Group #41007415 — subgroups

Subgroups — #41007415

Order 26973.

200 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 46 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	#22582661	C₃	1	2	yes	maximal, complemented
2	#22582601	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582525	C₃	1	1	yes	maximal, characteristic, complemented
4	#22582524	C₃	1	1	yes	maximal, characteristic, complemented
5	#22582512	C₃	1	2	yes	maximal, complemented
6	#22582511	C₃	1	2	yes	maximal, complemented
7	#22582506	C₃	1	1	yes	maximal, characteristic, complemented
8	#22582501	C₃	1	2	yes	maximal, complemented
9	C₃₇×#6911	C₃	1	1	yes	maximal, characteristic, complemented
10	#15688531	—	3	2	no	—
11	C₃×#11758608	—	3	1	no	—
12	#15688521	C₃²	1	2	yes	complemented
13	#15688521	C₃²	1	2	yes	complemented
14	#15688519	C₃²	1	2	yes	complemented
15	#15688516	—	3	2	no	—
16	#15688513	—	3	2	no	—
17	#15688506	C₃²	1	1	yes	characteristic, complemented
18	#15688505	C₃²	1	1	yes	characteristic, complemented
19	#15688504	—	3	2	no	—
20	C₉×#66524	C₃²	1	1	yes	characteristic, complemented
21	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
22	C₃×H(3,𝔽₃)×C₃₇	—	3	1	no	—
23	C₃₇×#732	—	3	2	no	—
24	C₃₇×#732	C₃²	1	2	yes	—
25	C₃₇×#732	C₃²	1	1	yes	characteristic
26	C₃³×C₃₇	—	9	1	no	—
27	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
28	3₋¹⁺²×C₃₇	—	9	1	no	—
29	H(3,𝔽₃)×C₃₇	—	3	2	no	—
30	H(3,𝔽₃)×C₃₇	—	3	1	no	—
31	H(3,𝔽₃)×C₃₇	—	9	2	no	—
32	C₃×C₉×C₃₇	H(3,𝔽₃)	1	2	yes	—
33	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
34	C₃²⋊(C₃₇⋊C₃)	—	3	2	no	—
35	#11758608	—	3	2	no	—
36	#11758608	—	3	2	no	—
37	#11758608	—	9	3	no	—
38	C₃×#66524	—	3	1	no	—
39	C₃×#66524	—	3	1	no	—
40	C₃×#66524	—	3	1	no	—
41	#11758606	—	9	2	no	—
42	#11758605	—	9	2	no	—
43	C₉×C₃₇⋊C₃	—	9	2	no	—
44	#10624851	—	37	1	no	maximal, Sylow
45	C₃²×C₃₇	—	3	1	no	—
46	C₃²×C₃₇	—	9	1	no	—
47	C₃²×C₃₇	—	9	2	no	—
48	C₃²×C₃₇	—	9	1	no	—
49	C₃²×C₃₇	—	27	3	no	—
50	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
51	C₃×C₃₇⋊C₃	—	9	2	no	—
52	C₉×C₃₇	—	3	2	no	—
53	C₉×C₃₇	—	3	1	no	—
54	#66524	—	3	3	no	—
55	#66524	—	9	1	no	—
56	#66524	—	9	1	no	—
57	#66524	—	9	2	no	—
58	#66524	—	9	2	no	—
59	#6913	—	37	2	no	—
60	#6911	C₃₇⋊C₃	1	1	yes	characteristic, complemented
61	#6900	—	37	1	no	—
62	#6883	—	37	1	no	—
63	#6883	—	37	1	no	—
64	#6881	—	37	2	no	—
65	#6881	—	37	2	no	—
66	#6879	—	37	1	no	—
67	#6878	—	37	2	no	—
68	C₃×C₃₇	—	3	1	no	—
69	C₃×C₃₇	—	9	1	no	—
70	C₃×C₃₇	—	27	2	no	—
71	C₃×C₃₇	—	27	1	no	—
72	C₃×C₃₇	#6911	1	1	yes	characteristic
73	C₃₇⋊C₃	—	27	2	no	—
74	#737	—	111	2	no	—
75	C₃×3₋¹⁺²	—	111	1	no	—
76	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
77	C₃×H(3,𝔽₃)	—	3	1	no	—
78	#733	—	37	2	no	—
79	#733	—	37	2	no	—
80	#732	—	3	2	no	—
81	#732	—	37	2	no	—
82	#732	C₃×C₃₇⋊C₃	1	2	yes	—
83	#732	C₃×C₃₇⋊C₃	1	1	yes	characteristic
84	#731	—	111	2	no	—
85	AΣL(1,𝔽₂₇)	—	111	2	no	—
86	C₉⋊C₉	—	37	1	no	—
87	C₉⋊C₉	—	37	1	no	—
88	C₉⋊C₉	—	111	2	no	—
89	C₉²	—	37	1	no	—
90	C₃₇	#10624851	1	1	yes	characteristic, complemented, Sylow
91	C₃³	—	9	1	no	—
92	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
93	3₋¹⁺²	—	9	1	no	—
94	3₋¹⁺²	—	333	2	no	—
95	3₋¹⁺²	—	333	2	no	—
96	3₋¹⁺²	—	333	3	no	—
97	3₋¹⁺²	—	111	2	no	—
98	3₋¹⁺²	—	111	2	no	—
99	H(3,𝔽₃)	—	3	2	no	—
100	H(3,𝔽₃)	—	3	1	no	—
101	H(3,𝔽₃)	—	9	2	no	—
102	H(3,𝔽₃)	—	111	2	no	—
103	C₃×C₉	—	333	2	no	—
104	C₃×C₉	—	111	1	no	—
105	C₃×C₉	—	111	1	no	—
106	C₃×C₉	—	111	1	no	—
107	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
108	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
109	C₃²	—	3	1	no	—
110	C₃²	—	9	1	no	—
111	C₃²	—	9	2	no	—
112	C₃²	—	9	1	no	—
113	C₃²	—	27	3	no	—
114	C₃²	—	333	2	no	—
115	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
116	C₉	—	3	2	no	—
117	C₉	—	3	1	no	—
118	C₉	—	333	1	no	—
119	C₉	—	333	1	no	—
120	C₉	—	333	2	no	—
121	C₉	—	333	2	no	—
122	C₉	—	111	3	no	—
123	C₃	—	3	1	no	—
124	C₃	—	9	1	no	—
125	C₃	—	27	2	no	—
126	C₃	—	27	1	no	—
127	C₃	—	999	2	no	—
128	C₃	#22582647	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22582661

