Group #28353854 — subgroups

Subgroups — #28353854

Order 13851.

494 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 64 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	2	yes	maximal, complemented
2	#17987421	C₃	1	1	yes	maximal, characteristic, complemented
3	#17987312	C₃	1	9	yes	maximal, complemented
4	C₁₉×C₃²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
5	C₃×#10587025	—	3	2	no	—
6	C₃×#10587025	—	3	2	no	—
7	C₃×#10587025	—	3	18	no	—
8	C₃×#10587025	—	3	1	no	—
9	C₃×#10587025	C₃²	1	6	yes	—
10	C₃²×#4294	—	3	2	no	—
11	C₃²⋊#4294	—	3	2	no	—
12	C₃²⋊#4294	C₃²	1	3	yes	—
13	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
14	C₃×C₁₉×H(3,𝔽₃)	—	3	1	no	—
15	C₃×C₁₉×H(3,𝔽₃)	—	3	2	no	—
16	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	3	yes	—
17	#10624608	—	19	1	no	maximal, Sylow
18	C₃³×C₁₉	—	3	6	no	—
19	C₃³×C₁₉	—	3	3	no	—
20	C₃³×C₁₉	—	9	1	no	—
21	C₃³×C₁₉	—	9	1	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	C₁₉×H(3,𝔽₃)	—	9	3	no	—
26	C₁₉×H(3,𝔽₃)	—	9	3	no	—
27	C₁₉×H(3,𝔽₃)	—	9	6	no	—
28	#10587025	—	3	18	no	—
29	#10587025	—	3	18	no	—
30	#10587025	—	3	18	no	—
31	#10587025	—	9	3	no	—
32	C₃×#4294	—	3	18	no	—
33	C₃×#4294	—	3	6	no	—
34	C₃×#4294	—	3	3	no	—
35	C₃²×3₋¹⁺²	—	19	2	no	—
36	#6891	—	19	1	no	—
37	C₃²⋊H(3,𝔽₃)	C₁₉⋊C₃	1	1	yes	characteristic
38	#6857	—	19	9	no	—
39	C₃²×C₁₉	—	3	2	no	—
40	C₃²×C₁₉	—	3	2	no	—
41	C₃²×C₁₉	—	3	2	no	—
42	C₃²×C₁₉	—	3	2	no	—
43	C₃²×C₁₉	—	3	2	no	—
44	C₃²×C₁₉	—	3	1	no	—
45	C₃²×C₁₉	—	3	1	no	—
46	C₃²×C₁₉	—	3	1	no	—
47	C₃²×C₁₉	—	3	1	no	—
48	C₃²×C₁₉	—	9	9	no	—
49	C₃²×C₁₉	—	9	6	no	—
50	C₃²×C₁₉	—	9	3	no	—
51	C₃²×C₁₉	—	9	3	no	—
52	C₃²×C₁₉	AΣL(1,𝔽₂₇)	1	6	yes	—
53	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
54	#4294	—	3	18	no	—
55	#4294	—	9	3	no	—
56	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
57	C₃×3₋¹⁺²	—	19	6	no	—
58	C₃×3₋¹⁺²	—	57	2	no	—
59	C₃×3₋¹⁺²	—	57	2	no	—
60	C₃×3₋¹⁺²	—	57	18	no	—
61	C₃×3₋¹⁺²	—	57	1	no	—
62	C₃×H(3,𝔽₃)	—	3	1	no	—
63	C₃×H(3,𝔽₃)	—	3	2	no	—
64	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	3	yes	—
65	C₃²×C₉	—	57	2	no	—
66	C₃²⋊C₉	—	19	3	no	—
67	C₃²⋊C₉	—	57	2	no	—
68	C₃×C₁₉	—	3	6	no	—
69	C₃×C₁₉	—	9	1	no	—
70	C₃×C₁₉	—	9	1	no	—
71	C₃×C₁₉	—	27	3	no	—
72	C₃×C₁₉	C₃×AΣL(1,𝔽₂₇)	1	2	yes	—
73	C₃×C₁₉	#6909	1	1	yes	characteristic
74	C₃×C₁₉	#6910	1	1	yes	characteristic
75	C₃³	—	3	6	no	—
76	C₃³	—	3	3	no	—
77	C₃³	—	9	1	no	—
78	C₃³	—	9	1	no	—
79	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
80	C₃³	C₃²⋊(C₁₉⋊C₃)	1	2	yes	—
81	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
82	3₋¹⁺²	—	57	18	no	—
83	3₋¹⁺²	—	57	18	no	—
84	3₋¹⁺²	—	57	18	no	—
85	3₋¹⁺²	—	171	3	no	—
86	H(3,𝔽₃)	—	9	3	no	—
87	H(3,𝔽₃)	—	9	3	no	—
88	H(3,𝔽₃)	—	9	6	no	—
89	C₃×C₉	—	57	18	no	—
90	C₃×C₉	—	57	6	no	—
91	C₃×C₉	—	57	3	no	—
92	C₁₉	#10624608	1	1	yes	characteristic, complemented, Sylow
93	C₃²	—	3	2	no	—
94	C₃²	—	3	2	no	—
95	C₃²	—	3	2	no	—
96	C₃²	—	3	2	no	—
97	C₃²	—	3	2	no	—
98	C₃²	—	3	1	no	—
99	C₃²	—	3	1	no	—
100	C₃²	—	3	1	no	—
101	C₃²	—	3	1	no	—
102	C₃²	—	9	9	no	—
103	C₃²	—	9	6	no	—
104	C₃²	—	9	3	no	—
105	C₃²	—	9	3	no	—
106	C₃²	#13095313	1	6	yes	—
107	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
108	C₉	—	57	18	no	—
109	C₉	—	171	3	no	—
110	C₃	—	3	6	no	—
111	C₃	—	9	1	no	—
112	C₃	—	9	1	no	—
113	C₃	—	27	3	no	—
114	C₃	C₃×#13095313	1	2	yes	—
115	C₃	#17987505	1	1	yes	characteristic
116	C₃	#17987509	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#10587025

