Group #41007666 — subgroups

Subgroups — #41007666

Order 26973.

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×#11758605	C₃	1	1	yes	maximal, characteristic
2	#22582605	C₃	1	9	yes	maximal, complemented
3	#22582597	C₃	1	18	yes	maximal, complemented
4	#22582578	C₃	1	2	yes	maximal, complemented
5	#22582575	C₃	1	3	yes	maximal, complemented
6	#22582553	C₃	1	3	yes	maximal
7	#22582535	C₃	1	3	yes	maximal, complemented
8	C₃₇×#6895	C₃	1	1	yes	maximal, characteristic, complemented
9	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
10	C₃×#11758605	—	3	9	no	—
11	C₃×#11758605	C₃²	1	6	yes	—
12	C₃×#11758605	C₃²	1	3	yes	—
13	C₃×#11758605	C₃²	1	9	yes	—
14	C₃²×#66524	C₃²	1	1	yes	characteristic
15	C₃²×#66524	C₃²	1	1	yes	characteristic
16	#15688506	C₃²	1	9	yes	complemented
17	#15688504	C₃²	1	18	yes	complemented
18	#15688504	C₃²	1	18	yes	complemented
19	#15688504	C₃²	1	18	yes	complemented
20	C₃²⋊#66524	C₃²	1	6	yes	—
21	#15688502	C₃²	1	18	yes	—
22	C₉×#66524	C₃²	1	9	yes	complemented
23	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
24	C₃₇×C₉⋊C₉	C₃²	1	9	yes	complemented
25	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
26	C₃³×C₃₇	C₃³	1	1	yes	characteristic
27	C₃²×C₃₇⋊C₃	—	3	3	no	—
28	C₃²×C₃₇⋊C₃	—	9	3	no	—
29	C₃²×C₃₇⋊C₃	C₃³	1	2	yes	—
30	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
31	C₃×C₉×C₃₇	—	3	3	no	—
32	C₃×C₉×C₃₇	C₃³	1	9	yes	—
33	C₃×C₉×C₃₇	C₃³	1	3	yes	—
34	C₃×#66524	—	3	3	no	—
35	C₃×#66524	3₋¹⁺²	1	9	yes	complemented
36	C₃×#66524	C₃³	1	18	yes	—
37	C₃×#66524	C₃³	1	3	yes	—
38	C₃×#66524	C₃³	1	3	yes	—
39	#11758605	—	3	18	no	—
40	#11758605	—	3	9	no	—
41	#11758605	—	9	9	no	—
42	#10624884	—	37	1	no	maximal, Sylow
43	C₃²×C₃₇	—	3	3	no	—
44	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
45	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
46	C₃×C₃₇⋊C₃	—	3	2	no	—
47	C₃×C₃₇⋊C₃	—	3	1	no	—
48	C₃×C₃₇⋊C₃	—	3	3	no	—
49	C₃×C₃₇⋊C₃	—	3	6	no	—
50	C₃×C₃₇⋊C₃	—	9	9	no	—
51	C₉×C₃₇	—	3	3	no	—
