Group #28356068 — subgroups

Subgroups — #28356068

Order 13851.

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

normal subgroups only

This group has 716 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×#10587022	C₃	1	1	yes	maximal, characteristic
2	#17987472	C₃	1	6	yes	maximal, complemented
3	#17987469	C₃	1	3	yes	maximal, complemented
4	#17987466	C₃	1	12	yes	maximal, complemented
5	#17987463	C₃	1	6	yes	maximal, complemented
6	#17987443	C₃	1	6	yes	maximal, complemented
7	#17987420	C₃	1	3	yes	maximal
8	#17987405	C₃	1	2	yes	maximal, complemented
9	C₁₉×#6887	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃³×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
11	C₃×#10587022	—	3	9	no	—
12	C₃×#10587022	C₃²	1	2	yes	—
13	C₃×#10587022	C₃²	1	4	yes	—
14	C₃×#10587022	C₃²	1	1	yes	characteristic
15	C₃×#10587022	C₃²	1	2	yes	—
16	C₃×#10587022	C₃²	1	9	yes	—
17	C₃²×#4294	C₃²	1	2	yes	—
18	#13095307	C₃²	1	6	yes	complemented
19	#13095306	C₃²	1	6	yes	complemented
20	#13095306	C₃²	1	18	yes	complemented
21	#13095305	C₃²	1	36	yes	complemented
22	C₃²⋊#4294	C₃²	1	6	yes	—
23	#13095303	C₃²	1	18	yes	—
24	C₉×#4294	C₃²	1	6	yes	complemented
25	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
26	C₁₉×C₉⋊C₉	C₃²	1	6	yes	complemented
27	C₁₉×C₃²⋊C₉	C₃²	1	3	yes	—
28	C₉²×C₁₉	C₃²	1	3	yes	complemented
29	#10624884	—	19	1	no	maximal, Sylow
30	C₃³×C₁₉	C₃³	1	1	yes	characteristic
31	C₃²×C₁₉⋊C₃	—	3	3	no	—
32	C₃²×C₁₉⋊C₃	—	9	3	no	—
33	C₃²×C₁₉⋊C₃	C₃³	1	2	yes	—
34	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
35	C₃×C₉×C₁₉	3₋¹⁺²	1	9	yes	complemented
36	C₃×C₉×C₁₉	C₃³	1	9	yes	—
37	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
38	C₃×C₉×C₁₉	C₃³	1	2	yes	—
39	C₃×#4294	—	3	6	no	—
40	C₃×#4294	C₃³	1	18	yes	—
41	C₃×#4294	C₃³	1	4	yes	—
42	C₃×#4294	C₃³	1	2	yes	—
43	#10587022	—	3	6	no	—
44	#10587022	—	3	12	no	—
45	#10587022	—	3	3	no	—
46	#10587022	—	3	6	no	—
47	#10587022	—	9	9	no	—
48	C₃²×3₋¹⁺²	—	19	1	no	—
49	#6900	—	19	6	no	—
50	#6900	—	19	3	no	—
51	#6899	—	19	12	no	—
52	#6899	—	19	6	no	—
53	#6895	—	19	6	no	—
54	#6891	—	19	3	no	—
55	#6887	—	19	2	no	—
56	#6887	C₁₉⋊C₃	1	1	yes	characteristic, complemented
57	C₃²×C₁₉	—	3	3	no	—
58	C₃²×C₁₉	C₃×3₋¹⁺²	1	3	yes	—
59	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
60	C₃×C₁₉⋊C₃	—	3	2	no	—
61	C₃×C₁₉⋊C₃	—	3	1	no	—
62	C₃×C₁₉⋊C₃	—	3	3	no	—
63	C₃×C₁₉⋊C₃	—	3	6	no	—
64	C₃×C₁₉⋊C₃	—	9	9	no	—
65	C₉×C₁₉	—	3	9	no	—
66	C₉×C₁₉	C₃×3₋¹⁺²	1	3	yes	—
67	C₉×C₁₉	C₃×3₋¹⁺²	1	6	yes	—
68	#4294	—	3	4	no	—
69	#4294	—	3	2	no	—
70	#4294	—	3	18	no	—
71	C₃⁴	—	19	1	no	—
72	C₃×3₋¹⁺²	—	19	4	no	—
73	C₃×3₋¹⁺²	—	19	2	no	—
74	C₃×3₋¹⁺²	—	19	2	no	—
75	C₃×3₋¹⁺²	—	19	1	no	—
76	C₃×3₋¹⁺²	—	19	9	no	—
77	C₃×3₋¹⁺²	—	57	9	no	—
78	C₃²×C₉	—	19	2	no	—
79	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
80	C₉⋊C₉	—	19	36	no	—
81	C₉⋊C₉	—	19	6	no	—
82	C₉⋊C₉	—	19	18	no	—
83	C₉⋊C₉	—	19	6	no	—
84	C₉⋊C₉	C₃×C₁₉⋊C₃	1	6	yes	complemented
85	C₃²⋊C₉	—	19	6	no	—
86	C₃²⋊C₉	—	19	18	no	—
87	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	—
88	C₉²	—	19	6	no	—
89	C₉²	C₃×C₁₉⋊C₃	1	3	yes	complemented
90	C₃×C₁₉	—	3	3	no	—
91	C₃×C₁₉	C₃²×3₋¹⁺²	1	1	yes	characteristic
92	C₃×C₁₉	3₋¹⁺⁴	1	3	yes	—
93	C₁₉⋊C₃	—	9	1	no	—
94	C₁₉⋊C₃	—	9	2	no	—
95	C₃³	—	19	2	no	—
96	C₃³	—	19	1	no	—
97	C₃³	—	57	3	no	—
98	C₃³	—	171	3	no	—
99	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
100	3₋¹⁺²	—	57	12	no	—
101	3₋¹⁺²	—	57	6	no	—
102	3₋¹⁺²	—	57	6	no	—
103	3₋¹⁺²	—	57	3	no	—
104	3₋¹⁺²	—	171	9	no	—
105	C₃×C₉	—	19	18	no	—
106	C₃×C₉	—	19	4	no	—
107	C₃×C₉	—	19	2	no	—
108	C₃×C₉	—	57	6	no	—
109	C₃×C₉	#10587022	1	9	yes	complemented
110	C₃×C₉	C₃²×C₁₉⋊C₃	1	9	yes	—
111	C₃×C₉	C₃²×C₁₉⋊C₃	1	2	yes	—
112	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
113	C₁₉	#10624884	1	1	yes	characteristic, complemented, Sylow
114	C₃²	—	3	3	no	—
115	C₃²	—	57	2	no	—
116	C₃²	—	57	1	no	—
117	C₃²	—	57	3	no	—
118	C₃²	—	57	6	no	—
119	C₃²	—	171	9	no	—
120	C₃²	C₃×#10587022	1	3	yes	—
121	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
122	C₉	—	3	9	no	—
123	C₉	—	57	4	no	—
124	C₉	—	57	2	no	—
125	C₉	—	57	18	no	—
126	C₉	C₃×#10587022	1	6	yes	—
127	C₉	C₃×#10587022	1	3	yes	—
128	C₃	—	3	3	no	—
129	C₃	—	171	1	no	—
130	C₃	—	171	2	no	—
131	C₃	C₃²×#10587022	1	1	yes	characteristic
132	C₃	#17987545	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#10587022

