Group #41004856 — subgroups

Subgroups — #41004856

Order 26973.

144 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 28 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×#15688521	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#15688519	C₃	1	1	yes	maximal, characteristic
3	#22582441	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃₇×#6868	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×#11758608	—	3	1	no	—
6	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
7	C₃²×#66524	—	3	1	no	—
8	#15688521	—	3	3	no	—
9	#15688519	—	3	3	no	—
10	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
11	C₃₇×C₃²⋊C₉	—	3	1	no	—
12	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
13	C₃²×C₃₇⋊C₃	—	9	1	no	—
14	C₃×C₉×C₃₇	—	3	1	no	—
15	C₃×C₉×C₃₇	—	3	1	no	—
16	C₃×C₉×C₃₇	—	3	1	no	—
17	C₃×C₉×C₃₇	—	3	1	no	—
18	C₃×C₉×C₃₇	—	9	1	no	—
19	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
20	C₃²⋊(C₃₇⋊C₃)	—	9	1	no	—
21	C₃²⋊(C₃₇⋊C₃)	—	9	1	no	—
22	#11758608	—	3	3	no	—
23	#11758608	—	9	1	no	—
24	#11758608	—	9	1	no	—
25	C₃×#66524	—	3	3	no	—
26	C₃×#66524	—	9	1	no	—
27	C₃×#66524	—	9	1	no	—
28	C₃×#66524	—	9	1	no	—
29	C₃×#66524	—	9	1	no	—
30	#10624481	—	37	1	no	maximal, Sylow
31	C₃²×C₃₇	—	3	1	no	—
32	C₃²×C₃₇	—	3	1	no	—
33	C₃²×C₃₇	—	3	1	no	—
34	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
35	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
36	C₃²×C₃₇	#732	1	1	yes	characteristic
37	C₃²×C₃₇	#732	1	1	yes	characteristic
38	C₃×C₃₇⋊C₃	—	9	3	no	—
39	C₃×C₃₇⋊C₃	—	27	1	no	—
40	C₃×C₃₇⋊C₃	—	27	1	no	—
41	C₃×C₃₇⋊C₃	—	27	1	no	—
42	C₉×C₃₇	—	9	1	no	—
43	C₉×C₃₇	—	27	1	no	—
44	#66524	—	9	3	no	—
45	#66524	—	9	3	no	—
46	C₃×#733	—	37	1	no	—
47	C₃×#732	—	37	1	no	—
48	#6868	C₃₇⋊C₃	1	1	yes	characteristic, complemented
49	#6855	—	37	1	no	—
50	C₃×C₃₇	—	3	1	no	—
51	C₃×C₃₇	—	3	1	no	—
52	C₃×C₃₇	—	3	1	no	—
53	C₃×C₃₇	#6856	1	1	yes	characteristic
54	C₃×C₃₇	#6878	1	1	yes	characteristic
55	C₃×C₃₇	#6881	1	1	yes	characteristic
56	C₃×C₃₇	#6883	1	1	yes	characteristic
57	C₃₇⋊C₃	—	27	3	no	—
58	C₃×3₋¹⁺²	—	111	1	no	—
59	C₃×H(3,𝔽₃)	—	111	1	no	—
60	C₃²×C₉	—	111	1	no	—
61	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
62	#733	—	111	3	no	—
63	#732	—	111	3	no	—
64	C₃²⋊C₉	—	3	1	no	—
65	C₃₇	#10624481	1	1	yes	characteristic, complemented, Sylow
66	C₃³	—	333	1	no	—
67	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
68	3₋¹⁺²	—	333	1	no	—
69	3₋¹⁺²	—	333	1	no	—
70	3₋¹⁺²	—	111	3	no	—
71	H(3,𝔽₃)	—	333	1	no	—
72	H(3,𝔽₃)	—	333	1	no	—
73	H(3,𝔽₃)	—	111	3	no	—
74	C₃×C₉	—	3	1	no	—
75	C₃×C₉	—	3	1	no	—
76	C₃×C₉	—	3	1	no	—
77	C₃×C₉	—	3	1	no	—
78	C₃×C₉	—	9	1	no	—
79	C₃×C₉	—	333	1	no	—
80	C₃×C₉	—	333	1	no	—
81	C₃×C₉	—	333	1	no	—
82	C₃×C₉	—	333	1	no	—
83	C₃×C₉	—	111	3	no	—
84	C₃²	—	3	1	no	—
85	C₃²	—	3	1	no	—
86	C₃²	—	3	1	no	—
87	C₃²	—	333	3	no	—
88	C₃²	—	999	1	no	—
89	C₃²	—	999	1	no	—
90	C₃²	—	999	1	no	—
91	C₃²	#15688514	1	1	yes	characteristic
92	C₃²	#15688514	1	1	yes	characteristic
93	C₃²	#15688518	1	1	yes	characteristic
94	C₃²	#15688519	1	1	yes	characteristic
95	C₉	—	9	1	no	—
96	C₉	—	27	1	no	—
97	C₉	—	333	3	no	—
98	C₉	—	333	3	no	—
99	C₃	—	3	1	no	—
100	C₃	—	3	1	no	—
101	C₃	—	3	1	no	—
102	C₃	—	999	3	no	—
103	C₃	#22582443	1	1	yes	characteristic
104	C₃	#22582504	1	1	yes	characteristic
105	C₃	#22582511	1	1	yes	characteristic
106	C₃	#22582522	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688521

