Group #28356689 — subgroups

Subgroups — #28356689

Order 13851.

350 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 48 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	C₃×3₋¹⁺²×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic
3	C₃²×#4296	—	3	3	no	—
4	C₃²⋊#4296	—	3	9	no	—
5	C₃³×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
6	C₃×#10587025	C₉	1	2	yes	complemented
7	C₃×#10587022	—	3	1	no	—
8	C₃×#10587023	—	3	1	no	—
9	3₋¹⁺²×C₁₉⋊C₃	—	3	9	no	—
10	C₃×C₉×C₁₉⋊C₃	—	3	1	no	—
11	C₃×C₁₉×3₋¹⁺²	C₉	1	1	yes	characteristic, complemented
12	#10624452	—	19	1	no	maximal, Sylow
13	C₃³×C₁₉	C₃×C₉	1	1	yes	characteristic
14	C₃²×C₁₉⋊C₃	—	3	3	no	—
15	C₃²×C₁₉⋊C₃	—	3	3	no	—
16	C₃²×C₁₉⋊C₃	—	3	3	no	—
17	C₃²×C₁₉⋊C₃	—	3	3	no	—
18	C₃²×C₁₉⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
19	C₃²×C₁₉⋊C₃	3₋¹⁺²	1	1	yes	characteristic
20	C₃²×C₁₉⋊C₃	3₋¹⁺²	1	1	yes	characteristic
21	C₃×#4296	—	3	9	no	—
22	C₃×#4296	—	9	3	no	—
23	C₃×#4296	—	9	3	no	—
24	C₃×#4296	—	9	3	no	—
25	C₁₉×3₋¹⁺²	—	3	3	no	—
26	C₃×C₉×C₁₉	—	3	1	no	—
27	#10587025	—	3	6	no	—
28	C₃×#4294	—	3	2	no	—
29	#10587023	—	3	9	no	—
30	#10587022	—	3	9	no	—
31	C₉×C₁₉⋊C₃	—	3	9	no	—
32	C₃²×3₋¹⁺²	—	19	1	no	—
33	C₃×C₃²⋊C₉	—	19	3	no	—
34	C₃²×C₁₉	—	3	1	no	—
35	C₃²×C₁₉	—	3	3	no	—
36	C₃²×C₁₉	C₃²⋊C₉	1	1	yes	characteristic
37	C₃×C₁₉⋊C₃	—	3	1	no	—
38	C₃×C₁₉⋊C₃	—	3	1	no	—
39	C₃×C₁₉⋊C₃	—	3	3	no	—
40	C₃×C₁₉⋊C₃	—	3	3	no	—
41	C₃×C₁₉⋊C₃	—	9	3	no	—
42	C₃×C₁₉⋊C₃	—	9	3	no	—
43	C₃×C₁₉⋊C₃	—	9	3	no	—
44	C₃×C₁₉⋊C₃	AΣL(1,𝔽₂₇)	1	3	yes	—
45	C₃×C₁₉⋊C₃	AΣL(1,𝔽₂₇)	1	3	yes	—
46	C₃×C₁₉⋊C₃	#731	1	3	yes	—
47	C₃×C₁₉⋊C₃	#733	1	3	yes	—
48	#4296	—	9	9	no	—
49	C₉×C₁₉	—	3	3	no	—
50	#4294	—	3	6	no	—
51	C₃⁴	—	19	1	no	—
52	C₃×3₋¹⁺²	—	19	2	no	—
53	C₃×3₋¹⁺²	—	57	1	no	—
54	C₃×3₋¹⁺²	—	57	1	no	—
55	C₃×3₋¹⁺²	—	57	9	no	—
56	C₃×3₋¹⁺²	#4296	1	1	yes	characteristic, complemented
57	C₃²×C₉	—	57	1	no	—
58	C₃²×C₉	—	57	3	no	—
59	C₃²⋊C₉	—	57	9	no	—
60	C₃×C₁₉	—	9	1	no	—
61	C₃×C₁₉	#6866	1	1	yes	characteristic
62	C₃×C₁₉	3₋¹⁺²⋊C₉	1	3	yes	—
63	C₁₉⋊C₃	—	9	1	no	—
64	C₁₉⋊C₃	—	9	1	no	—
65	C₁₉⋊C₃	#6857	1	3	yes	—
66	C₁₉⋊C₃	#6858	1	3	yes	—
67	C₁₉⋊C₃	#6860	1	3	yes	—
68	C₃³	—	19	1	no	—
69	C₃³	—	19	1	no	—
70	C₃³	—	19	1	no	—
71	C₃³	—	57	3	no	—
72	C₃³	—	57	3	no	—
73	C₃³	—	57	3	no	—
74	C₃³	—	57	3	no	—
75	C₃³	C₃×#4296	1	1	yes	characteristic
76	3₋¹⁺²	—	3	3	no	—
77	3₋¹⁺²	—	57	9	no	—
78	3₋¹⁺²	—	57	9	no	—
79	3₋¹⁺²	—	57	6	no	—
80	C₃×C₉	—	3	1	no	—
81	C₃×C₉	—	57	9	no	—
82	C₃×C₉	—	57	2	no	—
83	C₃×C₉	—	57	9	no	—
84	C₃×C₉	—	171	3	no	—
85	C₃×C₉	—	171	3	no	—
86	C₃×C₉	—	171	3	no	—
87	C₁₉	#10624452	1	1	yes	characteristic, complemented, Sylow
88	C₃²	—	3	1	no	—
89	C₃²	—	3	3	no	—
90	C₃²	—	19	3	no	—
91	C₃²	—	19	3	no	—
92	C₃²	—	19	3	no	—
93	C₃²	—	19	3	no	—
94	C₃²	—	57	1	no	—
95	C₃²	—	57	1	no	—
96	C₃²	—	57	3	no	—
97	C₃²	—	57	3	no	—
98	C₃²	—	171	3	no	—
99	C₃²	—	171	3	no	—
100	C₃²	—	171	3	no	—
101	C₃²	C₃²⋊#4296	1	1	yes	characteristic
102	C₉	—	3	3	no	—
103	C₉	—	57	6	no	—
104	C₉	—	171	9	no	—
105	C₃	—	9	1	no	—
106	C₃	—	19	3	no	—
107	C₃	—	19	3	no	—
108	C₃	—	19	3	no	—
109	C₃	—	171	1	no	—
110	C₃	—	171	1	no	—
111	C₃	#17987558	1	1	yes	characteristic
112	C₃	3₋¹⁺²⋊#4296	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₃²⋊#4296

