Group #28356071 — subgroups

Subgroups — #28356071

Order 13851.

716 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 274 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	#17987468	C₃	1	9	yes	maximal, complemented
3	#17987460	C₃	1	18	yes	maximal, complemented
4	#17987443	C₃	1	3	yes	maximal, complemented
5	#17987442	C₃	1	3	yes	maximal, complemented
6	#17987421	C₃	1	2	yes	maximal
7	#17987402	C₃	1	3	yes	maximal, complemented
8	C₁₉×#6891	C₃	1	1	yes	maximal, characteristic
9	C₃×#10587025	—	3	9	no	—
10	C₃×#10587025	C₃²	1	6	yes	—
11	C₃×#10587025	C₃²	1	3	yes	—
12	C₃×#10587025	C₃²	1	6	yes	—
13	C₃²×#4294	C₃²	1	1	yes	characteristic
14	C₃²×#4294	C₃²	1	1	yes	characteristic
15	C₃²×#4294	C₃²	1	1	yes	characteristic
16	#13095307	C₃²	1	9	yes	complemented
17	#13095306	C₃²	1	9	yes	complemented
18	#13095305	C₃²	1	18	yes	complemented
19	#13095305	C₃²	1	18	yes	complemented
20	#13095305	C₃²	1	18	yes	complemented
21	C₃²⋊#4294	C₃²	1	6	yes	—
22	C₃²⋊#4294	C₃²	1	12	yes	—
23	C₉×#4294	C₃²	1	9	yes	complemented
24	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
25	C₃×C₁₉×3₋¹⁺²	C₃²	1	3	yes	—
26	C₁₉×C₃²⋊C₉	C₃²	1	3	yes	—
27	C₁₉×C₃²⋊C₉	C₃²	1	6	yes	—
28	#10624884	—	19	1	no	maximal, Sylow
29	C₃³×C₁₉	—	3	3	no	—
30	C₃³×C₁₉	—	9	2	no	—
31	C₃³×C₁₉	—	9	1	no	—
32	C₃³×C₁₉	C₃³	1	2	yes	—
33	C₃³×C₁₉	C₃³	1	1	yes	characteristic
34	C₃³×C₁₉	C₃³	1	1	yes	characteristic
35	C₁₉×3₋¹⁺²	—	9	3	no	—
36	C₃×C₉×C₁₉	C₃³	1	9	yes	—
37	#10587025	—	3	18	no	—
38	#10587025	—	3	9	no	—
39	#10587025	—	9	6	no	—
40	C₃×#4294	—	3	3	no	—
41	C₃×#4294	—	3	3	no	—
42	C₃×#4294	3₋¹⁺²	1	9	yes	complemented
43	C₃×#4294	C₃³	1	18	yes	—
44	C₃×#4294	C₃³	1	3	yes	—
45	C₃×#4294	C₃³	1	3	yes	—
46	C₃×#4294	C₃³	1	3	yes	—
47	C₃²×3₋¹⁺²	—	19	1	no	—
48	#6900	—	19	9	no	—
49	#6899	—	19	18	no	—
50	#6895	—	19	3	no	—
51	#6895	—	19	3	no	—
52	#6891	—	19	2	no	—
53	#6891	C₁₉⋊C₃	1	1	yes	characteristic
54	#6887	—	19	3	no	—
55	C₃²×C₁₉	—	3	2	no	—
56	C₃²×C₁₉	—	3	1	no	—
57	C₃²×C₁₉	—	3	2	no	—
58	C₃²×C₁₉	—	3	1	no	—
59	C₃²×C₁₉	—	3	2	no	—
60	C₃²×C₁₉	—	3	1	no	—
61	C₃²×C₁₉	—	3	4	no	—
62	C₃²×C₁₉	—	3	2	no	—
63	C₃²×C₁₉	—	9	9	no	—