C₃

yes

maximal, complemented

#22582601

C₃

yes

maximal, characteristic, complemented

#22582525

C₃

yes

maximal, characteristic, complemented

#22582524

C₃

yes

maximal, characteristic, complemented

#22582512

C₃

yes

maximal, complemented

#22582511

C₃

yes

maximal, complemented

#22582506

C₃

yes

maximal, characteristic, complemented

#22582501

C₃

yes

maximal, complemented

C₃₇×#6911

C₃

yes

maximal, characteristic, complemented

#15688531

—

C₃×#11758608

—

#15688521

C₃²

yes

complemented

#15688521

C₃²

yes

complemented

#15688519

C₃²

yes

—

—

C₃²

yes

characteristic, complemented

#15688505

C₃²

yes

characteristic, complemented

#15688504

—

C₉×#66524

C₃²

yes

characteristic, complemented

C₃×3₋¹⁺²×C₃₇

C₃²

yes

characteristic

C₃×H(3,𝔽₃)×C₃₇

—

C₃₇×#732

—

C₃₇×#732

C₃²

yes

—

C₃₇×#732

C₃²

yes

characteristic

C₃³×C₃₇

—

C₃³×C₃₇

H(3,𝔽₃)

yes

characteristic

3₋¹⁺²×C₃₇

—

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×C₃₇

—

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

—

C₃×C₉×C₃₇

C₃³

yes

characteristic

C₃²⋊(C₃₇⋊C₃)

