Group #28355046 — subgroups

Subgroups — #28355046

Order 13851.

446 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 70 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	#17987518	C₃	1	3	yes	maximal, complemented
2	#17987511	C₃	1	3	yes	maximal, complemented
3	#17987506	C₃	1	3	yes	maximal, complemented
4	C₃×#13095314	C₃	1	3	yes	maximal, complemented
5	C₃²×C₁₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃×#10587025	C₃²	1	3	yes	—
7	C₃×#10587025	C₃²	1	3	yes	—
8	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	3	yes	—
9	#13095321	—	3	9	no	—
10	#13095318	—	3	9	no	—
11	#13095315	—	3	9	no	—
12	#13095314	—	3	3	no	—
13	#13095314	—	3	6	no	—
14	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
15	C₃×C₁₉×3₋¹⁺²	—	3	2	no	—
16	C₃×C₁₉×3₋¹⁺²	—	3	9	no	—
17	C₃×C₁₉×3₋¹⁺²	C₃²	1	3	yes	—
18	C₃²×C₉×C₁₉	—	3	1	no	—
19	#10624755	—	19	1	no	maximal, Sylow
20	C₃³×C₁₉	—	3	3	no	—
21	C₃³×C₁₉	—	9	3	no	—
22	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
23	C₃³×C₁₉	H(3,𝔽₃)	1	2	yes	—
24	C₃³×C₁₉	C₃³	1	1	yes	characteristic
25	C₃²×C₁₉⋊C₃	—	3	3	no	—
26	C₁₉×3₋¹⁺²	—	3	18	no	—
27	C₁₉×3₋¹⁺²	—	3	6	no	—
28	C₁₉×3₋¹⁺²	H(3,𝔽₃)	1	9	yes	—
29	C₃×C₉×C₁₉	—	3	9	no	—
30	C₃×C₉×C₁₉	—	3	3	no	—
31	C₃²⋊(C₁₉⋊C₃)	—	3	3	no	—
32	C₃²⋊(C₁₉⋊C₃)	—	3	3	no	—
33	C₃²⋊(C₁₉⋊C₃)	—	3	3	no	—
34	#10587025	—	3	3	no	—
35	#10587025	—	3	3	no	—
36	#10587025	—	3	3	no	—
37	#10587025	—	3	3	no	—
38	#10587025	—	3	3	no	—
39	#10587025	—	3	3	no	—
40	C₃×#4294	—	3	3	no	—
41	C₃×#4294	—	3	3	no	—
42	C₃²×3₋¹⁺²	C₁₉⋊C₃	1	1	yes	characteristic, complemented
43	#6912	—	19	3	no	—
44	#6910	—	19	3	no	—
45	#6909	—	19	3	no	—
46	C₃×AΣL(1,𝔽₂₇)	—	19	3	no	—
47	C₃²×C₁₉	—	3	3	no	—
48	C₃²×C₁₉	—	3	1	no	—
49	C₃²×C₁₉	—	3	2	no	—
50	C₃²×C₁₉	—	9	1	no	—
51	C₃²×C₁₉	—	9	9	no	—
52	C₃²×C₁₉	—	9	2	no	—
53	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	3	yes	—
54	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
55	C₃×C₁₉⋊C₃	—	9	3	no	—
56	C₃×C₁₉⋊C₃	—	27	3	no	—
57	C₉×C₁₉	—	3	9	no	—
58	#4294	—	9	3	no	—
59	#4294	—	9	3	no	—
60	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
61	C₃×3₋¹⁺²	—	3	2	no	—
62	C₃×3₋¹⁺²	—	3	9	no	—
63	C₃×3₋¹⁺²	—	19	3	no	—
64	C₃×3₋¹⁺²	—	19	3	no	—
65	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	3	yes	—
66	C₃×H(3,𝔽₃)	—	19	3	no	—
67	C₃²×C₉	—	3	1	no	—
68	#733	—	57	9	no	—
69	#731	—	57	9	no	—
70	AΣL(1,𝔽₂₇)	—	57	3	no	—
71	AΣL(1,𝔽₂₇)	—	57	6	no	—
72	AΣL(1,𝔽₂₇)	—	57	9	no	—
73	C₃×C₁₉	—	3	3	no	—
74	C₃×C₁₉	—	3	3	no	—
75	C₃×C₁₉	—	3	1	no	—
76	C₃×C₁₉	—	9	2	no	—
77	C₃×C₁₉	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
78	C₁₉⋊C₃	—	27	3	no	—
79	C₃³	—	3	3	no	—
80	C₃³	—	9	3	no	—
81	C₃³	—	57	3	no	—
82	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
83	C₃³	C₃²⋊(C₁₉⋊C₃)	1	2	yes	—
84	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
85	3₋¹⁺²	—	3	18	no	—
86	3₋¹⁺²	—	3	6	no	—
87	3₋¹⁺²	—	57	3	no	—
88	3₋¹⁺²	—	57	3	no	—
89	3₋¹⁺²	—	57	3	no	—
90	3₋¹⁺²	—	57	3	no	—
91	3₋¹⁺²	—	57	3	no	—
92	3₋¹⁺²	—	57	3	no	—
93	3₋¹⁺²	C₃²⋊(C₁₉⋊C₃)	1	9	yes	—
94	H(3,𝔽₃)	—	57	3	no	—
95	H(3,𝔽₃)	—	57	3	no	—
96	H(3,𝔽₃)	—	57	3	no	—
97	C₃×C₉	—	3	9	no	—
98	C₃×C₉	—	3	3	no	—
99	C₃×C₉	—	57	3	no	—
100	C₃×C₉	—	57	3	no	—
101	C₁₉	#10624755	1	1	yes	characteristic, complemented, Sylow
102	C₃²	—	513	3	no	—
103	C₃²	—	3	3	no	—
104	C₃²	—	3	1	no	—
105	C₃²	—	3	2	no	—
106	C₃²	—	9	1	no	—
107	C₃²	—	9	9	no	—
108	C₃²	—	9	2	no	—
109	C₃²	—	171	3	no	—
110	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
111	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	3	yes	—
112	C₉	—	3	9	no	—
113	C₉	—	171	3	no	—
114	C₉	—	171	3	no	—
115	C₃	—	513	3	no	—
116	C₃	—	3	3	no	—
117	C₃	—	3	1	no	—
118	C₃	—	3	3	no	—
119	C₃	—	9	2	no	—
120	C₃	#17987417	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17987518

