Group #28356868 — subgroups

Subgroups — #28356868

Order 13851.

374 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₃×#13095339	C₃	1	3	yes	maximal
2	C₃×#13095339	C₃	1	1	yes	maximal, characteristic
3	C₃×#13095338	C₃	1	1	yes	maximal, characteristic
4	C₃×C₉×#4296	C₃	1	1	yes	maximal, characteristic
5	(C₃×#4296)⋊C₉	C₃	1	6	yes	maximal
6	C₁₉⋊C₃×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic
7	C₃²×#4296	C₃²	1	3	yes	—
8	C₃²×#4296	C₃²	1	1	yes	characteristic
9	C₃²×#4296	C₃²	1	2	yes	—
10	C₃²×#4296	C₃²	1	3	yes	—
11	#13095339	—	3	9	no	—
12	#13095339	—	3	3	no	—
13	#13095338	—	3	3	no	—
14	C₉×#4296	—	3	3	no	—
15	C₃×C₉×C₁₉⋊C₃	C₃²	1	3	yes	—
16	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
17	#13095307	C₉	1	1	yes	characteristic, complemented
18	#13095306	C₉	1	1	yes	characteristic, complemented
19	#13095305	C₉	1	6	yes	complemented
20	C₁₉×C₉⋊C₉	C₉	1	1	yes	characteristic, complemented
21	#10624561	—	19	1	no	maximal, Sylow
22	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
23	C₃×#4296	—	3	3	no	—
24	C₃×#4296	—	3	6	no	—
25	C₃×#4296	—	3	3	no	—
26	C₃×#4296	—	3	3	no	—
27	C₃×#4296	—	3	2	no	—
28	C₃×#4296	—	3	3	no	—
29	C₃×#4296	—	3	1	no	—
30	C₃×#4296	—	3	6	no	—
31	C₃×#4296	—	3	6	no	—
32	C₃×#4296	3₋¹⁺²	1	9	yes	—
33	C₃×C₉×C₁₉	C₃×C₉	1	3	yes	—
34	C₃×C₉×C₁₉	C₃×C₉	1	1	yes	characteristic
35	C₃×#4294	C₃×C₉	1	3	yes	—
36	C₃×#4294	C₃×C₉	1	3	yes	—
37	C₃×#4294	C₃×C₉	1	2	yes	—
38	C₉×C₁₉⋊C₃	—	3	3	no	—
39	C₉×C₁₉⋊C₃	—	3	6	no	—
40	C₉×C₁₉⋊C₃	3₋¹⁺²	1	3	yes	—
41	C₉×C₁₉⋊C₃	3₋¹⁺²	1	3	yes	—
42	C₉×C₁₉⋊C₃	3₋¹⁺²	1	3	yes	—
43	C₃×C₉⋊C₉	—	19	1	no	—
44	C₃×C₉⋊C₉	—	19	1	no	—
45	C₃×C₉⋊C₉	—	19	1	no	—
46	C₃×C₉⋊C₉	—	19	3	no	—
47	C₃×C₉²	—	19	1	no	—
48	#6855	—	19	6	no	—
49	C₃²×C₁₉	C₃²×C₉	1	1	yes	characteristic
50	C₃×C₁₉⋊C₃	C₃×3₋¹⁺²	1	1	yes	characteristic
51	C₃×C₁₉⋊C₃	C₃×3₋¹⁺²	1	3	yes	—
52	C₃×C₁₉⋊C₃	C₃×3₋¹⁺²	1	1	yes	characteristic
53	C₃×C₁₉⋊C₃	C₃×3₋¹⁺²	1	1	yes	characteristic
