Group #41006377 — subgroups

Subgroups — C₉×#15688515 #41006377

Order 26973.

292 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 88 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×#15688515	C₃	1	1	yes	maximal, characteristic
2	C₉×#11758608	C₃	1	2	yes	maximal, complemented
3	C₃×C₉×#66524	C₃	1	1	yes	maximal, characteristic, complemented
4	#22582481	C₃	1	4	yes	maximal, complemented
5	H(3,𝔽₃)⋊#66524	C₃	1	2	yes	maximal
6	#22582469	C₃	1	2	yes	maximal, complemented
7	C₉×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic
8	C₃×#11758608	C₃²	1	2	yes	—
9	C₃²×#66524	C₃²	1	2	yes	—
10	C₃²×#66524	C₃²	1	1	yes	characteristic
11	#15688515	C₉	1	9	yes	direct product factor
12	#15688504	—	3	4	no	—
13	C₃²⋊#66524	C₃²	1	4	yes	—
14	C₉×#66524	—	3	2	no	—
15	C₉×#66524	—	3	2	no	—
16	C₉×#66524	—	3	1	no	—
17	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
18	C₃²×C₉×C₃₇	—	3	1	no	—
19	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
20	C₃₇×C₃²⋊C₉	—	3	2	no	—
21	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
22	C₃³×C₃₇	—	3	1	no	—
23	C₃³×C₃₇	C₃³	1	1	yes	characteristic
24	H(3,𝔽₃)×C₃₇	—	3	2	no	—
25	H(3,𝔽₃)×C₃₇	C₃×C₉	1	3	yes	—
26	C₃×C₉×C₃₇	—	3	3	no	—
27	C₃×C₉×C₃₇	—	3	2	no	—
28	C₃×C₉×C₃₇	—	9	3	no	—
29	C₃×C₉×C₃₇	H(3,𝔽₃)	1	2	yes	—
30	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
31	#11758608	—	3	4	no	—
32	#11758608	C₃×C₉	1	6	yes	complemented
33	C₃×#66524	—	3	1	no	—
34	C₃×#66524	—	3	2	no	—
35	C₃×#66524	—	3	2	no	—
36	C₃×#66524	—	3	2	no	—
37	C₃×#66524	—	3	6	no	—
38	C₃×#66524	—	3	4	no	—
39	C₃×#66524	C₃×C₉	1	3	yes	complemented
40	C₉×#731	—	37	1	no	maximal, Sylow
41	C₃²×C₃₇	—	3	1	no	—
42	C₃²×C₃₇	—	3	2	no	—
43	C₃²×C₃₇	—	3	3	no	—
44	C₃²×C₃₇	—	9	1	no	—
45	C₃²×C₃₇	—	9	2	no	—
46	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
47	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
48	C₃²×C₃₇	#737	1	2	yes	—
49	C₉×C₃₇	—	3	2	no	—
50	C₉×C₃₇	—	9	2	no	—
51	C₉×C₃₇	#731	1	3	yes	complemented
52	#66524	—	3	3	no	—
53	#66524	—	3	6	no	—
54	#66524	—	3	6	no	—
55	#66524	—	9	4	no	—
56	C₃×#731	—	37	1	no	—
57	C₉×3₋¹⁺²	—	37	2	no	—
58	C₉×H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic
59	C₃×C₉²	—	37	1	no	—
60	3₋¹⁺²⋊C₉	—	37	4	no	—
61	H(3,𝔽₃)⋊C₉	—	37	2	no	—
62	#6867	—	37	2	no	—
63	C₃×C₃₇	—	3	1	no	—
64	C₃×C₃₇	—	3	2	no	—
65	C₃×C₃₇	—	9	3	no	—
66	C₃×C₃₇	C₉×H(3,𝔽₃)	1	1	yes	characteristic
67	C₃×C₃₇	C₃×#731	1	1	yes	characteristic
68	C₃×C₃₇	#6908	1	2	yes	—
69	C₃×3₋¹⁺²	—	37	2	no	—
70	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
71	C₃²×C₉	—	3	1	no	—
72	C₃²×C₉	—	37	1	no	—
73	C₃²×C₉	—	37	2	no	—
74	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
75	#731	—	37	9	no	—
76	C₉⋊C₉	—	111	4	no	—
77	C₃²⋊C₉	—	3	2	no	—
78	C₃²⋊C₉	—	37	4	no	—
79	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	—
80	C₉²	—	111	2	no	—
81	C₉²	—	111	1	no	—
82	C₉²	—	111	2	no	—
83	C₃₇	C₉×#731	1	1	yes	characteristic, complemented, Sylow
84	C₃³	—	3	1	no	—
85	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
86	3₋¹⁺²	—	37	6	no	—
87	3₋¹⁺²	—	111	4	no	—
88	H(3,𝔽₃)	—	3	2	no	—
89	H(3,𝔽₃)	C₉×C₃₇⋊C₃	1	3	yes	—
90	C₃×C₉	—	3	3	no	—
91	C₃×C₉	—	3	2	no	—
92	C₃×C₉	—	9	3	no	—
93	C₃×C₉	—	37	3	no	—
94	C₃×C₉	—	111	4	no	—
95	C₃×C₉	—	111	1	no	—
96	C₃×C₉	—	111	2	no	—
97	C₃×C₉	—	111	2	no	—
98	C₃×C₉	—	111	2	no	—
99	C₃×C₉	—	111	6	no	—
100	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
101	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
102	C₃²	—	3	1	no	—
103	C₃²	—	3	2	no	—
104	C₃²	—	3	3	no	—
105	C₃²	—	9	1	no	—
106	C₃²	—	9	2	no	—
107	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
108	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
109	C₃²	#15688531	1	2	yes	—
110	C₉	—	3	2	no	—
111	C₉	—	9	2	no	—
112	C₉	—	333	4	no	—
113	C₉	—	111	3	no	—
114	C₉	—	111	6	no	—
115	C₉	—	111	6	no	—
116	C₉	#15688515	1	3	yes	direct product factor
117	C₃	—	3	1	no	—
118	C₃	—	3	2	no	—
119	C₃	—	9	3	no	—
120	C₃	C₉×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
121	C₃	C₃×#15688515	1	1	yes	characteristic
122	C₃	#22582630	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688515