C₃

yes

maximal, complemented

#17987511

C₃

yes

maximal, complemented

#17987506

C₃

yes

maximal, complemented

C₃×#13095314

C₃

yes

maximal, complemented

C₃²×C₁₉×3₋¹⁺²

C₃

yes

maximal, characteristic, complemented

C₃×#10587025

C₃²

yes

—

C₃×#10587025

C₃²

yes

—

C₃×C₃²⋊(C₁₉⋊C₃)

C₃²

yes

—

—

—

—

—

—

C₃⁴×C₁₉

C₃²

yes

characteristic

C₃×C₁₉×3₋¹⁺²

—

C₃×C₁₉×3₋¹⁺²

—

C₃×C₁₉×3₋¹⁺²

C₃²

yes

—

C₃²×C₉×C₁₉

—

#10624755

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

H(3,𝔽₃)

yes

characteristic

C₃³×C₁₉

H(3,𝔽₃)

yes

—

C₃³×C₁₉

C₃³

yes

characteristic

C₃²×C₁₉⋊C₃

—

C₁₉×3₋¹⁺²

—

C₁₉×3₋¹⁺²

—

C₁₉×3₋¹⁺²

H(3,𝔽₃)

yes

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

—

—

—

—

—

—

—

—

C₃²×3₋¹⁺²

C₁₉⋊C₃

yes

characteristic, complemented

#6912

—

#6910

—

#6909

—

C₃×AΣL(1,𝔽₂₇)

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

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

—

C₃×3₋¹⁺²

C₃×C₁₉⋊C₃

yes

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

#733

—

#731

—

AΣL(1,𝔽₂₇)

—

AΣL(1,𝔽₂₇)

—

AΣL(1,𝔽₂₇)

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

C₃²⋊H(3,𝔽₃)

yes

characteristic

C₁₉⋊C₃

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

—

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₃²⋊(C₁₉⋊C₃)

yes

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

100

C₃×C₉

—

101

C₁₉

#10624755

yes

characteristic, complemented, Sylow

102

C₃²

—

513

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

171

—

110

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

111

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

—

112

C₉

—

113

C₉

—

171

—

114

C₉

—

171

—

115

C₃

—

513

—

116

C₃

—

117

C₃

—

118

C₃

—

119

C₃

—

120

C₃

#17987417

yes

characteristic

	subgroup	G/H
center	C₃	#17987417
commutator, cosocle	C₃³×C₁₉	C₃³
Frattini	C₃³	C₃²×C₁₉⋊C₃
Fitting	C₃²×C₁₉×3₋¹⁺²	C₃
socle	C₃×C₁₉	C₃²⋊H(3,𝔽₃)
hypercenter	C₃²×3₋¹⁺²	C₁₉⋊C₃
hypocenter	C₁₉	#10624755
LCS	G ▷ C₃³×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃²×3₋¹⁺²

subgroup

G/H

center

C₃

#17987417

commutator, cosocle

C₃³×C₁₉

C₃³

Frattini

C₃³

C₃²×C₁₉⋊C₃

Fitting

C₃²×C₁₉×3₋¹⁺²

C₃

socle

C₃×C₁₉

C₃²⋊H(3,𝔽₃)

hypercenter

C₃²×3₋¹⁺²

C₁₉⋊C₃

hypocenter

C₁₉

#10624755

LCS

G ▷ C₃³×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃ ◁ C₃³ ◁ C₃²×3₋¹⁺²

N	H ≅ G/N	type
1. #17987518	C₃	semidirect
2. #17987511	C₃	semidirect
3. #17987506	C₃	semidirect
4. C₃×#13095314	C₃	semidirect
5. C₃²×C₁₉×3₋¹⁺²	C₃	semidirect
42. C₃²×3₋¹⁺²	C₁₉⋊C₃	semidirect
101. C₁₉	#10624755	semidirect

H ≅ G/N

type

1. #17987518

C₃

semidirect

2. #17987511

C₃

semidirect

3. #17987506

C₃

semidirect

4. C₃×#13095314

C₃

semidirect

5. C₃²×C₁₉×3₋¹⁺²

C₃

semidirect

42. C₃²×3₋¹⁺²

C₁₉⋊C₃

semidirect

101. C₁₉

#10624755

semidirect

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

subgroup	index
#17987518	3
#17987511	3
#17987506	3
C₃×#13095314	3
C₃²×C₁₉×3₋¹⁺²	3
#10624755	19

subgroup

index

C₃²×C₁₉×3₋¹⁺²

Subgroups — #28355046

Conjugacy classes of subgroups

Complements

Maximal subgroups