54	C₃×C₁₉⋊C₃	#737	1	6	yes	—
55	#4296	—	3	9	no	—
56	#4296	—	9	3	no	—
57	#4296	—	9	2	no	—
58	#4296	—	9	1	no	—
59	C₉×C₁₉	—	3	3	no	—
60	C₉×C₁₉	C₉⋊C₉	1	3	yes	complemented
61	#4294	—	3	3	no	—
62	#4294	—	3	3	no	—
63	#4294	C₉⋊C₉	1	6	yes	complemented
64	C₃²×C₉	—	19	1	no	—
65	C₃²×C₉	—	19	3	no	—
66	C₃²×C₉	—	19	3	no	—
67	C₃²×C₉	—	19	2	no	—
68	C₃²×C₉	—	19	3	no	—
69	C₃²×C₉	—	19	1	no	—
70	C₉⋊C₉	—	19	6	no	—
71	C₉⋊C₉	—	19	1	no	—
72	C₉⋊C₉	—	19	1	no	—
73	C₉⋊C₉	—	57	3	no	—
74	C₉⋊C₉	—	57	3	no	—
75	C₉⋊C₉	—	57	9	no	—
76	C₉⋊C₉	#4296	1	1	yes	characteristic, complemented
77	C₉²	—	57	3	no	—
78	C₃×C₁₉	C₃×C₉⋊C₉	1	1	yes	characteristic
79	C₃×C₁₉	C₉×3₋¹⁺²	1	3	yes	—
80	C₁₉⋊C₃	#6894	1	3	yes	—
81	C₁₉⋊C₃	#6896	1	3	yes	—
82	C₁₉⋊C₃	#6899	1	2	yes	—
83	C₁₉⋊C₃	#6900	1	1	yes	characteristic
84	C₃³	—	19	1	no	—
85	C₃×C₉	—	19	2	no	—
86	C₃×C₉	—	19	3	no	—
87	C₃×C₉	—	19	3	no	—
88	C₃×C₉	—	19	3	no	—
89	C₃×C₉	—	19	9	no	—
90	C₃×C₉	—	19	3	no	—
91	C₃×C₉	—	19	3	no	—
92	C₃×C₉	—	57	3	no	—
93	C₃×C₉	—	57	6	no	—
94	C₃×C₉	—	57	3	no	—
95	C₃×C₉	—	57	6	no	—
96	C₃×C₉	—	57	3	no	—
97	C₃×C₉	—	57	2	no	—
98	C₃×C₉	—	57	1	no	—
99	C₃×C₉	—	57	3	no	—
100	C₃×C₉	—	57	3	no	—
101	C₃×C₉	—	57	6	no	—
102	C₃×C₉	—	57	6	no	—
103	C₃×C₉	C₃×#4296	1	3	yes	—
104	C₃×C₉	C₃×#4296	1	1	yes	characteristic
105	C₁₉	#10624561	1	1	yes	characteristic, complemented, Sylow
106	C₃²	—	19	1	no	—
107	C₃²	—	19	1	no	—
108	C₃²	—	19	1	no	—
109	C₃²	—	19	3	no	—
110	C₃²	—	19	6	no	—
111	C₃²	C₃²×#4296	1	1	yes	characteristic
112	C₉	—	3	3	no	—
113	C₉	—	19	6	no	—
114	C₉	—	57	9	no	—
115	C₉	—	57	3	no	—
116	C₉	—	57	3	no	—
117	C₉	—	171	3	no	—
118	C₉	—	171	2	no	—
119	C₉	—	171	1	no	—
120	C₉	#13095339	1	3	yes	complemented
121	C₃	—	19	3	no	—
122	C₃	—	19	3	no	—
123	C₃	—	19	2	no	—
124	C₃	—	19	1	no	—
125	C₃	C₃×#13095339	1	1	yes	characteristic
126	C₃	3₋¹⁺²×#4296	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#13095339

