Group #28356575 — subgroups

Subgroups — C₃×#17987518 #28356575

Order 13851.

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

normal subgroups only

This group has 980 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
2	C₃²×#10587025	C₃	1	1	yes	maximal, characteristic
3	#17987518	C₃	1	27	yes	maximal, direct product factor
4	C₃×#13095321	C₃	1	9	yes	maximal, complemented
5	C₃²×C₁₉×3₋¹⁺²	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	1	no	—
10	C₃×#10587025	—	3	2	no	—
11	C₃×#10587025	—	3	6	no	—
12	C₃×#10587025	C₃²	1	6	yes	—
13	C₃×#10587025	C₃²	1	18	yes	—
14	C₃×#10587025	C₃²	1	9	yes	—
15	C₃²×#4294	—	3	2	no	—
16	C₃²×#4294	C₃²	1	3	yes	—
17	#13095321	C₃²	1	81	yes	complemented
18	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
19	C₃×C₁₉×3₋¹⁺²	—	3	2	no	—
20	C₃×C₁₉×3₋¹⁺²	—	3	6	no	—
21	C₃×C₁₉×3₋¹⁺²	C₃²	1	3	yes	—
22	C₃×C₁₉×3₋¹⁺²	C₃²	1	9	yes	—
23	C₃²×C₉×C₁₉	—	3	1	no	—
24	C₃×#6912	—	19	1	no	maximal, Sylow
25	C₃³×C₁₉	—	9	3	no	—
26	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
27	C₃³×C₁₉	H(3,𝔽₃)	1	2	yes	—
28	C₃³×C₁₉	H(3,𝔽₃)	1	6	yes	—
29	C₃³×C₁₉	C₃³	1	1	yes	characteristic
30	C₃³×C₁₉	C₃³	1	3	yes	—
31	C₁₉×3₋¹⁺²	—	3	6	no	—
32	C₁₉×3₋¹⁺²	—	3	18	no	—
33	C₁₉×3₋¹⁺²	C₃³	1	9	yes	—
34	C₃×C₉×C₁₉	—	3	9	no	—
35	C₃×C₉×C₁₉	—	3	3	no	—
36	#10587025	—	3	12	no	—
37	#10587025	—	3	18	no	—
38	#10587025	—	3	18	no	—
39	#10587025	—	3	18	no	—
40	#10587025	—	3	9	no	—
41	#10587025	C₃³	1	18	yes	—
42	C₃×#4294	—	3	18	no	—
43	C₃×#4294	—	3	6	no	—
44	C₃×#4294	—	3	6	no	—
45	C₃×#4294	—	3	3	no	—
46	C₃×#4294	C₃³	1	9	yes	—
47	C₃²×3₋¹⁺²	—	19	1	no	—
48	C₃²×3₋¹⁺²	—	19	2	no	—
49	C₃²×3₋¹⁺²	C₁₉⋊C₃	1	1	yes	characteristic
50	#6912	—	19	27	no	—
51	C₃×#733	—	19	9	no	—
52	C₃²×C₁₉	—	3	3	no	—
53	C₃²×C₁₉	—	3	6	no	—
54	C₃²×C₁₉	—	3	3	no	—
55	C₃²×C₁₉	—	9	6	no	—
56	C₃²×C₁₉	—	9	3	no	—
57	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	6	yes	—
58	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	2	yes	—
59	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	3	yes	—
60	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
61	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
62	C₉×C₁₉	—	3	9	no	—
63	#4294	—	3	18	no	—
64	#4294	—	3	9	no	—
65	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
66	C₃×3₋¹⁺²	—	3	6	no	—
67	C₃×3₋¹⁺²	—	3	2	no	—
68	C₃×3₋¹⁺²	—	19	9	no	—
69	C₃×3₋¹⁺²	—	19	18	no	—
70	C₃×3₋¹⁺²	—	19	6	no	—
71	C₃×3₋¹⁺²	—	57	6	no	—
72	C₃×3₋¹⁺²	—	57	12	no	—
73	C₃×3₋¹⁺²	—	57	1	no	—
74	C₃×3₋¹⁺²	—	57	2	no	—
75	C₃×3₋¹⁺²	—	57	2	no	—
76	C₃×3₋¹⁺²	—	57	2	no	—
77	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	9	yes	—
78	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	3	yes	—
79	C₃²×C₉	—	3	1	no	—
80	C₃²×C₉	—	19	3	no	—
81	C₃²×C₉	—	57	2	no	—
82	#733	—	19	81	no	—
83	C₃×C₁₉	—	3	6	no	—
84	C₃×C₁₉	—	3	3	no	—
85	C₃×C₁₉	—	3	2	no	—
86	C₃×C₁₉	—	3	1	no	—
87	C₃×C₁₉	#6912	1	3	yes	complemented
88	C₃×C₁₉	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
89	C₃³	—	9	3	no	—
90	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
91	C₃³	C₃²⋊(C₁₉⋊C₃)	1	2	yes	—
92	C₃³	C₃²⋊(C₁₉⋊C₃)	1	6	yes	—
93	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
94	C₃³	C₃²×C₁₉⋊C₃	1	3	yes	—
95	3₋¹⁺²	—	3	18	no	—
96	3₋¹⁺²	—	3	6	no	—
97	3₋¹⁺²	—	19	18	no	—
98	3₋¹⁺²	—	57	9	no	—
99	3₋¹⁺²	—	57	18	no	—
100	3₋¹⁺²	—	57	18	no	—
101	3₋¹⁺²	—	57	18	no	—
102	3₋¹⁺²	—	57	12	no	—
103	3₋¹⁺²	C₃²×C₁₉⋊C₃	1	9	yes	—
104	C₃×C₉	—	3	9	no	—
105	C₃×C₉	—	3	3	no	—
106	C₃×C₉	—	19	9	no	—
107	C₃×C₉	—	57	6	no	—
108	C₃×C₉	—	57	18	no	—
109	C₃×C₉	—	57	3	no	—
110	C₃×C₉	—	57	6	no	—
111	C₁₉	C₃×#6912	1	1	yes	characteristic, complemented, Sylow
112	C₃²	—	3	3	no	—
113	C₃²	—	3	6	no	—
114	C₃²	—	3	3	no	—
115	C₃²	—	9	6	no	—
116	C₃²	—	9	3	no	—
117	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	3	yes	—
118	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	2	yes	—
119	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	6	yes	—
120	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
121	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
122	C₉	—	3	9	no	—
123	C₉	—	57	9	no	—
124	C₉	—	57	18	no	—
125	C₃	—	3	3	no	—
126	C₃	—	3	6	no	—
127	C₃	—	3	2	no	—
128	C₃	—	3	1	no	—
129	C₃	#17987518	1	3	yes	direct product factor
130	C₃	C₃²×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#10587025

