Group #41007667 — subgroups

Subgroups — #41007667

Order 26973.

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

normal subgroups only

This group has 716 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	#22582601	C₃	1	9	yes	maximal, complemented
2	#22582594	C₃	1	18	yes	maximal, complemented
3	#22582578	C₃	1	6	yes	maximal, complemented
4	#22582554	C₃	1	3	yes	maximal
5	#22582538	C₃	1	3	yes	maximal, complemented
6	C₃²×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic
7	C₃×#11758608	C₃²	1	9	yes	—
8	#15688506	C₃²	1	9	yes	complemented
9	#15688505	C₃²	1	9	yes	complemented
10	#15688504	C₃²	1	54	yes	complemented
11	C₃²⋊#66524	C₃²	1	9	yes	—
12	C₃²⋊#66524	C₃²	1	18	yes	—
13	C₉×#66524	C₃²	1	9	yes	complemented
14	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
15	C₃×3₋¹⁺²×C₃₇	—	3	9	no	—
16	C₃×3₋¹⁺²×C₃₇	C₃²	1	6	yes	—
17	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
18	C₃²×C₉×C₃₇	C₃²	1	2	yes	—
19	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
20	C₃³×C₃₇	—	3	3	no	—
21	C₃³×C₃₇	—	9	3	no	—
22	C₃³×C₃₇	C₃³	1	2	yes	—
23	C₃³×C₃₇	C₃³	1	1	yes	characteristic
24	C₃³×C₃₇	C₃³	1	1	yes	characteristic
25	3₋¹⁺²×C₃₇	—	3	18	no	—
26	3₋¹⁺²×C₃₇	—	3	9	no	—
27	C₃×C₉×C₃₇	—	3	6	no	—
28	C₃×C₉×C₃₇	3₋¹⁺²	1	9	yes	complemented
29	C₃×C₉×C₃₇	C₃³	1	6	yes	—
30	C₃×C₉×C₃₇	C₃³	1	3	yes	—
31	#11758608	—	9	9	no	—
32	C₃×#66524	C₃³	1	27	yes	—
33	#10624884	—	37	1	no	maximal, Sylow
34	C₃²×C₃₇	—	3	3	no	—
35	C₃²×C₃₇	—	3	2	no	—
36	C₃²×C₃₇	—	3	1	no	—
37	C₃²×C₃₇	—	3	3	no	—
38	C₃²×C₃₇	—	3	6	no	—
39	C₃²×C₃₇	—	9	9	no	—
40	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
41	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
42	C₉×C₃₇	—	3	6	no	—
43	C₉×C₃₇	C₃×3₋¹⁺²	1	9	yes	—
44	#66524	—	3	27	no	—
45	C₃²×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic
46	#6900	—	37	9	no	—
47	#6899	—	37	18	no	—
48	#6895	—	37	6	no	—
49	#6891	—	37	3	no	—
50	#6887	—	37	3	no	—
51	C₃×C₃₇	—	3	3	no	—
52	C₃×C₃₇	—	9	1	no	—
53	C₃×C₃₇	—	9	2	no	—
54	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
55	C₃×C₃₇	3₋¹⁺⁴	1	3	yes	—
56	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
57	C₃×3₋¹⁺²	—	3	9	no	—
58	C₃×3₋¹⁺²	—	37	9	no	—
59	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	6	yes	—
60	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	3	yes	—
61	C₃²×C₉	C₃×C₃₇⋊C₃	1	2	yes	—
62	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
63	C₉⋊C₉	—	37	54	no	—
64	C₉⋊C₉	—	37	9	no	—
65	C₉⋊C₉	—	37	9	no	—
66	C₃²⋊C₉	—	37	9	no	—
67	C₃²⋊C₉	—	37	18	no	—
68	C₉²	—	37	9	no	—
69	C₃₇	#10624884	1	1	yes	characteristic, complemented, Sylow
70	C₃³	—	3	3	no	—
71	C₃³	—	9	3	no	—
72	C₃³	C₃²×C₃₇⋊C₃	1	2	yes	—
73	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
74	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
75	3₋¹⁺²	—	3	18	no	—
76	3₋¹⁺²	—	3	9	no	—
77	3₋¹⁺²	—	333	9	no	—
78	C₃×C₉	—	3	6	no	—
79	C₃×C₉	—	37	27	no	—
80	C₃×C₉	#11758608	1	9	yes	complemented
81	C₃×C₉	C₃²×C₃₇⋊C₃	1	6	yes	—
82	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
83	C₃²	—	3	3	no	—
84	C₃²	—	3	2	no	—
85	C₃²	—	3	1	no	—
86	C₃²	—	3	3	no	—
87	C₃²	—	3	6	no	—
88	C₃²	—	9	9	no	—
89	C₃²	C₃×#11758608	1	3	yes	—
90	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
91	C₉	—	3	6	no	—
92	C₉	—	111	27	no	—
93	C₉	C₃×#11758608	1	9	yes	—
94	C₃	—	3	3	no	—
95	C₃	—	9	1	no	—
96	C₃	—	9	2	no	—
97	C₃	C₃²×#11758608	1	1	yes	characteristic
98	C₃	#22582679	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22582601

