Group #41006510 — subgroups

Subgroups — #41006510

Order 26973.

596 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₃²×C₃²⋊(C₃₇⋊C₃)	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#15688514	C₃	1	9	yes	maximal, complemented
3	#22582554	C₃	1	1	yes	maximal, characteristic, complemented
4	#22582550	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃₇×#6891	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃³×C₃₇⋊C₃	C₃²	1	3	yes	—
7	C₃×#11758608	C₃²	1	3	yes	—
8	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
9	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	9	no	—
10	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
11	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	3	yes	—
12	#15688514	—	3	27	no	—
13	C₃²⋊#66524	—	3	3	no	—
14	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
15	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
16	C₃₇×C₃²⋊C₉	—	3	3	no	—
17	C₃³×C₃₇	—	3	3	no	—
18	C₃³×C₃₇	—	9	3	no	—
19	C₃³×C₃₇	H(3,𝔽₃)	1	3	yes	—
20	C₃³×C₃₇	C₃³	1	1	yes	characteristic
21	C₃²×C₃₇⋊C₃	—	3	6	no	—
22	C₃²×C₃₇⋊C₃	—	3	3	no	—
23	C₃²×C₃₇⋊C₃	—	3	3	no	—
24	C₃²×C₃₇⋊C₃	—	9	9	no	—
25	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	9	yes	complemented
26	3₋¹⁺²×C₃₇	—	3	9	no	—
27	C₃×C₉×C₃₇	—	3	3	no	—
28	C₃²⋊(C₃₇⋊C₃)	—	3	27	no	—
29	C₃²⋊(C₃₇⋊C₃)	—	3	9	no	—
30	C₃²⋊(C₃₇⋊C₃)	—	9	9	no	—
31	#11758608	—	3	9	no	—
32	C₃×#66524	—	3	3	no	—
33	#10624735	—	37	1	no	maximal, Sylow
34	C₃²×C₃₇	—	3	3	no	—
35	C₃²×C₃₇	—	3	3	no	—
36	C₃²×C₃₇	—	3	3	no	—
37	C₃²×C₃₇	—	3	6	no	—
38	C₃²×C₃₇	—	9	9	no	—
39	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
40	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
41	C₃×C₃₇⋊C₃	—	3	9	no	—
42	C₃×C₃₇⋊C₃	—	9	18	no	—
43	C₃×C₃₇⋊C₃	—	9	9	no	—
44	C₃×C₃₇⋊C₃	—	9	9	no	—
45	C₃×C₃₇⋊C₃	—	9	3	no	—
46	C₃×C₃₇⋊C₃	—	9	9	no	—
47	C₉×C₃₇	—	9	3	no	—
48	#66524	—	9	3	no	—
49	C₃²×H(3,𝔽₃)	—	37	1	no	—
50	C₃×AΣL(1,𝔽₂₇)	—	37	9	no	—
51	#6891	—	37	1	no	—
52	#6891	C₃₇⋊C₃	1	1	yes	characteristic, complemented
53	C₃²⋊H(3,𝔽₃)	—	37	1	no	—
54	C₃×C₃₇	—	3	3	no	—
55	C₃×C₃₇	—	9	3	no	—
56	C₃×C₃₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
57	C₃×C₃₇	#6909	1	3	yes	—
58	C₃₇⋊C₃	—	9	9	no	—
59	C₃₇⋊C₃	—	27	3	no	—
60	C₃⁴	—	37	3	no	—
61	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
62	C₃×3₋¹⁺²	—	37	3	no	—
63	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	3	yes	—
64	C₃×H(3,𝔽₃)	—	37	3	no	—
65	C₃×H(3,𝔽₃)	—	111	3	no	—
66	C₃×H(3,𝔽₃)	—	111	9	no	—
67	C₃×H(3,𝔽₃)	—	111	3	no	—
68	AΣL(1,𝔽₂₇)	—	111	27	no	—
69	C₃²⋊C₉	—	3	3	no	—
70	C₃²⋊C₉	—	111	3	no	—
71	C₃₇	#10624735	1	1	yes	characteristic, complemented, Sylow
72	C₃³	—	3	3	no	—
73	C₃³	—	9	3	no	—
74	C₃³	—	37	9	no	—
75	C₃³	—	333	9	no	—
76	C₃³	—	111	3	no	—
77	C₃³	—	111	6	no	—
78	C₃³	—	111	3	no	—
79	C₃³	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
80	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
81	3₋¹⁺²	—	3	9	no	—
82	3₋¹⁺²	—	111	9	no	—
83	H(3,𝔽₃)	—	333	9	no	—
84	H(3,𝔽₃)	—	111	27	no	—
85	H(3,𝔽₃)	—	111	9	no	—
86	C₃×C₉	—	3	3	no	—
87	C₃×C₉	—	111	3	no	—
88	C₃²	—	3	3	no	—
89	C₃²	—	3	3	no	—
90	C₃²	—	3	3	no	—
91	C₃²	—	3	6	no	—
92	C₃²	—	9	9	no	—
93	C₃²	—	333	9	no	—
94	C₃²	—	333	18	no	—
95	C₃²	—	333	9	no	—
96	C₃²	—	333	3	no	—
97	C₃²	—	333	9	no	—
98	C₃²	—	111	9	no	—
99	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	3	yes	—
100	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
101	C₉	—	9	3	no	—
102	C₉	—	333	3	no	—
103	C₃	—	3	3	no	—
104	C₃	—	9	3	no	—
105	C₃	—	333	9	no	—
106	C₃	—	999	3	no	—
107	C₃	(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	1	1	yes	characteristic
108	C₃	#22582639	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₃²⋊(C₃₇⋊C₃)

