Group #41007860 — subgroups

Subgroups — #41007860

Order 26973.

730 conjugacy classes of subgroups; only the 295 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 295 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×#15688530	C₃	1	1	yes	maximal, characteristic
2	#22582576	C₃	1	8	yes	maximal, complemented
3	#22582574	C₃	1	8	yes	maximal, complemented
4	#22582574	C₃	1	8	yes	maximal, complemented
5	#22582571	C₃	1	8	yes	maximal, complemented
6	H(3,𝔽₃)×#66524	C₃	1	1	yes	maximal, characteristic
7	H(3,𝔽₃)×#66524	C₃	1	1	yes	maximal, characteristic
8	C₃×#15688504	C₃	1	4	yes	maximal, complemented
9	C₉×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic
10	#15688530	—	3	9	no	—
11	C₃×#11758608	C₃²	1	8	yes	—
12	C₃²×#66524	C₃²	1	4	yes	—
13	C₃²×#66524	C₃²	1	4	yes	—
14	C₃²×#66524	C₃²	1	4	yes	—
15	#15688505	C₃²	1	24	yes	—
16	#15688504	C₃²	1	1	yes	characteristic
17	#15688504	C₃²	1	8	yes	—
18	#15688504	C₃²	1	8	yes	—
19	#15688504	C₃²	1	8	yes	—
20	#15688504	C₃²	1	8	yes	—
21	#15688504	C₃²	1	24	yes	—
22	C₃²⋊#66524	C₃²	1	8	yes	—
23	C₃²⋊#66524	C₃²	1	8	yes	—
24	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
25	C₃²×C₉×C₃₇	C₃²	1	4	yes	—
26	C₃₇×C₃²⋊C₉	C₃²	1	8	yes	—
27	C₃³×C₃₇	C₃³	1	4	yes	—
28	H(3,𝔽₃)×C₃₇	3₋¹⁺²	1	9	yes	—
29	C₃×C₉×C₃₇	—	9	4	no	—
30	C₃×C₉×C₃₇	C₃³	1	8	yes	—
31	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
32	#11758608	—	3	24	no	—
33	C₃×#66524	—	3	12	no	—
34	C₃×#66524	—	3	12	no	—
35	C₃×#66524	—	9	4	no	—
36	C₃×#66524	C₃³	1	8	yes	—
37	C₃×#66524	C₃³	1	8	yes	—
38	C₃×#66524	C₃³	1	1	yes	characteristic
39	C₃×#66524	C₃³	1	1	yes	characteristic
40	C₃×#66524	C₃³	1	8	yes	—
41	C₃×#66524	C₃³	1	1	yes	characteristic
42	#10624900	—	37	1	no	maximal, Sylow
43	C₃²×C₃₇	—	3	4	no	—
44	C₃²×C₃₇	—	3	4	no	—
45	C₃²×C₃₇	—	3	4	no	—
46	C₃²×C₃₇	C₃×3₋¹⁺²	1	12	yes	—
47	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
48	C₉×C₃₇	—	3	1	no	—
49	C₉×C₃₇	—	3	8	no	—
50	#66524	—	3	1	no	—
51	#66524	—	3	8	no	—
52	#66524	—	3	8	no	—
53	#66524	—	3	1	no	—
54	#66524	—	3	8	no	—
55	#66524	#737	1	3	yes	complemented
56	C₃×#737	—	37	1	no	—
57	#6895	—	37	8	no	—
58	#6894	—	37	8	no	—
59	#6894	—	37	8	no	—
60	#6894	—	37	8	no	—
61	C₉×H(3,𝔽₃)	—	37	1	no	—
62	C₉×H(3,𝔽₃)	—	37	1	no	—
63	C₉×H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic
64	C₃×C₉⋊C₉	—	37	4	no	—
65	C₃×C₃₇	—	3	12	no	—
66	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
67	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
68	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
69	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
70	#737	—	111	9	no	—
71	C₃×3₋¹⁺²	—	37	8	no	—
72	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
73	C₃²×C₉	—	37	4	no	—
74	C₃²×C₉	—	37	4	no	—
75	C₃²×C₉	—	37	4	no	—
76	C₃²×C₉	C₃×C₃₇⋊C₃	1	4	yes	—
77	C₉⋊C₉	—	37	1	no	—
78	C₉⋊C₉	—	37	8	no	—
79	C₉⋊C₉	—	37	8	no	—
80	C₉⋊C₉	—	37	8	no	—
81	C₉⋊C₉	—	37	8	no	—
82	C₉⋊C₉	—	37	24	no	—
83	C₉⋊C₉	—	37	24	no	—
84	C₃²⋊C₉	—	37	8	no	—
85	C₃²⋊C₉	—	37	8	no	—
86	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	8	yes	—
87	C₃₇	#10624900	1	1	yes	characteristic, complemented, Sylow
88	C₃³	C₃²×C₃₇⋊C₃	1	4	yes	—
89	3₋¹⁺²	—	111	24	no	—
90	H(3,𝔽₃)	#11758605	1	9	yes	—
91	C₃×C₉	—	9	4	no	—
92	C₃×C₉	—	37	1	no	—
93	C₃×C₉	—	37	8	no	—
94	C₃×C₉	—	37	8	no	—
95	C₃×C₉	—	37	1	no	—
96	C₃×C₉	—	37	1	no	—
97	C₃×C₉	—	37	8	no	—
98	C₃×C₉	—	333	4	no	—
99	C₃×C₉	—	111	12	no	—
100	C₃×C₉	—	111	12	no	—
101	C₃×C₉	C₃²×C₃₇⋊C₃	1	8	yes	—
102	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
103	C₃²	—	3	4	no	—
104	C₃²	—	3	4	no	—
105	C₃²	—	3	4	no	—
106	C₃²	C₃×#11758605	1	12	yes	—
107	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
108	C₉	—	3	1	no	—
109	C₉	—	3	8	no	—
110	C₉	—	37	3	no	—
111	C₉	—	111	1	no	—
112	C₉	—	111	8	no	—
113	C₉	—	111	8	no	—
114	C₉	—	111	1	no	—
115	C₉	—	111	8	no	—
116	C₃	—	3	12	no	—
117	C₃	C₃²×#11758605	1	1	yes	characteristic
118	C₃	#737×C₃₇⋊C₃	1	1	yes	characteristic
119	C₃	#22582677	1	1	yes	characteristic
120	C₃	#22582680	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688530