52	C₉×C₃₇	—	3	9	no	—
53	#66524	—	3	3	no	—
54	#66524	—	3	18	no	—
55	#66524	C₃×3₋¹⁺²	1	9	yes	—
56	C₃²×3₋¹⁺²	—	37	1	no	—
57	#6900	—	37	9	no	—
58	#6899	—	37	18	no	—
59	#6895	—	37	2	no	—
60	#6895	—	37	3	no	—
61	#6895	C₃₇⋊C₃	1	1	yes	characteristic, complemented
62	#6891	—	37	3	no	—
63	#6887	—	37	3	no	—
64	C₃×C₃₇	—	3	3	no	—
65	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
66	C₃×C₃₇	3₋¹⁺⁴	1	3	yes	—
67	C₃₇⋊C₃	—	9	1	no	—
68	C₃₇⋊C₃	—	9	2	no	—
69	C₃⁴	—	37	1	no	—
70	C₃×3₋¹⁺²	—	37	9	no	—
71	C₃×3₋¹⁺²	—	37	6	no	—
72	C₃×3₋¹⁺²	—	37	3	no	—
73	C₃×3₋¹⁺²	—	111	9	no	—
74	C₃²×C₉	—	37	1	no	—
75	C₃²×C₉	—	37	1	no	—
76	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
77	C₉⋊C₉	—	37	18	no	—
78	C₉⋊C₉	—	37	18	no	—
79	C₉⋊C₉	—	37	18	no	—
80	C₉⋊C₉	—	37	9	no	—
81	C₉⋊C₉	C₃×C₃₇⋊C₃	1	9	yes	complemented
82	C₃²⋊C₉	—	37	6	no	—
83	C₃²⋊C₉	—	37	18	no	—
84	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
85	C₉²	—	37	9	no	—
86	C₃₇	#10624884	1	1	yes	characteristic, complemented, Sylow
87	C₃³	—	37	2	no	—
88	C₃³	—	37	1	no	—
89	C₃³	—	333	3	no	—
90	C₃³	—	111	3	no	—
91	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
92	3₋¹⁺²	—	333	9	no	—
93	3₋¹⁺²	—	111	18	no	—
94	3₋¹⁺²	—	111	9	no	—
95	C₃×C₉	—	3	3	no	—
96	C₃×C₉	—	37	9	no	—
97	C₃×C₉	—	37	3	no	—
98	C₃×C₉	—	37	18	no	—
99	C₃×C₉	—	37	3	no	—
100	C₃×C₉	—	111	3	no	—
101	C₃×C₉	C₃²×C₃₇⋊C₃	1	9	yes	—
102	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
103	C₃²	—	3	3	no	—
104	C₃²	—	333	9	no	—
105	C₃²	—	111	2	no	—
106	C₃²	—	111	1	no	—
107	C₃²	—	111	3	no	—
108	C₃²	—	111	6	no	—
109	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	3	yes	—
110	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
111	C₉	—	3	3	no	—
112	C₉	—	3	9	no	—
113	C₉	—	37	9	no	—
114	C₉	—	111	3	no	—
115	C₉	—	111	18	no	—
116	C₃	—	3	3	no	—
117	C₃	—	333	1	no	—
118	C₃	—	333	2	no	—
119	C₃	C₃×3₋¹⁺²×C₃₇⋊C₃	1	1	yes	characteristic
120	C₃	#22582678	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758605