C₃

yes

maximal, characteristic

#17987472

C₃

yes

maximal, complemented

#17987469

C₃

yes

maximal, complemented

#17987466

C₃

yes

maximal, complemented

#17987463

C₃

yes

maximal, complemented

#17987443

C₃

yes

maximal, complemented

#17987420

C₃

yes

maximal

#17987405

C₃

yes

maximal, complemented

C₁₉×#6887

C₃

yes

maximal, characteristic, complemented

C₃³×C₁₉⋊C₃

C₃²

yes

characteristic

C₃×#10587022

—

C₃×#10587022

C₃²

yes

—

C₃×#10587022

C₃²

yes

—

C₃×#10587022

C₃²

yes

characteristic

C₃×#10587022

C₃²

yes

—

C₃×#10587022

C₃²

yes

—

C₃²×#4294

C₃²

yes

—

#13095307

C₃²

yes

complemented

#13095306

C₃²

yes

complemented

#13095306

C₃²

yes

complemented

#13095305

C₃²

yes

complemented

C₃²⋊#4294

C₃²

yes

—

#13095303

C₃²

yes

—

C₉×#4294

C₃²

yes

complemented

C₃²×C₉×C₁₉

C₃²

yes

characteristic

C₁₉×C₉⋊C₉

C₃²

yes

complemented

C₁₉×C₃²⋊C₉

C₃²

yes

—

C₉²×C₁₉

C₃²

yes

complemented

#10624884

—

maximal, Sylow

C₃³×C₁₉

C₃³

yes

characteristic

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

C₃³

yes

—

C₃²×C₁₉⋊C₃

C₃³

yes

characteristic

C₃×C₉×C₁₉

3₋¹⁺²

yes

complemented

C₃×C₉×C₁₉

C₃³

yes

—

C₃×C₉×C₁₉

C₃³

yes

characteristic

C₃×C₉×C₁₉

C₃³

yes

—

C₃×#4294

—

C₃×#4294

C₃³

yes

—

C₃×#4294

C₃³

yes

—

C₃×#4294

C₃³

yes

—

—

—

—

—

—

C₃²×3₋¹⁺²

—

—

—

—

—

—

—

—

C₁₉⋊C₃

yes

characteristic, complemented

C₃²×C₁₉

—

C₃²×C₁₉

C₃×3₋¹⁺²

yes

—

C₃²×C₁₉

C₃⁴

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₉×C₁₉

—

C₉×C₁₉

C₃×3₋¹⁺²

yes

—

C₉×C₁₉

C₃×3₋¹⁺²

yes

—

#4294

—

#4294

—

#4294

—

C₃⁴

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃²×C₉

—

C₃²×C₉

C₃×C₁₉⋊C₃

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

C₃×C₁₉⋊C₃

yes

complemented

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₁₉⋊C₃

yes

—

C₉²

—

C₉²

C₃×C₁₉⋊C₃

yes

complemented

C₃×C₁₉

—

C₃×C₁₉

C₃²×3₋¹⁺²

yes

characteristic

C₃×C₁₉

3₋¹⁺⁴

yes

—

C₁₉⋊C₃

—

C₁₉⋊C₃

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

171

—

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

100

3₋¹⁺²

—

101

3₋¹⁺²

—

102

3₋¹⁺²

—

103

3₋¹⁺²

—

104

3₋¹⁺²

—

171

—

105

C₃×C₉

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

#10587022

yes

complemented

110

C₃×C₉

C₃²×C₁₉⋊C₃

yes

—

111

C₃×C₉

C₃²×C₁₉⋊C₃

yes

—

112

C₃×C₉

C₃²×C₁₉⋊C₃

yes

characteristic

113

C₁₉

#10624884

yes

