Group #41006388 — subgroups

Subgroups — #41006388

Order 26973.

274 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 82 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×#15688521	C₃	1	1	yes	maximal, characteristic
2	#22582548	C₃	1	3	yes	maximal
3	C₃×C₉×#66524	C₃	1	1	yes	maximal, characteristic
4	#22582481	C₃	1	3	yes	maximal
5	#22582478	C₃	1	3	yes	maximal
6	#22582472	C₃	1	1	yes	maximal, characteristic
7	C₃₇×#6868	C₃	1	1	yes	maximal, characteristic
8	C₃×#11758608	C₃²	1	3	yes	—
9	C₃²×#66524	C₃²	1	1	yes	characteristic
10	C₃²×#66524	C₃²	1	1	yes	characteristic
11	C₃²×#66524	C₃²	1	1	yes	characteristic
12	#15688521	C₉	1	9	yes	complemented
13	#15688504	—	3	6	no	—
14	C₃²⋊#66524	C₃²	1	3	yes	—
15	C₉×#66524	—	3	3	no	—
16	C₉×#66524	—	3	2	no	—
17	C₉×#66524	—	3	1	no	—
18	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
19	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
20	C₃³×C₃₇	C₃³	1	1	yes	characteristic
21	C₃×C₉×C₃₇	—	3	1	no	—
22	C₃×C₉×C₃₇	—	3	2	no	—
23	C₃×C₉×C₃₇	—	3	3	no	—
24	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
25	#11758608	—	3	6	no	—
26	#11758608	C₃×C₉	1	9	yes	—
27	C₃×#66524	—	3	1	no	—
28	C₃×#66524	—	3	1	no	—
29	C₃×#66524	—	3	3	no	—
30	C₃×#66524	—	3	3	no	—
31	C₃×#66524	—	3	3	no	—
32	C₃×#66524	—	3	3	no	—
33	C₃×#66524	—	3	3	no	—
34	C₃×#66524	H(3,𝔽₃)	1	2	yes	—
35	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
36	C₉×#733	—	37	1	no	maximal, Sylow
37	C₃²×C₃₇	—	3	1	no	—
38	C₃²×C₃₇	—	3	1	no	—
39	C₃²×C₃₇	—	3	1	no	—
40	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
41	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
42	C₃²×C₃₇	#737	1	2	yes	—
43	C₉×C₃₇	—	3	3	no	—
44	C₉×C₃₇	—	9	3	no	—
45	#66524	—	3	9	no	—
46	#66524	—	3	2	no	—
47	#66524	—	3	3	no	—
48	#66524	—	3	3	no	—
49	#66524	—	9	3	no	—
50	#66524	#733	1	3	yes	complemented
51	C₃×#733	—	37	1	no	—
52	C₉×3₋¹⁺²	—	37	3	no	—
53	C₃×C₉²	—	37	1	no	—
54	3₋¹⁺²⋊C₉	—	37	3	no	—
55	3₋¹⁺²⋊C₉	—	37	3	no	—
56	#6868	—	37	1	no	—
57	#6868	C₃₇⋊C₃	1	1	yes	characteristic
58	C₃×C₃₇	—	3	1	no	—
59	C₃×C₃₇	—	3	2	no	—
60	C₃×C₃₇	C₉×H(3,𝔽₃)	1	1	yes	characteristic
61	C₃×C₃₇	C₃×#733	1	1	yes	characteristic
62	C₃×C₃₇	#6908	1	1	yes	characteristic
63	C₃×C₃₇	#6908	1	1	yes	characteristic
64	C₃×3₋¹⁺²	—	37	3	no	—
65	C₃²×C₉	—	37	1	no	—
66	C₃²×C₉	—	37	1	no	—
67	C₃²×C₉	—	37	1	no	—
68	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
69	#733	—	37	9	no	—
70	C₉⋊C₉	—	111	6	no	—
71	C₃²⋊C₉	—	37	3	no	—
72	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
73	C₉²	—	111	1	no	—
74	C₉²	—	111	2	no	—
75	C₉²	—	111	3	no	—
76	C₃₇	C₉×#733	1	1	yes	characteristic, complemented, Sylow
77	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
78	3₋¹⁺²	—	37	9	no	—
79	3₋¹⁺²	—	111	6	no	—
80	C₃×C₉	—	3	1	no	—
81	C₃×C₉	—	3	2	no	—
82	C₃×C₉	—	3	3	no	—
83	C₃×C₉	—	37	1	no	—
84	C₃×C₉	—	37	2	no	—
85	C₃×C₉	—	111	3	no	—
86	C₃×C₉	—	111	3	no	—
87	C₃×C₉	—	111	1	no	—
88	C₃×C₉	—	111	1	no	—
89	C₃×C₉	—	111	3	no	—
90	C₃×C₉	—	111	3	no	—
91	C₃×C₉	—	111	3	no	—
92	C₃×C₉	C₉×C₃₇⋊C₃	1	3	yes	—
93	C₃²	—	3	1	no	—
94	C₃²	—	3	1	no	—
95	C₃²	—	3	1	no	—
96	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
97	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
98	C₃²	#15688532	1	2	yes	—
99	C₉	—	3	3	no	—
100	C₉	—	9	3	no	—
101	C₉	—	37	3	no	—
102	C₉	—	333	3	no	—
103	C₉	—	111	9	no	—
104	C₉	—	111	2	no	—
105	C₉	—	111	3	no	—
106	C₉	—	111	3	no	—
107	C₃	—	3	1	no	—
108	C₃	—	3	2	no	—
109	C₃	#22582539	1	1	yes	characteristic
110	C₃	#733×C₃₇⋊C₃	1	1	yes	characteristic
111	C₃	#22582631	1	1	yes	characteristic
112	C₃	#22582634	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688521