64	C₃²×C₁₉	C₃×3₋¹⁺²	1	3	yes	—
65	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
66	C₉×C₁₉	—	3	9	no	—
67	#4294	—	3	3	no	—
68	#4294	—	3	3	no	—
69	#4294	—	3	18	no	—
70	#4294	C₃×3₋¹⁺²	1	9	yes	—
71	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
72	C₃×3₋¹⁺²	—	19	6	no	—
73	C₃×3₋¹⁺²	—	19	3	no	—
74	C₃×3₋¹⁺²	—	19	6	no	—
75	C₃×3₋¹⁺²	—	57	9	no	—
76	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	3	yes	—
77	C₃²×C₉	—	19	1	no	—
78	C₃²×C₉	—	19	1	no	—
79	C₃²×C₉	—	19	1	no	—
80	C₉⋊C₉	—	19	18	no	—
81	C₉⋊C₉	—	19	18	no	—
82	C₉⋊C₉	—	19	18	no	—
83	C₉⋊C₉	—	19	9	no	—
84	C₉⋊C₉	—	19	9	no	—
85	C₃²⋊C₉	—	19	6	no	—
86	C₃²⋊C₉	—	19	12	no	—
87	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	—
88	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	6	yes	—
89	C₉²	—	19	9	no	—
90	C₃×C₁₉	—	3	3	no	—
91	C₃×C₁₉	—	9	1	no	—
92	C₃×C₁₉	—	9	2	no	—
93	C₃×C₁₉	C₃²×3₋¹⁺²	1	1	yes	characteristic
94	C₃×C₁₉	3₋¹⁺⁴	1	2	yes	—
95	C₃×C₁₉	3₋¹⁺⁴	1	1	yes	characteristic
96	C₃³	—	3	3	no	—
97	C₃³	—	9	2	no	—
98	C₃³	—	9	1	no	—
99	C₃³	C₃²×C₁₉⋊C₃	1	2	yes	—
100	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
101	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
102	3₋¹⁺²	—	9	3	no	—
103	3₋¹⁺²	—	57	18	no	—
104	3₋¹⁺²	—	57	9	no	—
105	3₋¹⁺²	—	171	6	no	—
106	C₃×C₉	—	19	9	no	—
107	C₃×C₉	—	19	3	no	—
108	C₃×C₉	—	19	18	no	—
109	C₃×C₉	—	19	3	no	—
110	C₃×C₉	—	19	3	no	—
111	C₃×C₉	—	57	3	no	—
112	C₃×C₉	—	57	3	no	—
113	C₃×C₉	C₃²×C₁₉⋊C₃	1	9	yes	—
114	C₁₉	#10624884	1	1	yes	characteristic, complemented, Sylow
115	C₃²	—	3	2	no	—
116	C₃²	—	3	1	no	—
117	C₃²	—	3	2	no	—
118	C₃²	—	3	1	no	—
119	C₃²	—	3	2	no	—
120	C₃²	—	3	4	no	—
121	C₃²	—	3	1	no	—
122	C₃²	—	3	2	no	—
123	C₃²	—	9	9	no	—
124	C₃²	3₋¹⁺²×C₁₉⋊C₃	1	3	yes	—
125	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
126	C₉	—	3	9	no	—
127	C₉	—	19	9	no	—
128	C₉	—	57	3	no	—
129	C₉	—	57	3	no	—
130	C₉	—	57	18	no	—
131	C₃	—	3	3	no	—
132	C₃	—	9	1	no	—
133	C₃	—	9	2	no	—
134	C₃	C₃×3₋¹⁺²×C₁₉⋊C₃	1	1	yes	characteristic
135	C₃	#17987546	1	2	yes	—
136	C₃	#17987547	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#10587025

