Group #41006460 — subgroups

Subgroups — #41006460

Order 26973.

196 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 85 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×C₂₇×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
2	C₉×#11758606	C₃	1	1	yes	maximal, characteristic
3	#22582490	C₃	1	2	yes	maximal, complemented
4	#22582488	C₃	1	2	yes	maximal, complemented
5	#22582488	C₃	1	2	yes	maximal, complemented
6	#22582465	C₃	1	2	yes	maximal
7	#22582463	C₃	1	1	yes	maximal, characteristic, complemented
8	#22582462	C₃	1	1	yes	maximal, characteristic, complemented
9	C₃₇×#6864	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃×#11758606	C₉	1	1	yes	characteristic
11	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
12	C₃×#11758601	C₃²	1	2	yes	—
13	C₃×#11758601	C₃²	1	2	yes	—
14	C₃×#11758601	C₃²	1	2	yes	—
15	C₂₇×C₃₇⋊C₃	—	3	3	no	—
16	#15688506	C₉	1	2	yes	—
17	#15688504	C₉	1	2	yes	—
18	#15688504	C₉	1	2	yes	—
19	#15688502	C₉	1	2	yes	—
20	C₉×#66524	C₃²	1	1	yes	characteristic
21	C₉×#66524	C₃²	1	1	yes	characteristic
22	C₃×C₂₇×C₃₇	C₃²	1	2	yes	—
23	C₃×C₂₇×C₃₇	C₃²	1	1	yes	characteristic
24	C₉²×C₃₇	C₃²	1	1	yes	characteristic
25	C₃²×C₃₇⋊C₃	C₃×C₉	1	1	yes	characteristic
26	C₃×C₉×C₃₇	C₃×C₉	1	2	yes	—
27	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
28	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
29	C₃×#66524	C₃×C₉	1	2	yes	—
30	C₃×#66524	C₃×C₉	1	2	yes	—
31	C₃×#66524	C₃×C₉	1	1	yes	characteristic
32	C₃×#66524	C₃×C₉	1	1	yes	characteristic
33	C₃×#66524	C₃×C₉	1	2	yes	—
34	#11758606	—	3	3	no	—
35	C₉×C₃₇⋊C₃	—	3	1	no	—
36	C₉×C₃₇⋊C₃	—	3	2	no	—
37	C₂₇×C₃₇	—	3	2	no	—
38	C₂₇×C₃₇	3₋¹⁺²	1	3	yes	complemented
39	#11758601	—	3	2	no	—
40	#11758601	—	3	2	no	—
41	#11758601	—	3	2	no	—
42	#10624727	—	37	1	no	maximal, Sylow
43	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
44	C₃×C₃₇⋊C₃	—	3	1	no	—
45	C₃×C₃₇⋊C₃	—	3	2	no	—
46	C₃×C₃₇⋊C₃	C₂₇⋊C₃	1	3	yes	—
47	C₉×C₃₇	—	3	1	no	—
48	C₉×C₃₇	—	3	2	no	—
