Group #28356564 — subgroups

Subgroups — C₃×#17987509 #28356564

Order 13851.

1088 conjugacy classes of subgroups; only the 292 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 292 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	C₃²×#10587025	C₃	1	1	yes	maximal, characteristic, complemented
3	#17987509	C₃	1	27	yes	maximal, direct product factor
4	C₃×#13095316	C₃	1	9	yes	maximal, complemented
5	C₃²×C₁₉×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
6	C₃×#10587025	—	3	2	no	—
7	C₃×#10587025	—	3	2	no	—
8	C₃×#10587025	—	3	12	no	—
9	C₃×#10587025	—	3	2	no	—
10	C₃×#10587025	—	3	1	no	—
11	C₃×#10587025	—	3	6	no	—
12	C₃×#10587025	C₃²	1	6	yes	complemented
13	C₃×#10587025	C₃²	1	18	yes	complemented
14	C₃×#10587025	C₃²	1	9	yes	complemented
15	C₃²×#4294	—	3	2	no	—
16	C₃²×#4294	C₃²	1	3	yes	complemented
17	#13095316	C₃²	1	81	yes	complemented
18	C₃⁴×C₁₉	—	3	1	no	—
19	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
20	C₃×C₁₉×H(3,𝔽₃)	—	3	2	no	—
21	C₃×C₁₉×H(3,𝔽₃)	—	3	6	no	—
22	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	3	yes	—
23	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	9	yes	—
24	C₃×#6910	—	19	1	no	maximal, Sylow
25	C₃³×C₁₉	—	3	9	no	—
26	C₃³×C₁₉	—	3	3	no	—
27	C₃³×C₁₉	—	9	3	no	—
28	C₃³×C₁₉	—	9	9	no	—
29	C₃³×C₁₉	H(3,𝔽₃)	1	2	yes	—
30	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
31	C₃³×C₁₉	H(3,𝔽₃)	1	6	yes	—
32	C₃³×C₁₉	C₃³	1	1	yes	characteristic
33	C₃³×C₁₉	C₃³	1	3	yes	—
34	C₁₉×H(3,𝔽₃)	—	3	6	no	—
35	C₁₉×H(3,𝔽₃)	—	3	18	no	—
36	C₁₉×H(3,𝔽₃)	C₃³	1	9	yes	—
37	#10587025	—	3	12	no	—
38	#10587025	—	3	18	no	—
39	#10587025	—	3	18	no	—
40	#10587025	—	3	18	no	—
41	#10587025	—	3	9	no	—
42	#10587025	C₃³	1	18	yes	complemented
43	C₃×#4294	—	3	18	no	—
44	C₃×#4294	—	3	6	no	—
45	C₃×#4294	—	3	6	no	—
46	C₃×#4294	—	3	3	no	—
47	C₃×#4294	C₃³	1	9	yes	complemented
48	C₃²×3₋¹⁺²	—	19	1	no	—
49	C₃²×3₋¹⁺²	—	19	2	no	—
50	C₃²×H(3,𝔽₃)	C₁₉⋊C₃	1	1	yes	characteristic
51	#6910	—	19	27	no	—
52	C₃×#731	—	19	9	no	—
53	C₃²×C₁₉	—	3	3	no	—
54	C₃²×C₁₉	—	3	6	no	—
55	C₃²×C₁₉	—	3	3	no	—
56	C₃²×C₁₉	—	3	9	no	—
57	C₃²×C₁₉	—	9	6	no	—
58	C₃²×C₁₉	—	9	3	no	—
59	C₃²×C₁₉	—	9	27	no	—
60	C₃²×C₁₉	—	9	9	no	—
61	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	6	yes	—
62	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	2	yes	—
63	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	3	yes	—
64	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
65	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
66	#4294	—	3	18	no	—
67	#4294	—	3	9	no	—
68	C₃⁴	—	3	1	no	—
69	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
70	C₃×3₋¹⁺²	—	19	9	no	—
71	C₃×3₋¹⁺²	—	19	6	no	—
72	C₃×3₋¹⁺²	—	19	18	no	—
73	C₃×3₋¹⁺²	—	57	6	no	—
74	C₃×3₋¹⁺²	—	57	12	no	—
75	C₃×3₋¹⁺²	—	57	2	no	—
76	C₃×3₋¹⁺²	—	57	1	no	—
77	C₃×3₋¹⁺²	—	57	2	no	—
78	C₃×3₋¹⁺²	—	57	2	no	—
79	C₃×H(3,𝔽₃)	—	3	6	no	—
80	C₃×H(3,𝔽₃)	—	3	2	no	—
81	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	3	yes	—
82	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	9	yes	—
83	C₃²×C₉	—	19	3	no	—
84	C₃²×C₉	—	57	2	no	—
85	#731	—	19	81	no	—
86	C₃×C₁₉	—	3	3	no	—
87	C₃×C₁₉	—	3	6	no	—
88	C₃×C₁₉	—	3	2	no	—
89	C₃×C₁₉	—	3	1	no	—
90	C₃×C₁₉	—	9	9	no	—
91	C₃×C₁₉	#6910	1	3	yes	complemented
92	C₃×C₁₉	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
93	C₃³	—	3	9	no	—
94	C₃³	—	3	3	no	—
95	C₃³	—	9	3	no	—
96	C₃³	—	9	9	no	—
97	C₃³	C₃²⋊(C₁₉⋊C₃)	1	2	yes	—
98	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
99	C₃³	C₃²⋊(C₁₉⋊C₃)	1	6	yes	—
100	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
101	C₃³	C₃²×C₁₉⋊C₃	1	3	yes	—
102	3₋¹⁺²	—	19	18	no	—
103	3₋¹⁺²	—	57	18	no	—
104	3₋¹⁺²	—	57	9	no	—
105	3₋¹⁺²	—	57	12	no	—
106	3₋¹⁺²	—	57	18	no	—
107	3₋¹⁺²	—	57	18	no	—
108	H(3,𝔽₃)	—	3	6	no	—
109	H(3,𝔽₃)	—	3	18	no	—
110	H(3,𝔽₃)	C₃²×C₁₉⋊C₃	1	9	yes	—
111	C₃×C₉	—	19	9	no	—
112	C₃×C₉	—	57	6	no	—
113	C₃×C₉	—	57	18	no	—
114	C₃×C₉	—	57	3	no	—
115	C₃×C₉	—	57	6	no	—
116	C₁₉	C₃×#6910	1	1	yes	characteristic, complemented, Sylow
117	C₃²	—	3	3	no	—
118	C₃²	—	3	6	no	—
119	C₃²	—	3	3	no	—
120	C₃²	—	3	9	no	—
121	C₃²	—	9	6	no	—
122	C₃²	—	9	3	no	—
123	C₃²	—	9	27	no	—
124	C₃²	—	9	9	no	—
125	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	2	yes	—
126	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	6	yes	—
127	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	3	yes	—
128	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
129	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
130	C₉	—	57	9	no	—
131	C₉	—	57	18	no	—
132	C₃	—	3	3	no	—
133	C₃	—	3	6	no	—
134	C₃	—	3	2	no	—
135	C₃	—	3	1	no	—
136	C₃	—	9	9	no	—
137	C₃	#17987509	1	3	yes	direct product factor
138	C₃	C₃²×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#10587025