C₃

yes

maximal

C₃×#13095339

C₃

yes

maximal, characteristic

C₃×#13095338

C₃

yes

maximal, characteristic

C₃×C₉×#4296

C₃

yes

maximal, characteristic

(C₃×#4296)⋊C₉

C₃

yes

maximal

C₁₉⋊C₃×C₉⋊C₉

C₃

yes

maximal, characteristic

C₃²×#4296

C₃²

yes

—

C₃²×#4296

C₃²

yes

characteristic

C₃²×#4296

C₃²

yes

—

C₃²×#4296

C₃²

yes

—

—

—

—

—

C₃×C₉×C₁₉⋊C₃

C₃²

yes

—

C₃×C₉×C₁₉⋊C₃

C₃²

yes

characteristic

#13095307

C₉

yes

characteristic, complemented

#13095306

C₉

yes

characteristic, complemented

#13095305

C₉

yes

complemented

C₁₉×C₉⋊C₉

C₉

yes

characteristic, complemented

#10624561

—

maximal, Sylow

C₃²×C₁₉⋊C₃

C₃³

yes

—

—

—

—

—

—

—

—

—

3₋¹⁺²

yes

—

C₃×C₉×C₁₉

C₃×C₉

yes

—

C₃×C₉×C₁₉

C₃×C₉

yes

characteristic

C₃×#4294

C₃×C₉

yes

—

C₃×#4294

C₃×C₉

yes

—

C₃×#4294

C₃×C₉

yes

—

C₉×C₁₉⋊C₃

—

C₉×C₁₉⋊C₃

—

C₉×C₁₉⋊C₃

3₋¹⁺²

yes

—

C₉×C₁₉⋊C₃

3₋¹⁺²

yes

—

C₉×C₁₉⋊C₃

3₋¹⁺²

yes

—

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

—

C₃×C₉²

—

#6855

—

C₃²×C₁₉

C₃²×C₉

yes

characteristic

C₃×C₁₉⋊C₃

C₃×3₋¹⁺²

yes

characteristic

C₃×C₁₉⋊C₃

C₃×3₋¹⁺²

yes

—

C₃×C₁₉⋊C₃

C₃×3₋¹⁺²

yes

characteristic

C₃×C₁₉⋊C₃

C₃×3₋¹⁺²

yes

characteristic

C₃×C₁₉⋊C₃

#737

yes

—

—

—

—

—

C₉×C₁₉

—

C₉×C₁₉

C₉⋊C₉

yes

—

—

C₉⋊C₉

yes

complemented

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

#4296

yes

characteristic, complemented

C₉²

—

C₃×C₁₉

C₃×C₉⋊C₉

yes

characteristic

C₃×C₁₉

C₉×3₋¹⁺²

yes

—

C₁₉⋊C₃

#6894

yes

—

C₁₉⋊C₃

#6896

yes

—

C₁₉⋊C₃

#6899

yes

—

C₁₉⋊C₃

#6900

yes

characteristic

C₃³

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

100

C₃×C₉

—

101

C₃×C₉

—

102

C₃×C₉

—

103

C₃×C₉

C₃×#4296

yes

—

104

C₃×C₉

C₃×#4296

yes

characteristic

105

C₁₉

#10624561

yes

characteristic, complemented, Sylow

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

—

111

C₃²

C₃²×#4296

yes

characteristic

112

C₉

—

113

C₉

—

114

C₉

—

115

C₉

—

116

C₉

—

117

C₉

—

171

—

118

C₉

—

171

—

119

C₉

—

171

—

120

C₉

#13095339

yes

complemented

121

C₃

—

122

C₃

—

123

C₃

—

124

C₃

—

125

C₃

C₃×#13095339

yes

characteristic

126

C₃

3₋¹⁺²×#4296

yes

—

	subgroup	G/H
center, Frattini	C₃²	C₃²×#4296
commutator, socle	C₃²×C₁₉	C₃²×C₉
Fitting	C₁₉×C₉⋊C₉	C₉
prime radical, cosocle	C₃²×C₁₉⋊C₃	C₃³
hypercenter	C₉⋊C₉	#4296
hypocenter	C₁₉	#10624561
LCS	G ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₉⋊C₉

subgroup

G/H

center, Frattini

C₃²

C₃²×#4296

commutator, socle

C₃²×C₁₉

C₃²×C₉

Fitting

C₁₉×C₉⋊C₉

C₉

prime radical, cosocle

C₃²×C₁₉⋊C₃

C₃³

C₉⋊C₉

C₁₉

G ▷ C₃²×C₁₉ ▷ C₁₉

C₁ ◁ C₃² ◁ C₉⋊C₉

N	H ≅ G/N	type
17. #13095307	C₉	semidirect
18. #13095306	C₉	semidirect
19. #13095305	C₉	semidirect
20. C₁₉×C₉⋊C₉	C₉	semidirect
60. C₉×C₁₉	C₉⋊C₉	semidirect
63. #4294	C₉⋊C₉	semidirect
76. C₉⋊C₉	#4296	semidirect
105. C₁₉	#10624561	semidirect
120. C₉	#13095339	semidirect

H ≅ G/N

type

17. #13095307

C₉

semidirect

18. #13095306

C₉

semidirect

19. #13095305

C₉

semidirect

20. C₁₉×C₉⋊C₉

C₉

semidirect

60. C₉×C₁₉

C₉⋊C₉

semidirect

63. #4294

C₉⋊C₉

semidirect

76. C₉⋊C₉

#4296

semidirect

105. C₁₉

#10624561

semidirect

120. C₉

#13095339

semidirect

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

subgroup	index
C₃×#13095339	3
C₃×#13095338	3
C₃×C₉×#4296	3
(C₃×#4296)⋊C₉	3
C₁₉⋊C₃×C₉⋊C₉	3
#10624561	19

subgroup

index

C₃×C₉×#4296

C₁₉⋊C₃×C₉⋊C₉

Subgroups — #28356868

Conjugacy classes of subgroups

Complements

Maximal subgroups