Group #28355061 — subgroups

Subgroups — #28355061

Order 13851.

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

normal subgroups only

This group has 426 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	#17987514	C₃	1	3	yes	maximal, complemented
2	#17987506	C₃	1	6	yes	maximal, complemented
3	C₃×#13095322	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×#13095320	C₃	1	2	yes	maximal, complemented
5	C₃×C₁₉×#737	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃×#10587025	C₃²	1	3	yes	—
7	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	6	yes	—
8	#13095322	—	3	1	no	—
9	#13095322	—	3	2	no	—
10	#13095320	—	3	2	no	—
11	#13095320	—	3	4	no	—
12	#13095319	—	3	9	no	—
13	#13095315	—	3	18	no	—
14	C₁₉×#737	—	3	6	no	—
15	C₁₉×#737	—	3	3	no	—
16	C₃×C₁₉×3₋¹⁺²	—	3	2	no	—
17	C₃×C₁₉×3₋¹⁺²	C₃²	1	2	yes	—
18	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
19	C₃²×C₉×C₁₉	—	3	1	no	—
20	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
21	#10624757	—	19	1	no	maximal, Sylow
22	C₃³×C₁₉	—	3	1	no	—
23	C₃³×C₁₉	C₃³	1	1	yes	characteristic
24	C₃²×C₁₉⋊C₃	—	3	6	no	—
25	C₁₉×3₋¹⁺²	—	3	18	no	—
26	C₁₉×3₋¹⁺²	—	3	2	no	—
27	C₁₉×3₋¹⁺²	—	3	2	no	—
28	C₁₉×3₋¹⁺²	H(3,𝔽₃)	1	6	yes	—
29	C₁₉×H(3,𝔽₃)	—	3	2	no	—
30	C₁₉×H(3,𝔽₃)	H(3,𝔽₃)	1	3	yes	—
31	C₃×C₉×C₁₉	—	3	9	no	—
32	C₃×C₉×C₁₉	—	3	2	no	—
33	C₃×C₉×C₁₉	—	3	1	no	—
34	C₃×C₉×C₁₉	—	3	2	no	—
35	C₃×C₉×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
36	C₃×C₉×C₁₉	H(3,𝔽₃)	1	2	yes	—
37	C₃²⋊(C₁₉⋊C₃)	—	3	6	no	—
38	C₃²⋊(C₁₉⋊C₃)	—	3	6	no	—
39	C₃²⋊(C₁₉⋊C₃)	—	3	6	no	—
40	#10587025	—	3	6	no	—
41	#10587025	—	3	3	no	—
42	C₃×#4294	—	3	3	no	—
43	C₃×#737	C₁₉⋊C₃	1	1	yes	characteristic, complemented
44	#6911	—	19	3	no	—
45	#6909	—	19	6	no	—
46	C₃×#733	—	19	1	no	—
47	C₃×#732	—	19	2	no	—
48	C₃²×C₁₉	—	3	3	no	—
49	C₃²×C₁₉	—	9	1	no	—
50	C₃²×C₁₉	—	9	3	no	—
51	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
52	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
53	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	2	yes	—
54	C₃×C₁₉⋊C₃	—	9	6	no	—
55	C₃×C₁₉⋊C₃	—	27	6	no	—
56	C₉×C₁₉	—	3	1	no	—
57	C₉×C₁₉	—	3	6	no	—
58	C₉×C₁₉	—	3	2	no	—
59	#4294	—	9	3	no	—
60	#737	—	3	6	no	—
61	#737	—	3	3	no	—
62	C₃×3₋¹⁺²	—	3	2	no	—
63	C₃×3₋¹⁺²	—	19	3	no	—
64	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	2	yes	—
65	C₃×H(3,𝔽₃)	—	19	6	no	—
66	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	1	yes	characteristic
67	C₃²×C₉	—	3	1	no	—
68	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
69	#733	—	57	1	no	—
70	#733	—	57	2	no	—
71	#732	—	57	9	no	—
72	#732	—	57	2	no	—
73	#732	—	57	4	no	—
74	AΣL(1,𝔽₂₇)	—	57	18	no	—
75	C₃×C₁₉	—	3	2	no	—
76	C₃×C₁₉	—	3	1	no	—
77	C₃×C₁₉	—	3	1	no	—
78	C₃×C₁₉	—	9	3	no	—
79	C₃×C₁₉	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
80	C₁₉⋊C₃	—	27	6	no	—
81	C₃³	—	3	1	no	—
82	C₃³	—	57	6	no	—
83	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
84	3₋¹⁺²	—	3	18	no	—
85	3₋¹⁺²	—	3	2	no	—
86	3₋¹⁺²	—	3	2	no	—
87	3₋¹⁺²	—	57	6	no	—
88	3₋¹⁺²	—	57	3	no	—
89	3₋¹⁺²	C₃²⋊(C₁₉⋊C₃)	1	6	yes	—
90	H(3,𝔽₃)	—	3	2	no	—
91	H(3,𝔽₃)	—	57	6	no	—
92	H(3,𝔽₃)	—	57	6	no	—
93	H(3,𝔽₃)	—	57	6	no	—
94	H(3,𝔽₃)	C₃²⋊(C₁₉⋊C₃)	1	3	yes	—
95	C₃×C₉	—	3	9	no	—
96	C₃×C₉	—	3	2	no	—
97	C₃×C₉	—	3	1	no	—
98	C₃×C₉	—	3	2	no	—
99	C₃×C₉	—	57	3	no	—
100	C₃×C₉	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
101	C₃×C₉	C₃²⋊(C₁₉⋊C₃)	1	2	yes	—
102	C₁₉	#10624757	1	1	yes	characteristic, complemented, Sylow
103	C₃²	—	513	6	no	—
104	C₃²	—	3	3	no	—
105	C₃²	—	9	1	no	—
106	C₃²	—	9	3	no	—
107	C₃²	—	171	6	no	—
108	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
109	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
110	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	2	yes	—
111	C₉	—	3	1	no	—
112	C₉	—	3	6	no	—
113	C₉	—	3	2	no	—
114	C₉	—	171	3	no	—
115	C₃	—	513	6	no	—
116	C₃	—	3	2	no	—
117	C₃	—	3	1	no	—
118	C₃	—	3	1	no	—
119	C₃	—	9	3	no	—
120	C₃	#17987417	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17987514

