Group #41006338 — subgroups

Subgroups — #41006338

Order 26973.

198 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 106 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×#15688511	C₃	1	1	yes	maximal, characteristic
2	C₃×#15688511	C₃	1	1	yes	maximal, characteristic
3	#22582538	C₃	1	1	yes	maximal, characteristic
4	#22582495	C₃	1	9	yes	maximal, complemented
5	C₃×C₃₇×C₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic
6	#15688511	—	3	1	no	—
7	#15688511	—	3	1	no	—
8	#15688511	C₉	1	3	yes	complemented
9	#15688511	C₉	1	3	yes	complemented
10	#15688511	C₃²	1	3	yes	—
11	#15688511	C₃²	1	3	yes	—
12	C₃×#11758601	—	3	1	no	—
13	C₃×#11758601	C₉	1	3	yes	complemented
14	#15688506	—	3	1	no	—
15	#15688505	C₃²	1	3	yes	—
16	C₃²⋊#66524	—	3	1	no	—
17	C₉×#66524	—	3	1	no	—
18	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
19	C₃₇×C₂₇⋊C₃	—	3	1	no	—
20	C₃₇×C₂₇⋊C₃	—	3	1	no	—
21	C₃₇×C₂₇⋊C₃	C₃²	1	3	yes	—
22	C₃×C₂₇×C₃₇	—	3	1	no	—
23	C₃³×C₃₇	3₋¹⁺²	1	1	yes	characteristic
24	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
25	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
26	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
27	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
28	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
29	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
30	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
31	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
32	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
33	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
34	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
35	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
36	C₃×#66524	—	3	3	no	—
37	C₂₇×C₃₇	—	3	3	no	—
38	#11758601	—	3	3	no	—
39	#11758601	C₃×C₉	1	9	yes	—
40	#10624709	—	37	1	no	maximal, Sylow
41	C₃²×C₃₇	—	9	1	no	—
42	C₃²×C₃₇	C₉⋊C₉	1	1	yes	characteristic
43	C₃²×C₃₇	C₉⋊C₉	1	1	yes	characteristic
44	C₃²×C₃₇	C₉⋊C₉	1	1	yes	characteristic
45	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
46	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
47	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
48	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
49	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
50	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
51	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
52	C₉×C₃₇	C₃²×C₉	1	1	yes	characteristic
53	C₉×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
54	C₉×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
55	#66524	—	9	3	no	—
56	C₃×C₂₇⋊C₃	—	37	1	no	—
57	C₃×C₂₇⋊C₃	—	37	1	no	—
58	C₃×C₂₇⋊C₃	C₃₇⋊C₃	1	1	yes	characteristic
59	#6887	—	37	1	no	—
60	#6875	—	37	9	no	—
61	C₃×C₃₇	—	3	1	no	—
62	C₃×C₃₇	—	3	1	no	—
63	C₃×C₃₇	—	3	1	no	—
64	C₃×C₃₇	—	3	1	no	—
65	C₃×C₃₇	C₃×C₉⋊C₉	1	1	yes	characteristic
66	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
67	C₂₇⋊C₃	—	3	1	no	—
68	C₂₇⋊C₃	—	3	1	no	—
69	C₂₇⋊C₃	—	37	3	no	—
70	C₂₇⋊C₃	—	37	3	no	—
71	C₂₇⋊C₃	—	37	3	no	—
72	C₂₇⋊C₃	—	37	3	no	—
73	C₂₇⋊C₃	—	111	1	no	—
74	C₂₇⋊C₃	—	111	1	no	—
75	C₂₇⋊C₃	C₃×C₃₇⋊C₃	1	3	yes	—
76	C₃×C₂₇	—	3	1	no	—
77	C₃×C₂₇	—	37	3	no	—
78	C₃×C₂₇	—	111	1	no	—
79	C₉⋊C₉	—	37	3	no	—
80	C₉⋊C₉	—	111	1	no	—
81	C₃²⋊C₉	—	111	1	no	—
82	C₉²	—	111	1	no	—
83	C₃₇	#10624709	1	1	yes	characteristic, complemented, Sylow
84	C₃³	#11758605	1	1	yes	characteristic
85	C₃×C₉	—	111	3	no	—
86	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
87	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
88	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
89	C₃×C₉	#11758605	1	1	yes	characteristic
90	C₃×C₉	#11758605	1	1	yes	characteristic
91	C₃×C₉	#11758606	1	1	yes	characteristic
92	C₃×C₉	#11758606	1	1	yes	characteristic
93	C₃×C₉	#11758606	1	1	yes	characteristic
94	C₃×C₉	#11758608	1	1	yes	characteristic
95	C₃×C₉	#11758608	1	1	yes	characteristic
96	C₃×C₉	#11758608	1	1	yes	characteristic
97	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
98	C₂₇	—	3	3	no	—
99	C₂₇	—	37	9	no	—
100	C₂₇	—	111	3	no	—
101	C₃²	—	9	1	no	—
102	C₃²	#15688504	1	1	yes	characteristic
103	C₃²	#15688504	1	1	yes	characteristic
104	C₃²	#15688504	1	1	yes	characteristic
105	C₃²	C₃×#11758605	1	1	yes	characteristic
106	C₉	—	333	3	no	—
107	C₉	#15688504	1	1	yes	characteristic
108	C₉	#15688504	1	1	yes	characteristic
109	C₉	#15688504	1	1	yes	characteristic
110	C₉	#15688504	1	1	yes	characteristic
111	C₉	#15688504	1	1	yes	characteristic
112	C₉	#15688504	1	1	yes	characteristic
113	C₉	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
114	C₉	C₃×#11758606	1	1	yes	characteristic
115	C₉	C₃×#11758608	1	1	yes	characteristic
116	C₃	—	3	1	no	—
117	C₃	—	3	1	no	—
118	C₃	—	3	1	no	—
119	C₃	—	3	1	no	—
120	C₃	C₃×#15688504	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688511