C₃

yes

maximal, characteristic

#17987468

C₃

yes

maximal, complemented

#17987460

C₃

yes

maximal, complemented

#17987443

C₃

yes

maximal, complemented

#17987442

C₃

yes

maximal, complemented

#17987421

C₃

yes

maximal

#17987402

C₃

yes

maximal, complemented

C₁₉×#6891

C₃

yes

maximal, characteristic

C₃×#10587025

—

C₃×#10587025

C₃²

yes

—

C₃×#10587025

C₃²

yes

—

C₃×#10587025

C₃²

yes

—

C₃²×#4294

C₃²

yes

characteristic

C₃²×#4294

C₃²

yes

characteristic

C₃²×#4294

C₃²

yes

characteristic

#13095307

C₃²

yes

complemented

#13095306

C₃²

yes

complemented

#13095305

C₃²

yes

complemented

#13095305

C₃²

yes

complemented

#13095305

C₃²

yes

complemented

C₃²⋊#4294

C₃²

yes

—

C₃²⋊#4294

C₃²

yes

—

C₉×#4294

C₃²

yes

complemented

C₃⁴×C₁₉

C₃²

yes

characteristic

C₃×C₁₉×3₋¹⁺²

C₃²

yes

—

C₁₉×C₃²⋊C₉

C₃²

yes

—

C₁₉×C₃²⋊C₉

C₃²

yes

—

#10624884

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

C₃³

yes

—

C₃³×C₁₉

C₃³

yes

characteristic

C₃³×C₁₉

C₃³

yes

characteristic

C₁₉×3₋¹⁺²

—

C₃×C₉×C₁₉

C₃³

yes

—

—

—

—

—

—

3₋¹⁺²

yes

complemented

C₃×#4294

C₃³

yes

—

C₃×#4294

C₃³

yes

—

C₃×#4294

C₃³

yes

—

C₃×#4294

C₃³

yes

—

C₃²×3₋¹⁺²

—

—

—

—

—

—

C₁₉⋊C₃

yes

characteristic

#6887

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃×3₋¹⁺²

yes

—

C₃²×C₁₉

C₃⁴

yes

characteristic

C₉×C₁₉

—

—

—

—

C₃×3₋¹⁺²

yes

—

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×C₁₉⋊C₃

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₁₉⋊C₃

yes

—

C₃²⋊C₉

C₃×C₁₉⋊C₃

yes

—

C₉²

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

C₃²×3₋¹⁺²

yes

characteristic

C₃×C₁₉

3₋¹⁺⁴

yes

—

C₃×C₁₉

3₋¹⁺⁴

yes

characteristic

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²×C₁₉⋊C₃

yes

—

100

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

101

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

102

3₋¹⁺²

—

103

3₋¹⁺²

—

104

3₋¹⁺²

—

105

3₋¹⁺²

—

171

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

—

110

C₃×C₉

—

111

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

C₃²×C₁₉⋊C₃

yes

—

114

C₁₉

#10624884

yes

characteristic, complemented, Sylow

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

3₋¹⁺²×C₁₉⋊C₃

yes

—

125

C₃²

C₃³×C₁₉⋊C₃

yes

characteristic

126

C₉

—

127

C₉

—

128

C₉

—

129

C₉

—

130

C₉

—

131

C₃

—

132

C₃

—

133

C₃

—

134

C₃

C₃×3₋¹⁺²×C₁₉⋊C₃

yes

characteristic

135

C₃

#17987546

yes

—

136

C₃

#17987547

yes

characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₁₉⋊C₃
commutator, socle, cosocle	C₃²×C₁₉	C₃⁴
Fitting	C₁₉×#6891	C₃
prime radical	C₃⁴×C₁₉	C₃²
hypercenter	#6891	C₁₉⋊C₃
hypocenter	C₁₉	#10624884
LCS	G ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ #6891

subgroup

G/H

center, Frattini

C₃²

C₃³×C₁₉⋊C₃

commutator, socle, cosocle

C₃²×C₁₉

C₃⁴

Fitting

C₁₉×#6891

C₃

prime radical

C₃⁴×C₁₉

C₃²

C₁₉⋊C₃

C₁₉

G ▷ C₃²×C₁₉ ▷ C₁₉

C₁ ◁ C₃² ◁ #6891

N	H ≅ G/N	type
2. #17987468	C₃	semidirect
3. #17987460	C₃	semidirect
4. #17987443	C₃	semidirect
5. #17987442	C₃	semidirect
7. #17987402	C₃	semidirect
16. #13095307	C₃²	semidirect
17. #13095306	C₃²	semidirect
18. #13095305	C₃²	semidirect
19. #13095305	C₃²	semidirect
20. #13095305	C₃²	semidirect
23. C₉×#4294	C₃²	semidirect
42. C₃×#4294	3₋¹⁺²	semidirect
114. C₁₉	#10624884	semidirect

H ≅ G/N

type

2. #17987468

C₃

semidirect

3. #17987460

C₃

semidirect

4. #17987443

C₃

semidirect

5. #17987442

C₃

semidirect

7. #17987402

C₃

semidirect

16. #13095307

C₃²

semidirect

17. #13095306

C₃²

semidirect

18. #13095305

C₃²

semidirect

19. #13095305

C₃²

semidirect

20. #13095305

C₃²

semidirect

23. C₉×#4294

C₃²

semidirect

42. C₃×#4294

3₋¹⁺²

semidirect

114. C₁₉

#10624884

semidirect

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

subgroup	index
C₃²×#10587025	3
#17987468	3
#17987460	3
#17987443	3
#17987442	3
#17987421	3
#17987402	3
C₁₉×#6891	3
#10624884	19

subgroup

index

C₁₉×#6891

Subgroups — #28356071

Conjugacy classes of subgroups

Complements

Maximal subgroups