Group #41005286 — subgroups

Subgroups — C₃₇⋊C₃×#6860 #41005286

Order 26973.

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

normal subgroups only

This group has 476 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	C₃×3₋¹⁺²×C₃₇⋊C₃	C₃	1	2	yes	maximal
2	C₃²⋊C₉×C₃₇⋊C₃	C₃	1	2	yes	maximal
3	#22582455	C₃	1	4	yes	maximal, complemented
4	#22582454	C₃	1	4	yes	maximal, complemented
5	C₃₇×#6860	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
7	C₃×#11758608	C₃²	1	4	yes	—
8	C₃×#11758605	—	3	2	no	—
9	C₃×#11758606	—	3	2	no	—
10	3₋¹⁺²×C₃₇⋊C₃	—	3	18	no	—
11	C₃×C₉×C₃₇⋊C₃	—	3	2	no	—
12	C₃×C₉×C₃₇⋊C₃	—	3	2	no	—
13	C₃²⋊#66524	C₃²	1	4	yes	—
14	#15688502	—	3	2	no	—
15	#15688502	—	3	2	no	—
16	C₃×3₋¹⁺²×C₃₇	C₃²	1	2	yes	—
17	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
18	C₃³×C₃₇	C₃³	1	1	yes	characteristic
19	C₃²×C₃₇⋊C₃	—	3	6	no	—
20	C₃²×C₃₇⋊C₃	—	3	6	no	—
21	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
22	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
23	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
24	3₋¹⁺²×C₃₇	—	3	6	no	—
25	C₃×C₉×C₃₇	—	3	2	no	—
26	C₃×C₉×C₃₇	—	3	2	no	—
27	#11758608	—	3	12	no	—
28	C₃×#66524	—	3	4	no	—
29	C₃×#66524	—	3	4	no	—
30	#11758606	—	3	18	no	—
31	#11758605	—	3	18	no	—
32	C₉×C₃₇⋊C₃	—	3	18	no	—
33	C₉×C₃₇⋊C₃	—	9	6	no	—
34	C₃×#6860	—	37	1	no	maximal, Sylow
35	C₃²×C₃₇	—	3	2	no	—
36	C₃²×C₃₇	—	3	2	no	—
37	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
38	C₃×C₃₇⋊C₃	—	3	2	no	—
39	C₃×C₃₇⋊C₃	—	3	2	no	—
40	C₃×C₃₇⋊C₃	—	3	2	no	—
41	C₃×C₃₇⋊C₃	—	3	2	no	—
42	C₃×C₃₇⋊C₃	—	9	9	no	—
43	C₃×C₃₇⋊C₃	AΣL(1,𝔽₂₇)	1	6	yes	—
44	C₃×C₃₇⋊C₃	#733	1	6	yes	—
45	C₉×C₃₇	—	3	6	no	—
46	C₉×C₃₇	—	9	2	no	—
47	#66524	—	3	12	no	—
48	#66524	—	9	4	no	—
49	C₃²×3₋¹⁺²	—	37	2	no	—
50	C₃×C₃²⋊C₉	—	37	2	no	—
51	#6860	—	37	4	no	—
52	#6860	—	37	4	no	—
53	#6860	C₃₇⋊C₃	1	1	yes	characteristic, direct product factor
54	C₃×C₃₇	—	9	1	no	—
55	C₃×C₃₇	C₃×AΣL(1,𝔽₂₇)	1	2	yes	—
56	C₃×C₃₇	C₃×#733	1	2	yes	—
57	C₃₇⋊C₃	—	9	1	no	—
58	C₃₇⋊C₃	—	9	1	no	—
59	C₃₇⋊C₃	#6860	1	9	yes	direct product factor
60	C₃⁴	—	37	1	no	—
61	C₃×3₋¹⁺²	—	37	4	no	—
62	C₃×3₋¹⁺²	—	111	2	no	—
63	C₃×3₋¹⁺²	—	111	2	no	—
64	C₃×3₋¹⁺²	—	111	18	no	—
65	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	2	yes	—
66	C₃²×C₉	—	111	2	no	—
67	C₃²×C₉	—	111	2	no	—
68	C₃²⋊C₉	—	37	4	no	—
69	C₃²⋊C₉	—	111	2	no	—
70	C₃²⋊C₉	—	111	2	no	—
71	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	—
72	C₃₇	C₃×#6860	1	1	yes	characteristic, complemented, Sylow
73	C₃³	—	37	1	no	—
74	C₃³	—	37	1	no	—
75	C₃³	—	37	1	no	—
76	C₃³	—	111	6	no	—
77	C₃³	—	111	6	no	—
78	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
79	3₋¹⁺²	—	3	6	no	—
80	3₋¹⁺²	—	111	18	no	—
81	3₋¹⁺²	—	111	18	no	—
82	3₋¹⁺²	—	111	12	no	—
83	C₃×C₉	—	3	2	no	—
84	C₃×C₉	—	3	2	no	—
85	C₃×C₉	—	333	6	no	—
86	C₃×C₉	—	111	18	no	—
87	C₃×C₉	—	111	4	no	—
88	C₃×C₉	—	111	4	no	—
89	C₃²	—	3	2	no	—
90	C₃²	—	3	2	no	—
91	C₃²	—	37	6	no	—
92	C₃²	—	37	6	no	—
93	C₃²	—	333	9	no	—
94	C₃²	—	111	2	no	—
95	C₃²	—	111	2	no	—
96	C₃²	—	111	2	no	—
97	C₃²	—	111	2	no	—
98	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
99	C₉	—	3	6	no	—
100	C₉	—	9	2	no	—
101	C₉	—	333	4	no	—
102	C₉	—	111	12	no	—
103	C₃	—	9	1	no	—
104	C₃	—	37	9	no	—
105	C₃	—	333	1	no	—
106	C₃	—	333	1	no	—
107	C₃	AΣL(1,𝔽₂₇)×C₃₇⋊C₃	1	2	yes	—
108	C₃	#733×C₃₇⋊C₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×3₋¹⁺²×C₃₇⋊C₃