C₃

yes

maximal, characteristic

C₉×#11758608

C₃

yes

maximal, complemented

C₃×C₉×#66524

C₃

yes

maximal, characteristic, complemented

#22582481

C₃

yes

maximal, complemented

H(3,𝔽₃)⋊#66524

C₃

yes

maximal

#22582469

C₃

yes

maximal, complemented

C₉×H(3,𝔽₃)×C₃₇

C₃

yes

maximal, characteristic

C₃×#11758608

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

C₃²×#66524

C₃²

yes

characteristic

#15688515

C₉

yes

direct product factor

#15688504

—

C₃²⋊#66524

C₃²

yes

—

C₉×#66524

—

C₉×#66524

—

C₉×#66524

—

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

characteristic

C₃²×C₉×C₃₇

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×C₃²⋊C₉

—

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃³×C₃₇

—

C₃³×C₃₇

C₃³

yes

characteristic

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×C₃₇

C₃×C₉

yes

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

—

C₃×C₉×C₃₇

yes

—

C₃×C₉

yes

—

—

—

—

—

—

C₃×C₉

yes

complemented

C₉×#731

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃²×C₉

yes

characteristic

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

#737

yes

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

yes

—

—

—

—

—

C₉×3₋¹⁺²

—

C₉×H(3,𝔽₃)

C₃₇⋊C₃

yes

characteristic

C₃×C₉²

—

3₋¹⁺²⋊C₉

—

H(3,𝔽₃)⋊C₉

—

#6867

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₉×H(3,𝔽₃)

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

C₃×C₃₇⋊C₃

yes

characteristic

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

#731

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₉²

—

111

—

C₉²

—

111

—

C₉²

—

111

—

C₃₇

C₉×#731

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

111

—

H(3,𝔽₃)

—

H(3,𝔽₃)

C₉×C₃₇⋊C₃

yes

—

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

—

100

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

—

101

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

C₃×C₉×C₃₇⋊C₃

yes

characteristic

108

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

109

C₃²

#15688531

yes

—

110

C₉

—

111

C₉

—

112

C₉

—

333

—

113

C₉

—

111

—

114

C₉

—

111

—

115

C₉

—

111

—

116

C₉

#15688515

yes

direct product factor

117

C₃

—

118

C₃

—

119

C₃

—

120

C₃

C₉×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

121

C₃

C₃×#15688515

yes

characteristic

122

C₃

#22582630

yes

—

	subgroup	G/H
center	C₃×C₉	C₃²⋊(C₃₇⋊C₃)
commutator	C₃²×C₃₇	C₃²×C₉
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₉×H(3,𝔽₃)×C₃₇	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃×H(3,𝔽₃)×C₃₇	C₃²
cosocle	C₃³×C₃₇	C₃³
hypercenter	C₉×H(3,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	C₉×#731
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ C₉×H(3,𝔽₃)

subgroup

G/H

center

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

commutator

C₃²×C₃₇

C₃²×C₉

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₉×H(3,𝔽₃)×C₃₇

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃×H(3,𝔽₃)×C₃₇

C₃²

cosocle

C₃³×C₃₇

C₃³

hypercenter

C₉×H(3,𝔽₃)

C₃₇⋊C₃

hypocenter

C₃₇

C₉×#731

LCS

G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ C₉×H(3,𝔽₃)

N	H ≅ G/N	type
2. C₉×#11758608	C₃	semidirect
3. C₃×C₉×#66524	C₃	semidirect
4. #22582481	C₃	semidirect
6. #22582469	C₃	semidirect
11. #15688515	C₉	direct (N × H)
32. #11758608	C₃×C₉	semidirect
39. C₃×#66524	C₃×C₉	semidirect
51. C₉×C₃₇	#731	semidirect
83. C₃₇	C₉×#731	semidirect
116. C₉	#15688515	direct (N × H)

H ≅ G/N

type

2. C₉×#11758608

C₃

semidirect

3. C₃×C₉×#66524

C₃

semidirect

4. #22582481

C₃

semidirect

6. #22582469

C₃

semidirect

11. #15688515

C₉

direct (N × H)

32. #11758608

C₃×C₉

semidirect

39. C₃×#66524

C₃×C₉

semidirect

51. C₉×C₃₇

#731

semidirect

83. C₃₇

C₉×#731

semidirect

116. C₉

#15688515

direct (N × H)

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

subgroup	index
C₃×#15688515	3
C₉×#11758608	3
C₃×C₉×#66524	3
#22582481	3
H(3,𝔽₃)⋊#66524	3
#22582469	3
C₉×H(3,𝔽₃)×C₃₇	3
C₉×#731	37

subgroup

index

C₃×C₉×#66524

C₉×H(3,𝔽₃)×C₃₇

Subgroups — C₉×#15688515 #41006377

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C9×#15688515 #41006377

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#15688515 #41006377