Group #28354649 — subgroups

Subgroups — #28354649

Order 13851.

404 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 112 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	1	yes	maximal, characteristic
2	C₃×C₃²⋊#4294	C₃	1	2	yes	maximal
3	#17987338	C₃	1	9	yes	maximal, complemented
4	C₃×C₁₉×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
5	C₃×#10587025	—	3	2	no	—
6	C₃×#10587025	—	3	1	no	—
7	C₃×#10587025	—	3	3	no	—
8	C₃×#10587025	—	3	3	no	—
9	C₃×#10587025	C₉	1	9	yes	complemented
10	C₃²×#4294	—	3	2	no	—
11	C₃²×#4294	C₃²	1	3	yes	—
12	C₃²⋊#4294	—	3	2	no	—
13	C₃²⋊#4294	—	3	2	no	—
14	C₃²⋊#4294	C₃²	1	6	yes	—
15	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
16	C₃²×C₉×C₁₉	—	3	1	no	—
17	C₁₉×C₃²⋊C₉	—	3	2	no	—
18	C₁₉×C₃²⋊C₉	C₃²	1	3	yes	—
19	#10624691	—	19	1	no	maximal, Sylow
20	C₃³×C₁₉	—	9	2	no	—
21	C₃³×C₁₉	—	9	1	no	—
22	C₃³×C₁₉	C₃×C₉	1	3	yes	—
23	C₃³×C₁₉	H(3,𝔽₃)	1	2	yes	—
24	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
25	C₃³×C₁₉	3₋¹⁺²	1	3	yes	—
26	C₃³×C₁₉	3₋¹⁺²	1	3	yes	—
27	C₃³×C₁₉	C₃³	1	1	yes	characteristic
28	C₃×C₉×C₁₉	—	3	3	no	—
29	C₃×C₉×C₁₉	—	9	3	no	—
30	#10587025	—	3	18	no	—
31	#10587025	—	3	9	no	—
32	C₃×#4294	—	3	3	no	—
33	C₃×#4294	—	3	3	no	—
34	C₃×#4294	—	3	3	no	—
35	C₃×#4294	—	3	6	no	—
36	C₃×#4294	—	9	6	no	—
37	C₃×#4294	C₃×C₉	1	9	yes	complemented
38	C₃²×3₋¹⁺²	—	19	1	no	—
39	C₃×C₃²⋊C₉	—	19	2	no	—
40	C₃×C₃²⋊C₉	C₁₉⋊C₃	1	1	yes	characteristic
41	#6868	—	19	9	no	—
42	C₃²×C₁₉	—	3	4	no	—
43	C₃²×C₁₉	—	3	2	no	—
44	C₃²×C₁₉	—	3	2	no	—
45	C₃²×C₁₉	—	3	2	no	—
46	C₃²×C₁₉	—	3	1	no	—
47	C₃²×C₁₉	—	3	1	no	—
48	C₃²×C₁₉	—	9	3	no	—
49	C₃²×C₁₉	—	9	3	no	—
50	C₃²×C₁₉	—	9	3	no	—
51	C₃²×C₁₉	C₃²⋊C₉	1	6	yes	—
52	C₃²×C₁₉	C₃²⋊C₉	1	3	yes	—
53	C₃²×C₁₉	C₃²×C₉	1	1	yes	characteristic
54	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
55	C₃²×C₁₉	C₃×3₋¹⁺²	1	1	yes	characteristic
56	C₃²×C₁₉	C₃×3₋¹⁺²	1	1	yes	characteristic
57	C₉×C₁₉	—	9	3	no	—
58	#4294	—	3	9	no	—
