Group #28353552 — subgroups

Subgroups — #28353552

Order 13851.

608 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 166 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	1	yes	maximal, characteristic
2	C₃×C₃²⋊#4294	C₃	1	2	yes	maximal
3	#17987308	C₃	1	9	yes	maximal, complemented
4	C₃³×C₉×C₁₉	C₃	1	1	yes	maximal, characteristic
5	C₃×#10587025	—	3	3	no	—
6	C₃×#10587025	—	3	3	no	—
7	C₃×#10587025	—	3	3	no	—
8	C₃×#10587025	C₉	1	9	yes	complemented
9	C₃²×#4294	C₃²	1	6	yes	—
10	C₃²×#4294	C₃²	1	3	yes	—
11	C₃²⋊#4294	—	3	6	no	—
12	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
13	C₃²×C₉×C₁₉	—	3	3	no	—
14	C₃²×C₉×C₁₉	—	3	9	no	—
15	C₃²×C₉×C₁₉	C₃²	1	3	yes	—
16	#10624558	—	19	1	no	maximal, Sylow
17	C₃³×C₁₉	—	3	3	no	—
18	C₃³×C₁₉	—	3	6	no	—
19	C₃³×C₁₉	C₃×C₉	1	3	yes	—
20	C₃³×C₁₉	H(3,𝔽₃)	1	3	yes	—
21	C₃³×C₁₉	3₋¹⁺²	1	3	yes	—
22	C₃³×C₁₉	3₋¹⁺²	1	3	yes	—
23	C₃³×C₁₉	C₃³	1	1	yes	characteristic
24	C₃×C₉×C₁₉	—	3	27	no	—
25	C₃×C₉×C₁₉	—	3	3	no	—
26	C₃×C₉×C₁₉	—	3	6	no	—
27	C₃×C₉×C₁₉	3₋¹⁺²	1	9	yes	complemented
28	#10587025	—	3	27	no	—
29	C₃×#4294	—	3	3	no	—
30	C₃×#4294	—	3	3	no	—
31	C₃×#4294	—	3	3	no	—
32	C₃×#4294	—	3	6	no	—
33	C₃×#4294	—	3	18	no	—
34	C₃×#4294	C₃×C₉	1	9	yes	complemented
35	C₃²×3₋¹⁺²	—	19	1	no	—
36	C₃³×C₉	C₁₉⋊C₃	1	1	yes	characteristic
37	C₃×C₃²⋊C₉	—	19	2	no	—
38	#6855	—	19	9	no	—
39	C₃²×C₁₉	—	3	9	no	—
40	C₃²×C₁₉	—	3	3	no	—
41	C₃²×C₁₉	—	3	6	no	—
42	C₃²×C₁₉	—	3	9	no	—
43	C₃²×C₁₉	—	3	9	no	—
44	C₃²×C₁₉	C₃²⋊C₉	1	9	yes	—
45	C₃²×C₁₉	C₃²×C₉	1	1	yes	characteristic
46	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
47	C₃²×C₁₉	C₃×3₋¹⁺²	1	1	yes	characteristic
48	C₃²×C₁₉	C₃×3₋¹⁺²	1	1	yes	characteristic
49	C₃²×C₁₉	C₃×3₋¹⁺²	1	3	yes	—
50	C₃²×C₁₉	#737	1	6	yes	—
51	C₉×C₁₉	—	3	9	no	—
52	#4294	—	3	9	no	—