C₃

yes

maximal, characteristic

#22582548

C₃

yes

maximal

C₃×C₉×#66524

C₃

yes

maximal, characteristic

#22582481

C₃

yes

maximal

#22582478

C₃

yes

maximal

#22582472

C₃

yes

maximal, characteristic

C₃₇×#6868

C₃

yes

maximal, characteristic

C₃×#11758608

C₃²

yes

—

C₃²×#66524

C₃²

yes

characteristic

C₃²×#66524

C₃²

yes

characteristic

C₃²×#66524

C₃²

yes

characteristic

#15688521

C₉

yes

complemented

#15688504

—

C₃²⋊#66524

C₃²

yes

—

C₉×#66524

—

C₉×#66524

—

C₉×#66524

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃×C₉

yes

—

#11758608

—

#11758608

C₃×C₉

yes

—

—

—

—

—

—

—

—

yes

—

yes

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃²×C₉

yes

characteristic

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

#737

yes

—

C₉×C₃₇

—

C₉×C₃₇

—

—

—

—

—

—

yes

—

C₉×3₋¹⁺²

—

C₃×C₉²

—

3₋¹⁺²⋊C₉

—

3₋¹⁺²⋊C₉

—

#6868

—

#6868

C₃₇⋊C₃

yes

characteristic

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₉×H(3,𝔽₃)

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×3₋¹⁺²

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

#733

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₉²

—

111

—

C₉²

—

111

—

C₉²

—

111

—

C₃₇

C₉×#733

yes

characteristic, complemented, Sylow

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

111

—

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

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃×C₉×C₃₇⋊C₃

yes

characteristic

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

C₃²

#15688532

yes

—

C₉

—

100

C₉

—

101

C₉

—

102

C₉

—

333

—

103

C₉

—

111

—

104

C₉

—

111

—

105

C₉

—

111

—

106

C₉

—

111

—

107

C₃

—

108

C₃

—

109

C₃

#22582539

yes

characteristic

110

C₃

#733×C₃₇⋊C₃

yes

characteristic

111

C₃

#22582631

yes

characteristic

112

C₃

#22582634

yes

characteristic

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator	C₃²×C₃₇	C₃²×C₉
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6868	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical, cosocle	C₃³×C₃₇	C₃³
hypercenter	#6868	C₃₇⋊C₃
hypocenter	C₃₇	C₉×#733
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ #6868

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

commutator

C₃²×C₃₇

C₃²×C₉

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6868

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical, cosocle

C₃³×C₃₇

C₃³

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ #6868

N	H ≅ G/N	type
12. #15688521	C₉	semidirect
50. #66524	#733	semidirect
76. C₃₇	C₉×#733	semidirect

H ≅ G/N

type

12. #15688521

C₉

semidirect

50. #66524

#733

semidirect

76. C₃₇

C₉×#733

semidirect

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

subgroup	index
C₃×#15688521	3
#22582548	3
C₃×C₉×#66524	3
#22582481	3
#22582478	3
#22582472	3
C₃₇×#6868	3
C₉×#733	37

subgroup

index

C₃×C₉×#66524

C₃₇×#6868

Subgroups — #41006388

Conjugacy classes of subgroups

Complements

Maximal subgroups