Group #41006659 — subgroups

Subgroups — #41006659

Order 26973.

402 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 70 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	#22582651	C₃	1	6	yes	maximal, complemented
2	#22582642	C₃	1	3	yes	maximal
3	C₃×#15688521	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×#15688521	C₃	1	2	yes	maximal, complemented
5	C₃×C₃₇×#737	C₃	1	1	yes	maximal, characteristic
6	C₃×#11758608	C₃²	1	3	yes	—
7	C₃×#11758608	C₃²	1	6	yes	—
8	#15688521	—	3	1	no	—
9	#15688521	—	3	2	no	—
10	#15688521	—	3	2	no	—
11	#15688521	—	3	4	no	—
12	#15688520	—	3	18	no	—
13	#15688515	—	3	9	no	—
14	C₃₇×#737	—	3	6	no	—
15	C₃₇×#737	—	3	3	no	—
16	C₃×3₋¹⁺²×C₃₇	—	3	2	no	—
17	C₃×3₋¹⁺²×C₃₇	C₃²	1	2	yes	—
18	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
19	C₃²×C₉×C₃₇	—	3	1	no	—
20	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
21	C₃³×C₃₇	—	3	1	no	—
22	C₃³×C₃₇	C₃³	1	1	yes	characteristic
23	3₋¹⁺²×C₃₇	—	3	18	no	—
24	3₋¹⁺²×C₃₇	—	3	2	no	—
25	3₋¹⁺²×C₃₇	—	3	2	no	—
26	3₋¹⁺²×C₃₇	H(3,𝔽₃)	1	6	yes	—
27	H(3,𝔽₃)×C₃₇	—	3	2	no	—
28	H(3,𝔽₃)×C₃₇	H(3,𝔽₃)	1	3	yes	—
29	C₃×C₉×C₃₇	—	3	9	no	—
30	C₃×C₉×C₃₇	—	3	2	no	—
31	C₃×C₉×C₃₇	—	3	2	no	—
32	C₃×C₉×C₃₇	—	3	1	no	—
33	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
34	C₃×C₉×C₃₇	H(3,𝔽₃)	1	2	yes	—
35	#11758608	—	3	6	no	—
36	#11758608	—	3	6	no	—
37	#11758608	—	3	6	no	—
38	#11758608	—	3	3	no	—
39	#11758608	—	3	6	no	—
40	C₃×#66524	—	3	3	no	—
41	C₃×#66524	—	3	6	no	—
42	#10624758	—	37	1	no	maximal, Sylow
43	C₃²×C₃₇	—	3	3	no	—
44	C₃²×C₃₇	—	9	1	no	—
45	C₃²×C₃₇	—	9	3	no	—
46	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	2	yes	—
47	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
48	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
49	C₉×C₃₇	—	3	1	no	—
50	C₉×C₃₇	—	3	6	no	—
51	C₉×C₃₇	—	3	2	no	—
52	#66524	—	9	3	no	—
53	#66524	—	9	6	no	—
54	C₃×#737	C₃₇⋊C₃	1	1	yes	characteristic
55	#6912	—	37	6	no	—
56	#6910	—	37	3	no	—
57	C₃×#733	—	37	1	no	—
58	C₃×#733	—	37	2	no	—
59	C₃×C₃₇	—	3	1	no	—
60	C₃×C₃₇	—	3	2	no	—
61	C₃×C₃₇	—	3	1	no	—
62	C₃×C₃₇	—	9	3	no	—
63	C₃×C₃₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
64	#737	—	3	6	no	—
65	#737	—	3	3	no	—
66	C₃×3₋¹⁺²	—	3	2	no	—
67	C₃×3₋¹⁺²	—	37	3	no	—
68	C₃×3₋¹⁺²	—	37	6	no	—
69	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	2	yes	—
70	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
71	C₃²×C₉	—	3	1	no	—
72	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
73	#733	—	111	18	no	—
74	#733	—	111	1	no	—
75	#733	—	111	2	no	—
76	#733	—	111	2	no	—
77	#733	—	111	4	no	—
78	#731	—	111	9	no	—
79	C₃₇	#10624758	1	1	yes	characteristic, complemented, Sylow
80	C₃³	—	3	1	no	—
81	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
82	3₋¹⁺²	—	3	18	no	—
83	3₋¹⁺²	—	3	2	no	—
84	3₋¹⁺²	—	3	2	no	—
85	3₋¹⁺²	—	111	6	no	—
86	3₋¹⁺²	—	111	3	no	—
87	3₋¹⁺²	—	111	6	no	—
88	3₋¹⁺²	—	111	6	no	—
89	3₋¹⁺²	—	111	6	no	—
90	3₋¹⁺²	C₃²⋊(C₃₇⋊C₃)	1	6	yes	—
91	H(3,𝔽₃)	—	3	2	no	—
92	H(3,𝔽₃)	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
93	C₃×C₉	—	3	9	no	—
94	C₃×C₉	—	3	2	no	—
95	C₃×C₉	—	3	2	no	—
96	C₃×C₉	—	3	1	no	—
97	C₃×C₉	—	111	3	no	—
98	C₃×C₉	—	111	6	no	—
99	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
100	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
101	C₃²	—	3	3	no	—
102	C₃²	—	9	1	no	—
103	C₃²	—	9	3	no	—
104	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
105	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
106	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
107	C₉	—	3	1	no	—
108	C₉	—	3	6	no	—
109	C₉	—	3	2	no	—
110	C₉	—	333	3	no	—
111	C₉	—	333	6	no	—
112	C₃	—	3	1	no	—
113	C₃	—	3	2	no	—
114	C₃	—	3	1	no	—
115	C₃	—	9	3	no	—
116	C₃	#22582550	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22582651

