Group #41008461 — subgroups

Subgroups — #41008461

Order 26973.

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×#15688538	C₃	1	3	yes	maximal
2	C₃×#15688538	C₃	1	1	yes	maximal, characteristic
3	C₃×#15688537	C₃	1	1	yes	maximal, characteristic
4	C₃×C₉×#66526	C₃	1	1	yes	maximal, characteristic
5	(C₃×#66526)⋊C₉	C₃	1	6	yes	maximal
6	C₉⋊C₉×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
7	C₃²×#66526	C₃²	1	2	yes	—
8	C₃²×#66526	C₃²	1	3	yes	—
9	C₃²×#66526	C₃²	1	1	yes	characteristic
10	C₃²×#66526	C₃²	1	3	yes	—
11	#15688538	—	3	9	no	—
12	#15688538	—	3	3	no	—
13	#15688537	—	3	3	no	—
14	C₉×#66526	—	3	3	no	—
15	C₃×C₉×C₃₇⋊C₃	C₃²	1	3	yes	—
16	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
17	#15688506	C₉	1	1	yes	characteristic, complemented
18	#15688505	C₉	1	1	yes	characteristic, complemented
19	#15688504	C₉	1	6	yes	complemented
20	C₃₇×C₉⋊C₉	C₉	1	1	yes	characteristic, complemented
21	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
22	C₃×#66526	—	3	3	no	—
23	C₃×#66526	—	3	6	no	—
24	C₃×#66526	—	3	3	no	—
25	C₃×#66526	—	3	3	no	—
26	C₃×#66526	—	3	2	no	—
27	C₃×#66526	—	3	3	no	—
28	C₃×#66526	—	3	1	no	—
29	C₃×#66526	—	3	6	no	—
30	C₃×#66526	—	3	6	no	—
31	C₃×#66526	3₋¹⁺²	1	9	yes	—
32	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
33	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
34	C₃×#66524	C₃×C₉	1	3	yes	—
35	C₃×#66524	C₃×C₉	1	3	yes	—
36	C₃×#66524	C₃×C₉	1	2	yes	—
37	C₉×C₃₇⋊C₃	—	3	3	no	—
38	C₉×C₃₇⋊C₃	—	3	6	no	—
39	C₉×C₃₇⋊C₃	3₋¹⁺²	1	3	yes	—
40	C₉×C₃₇⋊C₃	3₋¹⁺²	1	3	yes	—
41	C₉×C₃₇⋊C₃	3₋¹⁺²	1	3	yes	—
42	#10624561	—	37	1	no	maximal, Sylow
43	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
44	C₃×C₃₇⋊C₃	C₃×3₋¹⁺²	1	1	yes	characteristic
45	C₃×C₃₇⋊C₃	C₃×3₋¹⁺²	1	3	yes	—
46	C₃×C₃₇⋊C₃	C₃×3₋¹⁺²	1	1	yes	characteristic
47	C₃×C₃₇⋊C₃	C₃×3₋¹⁺²	1	1	yes	characteristic
48	C₃×C₃₇⋊C₃	#737	1	6	yes	—
49	#66526	—	3	9	no	—
50	#66526	—	9	3	no	—
51	#66526	—	9	2	no	—