C₃

yes

maximal, complemented

#17987421

C₃

yes

maximal, characteristic, complemented

#17987312

C₃

yes

maximal, complemented

C₁₉×C₃²⋊H(3,𝔽₃)

C₃

yes

maximal, characteristic

—

—

—

—

C₃²

yes

—

C₃²×#4294

—

C₃²⋊#4294

—

C₃²⋊#4294

C₃²

yes

—

C₃⁴×C₁₉

C₃²

yes

characteristic

C₃×C₁₉×H(3,𝔽₃)

—

C₃×C₁₉×H(3,𝔽₃)

—

C₃×C₁₉×H(3,𝔽₃)

C₃²

yes

—

#10624608

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

H(3,𝔽₃)

yes

characteristic

C₃³×C₁₉

H(3,𝔽₃)

yes

—

C₃³×C₁₉

C₃³

yes

characteristic

C₁₉×H(3,𝔽₃)

—

C₁₉×H(3,𝔽₃)

—

C₁₉×H(3,𝔽₃)

—

—

—

—

—

—

—

—

C₃²×3₋¹⁺²

—

#6891

—

C₃²⋊H(3,𝔽₃)

C₁₉⋊C₃

yes

characteristic

#6857

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

#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×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×C₁₉⋊C₃

yes

—

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

C₃×AΣL(1,𝔽₂₇)

yes

—

C₃×C₁₉

yes

C₃×C₁₉

yes

C₃³

—

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

—

171

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₁₉

#10624608

yes

characteristic, complemented, Sylow

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

#13095313

yes

—

107

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

108

C₉

—

109

C₉

—

171

—

110

C₃

—

111

C₃

—

112

C₃

—

113

C₃

—

114

C₃

C₃×#13095313

yes

—

115

C₃

#17987505

yes

characteristic

116

C₃

#17987509

yes

characteristic

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)
commutator, cosocle	C₃³×C₁₉	C₃³
Frattini	C₃³	C₃²×C₁₉⋊C₃
Fitting, prime radical	C₁₉×C₃²⋊H(3,𝔽₃)	C₃
socle	C₃²×C₁₉	C₃×H(3,𝔽₃)
hypercenter	C₃²⋊H(3,𝔽₃)	C₁₉⋊C₃
hypocenter	C₁₉	#10624608
LCS	G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²⋊H(3,𝔽₃)

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

commutator, cosocle

C₃³×C₁₉

C₃³

Frattini

C₃³

C₃²×C₁₉⋊C₃

Fitting, prime radical

C₁₉×C₃²⋊H(3,𝔽₃)

C₃

socle

C₃²×C₁₉

C₃×H(3,𝔽₃)

hypercenter

C₃²⋊H(3,𝔽₃)

C₁₉⋊C₃

hypocenter

C₁₉

#10624608

LCS

G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²⋊H(3,𝔽₃)

N	H ≅ G/N	type
1. C₃²×#10587025	C₃	semidirect
2. #17987421	C₃	semidirect
3. #17987312	C₃	semidirect
92. C₁₉	#10624608	semidirect

H ≅ G/N

type

1. C₃²×#10587025

C₃

semidirect

2. #17987421

C₃

semidirect

3. #17987312

C₃

semidirect

92. C₁₉

#10624608

semidirect

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

subgroup	index
C₃²×#10587025	3
#17987421	3
#17987312	3
C₁₉×C₃²⋊H(3,𝔽₃)	3
#10624608	19

subgroup

index

C₁₉×C₃²⋊H(3,𝔽₃)

Subgroups — #28353854

Conjugacy classes of subgroups

Complements

Maximal subgroups