53	#4294	—	9	6	no	—
54	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
55	C₃×3₋¹⁺²	—	19	9	no	—
56	C₃×3₋¹⁺²	—	57	3	no	—
57	C₃×3₋¹⁺²	—	57	3	no	—
58	C₃×3₋¹⁺²	—	57	3	no	—
59	C₃²×C₉	—	3	3	no	—
60	C₃²×C₉	—	3	9	no	—
61	C₃²×C₉	—	19	3	no	—
62	C₃²×C₉	—	19	6	no	—
63	C₃²×C₉	C₃×C₁₉⋊C₃	1	3	yes	—
64	C₃²⋊C₉	—	57	6	no	—
65	C₃×C₁₉	—	3	9	no	—
66	C₃×C₁₉	C₃×C₃²⋊C₉	1	1	yes	characteristic
67	C₃×C₁₉	C₉×3₋¹⁺²	1	3	yes	—
68	C₃×C₁₉	#6891	1	1	yes	characteristic
69	C₃×C₁₉	#6893	1	2	yes	—
70	C₃×C₁₉	#6894	1	3	yes	—
71	C₃×C₁₉	#6894	1	3	yes	—
72	C₃³	—	3	3	no	—
73	C₃³	—	3	6	no	—
74	C₃³	C₉×C₁₉⋊C₃	1	3	yes	—
75	C₃³	#10587022	1	3	yes	—
76	C₃³	#10587023	1	3	yes	—
77	C₃³	C₃²⋊(C₁₉⋊C₃)	1	3	yes	—
78	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
79	3₋¹⁺²	—	57	27	no	—
80	C₃×C₉	—	3	27	no	—
81	C₃×C₉	—	3	3	no	—
82	C₃×C₉	—	3	6	no	—
83	C₃×C₉	—	19	9	no	—
84	C₃×C₉	—	57	3	no	—
85	C₃×C₉	—	57	3	no	—
86	C₃×C₉	—	57	3	no	—
87	C₃×C₉	—	57	6	no	—
88	C₃×C₉	—	57	18	no	—
89	C₃×C₉	#10587025	1	9	yes	complemented
90	C₁₉	#10624558	1	1	yes	characteristic, complemented, Sylow
91	C₃²	—	3	9	no	—
92	C₃²	—	3	3	no	—
93	C₃²	—	3	6	no	—
94	C₃²	—	3	9	no	—
95	C₃²	—	3	9	no	—
96	C₃²	#13095303	1	9	yes	—
97	C₃²	C₃×C₉×C₁₉⋊C₃	1	1	yes	characteristic
98	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
99	C₃²	C₃×#10587023	1	1	yes	characteristic
100	C₃²	C₃×#10587022	1	1	yes	characteristic
101	C₃²	C₃×#10587025	1	3	yes	—
102	C₃²	#13095332	1	6	yes	—
103	C₉	—	3	9	no	—
104	C₉	—	57	9	no	—
105	C₉	—	171	6	no	—
106	C₃	—	3	9	no	—
107	C₃	C₃×#13095303	1	1	yes	characteristic
108	C₃	C₉×#10587025	1	3	yes	—
109	C₃	#17987421	1	1	yes	characteristic
110	C₃	#17987431	1	2	yes	—
111	C₃	#17987438	1	3	yes	—
112	C₃	#17987439	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#10587025

