Group #41005108 — subgroups

Subgroups — #41005108

Order 26973.

374 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 166 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×#15688506	C₃	1	1	yes	maximal, characteristic
2	C₃×#15688505	C₃	1	1	yes	maximal, characteristic
3	C₃×#15688504	C₃	1	3	yes	maximal
4	C₃×C₉×#66524	C₃	1	1	yes	maximal, characteristic
5	#22582441	C₃	1	6	yes	maximal
6	C₃×C₃₇×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic
7	C₃²×#66524	C₃²	1	3	yes	—
8	C₃²×#66524	C₃²	1	3	yes	—
9	C₃²×#66524	C₃²	1	2	yes	—
10	C₃²×#66524	C₃²	1	1	yes	characteristic
11	#15688506	—	3	3	no	—
12	#15688505	—	3	3	no	—
13	#15688504	—	3	9	no	—
14	C₉×#66524	C₉	1	9	yes	complemented
15	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
16	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
17	C₃₇×C₉⋊C₉	—	3	3	no	—
18	C₃³×C₃₇	C₃³	1	1	yes	characteristic
19	C₃×C₉×C₃₇	—	3	3	no	—
20	C₃×C₉×C₃₇	—	3	3	no	—
21	C₃×C₉×C₃₇	—	3	6	no	—
22	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
23	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
24	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
25	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
26	C₃×#66524	—	3	6	no	—
27	C₃×#66524	—	3	3	no	—
28	C₃×#66524	—	3	3	no	—
29	C₃×#66524	—	3	3	no	—
30	C₃×#66524	—	3	3	no	—
31	C₃×#66524	—	3	6	no	—
32	C₃×#66524	—	3	6	no	—
33	C₃×#66524	C₃×C₉	1	6	yes	—
34	C₃×#66524	C₃×C₉	1	3	yes	—
35	C₃×#66524	3₋¹⁺²	1	9	yes	—
36	#10624550	—	37	1	no	maximal, Sylow
37	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
38	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
39	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
40	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
41	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
42	C₃²×C₃₇	#737	1	6	yes	—
43	C₉×C₃₇	—	9	3	no	—
44	C₉×C₃₇	C₉⋊C₉	1	9	yes	complemented
45	#66524	—	3	6	no	—
46	#66524	—	9	3	no	—
47	#66524	—	9	3	no	—
48	#66524	C₉⋊C₉	1	9	yes	complemented
49	C₃×C₉⋊C₉	—	37	3	no	—
50	C₃×C₉⋊C₉	—	37	1	no	—
51	C₃×C₉⋊C₉	—	37	1	no	—
52	C₃×C₉⋊C₉	C₃₇⋊C₃	1	1	yes	characteristic
53	C₃×C₉²	—	37	1	no	—
54	#6855	—	37	6	no	—
55	C₃×C₃₇	C₃×C₉⋊C₉	1	1	yes	characteristic
56	C₃×C₃₇	C₃×C₉⋊C₉	1	1	yes	characteristic
57	C₃×C₃₇	#6887	1	2	yes	—
58	C₃×C₃₇	#6896	1	3	yes	—
59	C₃×C₃₇	#6896	1	3	yes	—
60	C₃×C₃₇	#6896	1	3	yes	—
61	C₃²×C₉	—	37	1	no	—
62	C₃²×C₉	—	37	2	no	—
63	C₃²×C₉	—	37	3	no	—
64	C₃²×C₉	—	37	3	no	—
65	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	—
66	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
67	C₉⋊C₉	—	3	3	no	—
68	C₉⋊C₉	—	111	9	no	—
69	C₉⋊C₉	—	111	3	no	—
70	C₉⋊C₉	—	111	3	no	—
71	C₉²	—	37	9	no	—
72	C₃₇	#10624550	1	1	yes	characteristic, complemented, Sylow
73	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
74	C₃×C₉	—	3	3	no	—
75	C₃×C₉	—	3	3	no	—
76	C₃×C₉	—	3	6	no	—
77	C₃×C₉	—	37	3	no	—
78	C₃×C₉	—	37	9	no	—
79	C₃×C₉	—	37	6	no	—
80	C₃×C₉	—	111	3	no	—
81	C₃×C₉	—	111	3	no	—
82	C₃×C₉	—	111	3	no	—
83	C₃×C₉	—	111	3	no	—
84	C₃×C₉	—	111	6	no	—
85	C₃×C₉	—	111	6	no	—
86	C₃×C₉	—	111	6	no	—
87	C₃×C₉	C₉×C₃₇⋊C₃	1	3	yes	—
88	C₃×C₉	#11758605	1	3	yes	—
89	C₃×C₉	#11758606	1	3	yes	—
90	C₃×C₉	#11758608	1	3	yes	—
91	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
92	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	3	yes	—
93	C₃²	C₃×#11758606	1	1	yes	characteristic
94	C₃²	C₃×#11758605	1	1	yes	characteristic
95	C₃²	C₃×#11758608	1	1	yes	characteristic
96	C₃²	#15688532	1	6	yes	—
97	C₉	—	9	3	no	—
98	C₉	—	37	9	no	—
99	C₉	—	333	3	no	—
100	C₉	—	333	3	no	—
101	C₉	—	111	6	no	—
102	C₉	#15688504	1	9	yes	complemented
103	C₃	C₉⋊C₉×C₃₇⋊C₃	1	1	yes	characteristic
104	C₃	C₃×#15688504	1	1	yes	characteristic
105	C₃	#22582537	1	2	yes	—
106	C₃	#22582580	1	3	yes	—
107	C₃	#22582583	1	3	yes	—
108	C₃	#22582584	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688506