C₃

yes

maximal

C₃×3₋¹⁺²×C₁₉⋊C₃

C₃

yes

maximal, characteristic

C₃²×#4296

—

C₃²⋊#4296

—

C₃³×C₁₉⋊C₃

C₃²

yes

characteristic

C₃×#10587025

C₉

yes

complemented

C₃×#10587022

—

C₃×#10587023

—

3₋¹⁺²×C₁₉⋊C₃

—

C₃×C₉×C₁₉⋊C₃

—

C₃×C₁₉×3₋¹⁺²

C₉

yes

characteristic, complemented

#10624452

—

maximal, Sylow

C₃³×C₁₉

C₃×C₉

yes

characteristic

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

H(3,𝔽₃)

yes

characteristic

C₃²×C₁₉⋊C₃

3₋¹⁺²

yes

characteristic

C₃²×C₁₉⋊C₃

3₋¹⁺²

yes

—

—

—

—

C₁₉×3₋¹⁺²

—

C₃×C₉×C₁₉

—

—

—

—

—

C₉×C₁₉⋊C₃

—

C₃²×3₋¹⁺²

—

C₃×C₃²⋊C₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

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

—

C₃×C₁₉⋊C₃

AΣL(1,𝔽₂₇)

yes

—

C₃×C₁₉⋊C₃

AΣL(1,𝔽₂₇)

yes

—

C₃×C₁₉⋊C₃

#731

yes

—

C₃×C₁₉⋊C₃

#733

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

#4296

yes

characteristic, complemented

C₃²×C₉

—

C₃²×C₉

—

C₃²⋊C₉

—

C₃×C₁₉

—

C₃×C₁₉

#6866

yes

characteristic

C₃×C₁₉

3₋¹⁺²⋊C₉

yes

—

C₁₉⋊C₃

—

C₁₉⋊C₃

—

C₁₉⋊C₃

#6857

yes

—

C₁₉⋊C₃

#6858

yes

—

C₁₉⋊C₃

#6860

yes

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

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

—

171

—

C₃×C₉

—

171

—

C₁₉

#10624452

yes

characteristic, complemented, Sylow

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

171

—

C₃²

—

171

—

100

C₃²

—

171

—

101

C₃²

C₃²⋊#4296

yes

characteristic

102

C₉

—

103

C₉

—

104

C₉

—

171

—

105

C₃

—

106

C₃

—

107

C₃

—

108

C₃

—

109

C₃

—

171

—

110

C₃

—

171

—

111

C₃

#17987558

yes

characteristic

112

C₃

3₋¹⁺²⋊#4296

yes

—

	subgroup	G/H
center	C₃²	C₃²⋊#4296
commutator	C₃³×C₁₉	C₃×C₉
Frattini	C₃³	C₃×#4296
Fitting	C₃×C₁₉×3₋¹⁺²	C₉
socle	C₃²×C₁₉	C₃²⋊C₉
prime radical, cosocle	C₃³×C₁₉⋊C₃	C₃²
hypercenter	C₃×3₋¹⁺²	#4296
hypocenter	C₁₉	#10624452
LCS	G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×3₋¹⁺²

subgroup

G/H

center

C₃²

C₃²⋊#4296

commutator

C₃³×C₁₉

C₃×C₉

Frattini

C₃³

C₃×#4296

Fitting

C₃×C₁₉×3₋¹⁺²

C₉

socle

C₃²×C₁₉

C₃²⋊C₉

prime radical, cosocle

C₃³×C₁₉⋊C₃

C₃²

C₃×3₋¹⁺²

C₁₉

G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ C₃×3₋¹⁺²

N	H ≅ G/N	type
6. C₃×#10587025	C₉	semidirect
11. C₃×C₁₉×3₋¹⁺²	C₉	semidirect
56. C₃×3₋¹⁺²	#4296	semidirect
87. C₁₉	#10624452	semidirect

H ≅ G/N

type

6. C₃×#10587025

C₉

semidirect

11. C₃×C₁₉×3₋¹⁺²

C₉

semidirect

56. C₃×3₋¹⁺²

#4296

semidirect

87. C₁₉

#10624452

semidirect

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

subgroup	index
C₃×C₃²⋊#4296	3
C₃×3₋¹⁺²×C₁₉⋊C₃	3
#10624452	19

subgroup

index

C₃×C₃²⋊#4296

C₃×3₋¹⁺²×C₁₉⋊C₃

#10624452

Subgroups — #28356689

Conjugacy classes of subgroups

Complements

Maximal subgroups