C₃

yes

maximal

C₃²⋊C₉×C₃₇⋊C₃

C₃

yes

maximal

#22582455

C₃

yes

maximal, complemented

#22582454

C₃

yes

maximal, complemented

C₃₇×#6860

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

C₃²

yes

characteristic

C₃×#11758608

C₃²

yes

—

C₃×#11758605

—

C₃×#11758606

—

3₋¹⁺²×C₃₇⋊C₃

—

C₃×C₉×C₃₇⋊C₃

—

C₃×C₉×C₃₇⋊C₃

—

C₃²⋊#66524

C₃²

yes

—

#15688502

—

#15688502

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

yes

C₃²×C₃₇⋊C₃

yes

C₃²×C₃₇⋊C₃

yes

3₋¹⁺²×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

—

—

—

—

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₃×#6860

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

AΣL(1,𝔽₂₇)

yes

—

C₃×C₃₇⋊C₃

#733

yes

—

C₉×C₃₇

—

C₉×C₃₇

—

#66524

—

#66524

—

C₃²×3₋¹⁺²

—

C₃×C₃²⋊C₉

—

#6860

—

#6860

—

#6860

C₃₇⋊C₃

yes

characteristic, direct product factor

C₃×C₃₇

—

C₃×C₃₇

C₃×AΣL(1,𝔽₂₇)

yes

—

C₃×C₃₇

C₃×#733

yes

—

C₃₇⋊C₃

—

C₃₇⋊C₃

—

C₃₇⋊C₃

#6860

yes

direct product factor

C₃⁴

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

—

C₃²×C₉

—

111

—

C₃²×C₉

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₃₇

C₃×#6860

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

333

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

333

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

C₉

—

100

C₉

—

101

C₉

—

333

—

102

C₉

—

111

—

103

C₃

—

104

C₃

—

105

C₃

—

333

—

106

C₃

—

333

—

107

C₃

AΣL(1,𝔽₂₇)×C₃₇⋊C₃

yes

—

108

C₃

#733×C₃₇⋊C₃

yes

—

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6860	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	#6860	C₃₇⋊C₃
hypocenter	C₃₇	C₃×#6860
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ #6860

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6860

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃³×C₃₇⋊C₃

C₃²

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ #6860

N	H ≅ G/N	type
3. #22582455	C₃	semidirect
4. #22582454	C₃	semidirect
5. C₃₇×#6860	C₃	semidirect
53. #6860	C₃₇⋊C₃	direct (N × H)
59. C₃₇⋊C₃	#6860	direct (N × H)
72. C₃₇	C₃×#6860	semidirect

H ≅ G/N

type

3. #22582455

C₃

semidirect

4. #22582454

C₃

semidirect

5. C₃₇×#6860

C₃

semidirect

53. #6860

C₃₇⋊C₃

direct (N × H)

59. C₃₇⋊C₃

#6860

direct (N × H)

72. C₃₇

C₃×#6860

semidirect

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

subgroup	index
C₃×3₋¹⁺²×C₃₇⋊C₃	3
C₃²⋊C₉×C₃₇⋊C₃	3
#22582455	3
#22582454	3
C₃₇×#6860	3
C₃×#6860	37

subgroup

index

C₃×3₋¹⁺²×C₃₇⋊C₃

C₃²⋊C₉×C₃₇⋊C₃

#22582455

#22582454

C₃₇×#6860

C₃×#6860

Subgroups — C₃₇⋊C₃×#6860 #41005286

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C37⋊C3×#6860 #41005286

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇⋊C₃×#6860 #41005286