Group #41008521 — subgroups

Subgroups — #41008521

Order 26973.

312 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 84 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×#15688541	C₃	1	2	yes	maximal, complemented
2	C₃×#15688541	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582701	C₃	1	9	yes	maximal, complemented
4	C₃×#15688530	C₃	1	1	yes	maximal, characteristic
5	#15688541	—	3	2	no	—
6	#15688541	—	3	2	no	—
7	#15688541	—	3	1	no	—
8	#15688541	—	3	2	no	—
9	#15688541	C₃²	1	6	yes	complemented
10	C₃×#11758603	—	3	2	no	—
11	C₃×#11758603	C₃²	1	3	yes	complemented
12	#15688530	—	3	2	no	—
13	#15688530	—	3	4	no	—
14	#15688530	C₉	1	6	yes	—
15	#15688530	C₃²	1	3	yes	—
16	C₃×#11758608	—	3	2	no	—
17	C₃×#11758608	C₉	1	2	yes	—
18	C₃²×#66524	—	3	1	no	—
19	C₃²×#66524	C₃²	1	1	yes	characteristic
20	C₃×H(3,𝔽₃)×C₃₇	C₉	1	1	yes	characteristic
21	C₃³×C₃₇	—	3	1	no	—
22	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
23	H(3,𝔽₃)×C₃₇	—	3	2	no	—
24	H(3,𝔽₃)×C₃₇	C₃×C₉	1	3	yes	—
25	#11758608	—	3	4	no	—
26	#11758608	—	3	12	no	—
27	#11758608	—	3	6	no	—
28	#11758608	C₃×C₉	1	6	yes	—
29	C₃×#66524	—	3	2	no	—
30	C₃×#66524	—	3	6	no	—
31	C₃×#66524	—	3	3	no	—
32	C₃×#66524	C₃×C₉	1	2	yes	—
33	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
34	C₃×#66524	H(3,𝔽₃)	1	2	yes	—
35	C₃×#66524	3₋¹⁺²	1	2	yes	—
36	C₃×#66524	3₋¹⁺²	1	4	yes	—
37	C₃×#66524	C₃³	1	1	yes	characteristic
38	#11758603	—	3	6	no	—
39	#11758603	—	3	3	no	—
40	#10624704	—	37	1	no	maximal, Sylow
41	C₃²×C₃₇	—	3	3	no	—
42	C₃²×C₃₇	—	9	1	no	—
43	C₃²×C₃₇	—	9	3	no	—
44	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
45	C₃²×C₃₇	C₃²⋊C₉	1	2	yes	—
46	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
47	#66524	—	3	6	no	—
48	#66524	C₃²⋊C₉	1	2	yes	—
49	#66524	C₃²⋊C₉	1	4	yes	—
50	#66524	C₃×H(3,𝔽₃)	1	1	yes	characteristic
51	#66524	C₃×3₋¹⁺²	1	2	yes	—
52	C₃×#737	—	37	1	no	—
53	C₃×C₂₇⋊C₃	—	37	1	no	—
54	C₃×C₂₇⋊C₃	—	37	2	no	—
55	#6873	—	37	9	no	—
56	C₃×C₃₇	—	3	1	no	—
57	C₃×C₃₇	—	3	2	no	—
58	C₃×C₃₇	—	3	1	no	—
59	C₃×C₃₇	—	9	3	no	—
60	C₃×C₃₇	C₃×C₃²⋊C₉	1	1	yes	characteristic
61	#737	—	37	6	no	—
62	#737	—	37	3	no	—
63	#737	—	111	2	no	—
64	#737	—	111	4	no	—
65	C₃×3₋¹⁺²	—	37	2	no	—
66	C₃×3₋¹⁺²	—	111	2	no	—
67	C₃×H(3,𝔽₃)	#66526	1	1	yes	characteristic
68	C₃²×C₉	—	37	1	no	—
69	C₃²×C₉	—	111	1	no	—
70	C₂₇⋊C₃	—	37	6	no	—
71	C₂₇⋊C₃	—	111	2	no	—
72	C₂₇⋊C₃	—	111	1	no	—
73	C₂₇⋊C₃	—	111	2	no	—
74	C₂₇⋊C₃	—	111	2	no	—
75	C₃×C₂₇	—	37	3	no	—
76	C₃×C₂₇	—	111	2	no	—
77	C₃₇	#10624704	1	1	yes	characteristic, complemented, Sylow
78	C₃³	—	3	1	no	—
79	C₃³	C₃×#66526	1	1	yes	characteristic
80	3₋¹⁺²	—	37	6	no	—
81	3₋¹⁺²	—	111	12	no	—
82	3₋¹⁺²	—	111	6	no	—
83	3₋¹⁺²	—	111	4	no	—
84	H(3,𝔽₃)	—	3	2	no	—
85	H(3,𝔽₃)	C₃×#66526	1	3	yes	—
86	C₃×C₉	—	37	2	no	—
87	C₃×C₉	—	37	1	no	—
88	C₃×C₉	—	37	2	no	—
89	C₃×C₉	—	37	4	no	—
90	C₃×C₉	—	37	2	no	—
91	C₃×C₉	—	37	1	no	—
92	C₃×C₉	—	111	3	no	—
93	C₃×C₉	—	111	6	no	—
94	C₃×C₉	—	111	2	no	—
95	C₂₇	—	111	3	no	—
96	C₂₇	—	111	6	no	—
97	C₃²	—	3	3	no	—
98	C₃²	—	9	1	no	—
99	C₃²	—	9	3	no	—
100	C₃²	C₃²⋊#66526	1	1	yes	characteristic
101	C₃²	C₃²⋊#66526	1	2	yes	—
102	C₃²	C₃²×#66526	1	1	yes	characteristic
103	C₉	—	37	1	no	—
104	C₉	—	37	2	no	—
105	C₉	—	37	4	no	—
106	C₉	—	37	2	no	—
107	C₉	—	111	6	no	—
108	C₃	—	3	1	no	—
109	C₃	—	3	2	no	—
110	C₃	—	3	1	no	—
111	C₃	—	9	3	no	—
112	C₃	C₃×C₃²⋊#66526	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688541

