Group #41007652 — subgroups

Subgroups — #41007652

Order 26973.

1058 conjugacy classes of subgroups; only the 292 normal subgroups are shown.

normal subgroups only

This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 292 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×#15688531	C₃	1	3	yes	maximal, complemented
2	#22582564	C₃	1	6	yes	maximal, complemented
3	#22582555	C₃	1	27	yes	maximal, complemented
4	#22582554	C₃	1	2	yes	maximal, complemented
5	#22582550	C₃	1	1	yes	maximal, characteristic
6	C₃²×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
7	#15688531	—	3	27	no	—
8	C₃×#11758608	C₃²	1	6	yes	—
9	C₃×#11758605	C₃²	1	9	yes	complemented
10	C₃×#11758606	C₃²	1	9	yes	complemented
11	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	3	yes	—
12	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	9	yes	—
13	C₃×C₉×C₃₇⋊C₃	C₃²	1	9	yes	complemented
14	C₃²⋊#66524	C₃²	1	18	yes	—
15	#15688502	C₃²	1	54	yes	complemented
16	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
17	C₃×3₋¹⁺²×C₃₇	—	3	9	no	—
18	C₃×3₋¹⁺²×C₃₇	C₃²	1	9	yes	—
19	C₃²×C₉×C₃₇	C₃²	1	2	yes	—
20	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
21	C₃³×C₃₇	—	3	3	no	—
22	C₃³×C₃₇	—	9	3	no	—
23	C₃³×C₃₇	C₃³	1	3	yes	—
24	C₃³×C₃₇	C₃³	1	1	yes	characteristic
25	C₃²×C₃₇⋊C₃	C₃³	1	9	yes	—
26	3₋¹⁺²×C₃₇	—	3	27	no	—
27	C₃×C₉×C₃₇	—	3	6	no	—
28	C₃×C₉×C₃₇	H(3,𝔽₃)	1	9	yes	complemented
29	C₃×C₉×C₃₇	C₃³	1	6	yes	—
30	C₃×C₉×C₃₇	C₃³	1	3	yes	—
31	C₃²⋊(C₃₇⋊C₃)	—	3	9	no	—
32	C₃²⋊(C₃₇⋊C₃)	—	9	9	no	—
33	#11758608	—	3	18	no	—
34	C₃×#66524	C₃³	1	18	yes	—
35	#11758606	—	3	27	no	—
36	#11758605	—	3	27	no	—
37	C₉×C₃₇⋊C₃	—	3	27	no	—
38	#10624882	—	37	1	no	maximal, Sylow
39	C₃²×C₃₇	—	3	3	no	—
40	C₃²×C₃₇	—	3	3	no	—
41	C₃²×C₃₇	—	3	3	no	—
42	C₃²×C₃₇	—	3	6	no	—
43	C₃²×C₃₇	—	9	9	no	—
44	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
45	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
46	C₃×C₃₇⋊C₃	—	3	9	no	—
47	C₃×C₃₇⋊C₃	—	3	27	no	—
48	C₉×C₃₇	—	3	6	no	—
49	C₉×C₃₇	C₃×H(3,𝔽₃)	1	9	yes	—
50	#66524	—	3	18	no	—
51	C₃×#737	—	37	3	no	—
52	C₃²×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic, complemented
53	#6893	—	37	6	no	—
54	C₉⋊H(3,𝔽₃)	—	37	27	no	—
55	#6891	—	37	2	no	—
56	C₃²⋊H(3,𝔽₃)	—	37	1	no	—
57	C₃×C₃₇	—	3	3	no	—
58	C₃×C₃₇	—	9	3	no	—
59	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
60	C₃×C₃₇	3₋¹⁺⁴	1	3	yes	—
61	C₃₇⋊C₃	—	9	9	no	—
62	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
63	#737	—	111	27	no	—
64	C₃×3₋¹⁺²	—	3	9	no	—
65	C₃×3₋¹⁺²	—	37	6	no	—
66	C₃×3₋¹⁺²	—	37	9	no	—
67	C₃×3₋¹⁺²	—	37	9	no	—
68	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	9	yes	—
69	C₃×H(3,𝔽₃)	—	37	9	no	—
70	C₃×H(3,𝔽₃)	—	37	3	no	—
71	C₃²×C₉	—	37	9	no	—
72	C₃²×C₉	C₃×C₃₇⋊C₃	1	2	yes	—
73	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
74	C₃²⋊C₉	—	37	18	no	—
75	C₃²⋊C₉	—	37	54	no	—
76	C₃₇	#10624882	1	1	yes	characteristic, complemented, Sylow
77	C₃³	—	3	3	no	—
78	C₃³	—	9	3	no	—
79	C₃³	—	37	9	no	—
80	C₃³	C₃²×C₃₇⋊C₃	1	3	yes	—
81	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
82	3₋¹⁺²	—	3	27	no	—
83	3₋¹⁺²	—	111	18	no	—
84	3₋¹⁺²	—	111	27	no	—
85	3₋¹⁺²	—	111	27	no	—
86	H(3,𝔽₃)	—	333	9	no	—
87	H(3,𝔽₃)	—	111	9	no	—
88	C₃×C₉	—	3	6	no	—
89	C₃×C₉	—	37	18	no	—
90	C₃×C₉	—	111	27	no	—
91	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	9	yes	complemented
92	C₃×C₉	C₃²×C₃₇⋊C₃	1	6	yes	—
93	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
94	C₃²	—	3	3	no	—
95	C₃²	—	3	3	no	—
96	C₃²	—	3	3	no	—
97	C₃²	—	3	6	no	—
98	C₃²	—	9	9	no	—
99	C₃²	—	111	9	no	—
100	C₃²	—	111	27	no	—
101	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
102	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
103	C₉	—	3	6	no	—
104	C₉	—	111	18	no	—
105	C₉	C₃×C₃²⋊(C₃₇⋊C₃)	1	9	yes	—
106	C₃	—	3	3	no	—
107	C₃	—	9	3	no	—
108	C₃	—	333	9	no	—
109	C₃	C₃²×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
110	C₃	#22582679	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688531

