Group #41007677 — subgroups

Subgroups — #41007677

Order 26973.

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

normal subgroups only

This group has 728 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	C₃²×#11758608	C₃	1	1	yes	maximal, characteristic
2	#22582576	C₃	1	9	yes	maximal, complemented
3	#22582574	C₃	1	18	yes	maximal, complemented
4	#22582571	C₃	1	9	yes	maximal, complemented
5	C₃×C₃²⋊#66524	C₃	1	2	yes	maximal
6	C₃×C₃₇×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
7	C₃×#11758608	—	3	9	no	—
8	C₃×#11758608	C₃²	1	3	yes	—
9	C₃×#11758608	C₃²	1	6	yes	—
10	C₃²×#66524	C₃²	1	3	yes	—
11	C₃²×#66524	C₃²	1	6	yes	—
12	#15688505	C₃²	1	27	yes	complemented
13	#15688504	C₃²	1	54	yes	complemented
14	C₃²⋊#66524	C₃²	1	6	yes	—
15	C₃²⋊#66524	C₃²	1	6	yes	—
16	C₃²⋊#66524	C₃²	1	6	yes	—
17	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
18	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
19	C₃₇×C₃²⋊C₉	C₃²	1	6	yes	—
20	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
21	C₃³×C₃₇	—	9	2	no	—
22	C₃³×C₃₇	—	9	1	no	—
23	C₃³×C₃₇	3₋¹⁺²	1	9	yes	—
24	C₃³×C₃₇	C₃³	1	1	yes	characteristic
25	C₃³×C₃₇	C₃³	1	2	yes	—
26	C₃³×C₃₇	C₃³	1	1	yes	characteristic
27	C₃×C₉×C₃₇	—	9	3	no	—
28	C₃×C₉×C₃₇	C₃³	1	9	yes	—
29	#11758608	—	3	9	no	—
30	#11758608	—	3	18	no	—
31	C₃×#66524	—	3	9	no	—
32	C₃×#66524	—	9	6	no	—
33	C₃×#66524	C₃³	1	18	yes	—
34	C₃×#66524	C₃³	1	9	yes	—
35	#10624886	—	37	1	no	maximal, Sylow
36	C₃²×C₃₇	—	3	2	no	—
37	C₃²×C₃₇	—	3	1	no	—
38	C₃²×C₃₇	—	3	2	no	—
39	C₃²×C₃₇	—	3	2	no	—
40	C₃²×C₃₇	—	3	2	no	—
41	C₃²×C₃₇	—	3	1	no	—
42	C₃²×C₃₇	—	3	2	no	—
43	C₃²×C₃₇	—	9	9	no	—
44	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
45	C₃²×C₃₇	C₃×3₋¹⁺²	1	6	yes	—
46	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
47	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
48	C₉×C₃₇	—	3	9	no	—
49	#66524	—	3	9	no	—
50	#66524	—	3	18	no	—
51	C₃²×3₋¹⁺²	—	37	1	no	—
52	#6895	—	37	9	no	—
53	#6894	—	37	9	no	—
54	#6894	—	37	18	no	—
55	C₃×C₃²⋊C₉	—	37	2	no	—
56	C₃×C₃²⋊C₉	C₃₇⋊C₃	1	1	yes	characteristic
57	C₃×C₃₇	—	3	6	no	—
58	C₃×C₃₇	—	3	3	no	—
59	C₃×C₃₇	—	3	3	no	—
60	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
61	C₃×C₃₇	3₋¹⁺⁴	1	2	yes	—
62	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
63	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
64	C₃×3₋¹⁺²	—	37	3	no	—
65	C₃×3₋¹⁺²	—	37	6	no	—
66	C₃×3₋¹⁺²	—	111	9	no	—
67	C₃²×C₉	—	37	6	no	—
68	C₃²×C₉	—	37	3	no	—
69	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	—
70	C₉⋊C₉	—	37	54	no	—
71	C₉⋊C₉	—	37	27	no	—
72	C₃²⋊C₉	—	37	6	no	—
73	C₃²⋊C₉	—	37	6	no	—
74	C₃²⋊C₉	—	37	6	no	—
75	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	6	yes	—
76	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
77	C₃₇	#10624886	1	1	yes	characteristic, complemented, Sylow
78	C₃³	—	9	2	no	—
79	C₃³	—	9	1	no	—
80	C₃³	#11758605	1	9	yes	—
81	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
82	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
83	C₃³	C₃²×C₃₇⋊C₃	1	2	yes	—
84	3₋¹⁺²	—	111	9	no	—
85	3₋¹⁺²	—	111	18	no	—
86	C₃×C₉	—	9	3	no	—
87	C₃×C₉	—	37	18	no	—
88	C₃×C₉	—	37	9	no	—
89	C₃×C₉	—	333	6	no	—
90	C₃×C₉	—	111	9	no	—
91	C₃×C₉	C₃²×C₃₇⋊C₃	1	9	yes	—
92	C₃²	—	3	2	no	—
93	C₃²	—	3	2	no	—
94	C₃²	—	3	2	no	—
95	C₃²	—	3	2	no	—
96	C₃²	—	3	1	no	—
97	C₃²	—	3	1	no	—
98	C₃²	—	3	2	no	—
99	C₃²	—	9	9	no	—
100	C₃²	C₃×#11758605	1	3	yes	—
101	C₃²	C₃×#11758605	1	3	yes	—
102	C₃²	C₃×#11758605	1	6	yes	—
103	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
104	C₉	—	3	9	no	—
105	C₉	—	111	9	no	—
106	C₉	—	111	18	no	—
107	C₃	—	3	6	no	—
108	C₃	—	3	3	no	—
109	C₃	—	3	3	no	—
110	C₃	C₃²×#11758605	1	1	yes	characteristic
111	C₃	#22582679	1	2	yes	—
112	C₃	#22582680	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758608