C₃

yes

maximal, characteristic

C₃×C₃²⋊#4294

C₃

yes

maximal

#17987308

C₃

yes

maximal, complemented

C₃³×C₉×C₁₉

C₃

yes

maximal, characteristic

—

—

—

C₉

yes

complemented

C₃²×#4294

C₃²

yes

—

C₃²×#4294

C₃²

yes

—

C₃²⋊#4294

—

C₃⁴×C₁₉

C₃²

yes

characteristic

C₃²×C₉×C₁₉

—

C₃²×C₉×C₁₉

—

C₃²×C₉×C₁₉

C₃²

yes

—

#10624558

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

C₃×C₉

yes

—

C₃³×C₁₉

H(3,𝔽₃)

yes

—

C₃³×C₁₉

3₋¹⁺²

yes

—

C₃³×C₁₉

3₋¹⁺²

yes

—

C₃³×C₁₉

C₃³

yes

characteristic

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

3₋¹⁺²

yes

—

—

—

—

—

—

C₃×C₉

yes

complemented

C₃²×3₋¹⁺²

—

C₃³×C₉

C₁₉⋊C₃

yes

characteristic

C₃×C₃²⋊C₉

—

#6855

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃²⋊C₉

yes

—

C₃²×C₁₉

C₃²×C₉

yes

characteristic

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₉

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₁₉

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₁₉

C₃×3₋¹⁺²

yes

—

C₃²×C₁₉

#737

yes

—

C₉×C₁₉

—

#4294

—

#4294

—

C₃⁴

C₃×C₁₉⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₁₉⋊C₃

yes

—

C₃²⋊C₉

—

C₃×C₁₉

—

C₃×C₁₉

C₃×C₃²⋊C₉

yes

characteristic

C₃×C₁₉

C₉×3₋¹⁺²

yes

—

C₃×C₁₉

#6891

yes

characteristic

C₃×C₁₉

#6893

yes

—

C₃×C₁₉

#6894

yes

—

C₃×C₁₉

#6894

yes

—

C₃³

—

C₃³

—

C₃³

C₉×C₁₉⋊C₃

yes

—

C₃³

#10587022

yes

—

C₃³

#10587023

yes

—

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

—

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

#10587025

yes

complemented

C₁₉

#10624558

yes

characteristic, complemented, Sylow

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

#13095303

yes

—

C₃²

C₃×C₉×C₁₉⋊C₃

yes

characteristic

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

C₃²

C₃×#10587023

yes

characteristic

100

C₃²

C₃×#10587022

yes

characteristic

101

C₃²

C₃×#10587025

yes

—

102

C₃²

#13095332

yes

—

103

C₉

—

104

C₉

—

105

C₉

—

171

—

106

C₃

—

107

C₃

C₃×#13095303

yes

characteristic

108

C₃

C₉×#10587025

yes

—

109

C₃

#17987421

yes

characteristic

110

C₃

#17987431

yes

—

111

C₃

#17987438

yes

—

112

C₃

#17987439

yes

—

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₁₉⋊C₃
commutator	C₃²×C₁₉	C₃²×C₉
Fitting	C₃³×C₉×C₁₉	C₃
socle, cosocle	C₃³×C₁₉	C₃³
prime radical	C₃⁴×C₁₉	C₃²
hypercenter	C₃³×C₉	C₁₉⋊C₃
hypocenter	C₁₉	#10624558
LCS	G ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃³ ◁ C₃³×C₉

subgroup

G/H

center, Frattini

C₃³

C₃²×C₁₉⋊C₃

commutator

C₃²×C₁₉

C₃²×C₉

Fitting

C₃³×C₉×C₁₉

C₃

socle, cosocle

C₃³×C₁₉

C₃³

prime radical

C₃⁴×C₁₉

C₃²

hypercenter

C₃³×C₉

C₁₉⋊C₃

C₁₉

G ▷ C₃²×C₁₉ ▷ C₁₉

C₁ ◁ C₃³ ◁ C₃³×C₉

N	H ≅ G/N	type
3. #17987308	C₃	semidirect
8. C₃×#10587025	C₉	semidirect
27. C₃×C₉×C₁₉	3₋¹⁺²	semidirect
34. C₃×#4294	C₃×C₉	semidirect
89. C₃×C₉	#10587025	semidirect
90. C₁₉	#10624558	semidirect

H ≅ G/N

type

3. #17987308

C₃

semidirect

8. C₃×#10587025

C₉

semidirect

27. C₃×C₉×C₁₉

3₋¹⁺²

semidirect

34. C₃×#4294

C₃×C₉

semidirect

89. C₃×C₉

#10587025

semidirect

90. C₁₉

#10624558

semidirect

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

subgroup	index
C₃²×#10587025	3
C₃×C₃²⋊#4294	3
#17987308	3
C₃³×C₉×C₁₉	3
#10624558	19

subgroup

index

C₃²×#10587025

C₃×C₃²⋊#4294

#17987308

C₃³×C₉×C₁₉

#10624558

Subgroups — #28353552

Conjugacy classes of subgroups

Complements

Maximal subgroups