characteristic, complemented, Sylow

114

C₃²

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

—

119

C₃²

—

171

—

120

C₃²

C₃×#10587022

yes

—

121

C₃²

C₃³×C₁₉⋊C₃

yes

characteristic

122

C₉

—

123

C₉

—

124

C₉

—

125

C₉

—

126

C₉

C₃×#10587022

yes

—

127

C₉

C₃×#10587022

yes

—

128

C₃

—

129

C₃

—

171

—

130

C₃

—

171

—

131

C₃

C₃²×#10587022

yes

characteristic

132

C₃

#17987545

yes

—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₁₉⋊C₃
commutator, socle, cosocle	C₃²×C₁₉	C₃⁴
Fitting	C₁₉×#6887	C₃
prime radical	C₃³×C₁₉⋊C₃	C₃²
hypercenter	#6887	C₁₉⋊C₃
hypocenter	C₁₉	#10624884
LCS	G ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ #6887

subgroup

G/H

center, Frattini

C₃²

C₃³×C₁₉⋊C₃

commutator, socle, cosocle

C₃²×C₁₉

C₃⁴

Fitting

C₁₉×#6887

C₃

prime radical

C₃³×C₁₉⋊C₃

C₃²

C₁₉⋊C₃

C₁₉

G ▷ C₃²×C₁₉ ▷ C₁₉

C₁ ◁ C₃² ◁ #6887

N	H ≅ G/N	type
2. #17987472	C₃	semidirect
3. #17987469	C₃	semidirect
4. #17987466	C₃	semidirect
5. #17987463	C₃	semidirect
6. #17987443	C₃	semidirect
8. #17987405	C₃	semidirect
9. C₁₉×#6887	C₃	semidirect
18. #13095307	C₃²	semidirect
19. #13095306	C₃²	semidirect
20. #13095306	C₃²	semidirect
21. #13095305	C₃²	semidirect
24. C₉×#4294	C₃²	semidirect
26. C₁₉×C₉⋊C₉	C₃²	semidirect
28. C₉²×C₁₉	C₃²	semidirect
35. C₃×C₉×C₁₉	3₋¹⁺²	semidirect
56. #6887	C₁₉⋊C₃	semidirect
84. C₉⋊C₉	C₃×C₁₉⋊C₃	semidirect
89. C₉²	C₃×C₁₉⋊C₃	semidirect
109. C₃×C₉	#10587022	semidirect
113. C₁₉	#10624884	semidirect

H ≅ G/N

type

2. #17987472

C₃

semidirect

3. #17987469

C₃

semidirect

4. #17987466

C₃

semidirect

5. #17987463

C₃

semidirect

6. #17987443

C₃

semidirect

8. #17987405

C₃

semidirect

9. C₁₉×#6887

C₃

semidirect

18. #13095307

C₃²

semidirect

19. #13095306

C₃²

semidirect

20. #13095306

C₃²

semidirect

21. #13095305

C₃²

semidirect

24. C₉×#4294

C₃²

semidirect

26. C₁₉×C₉⋊C₉

C₃²

semidirect

28. C₉²×C₁₉

C₃²

semidirect

35. C₃×C₉×C₁₉

3₋¹⁺²

semidirect

56. #6887

C₁₉⋊C₃

semidirect

84. C₉⋊C₉

C₃×C₁₉⋊C₃

semidirect

89. C₉²

C₃×C₁₉⋊C₃

semidirect

109. C₃×C₉

#10587022

semidirect

113. C₁₉

#10624884

semidirect

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

subgroup	index
C₃²×#10587022	3
#17987472	3
#17987469	3
#17987466	3
#17987463	3
#17987443	3
#17987420	3
#17987405	3
C₁₉×#6887	3
#10624884	19

subgroup

index

C₁₉×#6887

Subgroups — #28356068

Conjugacy classes of subgroups

Complements

Maximal subgroups