C₃

yes

maximal, characteristic, complemented

C₃×#15688514

C₃

yes

maximal, complemented

#22582554

C₃

yes

maximal, characteristic, complemented

#22582550

C₃

yes

maximal, characteristic, complemented

C₃₇×#6891

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

C₃²

yes

—

C₃×#11758608

C₃²

yes

—

C₃×C₃²⋊(C₃₇⋊C₃)

—

C₃×C₃²⋊(C₃₇⋊C₃)

—

C₃×C₃²⋊(C₃₇⋊C₃)

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

—

#15688514

—

C₃²⋊#66524

—

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃₇×C₃²⋊C₉

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

H(3,𝔽₃)

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

H(3,𝔽₃)

yes

complemented

3₋¹⁺²×C₃₇

—

C₃×C₉×C₃₇

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

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

—

C₉×C₃₇

—

#66524

—

C₃²×H(3,𝔽₃)

—

C₃×AΣL(1,𝔽₂₇)

—

#6891

—

#6891

C₃₇⋊C₃

yes

characteristic, complemented

C₃²⋊H(3,𝔽₃)

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃²⋊H(3,𝔽₃)

yes

characteristic

C₃×C₃₇

#6909

yes

—

C₃₇⋊C₃

—

C₃₇⋊C₃

—

C₃⁴

—

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

111

—

C₃×H(3,𝔽₃)

—

111

—

C₃×H(3,𝔽₃)

—

111

—

AΣL(1,𝔽₂₇)

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃₇

#10624735

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

—

333

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

111

—

H(3,𝔽₃)

—

333

—

H(3,𝔽₃)

—

111

—

H(3,𝔽₃)

—

111

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

333

—

C₃²

—

333

—

C₃²

—

333

—

C₃²

—

333

—

C₃²

—

333

—

C₃²

—

111

—

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

—

100

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

101

C₉

—

102

C₉

—

333

—

103

C₃

—

104

C₃

—

105

C₃

—

333

—

106

C₃

—

999

—

107

C₃

(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)

yes

characteristic

108

C₃

#22582639

yes

—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6891	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
hypercenter	#6891	C₃₇⋊C₃
hypocenter	C₃₇	#10624735
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ #6891

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6891

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ #6891

N	H ≅ G/N	type
1. C₃²×C₃²⋊(C₃₇⋊C₃)	C₃	semidirect
2. C₃×#15688514	C₃	semidirect
3. #22582554	C₃	semidirect
4. #22582550	C₃	semidirect
5. C₃₇×#6891	C₃	semidirect
25. C₃²×C₃₇⋊C₃	H(3,𝔽₃)	semidirect
52. #6891	C₃₇⋊C₃	semidirect
71. C₃₇	#10624735	semidirect

H ≅ G/N

type

1. C₃²×C₃²⋊(C₃₇⋊C₃)

C₃

semidirect

2. C₃×#15688514

C₃

semidirect

3. #22582554

C₃

semidirect

4. #22582550

C₃

semidirect

5. C₃₇×#6891

C₃

semidirect

25. C₃²×C₃₇⋊C₃

H(3,𝔽₃)

semidirect

52. #6891

C₃₇⋊C₃

semidirect

71. C₃₇

#10624735

semidirect

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

subgroup	index
C₃²×C₃²⋊(C₃₇⋊C₃)	3
C₃×#15688514	3
#22582554	3
#22582550	3
C₃₇×#6891	3
#10624735	37

subgroup

index

C₃²×C₃²⋊(C₃₇⋊C₃)

C₃₇×#6891

Subgroups — #41006510

Conjugacy classes of subgroups

Complements

Maximal subgroups