49	C₉×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
50	C₉×C₃₇	#737	1	2	yes	—
51	#66524	—	3	2	no	—
52	#66524	—	3	1	no	—
53	#66524	—	3	1	no	—
54	#66524	—	3	2	no	—
55	#66524	—	3	2	no	—
56	C₃²×C₂₇	—	37	1	no	—
57	C₉×3₋¹⁺²	—	37	1	no	—
58	C₉⋊C₂₇	—	37	2	no	—
59	C₉⋊C₂₇	—	37	2	no	—
60	C₉⋊C₂₇	—	37	2	no	—
61	C₃²⋊C₂₇	—	37	2	no	—
62	#6864	—	37	1	no	—
63	#6864	—	37	1	no	—
64	#6864	C₃₇⋊C₃	1	1	yes	characteristic, complemented
65	C₃×C₃₇	C₉×3₋¹⁺²	1	1	yes	characteristic
66	C₃×C₃₇	C₃×C₂₇⋊C₃	1	1	yes	characteristic
67	C₃×C₃₇	#6903	1	2	yes	—
68	C₃₇⋊C₃	—	3	3	no	—
69	C₃×3₋¹⁺²	—	37	1	no	—
70	C₃²×C₉	—	37	1	no	—
71	C₃×C₂₇	—	37	2	no	—
72	C₃×C₂₇	—	37	2	no	—
73	C₃×C₂₇	—	37	2	no	—
74	C₃×C₂₇	—	111	3	no	—
75	C₃×C₂₇	C₃×C₃₇⋊C₃	1	2	yes	—
76	C₃×C₂₇	C₃×C₃₇⋊C₃	1	1	yes	characteristic
77	C₉⋊C₉	—	37	2	no	—
78	C₉⋊C₉	—	37	2	no	—
79	C₉⋊C₉	—	37	2	no	—
80	C₃²⋊C₉	—	37	2	no	—
81	C₉²	—	37	1	no	—
82	C₉²	—	37	1	no	—
83	C₉²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
84	C₃₇	#10624727	1	1	yes	characteristic, complemented, Sylow
85	C₃³	—	37	1	no	—
86	3₋¹⁺²	—	111	3	no	—
87	C₃×C₉	—	37	2	no	—
88	C₃×C₉	—	37	2	no	—
89	C₃×C₉	—	37	1	no	—
90	C₃×C₉	—	37	1	no	—
91	C₃×C₉	—	37	2	no	—
92	C₃×C₉	—	111	1	no	—
93	C₃×C₉	—	111	2	no	—
94	C₃×C₉	C₉×C₃₇⋊C₃	1	2	yes	—
95	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
96	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
97	C₂₇	—	3	2	no	—
98	C₂₇	—	111	2	no	—
99	C₂₇	—	111	2	no	—
100	C₂₇	—	111	2	no	—
101	C₂₇	#11758606	1	3	yes	complemented
102	C₃²	—	37	3	no	—
103	C₃²	—	111	1	no	—
104	C₃²	—	111	2	no	—
105	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
106	C₉	—	3	1	no	—
107	C₉	—	3	2	no	—
108	C₉	—	111	2	no	—
109	C₉	—	111	1	no	—
110	C₉	—	111	1	no	—
111	C₉	—	111	2	no	—
112	C₉	—	111	2	no	—
113	C₉	C₃×#11758606	1	1	yes	characteristic
114	C₉	#15688531	1	2	yes	—
115	C₃	—	111	3	no	—
116	C₃	C₉×#11758606	1	1	yes	characteristic
117	C₃	C₂₇⋊C₃×C₃₇⋊C₃	1	1	yes	characteristic
118	C₃	#22582614	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₂₇×C₃₇⋊C₃

