Group #28355803 — subgroups

Subgroups — #28355803

Order 13851.

198 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 52 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×#13095322	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#13095320	C₃	1	1	yes	maximal, characteristic
3	#17987405	C₃	1	1	yes	maximal, characteristic, complemented
4	#17987388	C₃	1	9	yes	maximal, complemented
5	C₃×C₁₉×#733	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃×#10587025	—	3	1	no	—
7	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	1	no	—
8	#13095322	—	3	1	no	—
9	#13095322	—	3	1	no	—
10	#13095322	C₃²	1	3	yes	complemented
11	#13095320	—	3	1	no	—
12	#13095320	—	3	1	no	—
13	#13095320	C₃²	1	3	yes	—
14	#13095307	C₃²	1	3	yes	complemented
15	#13095306	—	3	1	no	—
16	C₃²⋊#4294	—	3	1	no	—
17	C₉×#4294	—	3	1	no	—
18	C₃×C₁₉×3₋¹⁺²	—	3	1	no	—
19	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
20	C₁₉×#733	—	3	1	no	—
21	C₁₉×#733	—	3	1	no	—
22	C₁₉×#733	C₃²	1	3	yes	complemented
23	#10624848	—	19	1	no	maximal, Sylow
24	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
25	C₃²×C₁₉⋊C₃	—	9	1	no	—
26	C₁₉×3₋¹⁺²	—	3	3	no	—
27	C₁₉×3₋¹⁺²	—	9	1	no	—
28	C₁₉×3₋¹⁺²	—	9	1	no	—
29	C₃×C₉×C₁₉	—	3	1	no	—
30	C₃×C₉×C₁₉	—	3	1	no	—
31	C₃×C₉×C₁₉	—	3	1	no	—
32	C₃×C₉×C₁₉	—	9	1	no	—
33	C₃×C₉×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
34	C₃×C₉×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
35	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
36	C₃²⋊(C₁₉⋊C₃)	—	3	3	no	—
37	C₃²⋊(C₁₉⋊C₃)	—	9	1	no	—
38	C₃²⋊(C₁₉⋊C₃)	—	9	1	no	—
39	#10587025	—	3	3	no	—
40	#10587025	—	9	1	no	—
41	#10587025	—	9	1	no	—
42	C₃×#4294	—	3	3	no	—
43	C₃×#4294	—	9	1	no	—
44	C₃×#733	—	19	1	no	—
45	C₃×#733	C₁₉⋊C₃	1	1	yes	characteristic, complemented
46	C₃×#732	—	19	1	no	—
47	#6887	—	19	1	no	—
48	#6883	—	19	9	no	—
49	C₃²×C₁₉	—	9	1	no	—
50	C₃²×C₁₉	#732	1	1	yes	characteristic
51	C₃²×C₁₉	#732	1	1	yes	characteristic
52	C₃²×C₁₉	#732	1	1	yes	characteristic
53	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
54	C₃×C₁₉⋊C₃	—	9	3	no	—
55	C₃×C₁₉⋊C₃	—	27	3	no	—
56	C₉×C₁₉	—	3	1	no	—
57	C₉×C₁₉	—	3	1	no	—
58	C₉×C₁₉	—	3	1	no	—
59	C₉×C₁₉	—	9	3	no	—
60	#4294	—	9	3	no	—
61	#4294	—	9	3	no	—
62	C₃×3₋¹⁺²	—	3	1	no	—
63	C₃×3₋¹⁺²	—	57	1	no	—
64	C₃×H(3,𝔽₃)	—	57	1	no	—
65	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
66	#733	—	3	1	no	—
67	#733	—	3	1	no	—
68	#733	—	19	3	no	—
69	#733	—	57	1	no	—
70	#733	—	57	1	no	—
71	#733	C₃×C₁₉⋊C₃	1	3	yes	complemented
72	#732	—	19	3	no	—
73	#732	—	57	1	no	—
74	#732	—	57	1	no	—
75	C₉⋊C₉	—	19	3	no	—
76	C₉⋊C₉	—	57	1	no	—
77	C₃²⋊C₉	—	57	1	no	—
78	C₉²	—	57	1	no	—
79	C₃×C₁₉	—	3	1	no	—
80	C₃×C₁₉	—	3	1	no	—
81	C₃×C₁₉	—	3	1	no	—
82	C₃×C₁₉	—	3	1	no	—
83	C₃×C₁₉	C₃×#732	1	1	yes	characteristic
84	C₁₉⋊C₃	—	27	3	no	—
85	C₃³	—	171	1	no	—
86	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
87	3₋¹⁺²	—	3	3	no	—
88	3₋¹⁺²	—	9	1	no	—
89	3₋¹⁺²	—	9	1	no	—
90	3₋¹⁺²	—	57	3	no	—
91	3₋¹⁺²	—	171	1	no	—
92	3₋¹⁺²	—	171	1	no	—
93	H(3,𝔽₃)	—	57	3	no	—
94	H(3,𝔽₃)	—	171	1	no	—
95	H(3,𝔽₃)	—	171	1	no	—
96	C₃×C₉	—	3	1	no	—
97	C₃×C₉	—	3	1	no	—
98	C₃×C₉	—	3	1	no	—
99	C₃×C₉	—	9	1	no	—
100	C₃×C₉	—	57	3	no	—
101	C₃×C₉	—	171	1	no	—
102	C₃×C₉	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
103	C₃×C₉	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
104	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
105	C₁₉	#10624848	1	1	yes	characteristic, complemented, Sylow
106	C₃²	—	513	3	no	—
107	C₃²	—	9	1	no	—
108	C₃²	—	171	3	no	—
109	C₃²	#13095319	1	1	yes	characteristic
110	C₃²	#13095319	1	1	yes	characteristic
111	C₃²	#13095319	1	1	yes	characteristic
112	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
113	C₉	—	3	1	no	—
114	C₉	—	3	1	no	—
115	C₉	—	3	1	no	—
116	C₉	—	9	3	no	—
117	C₉	—	171	3	no	—
118	C₉	—	171	3	no	—
119	C₃	—	513	3	no	—
120	C₃	—	3	1	no	—
121	C₃	—	3	1	no	—
122	C₃	—	3	1	no	—
123	C₃	—	3	1	no	—
124	C₃	C₃×#13095319	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#13095322