C₃

yes

maximal, complemented

C₃²×#10587025

C₃

yes

maximal, characteristic, complemented

#17987509

C₃

yes

maximal, direct product factor

C₃×#13095316

C₃

yes

maximal, complemented

C₃²×C₁₉×H(3,𝔽₃)

C₃

yes

maximal, characteristic

—

—

—

—

—

—

C₃²

yes

complemented

C₃×#10587025

C₃²

yes

complemented

C₃×#10587025

C₃²

yes

complemented

C₃²×#4294

—

C₃²×#4294

C₃²

yes

complemented

#13095316

C₃²

yes

complemented

C₃⁴×C₁₉

—

C₃⁴×C₁₉

C₃²

yes

characteristic

C₃×C₁₉×H(3,𝔽₃)

—

C₃×C₁₉×H(3,𝔽₃)

—

C₃×C₁₉×H(3,𝔽₃)

C₃²

yes

—

C₃×C₁₉×H(3,𝔽₃)

C₃²

yes

—

C₃×#6910

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

H(3,𝔽₃)

yes

—

C₃³×C₁₉

H(3,𝔽₃)

yes

characteristic

C₃³×C₁₉

H(3,𝔽₃)

yes

—

C₃³×C₁₉

C₃³

yes

characteristic

C₃³×C₁₉

C₃³

yes

—

C₁₉×H(3,𝔽₃)

—

C₁₉×H(3,𝔽₃)

—

C₁₉×H(3,𝔽₃)

C₃³

yes

—

—

—

—

—

—

C₃³

yes

—

—

—

—

C₃³

yes

complemented

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

—

C₃²×H(3,𝔽₃)