59	#4294	—	9	6	no	—
60	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
61	C₃×3₋¹⁺²	—	19	9	no	—
62	C₃×3₋¹⁺²	—	57	3	no	—
63	C₃×3₋¹⁺²	—	57	3	no	—
64	C₃×3₋¹⁺²	—	57	2	no	—
65	C₃×3₋¹⁺²	—	57	1	no	—
66	C₃²×C₉	—	3	1	no	—
67	C₃²×C₉	—	19	3	no	—
68	C₃²×C₉	—	57	2	no	—
69	C₃²⋊C₉	—	3	2	no	—
70	C₃²⋊C₉	—	19	6	no	—
71	C₃²⋊C₉	—	57	2	no	—
72	C₃²⋊C₉	—	57	2	no	—
73	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	—
74	C₃×C₁₉	—	3	6	no	—
75	C₃×C₁₉	—	3	3	no	—
76	C₃×C₁₉	—	3	1	no	—
77	C₃×C₁₉	—	3	1	no	—
78	C₃×C₁₉	—	3	1	no	—
79	C₃×C₁₉	C₃×C₃²⋊C₉	1	1	yes	characteristic
80	C₃×C₁₉	#6911	1	1	yes	characteristic
81	C₃×C₁₉	#6912	1	2	yes	—
82	C₃³	—	9	1	no	—
83	C₃³	—	9	2	no	—
84	C₃³	C₉×C₁₉⋊C₃	1	3	yes	—
85	C₃³	#10587022	1	3	yes	—
86	C₃³	#10587023	1	3	yes	—
87	C₃³	C₃²⋊(C₁₉⋊C₃)	1	2	yes	—
88	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
89	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
90	3₋¹⁺²	—	57	18	no	—
91	3₋¹⁺²	—	57	9	no	—
92	C₃×C₉	—	3	3	no	—
93	C₃×C₉	—	9	3	no	—
94	C₃×C₉	—	19	9	no	—
95	C₃×C₉	—	57	3	no	—
96	C₃×C₉	—	57	3	no	—
97	C₃×C₉	—	57	3	no	—
98	C₃×C₉	—	57	6	no	—
99	C₃×C₉	—	171	6	no	—
100	C₁₉	#10624691	1	1	yes	characteristic, complemented, Sylow
101	C₃²	—	3	1	no	—
102	C₃²	—	3	2	no	—
103	C₃²	—	3	1	no	—
104	C₃²	—	3	4	no	—
105	C₃²	—	3	2	no	—
106	C₃²	—	3	2	no	—
107	C₃²	—	9	3	no	—
108	C₃²	—	9	3	no	—
109	C₃²	—	9	3	no	—
110	C₃²	#13095303	1	6	yes	—
111	C₃²	#13095303	1	3	yes	—
112	C₃²	C₃×C₉×C₁₉⋊C₃	1	1	yes	characteristic
113	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
114	C₃²	C₃×#10587023	1	1	yes	characteristic
115	C₃²	C₃×#10587022	1	1	yes	characteristic
116	C₉	—	9	3	no	—
117	C₉	—	57	9	no	—
118	C₉	—	171	6	no	—
119	C₃	—	3	3	no	—
120	C₃	—	3	6	no	—
121	C₃	—	3	1	no	—
122	C₃	—	3	1	no	—
123	C₃	—	3	1	no	—
124	C₃	C₃×#13095303	1	1	yes	characteristic
125	C₃	#17987514	1	1	yes	characteristic
126	C₃	#17987518	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#10587025