C₃

yes

maximal, characteristic

#22582576

C₃

yes

maximal, complemented

#22582574

C₃

yes

maximal, complemented

#22582574

C₃

yes

maximal, complemented

#22582571

C₃

yes

maximal, complemented

H(3,𝔽₃)×#66524

C₃

yes

maximal, characteristic

H(3,𝔽₃)×#66524

C₃

yes

maximal, characteristic

C₃×#15688504

C₃

yes

maximal, complemented

C₉×H(3,𝔽₃)×C₃₇

C₃

yes

maximal, characteristic

#15688530

—

C₃×#11758608

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

#15688505

C₃²

yes

—

#15688504

C₃²

yes

characteristic

#15688504

C₃²

yes

—

#15688504

C₃²

yes

—

#15688504

C₃²

yes

—

#15688504

C₃²

yes

—

#15688504

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

characteristic

C₃²×C₉×C₃₇

C₃²

yes

—

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃³×C₃₇

C₃³

yes

—

H(3,𝔽₃)×C₃₇

3₋¹⁺²

yes

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃³

yes

—

C₃×C₉×C₃₇

C₃³

yes

—

—

—

—

C₃³

yes

—

C₃×#66524

C₃³

yes

—

C₃×#66524

C₃³

yes

characteristic

C₃×#66524

C₃³

yes

characteristic

C₃×#66524

C₃³

yes

—

C₃×#66524

C₃³

yes

characteristic

#10624900

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₃²×C₃₇

C₃⁴

yes

characteristic

C₉×C₃₇

—

C₉×C₃₇

—

—

—

—

—

—

yes

—

—

—

—

—

C₉×H(3,𝔽₃)

—

C₉×H(3,𝔽₃)

—

C₉×H(3,𝔽₃)

C₃₇⋊C₃

yes

characteristic

C₃×C₉⋊C₉

—

C₃×C₃₇

—

C₃×C₃₇

C₃²×3₋¹⁺²

yes

characteristic

C₃×C₃₇

C₃×#737

yes

characteristic

C₃×C₃₇

3₋¹⁺⁴

yes

characteristic

C₃×C₃₇

3₋¹⁺⁴

yes

characteristic

#737

—

111

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

C₃×C₃₇⋊C₃

yes

characteristic

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₃₇

#10624900

yes

characteristic, complemented, Sylow

C₃³

C₃²×C₃₇⋊C₃

yes

—

3₋¹⁺²

—

111

—

H(3,𝔽₃)

#11758605

yes

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

333

—

C₃×C₉

—

111

—

100

C₃×C₉

—

111

—

101

C₃×C₉

C₃²×C₃₇⋊C₃

yes

—

102

C₃×C₉

C₃²×C₃₇⋊C₃

yes

characteristic

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

C₃×#11758605

yes

—

107

C₃²

C₃³×C₃₇⋊C₃

yes

characteristic

108

C₉

—

109

C₉

—

110

C₉

—

111

C₉

—

111

—

112

C₉

—

111

—

113

C₉

—

111

—

114

C₉

—

111

—

115

C₉

—

111

—

116

C₃

—

117

C₃

C₃²×#11758605

yes

characteristic

118

C₃

#737×C₃₇⋊C₃

yes

characteristic

119

C₃

#22582677

yes

characteristic

120

C₃

#22582680

yes

characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting	C₉×H(3,𝔽₃)×C₃₇	C₃
prime radical	C₃×H(3,𝔽₃)×C₃₇	C₃²
hypercenter	C₉×H(3,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	#10624900
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₉×H(3,𝔽₃)

subgroup

G/H

center, Frattini

C₃²

C₃³×C₃₇⋊C₃

commutator, socle, cosocle

C₃²×C₃₇

C₃⁴

Fitting

C₉×H(3,𝔽₃)×C₃₇

C₃

prime radical

C₃×H(3,𝔽₃)×C₃₇

C₃²

hypercenter

C₉×H(3,𝔽₃)

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃₇

C₁ ◁ C₃² ◁ C₉×H(3,𝔽₃)

N	H ≅ G/N	type
2. #22582576	C₃	semidirect
3. #22582574	C₃	semidirect
4. #22582574	C₃	semidirect
5. #22582571	C₃	semidirect
8. C₃×#15688504	C₃	semidirect
55. #66524	#737	semidirect
87. C₃₇	#10624900	semidirect

H ≅ G/N

type

2. #22582576

C₃

semidirect

3. #22582574

C₃

semidirect

4. #22582574

C₃

semidirect

5. #22582571

C₃

semidirect

8. C₃×#15688504

C₃

semidirect

55. #66524

#737

semidirect

87. C₃₇

#10624900

semidirect

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

subgroup	index
C₃×#15688530	3
#22582576	3
#22582574	3
#22582571	3
H(3,𝔽₃)×#66524	3
C₃×#15688504	3
C₉×H(3,𝔽₃)×C₃₇	3
#10624900	37

subgroup

index

C₉×H(3,𝔽₃)×C₃₇

Subgroups — #41007860

Conjugacy classes of subgroups

Complements

Maximal subgroups