C₃

yes

maximal, characteristic

C₃×#15688511

C₃

yes

maximal, characteristic

#22582538

C₃

yes

maximal, characteristic

#22582495

C₃

yes

maximal, complemented

C₃×C₃₇×C₂₇⋊C₃

C₃

yes

maximal, characteristic

#15688511

—

#15688511

—

#15688511

C₉

yes

complemented

#15688511

C₉

yes

complemented

#15688511

C₃²

yes

—

#15688511

C₃²

yes

—

C₃×#11758601

—

C₃×#11758601

C₉

yes

complemented

#15688506

—

#15688505

C₃²

yes

—

C₃²⋊#66524

—

C₉×#66524

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×C₂₇⋊C₃

—

C₃₇×C₂₇⋊C₃

—

C₃₇×C₂₇⋊C₃

C₃²

yes

—

C₃×C₂₇×C₃₇

—

C₃³×C₃₇

3₋¹⁺²

yes

characteristic

C₃×C₉×C₃₇

C₃×C₉

yes

characteristic

C₃×C₉×C₃₇

C₃×C₉

yes

characteristic

C₃×C₉×C₃₇

C₃×C₉

yes

characteristic

C₃×C₉×C₃₇

3₋¹⁺²

yes

characteristic

C₃×C₉×C₃₇

3₋¹⁺²

yes

characteristic

C₃×C₉×C₃₇

3₋¹⁺²

yes

characteristic

C₃×C₉×C₃₇

3₋¹⁺²

yes

characteristic

C₃×C₉×C₃₇

3₋¹⁺²

yes

characteristic

C₃×C₉×C₃₇

3₋¹⁺²

yes

characteristic

C₃×C₉×C₃₇

3₋¹⁺²

yes

characteristic

C₃×C₉×C₃₇

3₋¹⁺²

yes

characteristic

C₃×C₉×C₃₇

C₃³

yes

characteristic

C₃×#66524

—

C₂₇×C₃₇

—

#11758601

—

#11758601

C₃×C₉

yes

—

#10624709

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

C₉⋊C₉

yes

characteristic

C₃²×C₃₇

C₉⋊C₉

yes

characteristic

C₃²×C₃₇

C₉⋊C₉

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₉×C₃₇

C₉⋊C₉

yes

characteristic

C₉×C₃₇

C₉⋊C₉

yes

characteristic

C₉×C₃₇

C₉⋊C₉

yes

characteristic

C₉×C₃₇

C₉⋊C₉

yes

characteristic

C₉×C₃₇

C₉⋊C₉

yes

characteristic

C₉×C₃₇

C₉⋊C₉

yes

characteristic

C₉×C₃₇

C₃²×C₉

yes

characteristic

C₉×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₉×C₃₇

C₃×3₋¹⁺²

yes

characteristic

#66524

—

C₃×C₂₇⋊C₃

—

C₃×C₂₇⋊C₃

—

C₃×C₂₇⋊C₃

C₃₇⋊C₃

yes

characteristic

#6887

—

#6875

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃×C₉⋊C₉

yes

characteristic

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

C₃×C₃₇⋊C₃

yes

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

111

—

C₉⋊C₉

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₉²

—

111

—

C₃₇

#10624709

yes

characteristic, complemented, Sylow

C₃³

#11758605

yes

characteristic

C₃×C₉

—

111

—

C₃×C₉