C₃

yes

maximal, complemented

#22582642

C₃

yes

maximal

C₃×#15688521

C₃

yes

maximal, characteristic, complemented

C₃×#15688521

C₃

yes

maximal, complemented

C₃×C₃₇×#737

C₃

yes

maximal, characteristic

C₃×#11758608

C₃²

yes

—

C₃×#11758608

C₃²

yes

—

—

—

—

—

—

—

C₃₇×#737

—

C₃₇×#737

—

C₃×3₋¹⁺²×C₃₇

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

characteristic

C₃²×C₉×C₃₇

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃³×C₃₇

—

C₃³×C₃₇

C₃³

yes

characteristic

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

H(3,𝔽₃)

yes

—

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×C₃₇

H(3,𝔽₃)

yes

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

characteristic

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

—

—

—

—

—

—

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

#66524

—

#66524

—

C₃×#737

C₃₇⋊C₃

yes

—

—

—

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃²⋊H(3,𝔽₃)

yes

characteristic

#737

—

#737

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

—

C₃×H(3,𝔽₃)

C₃×C₃₇⋊C₃

yes

characteristic

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

#733

—

111

—

#733

—

111

—

#733

—

111

—

#733

—

111

—

#733

—

111

—

#731

—

111

—

C₃₇

#10624758

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

3₋¹⁺²

C₃²⋊(C₃₇⋊C₃)

yes

—

H(3,𝔽₃)

—

H(3,𝔽₃)

C₃²⋊(C₃₇⋊C₃)

yes

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

100

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

105

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

—

106

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

107

C₉

—

108

C₉

—

109

C₉

—

110

C₉

—

333

—

111

C₉

—

333

—

112

C₃

—

113

C₃

—

114

C₃

—

115

C₃

—

116

C₃

#22582550

yes

characteristic

	subgroup	G/H
center	C₃	#22582550
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃×C₃₇×#737	C₃
socle	C₃×C₃₇	C₃²⋊H(3,𝔽₃)
prime radical	C₃×H(3,𝔽₃)×C₃₇	C₃²
hypercenter	C₃×#737	C₃₇⋊C₃
hypocenter	C₃₇	#10624758
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×#737

subgroup

G/H

center

C₃

#22582550

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃×C₃₇×#737

C₃

socle

C₃×C₃₇

C₃²⋊H(3,𝔽₃)

prime radical

C₃×H(3,𝔽₃)×C₃₇

C₃²

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃ ◁ C₃³ ◁ C₃×#737

N	H ≅ G/N	type
1. #22582651	C₃	semidirect
3. C₃×#15688521	C₃	semidirect
4. C₃×#15688521	C₃	semidirect
79. C₃₇	#10624758	semidirect

H ≅ G/N

type

1. #22582651

C₃

semidirect

3. C₃×#15688521

C₃

semidirect

4. C₃×#15688521

C₃

semidirect

79. C₃₇

#10624758

semidirect

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

subgroup	index
#22582651	3
#22582642	3
C₃×#15688521	3
C₃×C₃₇×#737	3
#10624758	37

subgroup

index

C₃×C₃₇×#737

Subgroups — #41006659

Conjugacy classes of subgroups

Complements

Maximal subgroups