C₃

yes

maximal, characteristic

#22582576

C₃

yes

maximal, complemented

#22582574

C₃

yes

maximal, complemented

#22582571

C₃

yes

maximal, complemented

C₃×C₃²⋊#66524

C₃

yes

maximal

C₃×C₃₇×C₃²⋊C₉

C₃

yes

maximal, characteristic

C₃×#11758608

—

C₃×#11758608

C₃²

yes

—

C₃×#11758608

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

#15688505

C₃²

yes

complemented

#15688504

C₃²

yes

complemented

C₃²⋊#66524

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃²×C₉×C₃₇

C₃²

yes

—

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

3₋¹⁺²

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃³×C₃₇

C₃³

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃³

yes

—

—

—

—

—

C₃³

yes

—

C₃×#66524

C₃³

yes

—

#10624886

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₃²×C₃₇

C₃⁴

yes

characteristic

C₉×C₃₇

—

#66524

—

#66524

—

C₃²×3₋¹⁺²

—

#6895

—

#6894

—

#6894

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

C₃₇⋊C₃

yes

characteristic

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃²×3₋¹⁺²

yes

characteristic

C₃×C₃₇

3₋¹⁺⁴

yes

—

C₃×C₃₇

3₋¹⁺⁴

yes

characteristic

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₃₇

#10624886

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

#11758605

yes

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

C₃³

C₃²×C₃₇⋊C₃

yes

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

333

—

C₃×C₉

—

111

—

C₃×C₉

C₃²×C₃₇⋊C₃

yes

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

C₃×#11758605

yes

—

101

C₃²

C₃×#11758605

yes

—

102

C₃²

C₃×#11758605

yes

—

103

C₃²

C₃³×C₃₇⋊C₃

yes

characteristic

104

C₉

—

105

C₉

—

111

—

106

C₉

—

111

—

107

C₃

—

108

C₃

—

109

C₃

—

110

C₃

C₃²×#11758605

yes

characteristic

111

C₃

#22582679

yes

—

112

C₃

#22582680

yes

characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting	C₃×C₃₇×C₃²⋊C₉	C₃
prime radical	C₃⁴×C₃₇	C₃²
hypercenter	C₃×C₃²⋊C₉	C₃₇⋊C₃
hypocenter	C₃₇	#10624886
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃×C₃²⋊C₉

subgroup

G/H

center, Frattini

C₃²

C₃³×C₃₇⋊C₃

commutator, socle, cosocle

C₃²×C₃₇

C₃⁴

Fitting

C₃×C₃₇×C₃²⋊C₉

C₃

prime radical

C₃⁴×C₃₇

C₃²

hypercenter

C₃×C₃²⋊C₉

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃₇

C₁ ◁ C₃² ◁ C₃×C₃²⋊C₉

N	H ≅ G/N	type
2. #22582576	C₃	semidirect
3. #22582574	C₃	semidirect
4. #22582571	C₃	semidirect
12. #15688505	C₃²	semidirect
13. #15688504	C₃²	semidirect
77. C₃₇	#10624886	semidirect

H ≅ G/N

type

2. #22582576

C₃

semidirect

3. #22582574

C₃

semidirect

4. #22582571

C₃

semidirect

12. #15688505

C₃²

semidirect

13. #15688504

C₃²

semidirect

77. C₃₇

#10624886

semidirect

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

subgroup	index
C₃²×#11758608	3
#22582576	3
#22582574	3
#22582571	3
C₃×C₃²⋊#66524	3
C₃×C₃₇×C₃²⋊C₉	3
#10624886	37

subgroup

index

C₃×C₃²⋊#66524

C₃×C₃₇×C₃²⋊C₉

#10624886

Subgroups — #41007677

Conjugacy classes of subgroups

Complements

Maximal subgroups