C₃

yes

maximal, characteristic

C₃×C₃²⋊#4294

C₃

yes

maximal

#17987338

C₃

yes

maximal, complemented

C₃×C₁₉×C₃²⋊C₉

C₃

yes

maximal, characteristic

—

—

—

—

C₉

yes

complemented

C₃²×#4294

—

C₃²×#4294

C₃²

yes

—

C₃²⋊#4294

—

C₃²⋊#4294

—

C₃²⋊#4294

C₃²

yes

—

C₃⁴×C₁₉

C₃²

yes

characteristic

C₃²×C₉×C₁₉

—

C₁₉×C₃²⋊C₉

—

C₁₉×C₃²⋊C₉

C₃²

yes

—

#10624691

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

C₃×C₉

yes

—

C₃³×C₁₉

H(3,𝔽₃)

yes

—

C₃³×C₁₉

H(3,𝔽₃)

yes

characteristic

C₃³×C₁₉

3₋¹⁺²

yes

—

C₃³×C₁₉

3₋¹⁺²

yes

—

C₃³×C₁₉

C₃³

yes

characteristic

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

—

—

—

—

—

—

—

C₃×C₉

yes

complemented

C₃²×3₋¹⁺²

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

C₁₉⋊C₃

yes

characteristic

#6868

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃²⋊C₉

yes

—

C₃²×C₁₉

C₃²⋊C₉

yes

—

C₃²×C₁₉

C₃²×C₉

yes

characteristic

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₉

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₁₉

C₃×3₋¹⁺²

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

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₁₉⋊C₃

yes

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

C₃×C₃²⋊C₉

yes

C₃×C₁₉

yes

C₃×C₁₉

yes

—

C₃³

—

C₃³

—

C₃³

C₉×C₁₉⋊C₃

yes

—

C₃³

#10587022

yes

—

C₃³

#10587023

yes

—

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

—

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

171

—

100

C₁₉

#10624691

yes

characteristic, complemented, Sylow

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

#13095303

yes

—

111

C₃²

#13095303

yes

—

112

C₃²

C₃×C₉×C₁₉⋊C₃

yes

characteristic

113

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

114

C₃²

C₃×#10587023

yes

characteristic

115

C₃²

C₃×#10587022

yes

characteristic

116

C₉

—

117

C₉

—

118

C₉

—

171

—

119

C₃

—

120

C₃

—

121

C₃

—

122

C₃

—

123

C₃

—

124

C₃

C₃×#13095303

yes

characteristic

125

C₃

#17987514

yes

characteristic

126

C₃

#17987518

yes

—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)
commutator	C₃²×C₁₉	C₃²×C₉
Frattini	C₃³	C₃²×C₁₉⋊C₃
Fitting	C₃×C₁₉×C₃²⋊C₉	C₃
socle	C₃²×C₁₉	C₃×H(3,𝔽₃)
prime radical	C₃⁴×C₁₉	C₃²
cosocle	C₃³×C₁₉	C₃³
hypercenter	C₃×C₃²⋊C₉	C₁₉⋊C₃
hypocenter	C₁₉	#10624691
LCS	G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃×C₃²⋊C₉

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

commutator

C₃²×C₁₉

C₃²×C₉

Frattini

C₃³

C₃²×C₁₉⋊C₃

Fitting

C₃×C₁₉×C₃²⋊C₉

C₃

socle

C₃²×C₁₉

C₃×H(3,𝔽₃)

prime radical

C₃⁴×C₁₉

C₃²

cosocle

C₃³×C₁₉

C₃³

hypercenter

C₃×C₃²⋊C₉

C₁₉⋊C₃

hypocenter

C₁₉

#10624691

LCS

G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃×C₃²⋊C₉

N	H ≅ G/N	type
3. #17987338	C₃	semidirect
9. C₃×#10587025	C₉	semidirect
37. C₃×#4294	C₃×C₉	semidirect
100. C₁₉	#10624691	semidirect

H ≅ G/N

type

3. #17987338

C₃

semidirect

9. C₃×#10587025

C₉

semidirect

37. C₃×#4294

C₃×C₉

semidirect

100. C₁₉

#10624691

semidirect

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

subgroup	index
C₃²×#10587025	3
C₃×C₃²⋊#4294	3
#17987338	3
C₃×C₁₉×C₃²⋊C₉	3
#10624691	19

subgroup

index

C₃²×#10587025

C₃×C₃²⋊#4294

#17987338

C₃×C₁₉×C₃²⋊C₉

#10624691

Subgroups — #28354649

Conjugacy classes of subgroups

Complements

Maximal subgroups