C₃

yes

maximal, characteristic

C₉×#11758606

C₃

yes

maximal, characteristic

#22582490

C₃

yes

maximal, complemented

#22582488

C₃

yes

maximal, complemented

#22582488

C₃

yes

maximal, complemented

#22582465

C₃

yes

maximal

#22582463

C₃

yes

maximal, characteristic, complemented

#22582462

C₃

yes

maximal, characteristic, complemented

C₃₇×#6864

C₃

yes

maximal, characteristic, complemented

C₃×#11758606

C₉

yes

characteristic

C₃×C₉×C₃₇⋊C₃

C₃²

yes

characteristic

C₃×#11758601

C₃²

yes

—

C₃×#11758601

C₃²

yes

—

C₃×#11758601

C₃²

yes

—

C₂₇×C₃₇⋊C₃

—

#15688506

C₉

yes

—

#15688504

C₉

yes

—

#15688504

C₉

yes

—

#15688502

C₉

yes

—

C₉×#66524

C₃²

yes

characteristic

C₉×#66524

C₃²

yes

characteristic

C₃×C₂₇×C₃₇

C₃²

yes

—

C₃×C₂₇×C₃₇

C₃²

yes

characteristic

C₉²×C₃₇

C₃²

yes

characteristic

C₃²×C₃₇⋊C₃

C₃×C₉

yes

characteristic

C₃×C₉×C₃₇

C₃×C₉

yes

—

C₃×C₉×C₃₇

C₃×C₉

yes

characteristic

C₃×C₉×C₃₇

C₃³

yes

characteristic

C₃×#66524

C₃×C₉

yes

—

C₃×#66524

C₃×C₉

yes

—

C₃×#66524

C₃×C₉

yes

characteristic

C₃×#66524

C₃×C₉

yes

characteristic

C₃×#66524

C₃×C₉

yes

—

#11758606

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₂₇×C₃₇

—

C₂₇×C₃₇

3₋¹⁺²

yes

—

—

—

—

C₃²×C₃₇

C₃²×C₉

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

C₂₇⋊C₃

yes

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₉×C₃₇

#737

yes

—

—

—

—

—

—

C₃²×C₂₇

—

C₉×3₋¹⁺²

—

C₉⋊C₂₇

—

C₉⋊C₂₇

—

C₉⋊C₂₇

—

C₃²⋊C₂₇

—

#6864

—

#6864

—

#6864

C₃₇⋊C₃

yes

characteristic, complemented

C₃×C₃₇

C₉×3₋¹⁺²

yes

characteristic

C₃×C₃₇

C₃×C₂₇⋊C₃

yes

characteristic

C₃×C₃₇

#6903

yes

—

C₃₇⋊C₃

—

C₃×3₋¹⁺²

—

C₃²×C₉

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

111

—

C₃×C₂₇

C₃×C₃₇⋊C₃

yes

—

C₃×C₂₇

C₃×C₃₇⋊C₃

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₉²

—

C₉²

—

C₉²

C₃×C₃₇⋊C₃

yes

characteristic

C₃₇

#10624727

yes

characteristic, complemented, Sylow

C₃³

—

3₋¹⁺²

—

111

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

C₉×C₃₇⋊C₃

yes

—

C₃×C₉

C₉×C₃₇⋊C₃

yes

characteristic

C₃×C₉

C₃²×C₃₇⋊C₃

yes

characteristic

C₂₇

—

C₂₇

—

111

—

C₂₇

—

111

—

100

C₂₇

—

111

—

101

C₂₇

#11758606

yes

complemented

102

C₃²

—

103

C₃²

—

111

—

104

C₃²

—

111

—

105

C₃²

C₃×C₉×C₃₇⋊C₃

yes

characteristic

106

C₉

—

107

C₉

—

108

C₉

—

111

—

109

C₉

—

111

—

110

C₉

—

111

—

111

C₉

—

111

—

112

C₉

—

111

—

113

C₉

C₃×#11758606

yes

characteristic

114

C₉

#15688531

yes

—

115

C₃

—

111

—

116

C₃

C₉×#11758606

yes

characteristic

117

C₃

C₂₇⋊C₃×C₃₇⋊C₃

yes

characteristic

118

C₃

#22582614

yes

—

	subgroup	G/H
center, Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
commutator, socle	C₃²×C₃₇	C₃²×C₉
Fitting	C₃₇×#6864	C₃
prime radical	C₃²×C₃₇⋊C₃	C₃×C₉
cosocle	C₃×C₉×C₃₇	C₃³
hypercenter	#6864	C₃₇⋊C₃
hypocenter	C₃₇	#10624727
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₉ ◁ #6864

subgroup

G/H

center, Frattini

C₃×C₉

C₃²×C₃₇⋊C₃

commutator, socle

C₃²×C₃₇

C₃²×C₉

Fitting

C₃₇×#6864

C₃

prime radical

C₃²×C₃₇⋊C₃

C₃×C₉

cosocle

C₃×C₉×C₃₇

C₃³

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃₇

C₁ ◁ C₃×C₉ ◁ #6864

N	H ≅ G/N	type
3. #22582490	C₃	semidirect
4. #22582488	C₃	semidirect
5. #22582488	C₃	semidirect
7. #22582463	C₃	semidirect
8. #22582462	C₃	semidirect
9. C₃₇×#6864	C₃	semidirect
38. C₂₇×C₃₇	3₋¹⁺²	semidirect
64. #6864	C₃₇⋊C₃	semidirect
84. C₃₇	#10624727	semidirect
101. C₂₇	#11758606	semidirect

H ≅ G/N

type

3. #22582490

C₃

semidirect

4. #22582488

C₃

semidirect

5. #22582488

C₃

semidirect

7. #22582463

C₃

semidirect

8. #22582462

C₃

semidirect

9. C₃₇×#6864

C₃

semidirect

38. C₂₇×C₃₇

3₋¹⁺²

semidirect

64. #6864

C₃₇⋊C₃

semidirect

84. C₃₇

#10624727

semidirect

101. C₂₇

#11758606

semidirect

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

subgroup	index
C₃×C₂₇×C₃₇⋊C₃	3
C₉×#11758606	3
#22582490	3
#22582488	3
#22582465	3
#22582463	3
#22582462	3
C₃₇×#6864	3
#10624727	37

subgroup

index

C₃×C₂₇×C₃₇⋊C₃

C₃₇×#6864

Subgroups — #41006460

Conjugacy classes of subgroups

Complements

Maximal subgroups