C₃

yes

maximal, complemented

#22582594

C₃

yes

maximal, complemented

#22582578

C₃

yes

maximal, complemented

#22582554

C₃

yes

maximal

#22582538

C₃

yes

maximal, complemented

C₃²×3₋¹⁺²×C₃₇

C₃

yes

maximal, characteristic

C₃×#11758608

C₃²

yes

—

#15688506

C₃²

yes

complemented

#15688505

C₃²

yes

complemented

#15688504

C₃²

yes

complemented

C₃²⋊#66524

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

C₉×#66524

C₃²

yes

complemented

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₇

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

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

C₃³×C₃₇

C₃³

yes

characteristic

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

3₋¹⁺²

yes

complemented

C₃×C₉×C₃₇

C₃³

yes

—

C₃×C₉×C₃₇

C₃³

yes

—

#11758608

—

C₃×#66524

C₃³

yes

—

#10624884

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₃²×C₃₇

C₃⁴

yes

characteristic

C₉×C₃₇

—

C₉×C₃₇

C₃×3₋¹⁺²

yes

—

#66524

—

C₃²×3₋¹⁺²

C₃₇⋊C₃

yes

—

—

—

—

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃²×3₋¹⁺²

yes

characteristic

C₃×C₃₇

3₋¹⁺⁴

yes

—

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₉²

—

C₃₇

#10624884

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

C₃²×C₃₇⋊C₃

yes

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

333

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

#11758608

yes

complemented

C₃×C₉

C₃²×C₃₇⋊C₃

yes

—

C₃×C₉

C₃²×C₃₇⋊C₃

yes

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃×#11758608

yes

—

C₃²

C₃³×C₃₇⋊C₃

yes

characteristic

C₉

—

C₉

—

111

—

C₉

C₃×#11758608

yes

—

C₃

—

C₃

—

C₃

—

C₃

C₃²×#11758608

yes

characteristic

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	C₃⁴×C₃₇	C₃²
hypercenter	C₃²×3₋¹⁺²	C₃₇⋊C₃
hypocenter	C₃₇	#10624884
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

C₃⁴×C₃₇

C₃²

hypercenter

C₃²×3₋¹⁺²

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃₇

C₁ ◁ C₃² ◁ C₃²×3₋¹⁺²

N	H ≅ G/N	type
1. #22582601	C₃	semidirect
2. #22582594	C₃	semidirect
3. #22582578	C₃	semidirect
5. #22582538	C₃	semidirect
8. #15688506	C₃²	semidirect
9. #15688505	C₃²	semidirect
10. #15688504	C₃²	semidirect
13. C₉×#66524	C₃²	semidirect
28. C₃×C₉×C₃₇	3₋¹⁺²	semidirect
69. C₃₇	#10624884	semidirect
80. C₃×C₉	#11758608	semidirect

H ≅ G/N

type

1. #22582601

C₃

semidirect

2. #22582594

C₃

semidirect

3. #22582578

C₃

semidirect

5. #22582538

C₃

semidirect

8. #15688506

C₃²

semidirect

9. #15688505

C₃²

semidirect

10. #15688504

C₃²

semidirect

13. C₉×#66524

C₃²

semidirect

28. C₃×C₉×C₃₇

3₋¹⁺²

semidirect

69. C₃₇

#10624884

semidirect

80. C₃×C₉

#11758608

semidirect

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

subgroup	index
#22582601	3
#22582594	3
#22582578	3
#22582554	3
#22582538	3
C₃²×3₋¹⁺²×C₃₇	3
#10624884	37

subgroup

index

C₃²×3₋¹⁺²×C₃₇

Subgroups — #41007667

Conjugacy classes of subgroups

Complements

Maximal subgroups