C₃

yes

maximal, characteristic

C₃×#15688505

C₃

yes

maximal, characteristic

C₃×#15688504

C₃

yes

maximal

C₃×C₉×#66524

C₃

yes

maximal, characteristic

#22582441

C₃

yes

maximal

C₃×C₃₇×C₉⋊C₉

C₃

yes

maximal, characteristic

C₃²×#66524

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

—

—

C₉

yes

complemented

C₃²×C₉×C₃₇

C₃²

yes

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×C₉⋊C₉

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃×C₉

yes

—

C₃×C₉×C₃₇

3₋¹⁺²

yes

—

C₃×C₉×C₃₇

3₋¹⁺²

yes

—

C₃×C₉×C₃₇

3₋¹⁺²

yes

—

—

—

—

—

—

—

—

C₃×C₉

yes

—

C₃×#66524

C₃×C₉

yes

—

C₃×#66524

3₋¹⁺²

yes

—

#10624550

—

maximal, Sylow

C₃²×C₃₇

C₃²×C₉

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₃²×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₃₇

#737

yes

—

C₉×C₃₇

—

C₉×C₃₇

C₉⋊C₉

yes

—

—

—

C₉⋊C₉

yes

complemented

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

C₃₇⋊C₃

yes

characteristic

C₃×C₉²

—

#6855

—

C₃×C₃₇

C₃×C₉⋊C₉

yes

characteristic

C₃×C₃₇

C₃×C₉⋊C₉

yes

characteristic

C₃×C₃₇

#6887

yes

—

C₃×C₃₇

#6896

yes

—

C₃×C₃₇

#6896

yes

—

C₃×C₃₇

#6896

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₉²

—

C₃₇

#10624550

yes

characteristic, complemented, Sylow

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

C₉×C₃₇⋊C₃

yes

—

C₃×C₉

#11758605

yes

—

C₃×C₉

#11758606

yes

—

C₃×C₉

#11758608

yes

—

C₃²

C₃×C₉×C₃₇⋊C₃

yes

characteristic

C₃²

3₋¹⁺²×C₃₇⋊C₃

yes

—

C₃²

yes

C₃²

yes

C₃²

yes

C₃²

yes

—

C₉

—

C₉

—

C₉

—

333

—

100

C₉

—

333

—

101

C₉

—

111

—

102

C₉

#15688504

yes

complemented

103

C₃

C₉⋊C₉×C₃₇⋊C₃

yes

characteristic

104

C₃

C₃×#15688504

yes

characteristic

105

C₃

#22582537

yes

—

106

C₃

#22582580

yes

—

107

C₃

#22582583

yes

—

108

C₃

#22582584

yes

—

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₃₇⋊C₃
commutator	C₃²×C₃₇	C₃²×C₉
Fitting	C₃×C₃₇×C₉⋊C₉	C₃
socle, prime radical, cosocle	C₃³×C₃₇	C₃³
hypercenter	C₃×C₉⋊C₉	C₃₇⋊C₃
hypocenter	C₃₇	#10624550
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ C₃×C₉⋊C₉

subgroup

G/H

center, Frattini

C₃³

C₃²×C₃₇⋊C₃

commutator

C₃²×C₃₇

C₃²×C₉

Fitting

C₃×C₃₇×C₉⋊C₉

C₃

socle, prime radical, cosocle

C₃³×C₃₇

C₃³

hypercenter

C₃×C₉⋊C₉

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃₇

C₁ ◁ C₃³ ◁ C₃×C₉⋊C₉

N	H ≅ G/N	type
14. C₉×#66524	C₉	semidirect
44. C₉×C₃₇	C₉⋊C₉	semidirect
48. #66524	C₉⋊C₉	semidirect
72. C₃₇	#10624550	semidirect
102. C₉	#15688504	semidirect

H ≅ G/N

type

14. C₉×#66524

C₉

semidirect

44. C₉×C₃₇

C₉⋊C₉

semidirect

48. #66524

C₉⋊C₉

semidirect

72. C₃₇

#10624550

semidirect

102. C₉

#15688504

semidirect

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

subgroup	index
C₃×#15688506	3
C₃×#15688505	3
C₃×#15688504	3
C₃×C₉×#66524	3
#22582441	3
C₃×C₃₇×C₉⋊C₉	3
#10624550	37

subgroup

index

C₃×C₉×#66524

C₃×C₃₇×C₉⋊C₉

Subgroups — #41005108

Conjugacy classes of subgroups

Complements

Maximal subgroups