C₃

yes

maximal

#17987559

C₃

yes

maximal, complemented

C₃×3₋¹⁺²×C₁₉⋊C₃

C₃

yes

maximal, characteristic

C₃²×#4296

—

C₃²⋊#4296

—

C₃²⋊#4296

C₃²

yes

—

C₃³×C₁₉⋊C₃

C₃²

yes

characteristic

C₃×#10587025

C₉

yes

complemented

C₃×#10587022

C₉

yes

complemented

C₃×#10587023

C₉

yes

complemented

3₋¹⁺²×C₁₉⋊C₃

—

3₋¹⁺²×C₁₉⋊C₃

—

C₃×C₉×C₁₉⋊C₃

C₃²

yes

—

C₃×C₁₉×3₋¹⁺²

C₉

yes

characteristic, complemented

#10624690

—

maximal, Sylow

C₃³×C₁₉

C₃×C₉

yes

characteristic

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

C₃×C₉

yes

characteristic

C₃²×C₁₉⋊C₃

C₃×C₉

yes

characteristic

C₃²×C₁₉⋊C₃

H(3,𝔽₃)

yes

—

C₃²×C₁₉⋊C₃

3₋¹⁺²

yes

—

C₃²×C₁₉⋊C₃

C₃³

yes

characteristic

C₃×#4296

—

C₃×#4296

—

C₁₉×3₋¹⁺²

—

C₃×C₉×C₁₉

C₃×C₉

yes

complemented

#10587025

—

C₃×#4294

C₃×C₉

yes

complemented

#10587023

—

#10587022

—

C₉×C₁₉⋊C₃

—

C₉×C₁₉⋊C₃

—

C₃²×3₋¹⁺²

—

C₃×C₃²⋊C₉

—

#6867

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃²⋊C₉

yes

—

C₃²×C₁₉

C₃²×C₉

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

C₃²⋊C₉

yes

—

C₃×C₁₉⋊C₃

C₃²⋊C₉

yes

—

C₃×C₁₉⋊C₃

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₁₉⋊C₃

C₃×3₋¹⁺²

yes

—

#4296

—

C₉×C₁₉

—

#4294

—

C₃⁴

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

#4296

yes

characteristic, complemented

C₃²×C₉

—

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

C₃×C₃²⋊C₉

yes

characteristic

C₁₉⋊C₃

—

C₁₉⋊C₃

—

C₁₉⋊C₃

—

C₁₉⋊C₃

#6910

yes

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

171

—

C₃³

C₃×#4296

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

171

—

C₃×C₉

C₃×#4296

yes

complemented

C₁₉

#10624690

yes

characteristic, complemented, Sylow

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

171

—

C₃²

—

171

—

C₃²

—

171

—

C₃²

C₃²⋊#4296

yes

—

100

C₃²

C₃²×#4296

yes

characteristic

101

C₉

—

102

C₉

—

103

C₉

—

171

—

104

C₃

—

105

C₃

—

106

C₃

—

107

C₃

—

108

C₃

—

109

C₃

—

110

C₃

C₃×C₃²⋊#4296

yes

characteristic

	subgroup	G/H
center	C₃	C₃×C₃²⋊#4296
commutator	C₃²×C₁₉	C₃²×C₉
Frattini	C₃²	C₃²×#4296
Fitting	C₃×C₁₉×3₋¹⁺²	C₉
socle	C₃×C₁₉	C₃×C₃²⋊C₉
prime radical	C₃³×C₁₉⋊C₃	C₃²
cosocle	C₃²×C₁₉⋊C₃	C₃³
hypercenter	C₃×3₋¹⁺²	#4296
hypocenter	C₁₉	#10624690
LCS	G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×3₋¹⁺²

subgroup

G/H

center

C₃

C₃×C₃²⋊#4296

commutator

C₃²×C₁₉

C₃²×C₉

Frattini

C₃²

C₃²×#4296

Fitting

C₃×C₁₉×3₋¹⁺²

C₉

socle

C₃×C₁₉

C₃×C₃²⋊C₉

prime radical

C₃³×C₁₉⋊C₃

C₃²

cosocle

C₃²×C₁₉⋊C₃

C₃³

C₃×3₋¹⁺²

C₁₉

G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃ ◁ C₃³ ◁ C₃×3₋¹⁺²

N	H ≅ G/N	type
2. #17987559	C₃	semidirect
8. C₃×#10587025	C₉	semidirect
9. C₃×#10587022	C₉	semidirect
10. C₃×#10587023	C₉	semidirect
14. C₃×C₁₉×3₋¹⁺²	C₉	semidirect
26. C₃×C₉×C₁₉	C₃×C₉	semidirect
28. C₃×#4294	C₃×C₉	semidirect
58. C₃×3₋¹⁺²	#4296	semidirect
86. C₃×C₉	C₃×#4296	semidirect
87. C₁₉	#10624690	semidirect

H ≅ G/N

type

2. #17987559

C₃

semidirect

8. C₃×#10587025

C₉

semidirect

9. C₃×#10587022

C₉

semidirect

10. C₃×#10587023

C₉

semidirect

14. C₃×C₁₉×3₋¹⁺²

C₉

semidirect

26. C₃×C₉×C₁₉

C₃×C₉

semidirect

28. C₃×#4294

C₃×C₉

semidirect

58. C₃×3₋¹⁺²

#4296

semidirect

86. C₃×C₉

C₃×#4296

semidirect

87. C₁₉

#10624690

semidirect

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

subgroup	index
C₃×C₃²⋊#4296	3
#17987559	3
C₃×3₋¹⁺²×C₁₉⋊C₃	3
#10624690	19

subgroup

index

C₃×C₃²⋊#4296

#17987559

C₃×3₋¹⁺²×C₁₉⋊C₃

#10624690

Subgroups — #28356899

Conjugacy classes of subgroups

Complements

Maximal subgroups