C₃

yes

maximal

C₃²×#10587025

C₃

yes

maximal, characteristic

#17987518

C₃

yes

maximal, direct product factor

C₃×#13095321

C₃

yes

maximal, complemented

C₃²×C₁₉×3₋¹⁺²

C₃

yes

maximal, characteristic

—

—

—

—

—

—

C₃²

yes

—

C₃×#10587025

C₃²

yes

—

C₃×#10587025

C₃²

yes

—

C₃²×#4294

—

C₃²×#4294

C₃²

yes

—

#13095321

C₃²

yes

complemented

C₃⁴×C₁₉

C₃²

yes

characteristic

C₃×C₁₉×3₋¹⁺²

—

C₃×C₁₉×3₋¹⁺²

—

C₃×C₁₉×3₋¹⁺²

C₃²

yes

—

C₃×C₁₉×3₋¹⁺²

C₃²

yes

—

C₃²×C₉×C₁₉

—

C₃×#6912

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

H(3,𝔽₃)

yes

characteristic

C₃³×C₁₉

H(3,𝔽₃)

yes

—

C₃³×C₁₉

H(3,𝔽₃)

yes

—

C₃³×C₁₉

C₃³

yes

characteristic

C₃³×C₁₉

C₃³

yes

—

C₁₉×3₋¹⁺²

—

C₁₉×3₋¹⁺²

—

C₁₉×3₋¹⁺²

C₃³

yes

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

—

—

—

—

—

C₃³

yes

—

—

—

—

—

C₃³

yes

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₁₉⋊C₃

yes

characteristic

#6912

—

C₃×#733

—

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

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

—

C₃×3₋¹⁺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×C₁₉⋊C₃

yes

—

C₃×3₋¹⁺²

C₃×C₁₉⋊C₃

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

#733

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

#6912

yes

complemented

C₃×C₁₉

C₃²×H(3,𝔽₃)

yes

characteristic

C₃³

—

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

—

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

—

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

C₃³

C₃²×C₁₉⋊C₃

yes

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

100

3₋¹⁺²

—

101

3₋¹⁺²

—

102

3₋¹⁺²

—

103

3₋¹⁺²

C₃²×C₁₉⋊C₃

yes

—

104

C₃×C₉

—

105

C₃×C₉

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

—

110

C₃×C₉

—

111

C₁₉

C₃×#6912

yes

characteristic, complemented, Sylow

112

C₃²

—

113

C₃²

—

114

C₃²

—

115

C₃²

—

116

C₃²

—

117

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

—

118

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

—

119

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

—

120

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

121

C₃²

C₃³×C₁₉⋊C₃

yes

characteristic

122

C₉

—

123

C₉

—

124

C₉

—

125

C₃

—

126

C₃

—

127

C₃

—

128

C₃

—

129

C₃

#17987518

yes

direct product factor

130

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	C₃²×C₁₉×3₋¹⁺²	C₃
socle	C₃²×C₁₉	C₃×H(3,𝔽₃)
prime radical	C₃⁴×C₁₉	C₃²
hypercenter	C₃²×3₋¹⁺²	C₁₉⋊C₃
hypocenter	C₁₉	C₃×#6912
LCS	G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²×3₋¹⁺²

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

commutator, cosocle

C₃²×C₁₉

C₃⁴

Frattini

C₃²

C₃³×C₁₉⋊C₃

Fitting

C₃²×C₁₉×3₋¹⁺²

C₃

socle

C₃²×C₁₉

C₃×H(3,𝔽₃)

prime radical

C₃⁴×C₁₉

C₃²

hypercenter

C₃²×3₋¹⁺²

C₁₉⋊C₃

hypocenter

C₁₉

C₃×#6912

LCS

G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²×3₋¹⁺²

N	H ≅ G/N	type
3. #17987518	C₃	direct (N × H)
4. C₃×#13095321	C₃	semidirect
17. #13095321	C₃²	semidirect
87. C₃×C₁₉	#6912	semidirect
111. C₁₉	C₃×#6912	semidirect
129. C₃	#17987518	direct (N × H)

H ≅ G/N

type

3. #17987518

C₃

direct (N × H)

4. C₃×#13095321

C₃

semidirect

17. #13095321

C₃²

semidirect

87. C₃×C₁₉

#6912

semidirect

111. C₁₉

C₃×#6912

semidirect

129. C₃

#17987518

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
#17987518	3
C₃×#13095321	3
C₃²×C₁₉×3₋¹⁺²	3
C₃×#6912	19

subgroup

index

C₃²×C₁₉×3₋¹⁺²

Subgroups — C₃×#17987518 #28356575

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#17987518 #28356575

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#17987518 #28356575