C₃

yes

maximal, characteristic, complemented

C₃×#13095320

C₃

yes

maximal, characteristic

#17987405

C₃

yes

maximal, characteristic, complemented

#17987388

C₃

yes

maximal, complemented

C₃×C₁₉×#733

C₃

yes

maximal, characteristic, complemented

C₃×#10587025

—

C₃×C₃²⋊(C₁₉⋊C₃)

—

#13095322

—

#13095322

—

#13095322

C₃²

yes

—

—

C₃²

yes

—

#13095307

C₃²

yes

complemented

#13095306

—

C₃²⋊#4294

—

C₉×#4294

—

C₃×C₁₉×3₋¹⁺²

—

C₃²×C₉×C₁₉

C₃²

yes

characteristic

C₁₉×#733

—

C₁₉×#733

—

C₁₉×#733

C₃²

yes

—

C₃³×C₁₉

yes

C₃²×C₁₉⋊C₃

—

C₁₉×3₋¹⁺²

—

C₁₉×3₋¹⁺²

—

C₁₉×3₋¹⁺²

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

yes

C₃×C₉×C₁₉

yes

C₃×C₉×C₁₉

C₃³

yes

characteristic

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

—

—

—

—

—

—

C₁₉⋊C₃

yes

characteristic, complemented

C₃×#732

—

#6887

—

#6883

—

C₃²×C₁₉

—

C₃²×C₁₉

yes

C₃²×C₁₉

yes

C₃²×C₁₉

yes

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₉×C₁₉

—

C₉×C₁₉

—

C₉×C₁₉

—

C₉×C₁₉

—

#4294

—

#4294

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

C₃×C₁₉⋊C₃

yes

—

—

—

—

—

C₃×C₁₉⋊C₃

yes

—

—

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₉²

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

C₃×#732

yes

characteristic

C₁₉⋊C₃

—

C₃³

—

171

—

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

171

—

3₋¹⁺²

—

171

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

171

—

H(3,𝔽₃)