C₃

yes

maximal, complemented

#17987506

C₃

yes

maximal, complemented

C₃×#13095322

C₃

yes

maximal, characteristic, complemented

C₃×#13095320

C₃

yes

maximal, complemented

C₃×C₁₉×#737

C₃

yes

maximal, characteristic, complemented

C₃×#10587025

C₃²

yes

—

C₃×C₃²⋊(C₁₉⋊C₃)

C₃²

yes

—

—

—

—

—

—

—

C₁₉×#737

—

C₁₉×#737

—

C₃×C₁₉×3₋¹⁺²

—

C₃×C₁₉×3₋¹⁺²

C₃²

yes

—

C₃×C₁₉×H(3,𝔽₃)

C₃²

yes

characteristic

C₃²×C₉×C₁₉

—

C₃²×C₉×C₁₉

C₃²

yes

characteristic

#10624757

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

C₃³

yes

characteristic

C₃²×C₁₉⋊C₃

—

C₁₉×3₋¹⁺²

—

C₁₉×3₋¹⁺²

—

C₁₉×3₋¹⁺²

—

C₁₉×3₋¹⁺²

H(3,𝔽₃)

yes

—

C₁₉×H(3,𝔽₃)

—

C₁₉×H(3,𝔽₃)

H(3,𝔽₃)

yes

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

H(3,𝔽₃)

yes

characteristic

C₃×C₉×C₁₉

H(3,𝔽₃)

yes

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

—

—

—

C₁₉⋊C₃

yes

characteristic, complemented

—

—

—

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₉×C₁₉

—

C₉×C₁₉

—

C₉×C₁₉

—

#4294

—

#737

—

#737

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₁₉⋊C₃

yes

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×C₁₉⋊C₃

yes

characteristic

C₃²×C₉

—

C₃²×C₉

C₃×C₁₉⋊C₃

yes

—

—

—

—

—

AΣL(1,𝔽₂₇)

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

C₃²⋊H(3,𝔽₃)

yes

characteristic

C₁₉⋊C₃

—

C₃³

—

C₃³

—

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₃²⋊(C₁₉⋊C₃)

yes

—

—

—

—

—

C₃²⋊(C₁₉⋊C₃)

yes

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

100

C₃×C₉

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

101

C₃×C₉

C₃²⋊(C₁₉⋊C₃)

yes

—

102

C₁₉

#10624757

yes

characteristic, complemented, Sylow

103

C₃²

—

513

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

171

—

108

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

109

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

110

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

—

111

C₉

—

112

C₉

—

113

C₉

—

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	C₃×C₁₉×#737	C₃
socle	C₃×C₁₉	C₃²⋊H(3,𝔽₃)
hypercenter	C₃×#737	C₁₉⋊C₃
hypocenter	C₁₉	#10624757
LCS	G ▷ C₃³×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×#737

subgroup

G/H

center

C₃

#17987417

commutator, cosocle

C₃³×C₁₉

C₃³

Frattini

C₃³

C₃²×C₁₉⋊C₃

Fitting

C₃×C₁₉×#737

C₃

socle

C₃×C₁₉

C₃²⋊H(3,𝔽₃)

C₁₉⋊C₃

C₁₉

G ▷ C₃³×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃ ◁ C₃³ ◁ C₃×#737

N	H ≅ G/N	type
1. #17987514	C₃	semidirect
2. #17987506	C₃	semidirect
3. C₃×#13095322	C₃	semidirect
4. C₃×#13095320	C₃	semidirect
5. C₃×C₁₉×#737	C₃	semidirect
43. C₃×#737	C₁₉⋊C₃	semidirect
102. C₁₉	#10624757	semidirect

H ≅ G/N

type

1. #17987514

C₃

semidirect

2. #17987506

C₃

semidirect

3. C₃×#13095322

C₃

semidirect

4. C₃×#13095320

C₃

semidirect

5. C₃×C₁₉×#737

C₃

semidirect

43. C₃×#737

C₁₉⋊C₃

semidirect

102. C₁₉

#10624757

semidirect

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

subgroup	index
#17987514	3
#17987506	3
C₃×#13095322	3
C₃×#13095320	3
C₃×C₁₉×#737	3
#10624757	19

subgroup

index

C₃×C₁₉×#737

Subgroups — #28355061

Conjugacy classes of subgroups

Complements

Maximal subgroups