52	#66526	—	9	1	no	—
53	C₉×C₃₇	—	3	3	no	—
54	C₉×C₃₇	C₉⋊C₉	1	3	yes	complemented
55	#66524	—	3	3	no	—
56	#66524	—	3	3	no	—
57	#66524	C₉⋊C₉	1	6	yes	complemented
58	C₃×C₉⋊C₉	—	37	1	no	—
59	C₃×C₉⋊C₉	—	37	1	no	—
60	C₃×C₉⋊C₉	—	37	1	no	—
61	C₃×C₉⋊C₉	—	37	3	no	—
62	C₃×C₉²	—	37	1	no	—
63	#6855	—	37	6	no	—
64	C₃×C₃₇	C₃×C₉⋊C₉	1	1	yes	characteristic
65	C₃×C₃₇	C₉×3₋¹⁺²	1	3	yes	—
66	C₃₇⋊C₃	#6894	1	3	yes	—
67	C₃₇⋊C₃	#6896	1	3	yes	—
68	C₃₇⋊C₃	#6899	1	2	yes	—
69	C₃₇⋊C₃	#6900	1	1	yes	characteristic
70	C₃²×C₉	—	37	1	no	—
71	C₃²×C₉	—	37	3	no	—
72	C₃²×C₉	—	37	3	no	—
73	C₃²×C₉	—	37	2	no	—
74	C₃²×C₉	—	37	3	no	—
75	C₃²×C₉	—	37	1	no	—
76	C₉⋊C₉	—	37	6	no	—
77	C₉⋊C₉	—	37	1	no	—
78	C₉⋊C₉	—	37	1	no	—
79	C₉⋊C₉	—	111	3	no	—
80	C₉⋊C₉	—	111	3	no	—
81	C₉⋊C₉	—	111	9	no	—
82	C₉⋊C₉	#66526	1	1	yes	characteristic, complemented
83	C₉²	—	111	3	no	—
84	C₃₇	#10624561	1	1	yes	characteristic, complemented, Sylow
85	C₃³	—	37	1	no	—
86	C₃×C₉	—	37	2	no	—
87	C₃×C₉	—	37	3	no	—
88	C₃×C₉	—	37	3	no	—
89	C₃×C₉	—	37	3	no	—
90	C₃×C₉	—	37	9	no	—
91	C₃×C₉	—	37	3	no	—
92	C₃×C₉	—	37	3	no	—
93	C₃×C₉	—	111	3	no	—
94	C₃×C₉	—	111	6	no	—
95	C₃×C₉	—	111	3	no	—
96	C₃×C₉	—	111	2	no	—
97	C₃×C₉	—	111	1	no	—
98	C₃×C₉	—	111	3	no	—
99	C₃×C₉	—	111	3	no	—
100	C₃×C₉	—	111	6	no	—
101	C₃×C₉	—	111	6	no	—
102	C₃×C₉	—	111	3	no	—
103	C₃×C₉	—	111	6	no	—
104	C₃×C₉	C₃×#66526	1	3	yes	—
105	C₃×C₉	C₃×#66526	1	1	yes	characteristic
106	C₃²	—	37	1	no	—
107	C₃²	—	37	1	no	—
108	C₃²	—	37	1	no	—
109	C₃²	—	37	3	no	—
110	C₃²	—	37	6	no	—
111	C₃²	C₃²×#66526	1	1	yes	characteristic
112	C₉	—	3	3	no	—
113	C₉	—	37	6	no	—
114	C₉	—	333	3	no	—
115	C₉	—	333	2	no	—
116	C₉	—	333	1	no	—
117	C₉	—	111	9	no	—
118	C₉	—	111	3	no	—
119	C₉	—	111	3	no	—
120	C₉	#15688538	1	3	yes	complemented
121	C₃	—	37	3	no	—
122	C₃	—	37	2	no	—
123	C₃	—	37	1	no	—
124	C₃	—	37	3	no	—
125	C₃	C₃×#15688538	1	1	yes	characteristic
126	C₃	3₋¹⁺²×#66526	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688538