C₃

yes

maximal, complemented

C₃×#15688541

C₃

yes

maximal, characteristic, complemented

#22582701

C₃

yes

maximal, complemented

C₃×#15688530

C₃

yes

maximal, characteristic

—

—

—

—

C₃²

yes

complemented

C₃×#11758603

—

C₃×#11758603

C₃²

yes

—

—

C₉

yes

—

#15688530

C₃²

yes

—

C₃×#11758608

—

C₃×#11758608

C₉

yes

—

C₃²×#66524

—

C₃²×#66524

C₃²

yes

characteristic

C₃×H(3,𝔽₃)×C₃₇

C₉

yes

characteristic

C₃³×C₃₇

—

C₃³×C₃₇

C₃×C₉

yes

characteristic

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×C₃₇

C₃×C₉

yes

—

—

—

—

C₃×C₉

yes

—

—

—

—

C₃×C₉

yes

—

yes

yes

—

C₃×#66524

3₋¹⁺²

yes

—

C₃×#66524

3₋¹⁺²

yes

—

C₃×#66524

C₃³

yes

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃²⋊C₉

yes

characteristic

C₃²×C₃₇

C₃²⋊C₉

yes

—

C₃²×C₃₇

C₃²×C₉

yes

characteristic

#66524

—

#66524

C₃²⋊C₉

yes

—

#66524

C₃²⋊C₉

yes

—

#66524

C₃×H(3,𝔽₃)

yes

characteristic

#66524

C₃×3₋¹⁺²

yes

—

C₃×#737

—

C₃×C₂₇⋊C₃

—

C₃×C₂₇⋊C₃

—

#6873

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃×C₃²⋊C₉

yes

—

—

—

111

—

#737

—

111

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×H(3,𝔽₃)

#66526

yes

characteristic

C₃²×C₉

—

C₃²×C₉

—

111

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

—

111

—

C₃×C₂₇

—

C₃×C₂₇

—

111

—

C₃₇

#10624704

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

C₃×#66526

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

H(3,𝔽₃)

—

H(3,𝔽₃)

C₃×#66526

yes

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₂₇

—

111

—

C₂₇

—

111

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

C₃²⋊#66526

yes

characteristic

101

C₃²

C₃²⋊#66526

yes

—

102

C₃²

C₃²×#66526

yes

characteristic

103

C₉

—

104

C₉

—

105

C₉

—

106

C₉

—

107

C₉

—

111

—

108

C₃

—

109

C₃

—

110

C₃

—

111

C₃

—

112

C₃

C₃×C₃²⋊#66526

yes

characteristic

	subgroup	G/H
center	C₃	C₃×C₃²⋊#66526
commutator	C₃²×C₃₇	C₃²×C₉
Frattini	C₃²	C₃²×#66526
Fitting, prime radical	C₃×H(3,𝔽₃)×C₃₇	C₉
socle	C₃×C₃₇	C₃×C₃²⋊C₉
cosocle	C₃×#66524	C₃³
hypercenter	C₃×H(3,𝔽₃)	#66526
hypocenter	C₃₇	#10624704
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×H(3,𝔽₃)

subgroup

G/H

center

C₃

C₃×C₃²⋊#66526

commutator

C₃²×C₃₇

C₃²×C₉

Frattini

C₃²

C₃²×#66526

Fitting, prime radical

C₃×H(3,𝔽₃)×C₃₇

C₉

socle

C₃×C₃₇

C₃×C₃²⋊C₉

C₃³

C₃×H(3,𝔽₃)

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃ ◁ C₃³ ◁ C₃×H(3,𝔽₃)

N	H ≅ G/N	type
1. C₃×#15688541	C₃	semidirect
2. C₃×#15688541	C₃	semidirect
3. #22582701	C₃	semidirect
9. #15688541	C₃²	semidirect
11. C₃×#11758603	C₃²	semidirect
77. C₃₇	#10624704	semidirect

H ≅ G/N

type

1. C₃×#15688541

C₃

semidirect

2. C₃×#15688541

C₃

semidirect

3. #22582701

C₃

semidirect

9. #15688541

C₃²

semidirect

11. C₃×#11758603

C₃²

semidirect

77. C₃₇

#10624704

semidirect

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

subgroup	index
C₃×#15688541	3
#22582701	3
C₃×#15688530	3
#10624704	37

subgroup

index

Subgroups — #41008521

Conjugacy classes of subgroups

Complements

Maximal subgroups