C₃

yes

maximal, characteristic, complemented

C₃×#15688519

C₃

yes

maximal, characteristic

#22582441

C₃

yes

maximal, characteristic, complemented

C₃₇×#6868

C₃

yes

maximal, characteristic, complemented

C₃×#11758608

—

C₃×C₃²⋊(C₃₇⋊C₃)

—

C₃²×#66524

—

#15688521

—

#15688519

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×C₃²⋊C₉

—

C₃³×C₃₇

H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

—

—

—

—

—

—

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

AΣL(1,𝔽₂₇)

yes

characteristic

C₃²×C₃₇

AΣL(1,𝔽₂₇)

yes

C₃²×C₃₇

yes

C₃²×C₃₇

yes

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

—

—

—

—

—

C₃₇⋊C₃

yes

characteristic, complemented

#6855

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃₇⋊C₃

—

C₃×3₋¹⁺²

—

111

—

C₃×H(3,𝔽₃)

—

111

—

C₃²×C₉

—

111

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

#733

—

111

—

#732

—

111

—

C₃²⋊C₉

—

C₃₇

#10624481

yes

characteristic, complemented, Sylow

C₃³

—

333

—

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

3₋¹⁺²

—

333

—

3₋¹⁺²

—

333

—

3₋¹⁺²

—

111

—

H(3,𝔽₃)

—

333

—

H(3,𝔽₃)

—

333

—

H(3,𝔽₃)

—

111

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

333

—

C₃×C₉

—

333

—

C₃×C₉

—

333

—

C₃×C₉

—

333

—

C₃×C₉

—

111

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

333

—

C₃²

—

999

—

C₃²

—

999

—

C₃²

—

999

—

C₃²

yes

C₃²

yes

C₃²

yes

C₃²

yes

C₉

—

C₉

—

C₉

—

333

—

C₉

—

333

—

C₃

—

100

C₃

—

101

C₃

—

102

C₃

—

999

—

103

C₃

#22582443

yes

characteristic

104

C₃

#22582504

yes

characteristic

105

C₃

#22582511

yes

characteristic

106

C₃

#22582522

yes

characteristic

	subgroup	G/H
center	C₃²	#15688518
commutator, cosocle	C₃²×C₉×C₃₇	C₃²
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃₇×#6868	C₃
socle	C₃²×C₃₇	#732
prime radical	C₃×#15688519	C₃
hypercenter	#6868	C₃₇⋊C₃
hypocenter	C₃₇	#10624481
LCS	G ▷ C₃²×C₉×C₃₇ ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²×C₉ ◁ #6868

subgroup

G/H

center

C₃²

#15688518

commutator, cosocle

C₃²×C₉×C₃₇

C₃²

Frattini

C₃²×C₉

C₃×C₃₇⋊C₃

Fitting

C₃₇×#6868

C₃

C₃²×C₃₇

C₃

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₉×C₃₇ ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ C₃²×C₉ ◁ #6868

N	H ≅ G/N	type
1. C₃×#15688521	C₃	semidirect
3. #22582441	C₃	semidirect
4. C₃₇×#6868	C₃	semidirect
48. #6868	C₃₇⋊C₃	semidirect
65. C₃₇	#10624481	semidirect

H ≅ G/N

type

1. C₃×#15688521

C₃

semidirect

3. #22582441

C₃

semidirect

4. C₃₇×#6868

C₃

semidirect

48. #6868

C₃₇⋊C₃

semidirect

65. C₃₇

#10624481

semidirect

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

subgroup	index
C₃×#15688521	3
C₃×#15688519	3
#22582441	3
C₃₇×#6868	3
#10624481	37

subgroup

index

C₃₇×#6868

Subgroups — #41004856

Conjugacy classes of subgroups

Complements

Maximal subgroups