C₃

yes

maximal, characteristic

#22582605

C₃

yes

maximal, complemented

#22582597

C₃

yes

maximal, complemented

#22582578

C₃

yes

maximal, complemented

#22582575

C₃

yes

maximal, complemented

#22582553

C₃

yes

maximal

#22582535

C₃

yes

maximal, complemented

C₃₇×#6895

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

C₃²

yes

characteristic

C₃×#11758605

—

C₃×#11758605

C₃²

yes

—

C₃×#11758605

C₃²

yes

—

C₃×#11758605

C₃²

yes

—

C₃²×#66524

C₃²

yes

characteristic

C₃²×#66524

C₃²

yes

characteristic

#15688506

C₃²

yes

complemented

#15688504

C₃²

yes

complemented

#15688504

C₃²

yes

complemented

#15688504

C₃²

yes

complemented

C₃²⋊#66524

C₃²

yes

—

#15688502

C₃²

yes

—

C₉×#66524

C₃²

yes

complemented

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×C₉⋊C₉

C₃²

yes

complemented

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

C₃³

yes

—

C₃²×C₃₇⋊C₃

C₃³

yes

characteristic

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃³

yes

—

C₃×C₉×C₃₇

C₃³

yes

—

C₃×#66524

—

C₃×#66524

3₋¹⁺²

yes

complemented

C₃×#66524

C₃³

yes

—

C₃×#66524

C₃³

yes

—

C₃×#66524

C₃³

yes

—

—

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₃²×C₃₇

C₃⁴

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

—

#66524

—

#66524

—

#66524

C₃×3₋¹⁺²

yes

—

C₃²×3₋¹⁺²

—

—

—

—

—

C₃₇⋊C₃

yes

characteristic, complemented

#6891

—

#6887

—

C₃×C₃₇

—

C₃×C₃₇

C₃²×3₋¹⁺²

yes

characteristic

C₃×C₃₇

3₋¹⁺⁴

yes

—

C₃₇⋊C₃

—

C₃₇⋊C₃

—

C₃⁴

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

C₃×C₃₇⋊C₃

yes

complemented

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₉²

—

C₃₇

#10624884

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

333

—

C₃³

—

111

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

333

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

100

C₃×C₉

—

111

—

101

C₃×C₉

C₃²×C₃₇⋊C₃

yes

—

102

C₃×C₉

C₃²×C₃₇⋊C₃

yes

—

103

C₃²

—

104

C₃²

—

333

—

105

C₃²

—

111

—

106

C₃²

—

111

—

107

C₃²

—

111

—

108

C₃²

—

111

—

109

C₃²

3₋¹⁺²×C₃₇⋊C₃

yes

—

110

C₃²

C₃³×C₃₇⋊C₃

yes

characteristic

111

C₉

—

112

C₉

—

113

C₉

—

114

C₉

—

111

—

115

C₉

—

111

—

116

C₃

—

117

C₃

—

333

—

118

C₃

—

333

—

119

C₃

C₃×3₋¹⁺²×C₃₇⋊C₃

yes

characteristic

120

C₃

#22582678

yes

—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting	C₃₇×#6895	C₃
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	#6895	C₃₇⋊C₃
hypocenter	C₃₇	#10624884
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ #6895

subgroup

G/H

center, Frattini

C₃²

C₃³×C₃₇⋊C₃

commutator, socle, cosocle

C₃²×C₃₇

C₃⁴

Fitting

C₃₇×#6895

C₃

prime radical

C₃³×C₃₇⋊C₃

C₃²

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃₇

C₁ ◁ C₃² ◁ #6895

N	H ≅ G/N	type
2. #22582605	C₃	semidirect
3. #22582597	C₃	semidirect
4. #22582578	C₃	semidirect
5. #22582575	C₃	semidirect
7. #22582535	C₃	semidirect
8. C₃₇×#6895	C₃	semidirect
16. #15688506	C₃²	semidirect
17. #15688504	C₃²	semidirect
18. #15688504	C₃²	semidirect
19. #15688504	C₃²	semidirect
22. C₉×#66524	C₃²	semidirect
24. C₃₇×C₉⋊C₉	C₃²	semidirect
35. C₃×#66524	3₋¹⁺²	semidirect
61. #6895	C₃₇⋊C₃	semidirect
81. C₉⋊C₉	C₃×C₃₇⋊C₃	semidirect
86. C₃₇	#10624884	semidirect

H ≅ G/N

type

2. #22582605

C₃

semidirect

3. #22582597

C₃

semidirect

4. #22582578

C₃

semidirect

5. #22582575

C₃

semidirect

7. #22582535

C₃

semidirect

8. C₃₇×#6895

C₃

semidirect

16. #15688506

C₃²

semidirect

17. #15688504

C₃²

semidirect

18. #15688504

C₃²

semidirect

19. #15688504

C₃²

semidirect

22. C₉×#66524

C₃²

semidirect

24. C₃₇×C₉⋊C₉

C₃²

semidirect

35. C₃×#66524

3₋¹⁺²

semidirect

61. #6895

C₃₇⋊C₃

semidirect

81. C₉⋊C₉

C₃×C₃₇⋊C₃

semidirect

86. C₃₇

#10624884

semidirect

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

subgroup	index
C₃²×#11758605	3
#22582605	3
#22582597	3
#22582578	3
#22582575	3
#22582553	3
#22582535	3
C₃₇×#6895	3
#10624884	37

subgroup

index

C₃₇×#6895

Subgroups — #41007666

Conjugacy classes of subgroups

Complements

Maximal subgroups