C₃

yes

maximal

C₃×#15688538

C₃

yes

maximal, characteristic

C₃×#15688537

C₃

yes

maximal, characteristic

C₃×C₉×#66526

C₃

yes

maximal, characteristic

(C₃×#66526)⋊C₉

C₃

yes

maximal

C₉⋊C₉×C₃₇⋊C₃

C₃

yes

maximal, characteristic

C₃²×#66526

C₃²

yes

—

C₃²×#66526

C₃²

yes

—

C₃²×#66526

C₃²

yes

characteristic

C₃²×#66526

C₃²

yes

—

—

—

—

—

C₃×C₉×C₃₇⋊C₃

C₃²

yes

—

C₃×C₉×C₃₇⋊C₃

C₃²

yes

characteristic

#15688506

C₉

yes

characteristic, complemented

#15688505

C₉

yes

characteristic, complemented

#15688504

C₉

yes

complemented

C₃₇×C₉⋊C₉

C₉

yes

characteristic, complemented

C₃²×C₃₇⋊C₃

C₃³

yes

—

—

—

—

—

—

—

—

—

3₋¹⁺²

yes

—

C₃×C₉×C₃₇

C₃×C₉

yes

—

C₃×C₉×C₃₇

C₃×C₉

yes

characteristic

C₃×#66524

C₃×C₉

yes

—

C₃×#66524

C₃×C₉

yes

—

C₃×#66524

C₃×C₉

yes

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

3₋¹⁺²

yes

—

C₉×C₃₇⋊C₃

3₋¹⁺²

yes

—

C₉×C₃₇⋊C₃

3₋¹⁺²

yes

—

#10624561

—

maximal, Sylow

C₃²×C₃₇

C₃²×C₉

yes

characteristic

C₃×C₃₇⋊C₃

C₃×3₋¹⁺²

yes

characteristic

C₃×C₃₇⋊C₃

C₃×3₋¹⁺²

yes

—

C₃×C₃₇⋊C₃

C₃×3₋¹⁺²

yes

characteristic

C₃×C₃₇⋊C₃

C₃×3₋¹⁺²

yes

characteristic

C₃×C₃₇⋊C₃

#737

yes

—

—

—

—

—

C₉×C₃₇

—

C₉×C₃₇

C₉⋊C₉

yes

—

—

C₉⋊C₉

yes

complemented

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

—

C₃×C₉²

—

#6855

—

C₃×C₃₇

C₃×C₉⋊C₉

yes

characteristic

C₃×C₃₇

C₉×3₋¹⁺²

yes

—

C₃₇⋊C₃

#6894

yes

—

C₃₇⋊C₃

#6896

yes

—

C₃₇⋊C₃

#6899

yes

—

C₃₇⋊C₃

#6900

yes

characteristic

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₉⋊C₉

#66526

yes

characteristic, complemented

C₉²

—

111

—

C₃₇

#10624561

yes

characteristic, complemented, Sylow

C₃³

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

100

C₃×C₉

—

111

—

101

C₃×C₉

—

111

—

102

C₃×C₉

—

111

—

103

C₃×C₉

—

111

—

104

C₃×C₉

C₃×#66526

yes

—

105

C₃×C₉

C₃×#66526

yes

characteristic

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

—

111

C₃²

C₃²×#66526

yes

characteristic

112

C₉

—

113

C₉

—

114

C₉

—

333

—

115

C₉

—

333

—

116

C₉

—

333

—

117

C₉

—

111

—

118

C₉

—

111

—

119

C₉

—

111

—

120

C₉

#15688538

yes

complemented

121

C₃

—

122

C₃

—

123

C₃

—

124

C₃

—

125

C₃

C₃×#15688538

yes

characteristic

126

C₃

3₋¹⁺²×#66526

yes

—

	subgroup	G/H
center, Frattini	C₃²	C₃²×#66526
commutator, socle	C₃²×C₃₇	C₃²×C₉
Fitting	C₃₇×C₉⋊C₉	C₉
prime radical, cosocle	C₃²×C₃₇⋊C₃	C₃³
hypercenter	C₉⋊C₉	#66526
hypocenter	C₃₇	#10624561
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₉⋊C₉

subgroup

G/H

center, Frattini

C₃²

C₃²×#66526

commutator, socle

C₃²×C₃₇

C₃²×C₉

Fitting

C₃₇×C₉⋊C₉

C₉

prime radical, cosocle

C₃²×C₃₇⋊C₃

C₃³

C₉⋊C₉

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃₇

C₁ ◁ C₃² ◁ C₉⋊C₉

N	H ≅ G/N	type
17. #15688506	C₉	semidirect
18. #15688505	C₉	semidirect
19. #15688504	C₉	semidirect
20. C₃₇×C₉⋊C₉	C₉	semidirect
54. C₉×C₃₇	C₉⋊C₉	semidirect
57. #66524	C₉⋊C₉	semidirect
82. C₉⋊C₉	#66526	semidirect
84. C₃₇	#10624561	semidirect
120. C₉	#15688538	semidirect

H ≅ G/N

type

17. #15688506

C₉

semidirect

18. #15688505

C₉

semidirect

19. #15688504

C₉

semidirect

20. C₃₇×C₉⋊C₉

C₉

semidirect

54. C₉×C₃₇

C₉⋊C₉

semidirect

57. #66524

C₉⋊C₉

semidirect

82. C₉⋊C₉

#66526

semidirect

84. C₃₇

#10624561

semidirect

120. C₉

#15688538

semidirect

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

subgroup	index
C₃×#15688538	3
C₃×#15688537	3
C₃×C₉×#66526	3
(C₃×#66526)⋊C₉	3
C₉⋊C₉×C₃₇⋊C₃	3
#10624561	37

subgroup

index

C₃×C₉×#66526

C₉⋊C₉×C₃₇⋊C₃

Subgroups — #41008461

Conjugacy classes of subgroups

Complements

Maximal subgroups