—

—

—

—

—

—

—

—

—

C₉×C₃₇⋊C₃

—

#10624851

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

—

—

—

—

—

—

—

C₃₇⋊C₃

yes

characteristic, complemented

—

—

—

—

—

—

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

#6911

yes

characteristic

C₃₇⋊C₃

—

#737

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

characteristic

C₃×H(3,𝔽₃)

—

—

—

—

—

C₃×C₃₇⋊C₃

yes

—

#732

C₃×C₃₇⋊C₃

yes

characteristic

#731

—

111

—

AΣL(1,𝔽₂₇)

—

111

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

111

—

C₉²

—

C₃₇

#10624851

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

333

—

3₋¹⁺²

—

333

—

3₋¹⁺²

—

333

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

H(3,𝔽₃)

—

100

H(3,𝔽₃)

—

101

H(3,𝔽₃)

—

102

H(3,𝔽₃)

—

111

—

103

C₃×C₉

—

333

—

104

C₃×C₉

—

111

—

105

C₃×C₉

—

111

—

106

C₃×C₉

—

111

—

107

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

—

108

C₃×C₉

C₃²×C₃₇⋊C₃

yes

characteristic

109

C₃²

—

110

C₃²

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

333

—

115

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

116

C₉

—

117

C₉

—

118

C₉

—

333

—

119

C₉

—

333

—

120

C₉

—

333

—

121

C₉

—

333

—

122

C₉

—

111

—

123

C₃

—

124

C₃

—

125

C₃

—

126

C₃

—

127

C₃

—

999

—

128

C₃

#22582647

yes

characteristic

	subgroup	G/H
center	C₃	#22582647
commutator, cosocle	C₃×C₉×C₃₇	C₃³
Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6911	C₃
socle	C₃×C₃₇	#6911
hypercenter	#6911	C₃₇⋊C₃
hypocenter	C₃₇	#10624851
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×3₋¹⁺² ◁ #6911

subgroup

G/H

center

C₃

#22582647

commutator, cosocle

C₃×C₉×C₃₇

C₃³

Frattini

C₃×C₉

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6911

C₃

C₃×C₃₇

C₃₇⋊C₃

C₃₇

G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃ ◁ C₃² ◁ C₃×3₋¹⁺² ◁ #6911

N	H ≅ G/N	type
1. #22582661	C₃	semidirect
2. #22582601	C₃	semidirect
3. #22582525	C₃	semidirect
4. #22582524	C₃	semidirect
5. #22582512	C₃	semidirect
6. #22582511	C₃	semidirect
7. #22582506	C₃	semidirect
8. #22582501	C₃	semidirect
9. C₃₇×#6911	C₃	semidirect
12. #15688521	C₃²	semidirect
13. #15688521	C₃²	semidirect
14. #15688519	C₃²	semidirect
17. #15688506	C₃²	semidirect
18. #15688505	C₃²	semidirect
20. C₉×#66524	C₃²	semidirect
60. #6911	C₃₇⋊C₃	semidirect
90. C₃₇	#10624851	semidirect

H ≅ G/N

type

1. #22582661

C₃

semidirect

2. #22582601

C₃

semidirect

3. #22582525

C₃

semidirect

4. #22582524

C₃

semidirect

5. #22582512

C₃

semidirect

6. #22582511

C₃

semidirect

7. #22582506

C₃

semidirect

8. #22582501

C₃

semidirect

9. C₃₇×#6911

C₃

semidirect

12. #15688521

C₃²

semidirect

13. #15688521

C₃²

semidirect

14. #15688519

C₃²

semidirect

17. #15688506

C₃²

semidirect

18. #15688505

C₃²

semidirect

20. C₉×#66524

C₃²

semidirect

60. #6911

C₃₇⋊C₃

semidirect

90. C₃₇

#10624851

semidirect

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

subgroup	index
#22582661	3
#22582601	3
#22582525	3
#22582524	3
#22582512	3
#22582511	3
#22582506	3
#22582501	3
C₃₇×#6911	3
#10624851	37

subgroup

index

C₃₇×#6911

Subgroups — #41007415

Conjugacy classes of subgroups

Complements

Maximal subgroups