C₃

yes

maximal, complemented

#22582564

C₃

yes

maximal, complemented

#22582555

C₃

yes

maximal, complemented

#22582554

C₃

yes

maximal, complemented

#22582550

C₃

yes

maximal, characteristic

C₃²×3₋¹⁺²×C₃₇

C₃

yes

maximal, characteristic, complemented

#15688531

—

C₃×#11758608

C₃²

yes

—

C₃×#11758605

C₃²

yes

complemented

C₃×#11758606

C₃²

yes

complemented

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

—

C₃×C₉×C₃₇⋊C₃

C₃²

yes

complemented

C₃²⋊#66524

C₃²

yes

—

#15688502

C₃²

yes

complemented

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₇

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃²×C₉×C₃₇

C₃²

yes

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

C₃³

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

C₃³

yes

—

3₋¹⁺²×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

complemented

C₃×C₉×C₃₇

C₃³

yes

—

C₃×C₉×C₃₇

C₃³

yes

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

#11758608

—

C₃×#66524

C₃³

yes

—

#11758606

—

#11758605

—

C₉×C₃₇⋊C₃

—

#10624882

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₇

C₃⁴

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

C₃×H(3,𝔽₃)

yes

—

#66524

—

C₃×#737

—

C₃²×3₋¹⁺²

C₃₇⋊C₃

yes

characteristic, complemented

#6893

—

C₉⋊H(3,𝔽₃)

—

#6891

—

C₃²⋊H(3,𝔽₃)

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃²×H(3,𝔽₃)

yes

characteristic

C₃×C₃₇

3₋¹⁺⁴

yes

—

C₃₇⋊C₃

—

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

#737

—

111

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃₇

#10624882

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²×C₃₇⋊C₃

yes

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

H(3,𝔽₃)

—

333

—

H(3,𝔽₃)

—

111

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

complemented

C₃×C₉

C₃²×C₃₇⋊C₃

yes

—

C₃×C₉

C₃²×C₃₇⋊C₃

yes

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

111

—

100

C₃²

—

111

—

101

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

—

102

C₃²

C₃³×C₃₇⋊C₃

yes

characteristic

103

C₉

—

104

C₉

—

111

—

105

C₉

C₃×C₃²⋊(C₃₇⋊C₃)

yes

—

106

C₃

—

107

C₃

—

108

C₃

—

333

—

109

C₃

C₃²×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

110

C₃

#22582679

yes

—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting	C₃²×3₋¹⁺²×C₃₇	C₃
prime radical	#22582550	C₃
hypercenter	C₃²×3₋¹⁺²	C₃₇⋊C₃
hypocenter	C₃₇	#10624882
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²×3₋¹⁺²

subgroup

G/H

center, Frattini

C₃²

C₃³×C₃₇⋊C₃

commutator, socle, cosocle

C₃²×C₃₇

C₃⁴

Fitting

C₃²×3₋¹⁺²×C₃₇

C₃

prime radical

#22582550

C₃

hypercenter

C₃²×3₋¹⁺²

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃₇

C₁ ◁ C₃² ◁ C₃²×3₋¹⁺²

N	H ≅ G/N	type
1. C₃×#15688531	C₃	semidirect
2. #22582564	C₃	semidirect
3. #22582555	C₃	semidirect
4. #22582554	C₃	semidirect
6. C₃²×3₋¹⁺²×C₃₇	C₃	semidirect
9. C₃×#11758605	C₃²	semidirect
10. C₃×#11758606	C₃²	semidirect
13. C₃×C₉×C₃₇⋊C₃	C₃²	semidirect
15. #15688502	C₃²	semidirect
28. C₃×C₉×C₃₇	H(3,𝔽₃)	semidirect
52. C₃²×3₋¹⁺²	C₃₇⋊C₃	semidirect
76. C₃₇	#10624882	semidirect
91. C₃×C₉	C₃²⋊(C₃₇⋊C₃)	semidirect

H ≅ G/N

type

1. C₃×#15688531

C₃

semidirect

2. #22582564

C₃

semidirect

3. #22582555

C₃

semidirect

4. #22582554

C₃

semidirect

6. C₃²×3₋¹⁺²×C₃₇

C₃

semidirect

9. C₃×#11758605

C₃²

semidirect

10. C₃×#11758606

C₃²

semidirect

13. C₃×C₉×C₃₇⋊C₃

C₃²

semidirect

15. #15688502

C₃²

semidirect

28. C₃×C₉×C₃₇

H(3,𝔽₃)

semidirect

52. C₃²×3₋¹⁺²

C₃₇⋊C₃

semidirect

76. C₃₇

#10624882

semidirect

91. C₃×C₉

C₃²⋊(C₃₇⋊C₃)

semidirect

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

subgroup	index
C₃×#15688531	3
#22582564	3
#22582555	3
#22582554	3
#22582550	3
C₃²×3₋¹⁺²×C₃₇	3
#10624882	37

subgroup

index

C₃²×3₋¹⁺²×C₃₇

Subgroups — #41007652

Conjugacy classes of subgroups

Complements

Maximal subgroups