C₁₉⋊C₃

yes

characteristic

#6910

—

C₃×#731

—

C₃²×C₁₉

—

C₃²×C₁₉

—

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

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₉

C₃⁴

yes

characteristic

#4294

—

#4294

—

C₃⁴

—

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

—

C₃×3₋¹⁺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

—

C₃×H(3,𝔽₃)

C₃×C₁₉⋊C₃

yes

—

C₃²×C₉

—

C₃²×C₉

—

#731

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

#6910

yes

complemented

C₃×C₁₉

C₃²×H(3,𝔽₃)

yes

characteristic

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

—

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

—

100

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

101

C₃³

C₃²×C₁₉⋊C₃

yes

—

102

3₋¹⁺²

—

103

3₋¹⁺²

—

104

3₋¹⁺²

—

105

3₋¹⁺²

—

106

3₋¹⁺²

—

107

3₋¹⁺²

—

108

H(3,𝔽₃)

—

109

H(3,𝔽₃)

—

110

H(3,𝔽₃)

C₃²×C₁₉⋊C₃

yes

—

111

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

—

114

C₃×C₉

—

115

C₃×C₉

—

116

C₁₉

C₃×#6910

yes

characteristic, complemented, Sylow

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

—

125

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

—

126

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

—

127

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

—

128

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

129

C₃²

C₃³×C₁₉⋊C₃

yes

characteristic

130

C₉

—

131

C₉

—

132

C₃

—

133

C₃

—

134

C₃

—

135

C₃

—

136

C₃

—

137

C₃

#17987509

yes

direct product factor

138

C₃

C₃²×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)
commutator, cosocle	C₃²×C₁₉	C₃⁴
Frattini	C₃²	C₃³×C₁₉⋊C₃
Fitting, prime radical	C₃²×C₁₉×H(3,𝔽₃)	C₃
socle	C₃²×C₁₉	C₃×H(3,𝔽₃)
hypercenter	C₃²×H(3,𝔽₃)	C₁₉⋊C₃
hypocenter	C₁₉	C₃×#6910
LCS	G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²×H(3,𝔽₃)

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

commutator, cosocle

C₃²×C₁₉

C₃⁴

Frattini

C₃²

C₃³×C₁₉⋊C₃

Fitting, prime radical

C₃²×C₁₉×H(3,𝔽₃)

C₃

socle

C₃²×C₁₉

C₃×H(3,𝔽₃)

hypercenter

C₃²×H(3,𝔽₃)

C₁₉⋊C₃

hypocenter

C₁₉

C₃×#6910

LCS

G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²×H(3,𝔽₃)

N	H ≅ G/N	type
1. C₃²×#10587025	C₃	semidirect
2. C₃²×#10587025	C₃	semidirect
3. #17987509	C₃	direct (N × H)
4. C₃×#13095316	C₃	semidirect
12. C₃×#10587025	C₃²	semidirect
13. C₃×#10587025	C₃²	semidirect
14. C₃×#10587025	C₃²	semidirect
16. C₃²×#4294	C₃²	semidirect
17. #13095316	C₃²	semidirect
42. #10587025	C₃³	semidirect
47. C₃×#4294	C₃³	semidirect
91. C₃×C₁₉	#6910	semidirect
116. C₁₉	C₃×#6910	semidirect
137. C₃	#17987509	direct (N × H)

H ≅ G/N

type

1. C₃²×#10587025

C₃

semidirect

2. C₃²×#10587025

C₃

semidirect

3. #17987509

C₃

direct (N × H)

4. C₃×#13095316

C₃

semidirect

12. C₃×#10587025

C₃²

semidirect

13. C₃×#10587025

C₃²

semidirect

14. C₃×#10587025

C₃²

semidirect

16. C₃²×#4294

C₃²

semidirect

17. #13095316

C₃²

semidirect

42. #10587025

C₃³

semidirect

47. C₃×#4294

C₃³

semidirect

91. C₃×C₁₉

#6910

semidirect

116. C₁₉

C₃×#6910

semidirect

137. C₃

#17987509

direct (N × H)

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

subgroup	index
C₃²×#10587025	3
#17987509	3
C₃×#13095316	3
C₃²×C₁₉×H(3,𝔽₃)	3
C₃×#6910	19

subgroup

index

C₃²×C₁₉×H(3,𝔽₃)

Subgroups — C₃×#17987509 #28356564

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#17987509 #28356564

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#17987509 #28356564