—

171

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

100

C₃×C₉

—

101

C₃×C₉

—

171

—

102

C₃×C₉

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

103

C₃×C₉

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

104

C₃×C₉

C₃²×C₁₉⋊C₃

yes

characteristic

105

C₁₉

#10624848

yes

characteristic, complemented, Sylow

106

C₃²

—

513

—

107

C₃²

—

108

C₃²

—

171

—

109

C₃²

#13095319

yes

characteristic

110

C₃²

#13095319

yes

characteristic

111

C₃²

#13095319

yes

characteristic

112

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

113

C₉

—

114

C₉

—

115

C₉

—

116

C₉

—

117

C₉

—

171

—

118

C₉

—

171

—

119

C₃

—

513

—

120

C₃

—

121

C₃

—

122

C₃

—

123

C₃

—

124

C₃

C₃×#13095319

yes

characteristic

	subgroup	G/H
center	C₃	C₃×#13095319
commutator, cosocle	C₃×C₉×C₁₉	C₃³
Frattini	C₃×C₉	C₃²×C₁₉⋊C₃
Fitting	C₃×C₁₉×#733	C₃
socle	C₃×C₁₉	C₃×#732
prime radical	C₃×#13095320	C₃
hypercenter	C₃×#733	C₁₉⋊C₃
hypocenter	C₁₉	#10624848
LCS	G ▷ C₃×C₉×C₁₉ ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃²×C₉ ◁ C₃×#733

subgroup

G/H

center

C₃

C₃×#13095319

commutator, cosocle

C₃×C₉×C₁₉

C₃³

Frattini

C₃×C₉

C₃²×C₁₉⋊C₃

Fitting

C₃×C₁₉×#733

C₃

C₃×C₁₉

C₃

C₁₉⋊C₃

C₁₉

G ▷ C₃×C₉×C₁₉ ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃ ◁ C₃³ ◁ C₃²×C₉ ◁ C₃×#733

N	H ≅ G/N	type
1. C₃×#13095322	C₃	semidirect
3. #17987405	C₃	semidirect
4. #17987388	C₃	semidirect
5. C₃×C₁₉×#733	C₃	semidirect
10. #13095322	C₃²	semidirect
14. #13095307	C₃²	semidirect
22. C₁₉×#733	C₃²	semidirect
45. C₃×#733	C₁₉⋊C₃	semidirect
71. #733	C₃×C₁₉⋊C₃	semidirect
105. C₁₉	#10624848	semidirect

H ≅ G/N

type

1. C₃×#13095322

C₃

semidirect

3. #17987405

C₃

semidirect

4. #17987388

C₃

semidirect

5. C₃×C₁₉×#733

C₃

semidirect

10. #13095322

C₃²

semidirect

14. #13095307

C₃²

semidirect

22. C₁₉×#733

C₃²

semidirect

45. C₃×#733

C₁₉⋊C₃

semidirect

71. #733

C₃×C₁₉⋊C₃

semidirect

105. C₁₉

#10624848

semidirect

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

subgroup	index
C₃×#13095322	3
C₃×#13095320	3
#17987405	3
#17987388	3
C₃×C₁₉×#733	3
#10624848	19

subgroup

index

C₃×C₁₉×#733

Subgroups — #28355803

Conjugacy classes of subgroups

Complements

Maximal subgroups