C₉×C₃₇⋊C₃

yes

characteristic

C₃×C₉

C₉×C₃₇⋊C₃

yes

characteristic

C₃×C₉

C₉×C₃₇⋊C₃

yes

C₃×C₉

yes

C₃×C₉

yes

C₃×C₉

yes

C₃×C₉

yes

C₃×C₉

yes

C₃×C₉

yes

C₃×C₉

yes

C₃×C₉

yes

C₃×C₉

C₃²×C₃₇⋊C₃

yes

characteristic

C₂₇

—

C₂₇

—

100

C₂₇

—

111

—

101

C₃²

—

102

C₃²

#15688504

yes

characteristic

103

C₃²

#15688504

yes

characteristic

104

C₃²

#15688504

yes

characteristic

105

C₃²

C₃×#11758605

yes

characteristic

106

C₉

—

333

—

107

C₉

#15688504

yes

characteristic

108

C₉

#15688504

yes

characteristic

109

C₉

#15688504

yes

characteristic

110

C₉

#15688504

yes

characteristic

111

C₉

#15688504

yes

characteristic

112

C₉

#15688504

yes

characteristic

113

C₉

C₃×C₉×C₃₇⋊C₃

yes

characteristic

114

C₉

C₃×#11758606

yes

characteristic

115

C₉

C₃×#11758608

yes

characteristic

116

C₃

—

117

C₃

—

118

C₃

—

119

C₃

—

120

C₃

C₃×#15688504

yes

characteristic

	subgroup	G/H
center	C₉	#15688504
commutator	C₉×C₃₇	C₃²×C₉
Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
Fitting	C₃×C₃₇×C₂₇⋊C₃	C₃
socle	C₃×C₃₇	C₃×C₉⋊C₉
prime radical	C₃³×C₃₇	3₋¹⁺²
cosocle	C₃×C₉×C₃₇	C₃³
hypercenter	C₃×C₂₇⋊C₃	C₃₇⋊C₃
hypocenter	C₃₇	#10624709
LCS	G ▷ C₉×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₉ ◁ C₃²×C₉ ◁ C₃×C₂₇⋊C₃

subgroup

G/H

center

C₉

#15688504

commutator

C₉×C₃₇

C₃²×C₉

Frattini

C₃×C₉

C₃²×C₃₇⋊C₃

Fitting

C₃×C₃₇×C₂₇⋊C₃

C₃

socle

C₃×C₃₇

C₃×C₉⋊C₉

prime radical

C₃³×C₃₇

3₋¹⁺²

cosocle

C₃×C₉×C₃₇

C₃³

hypercenter

C₃×C₂₇⋊C₃

C₃₇⋊C₃

hypocenter

C₃₇

#10624709

LCS

G ▷ C₉×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₉ ◁ C₃²×C₉ ◁ C₃×C₂₇⋊C₃

N	H ≅ G/N	type
4. #22582495	C₃	semidirect
8. #15688511	C₉	semidirect
9. #15688511	C₉	semidirect
13. C₃×#11758601	C₉	semidirect
83. C₃₇	#10624709	semidirect

H ≅ G/N

type

4. #22582495

C₃

semidirect

8. #15688511

C₉

semidirect

9. #15688511

C₉

semidirect

13. C₃×#11758601

C₉

semidirect

83. C₃₇

#10624709

semidirect

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

subgroup	index
C₃×#15688511	3
#22582538	3
#22582495	3
C₃×C₃₇×C₂₇⋊C₃	3
#10624709	37

subgroup

index

C₃×C₃₇×C₂₇⋊C₃

Subgroups — #41006338

Conjugacy classes of subgroups

Complements

Maximal subgroups