Group #41004954 — subgroups

Subgroups — #41004954

Order 26973.

350 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 31 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
2	#22582468	C₃	1	2	yes	maximal, complemented
3	C₃₇×#6866	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃³×C₃₇⋊C₃	C₉	1	3	yes	complemented
5	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
6	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
7	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
8	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
9	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	6	no	—
10	C₃²⋊#66524	—	3	2	no	—
11	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
12	C₃₇×C₃²⋊C₉	—	3	1	no	—
13	C₃³×C₃₇	—	3	1	no	—
14	C₃³×C₃₇	—	3	2	no	—
15	C₃³×C₃₇	—	9	3	no	—
16	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
17	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
18	C₃³×C₃₇	3₋¹⁺²	1	1	yes	characteristic
19	C₃³×C₃₇	3₋¹⁺²	1	1	yes	characteristic
20	C₃²×C₃₇⋊C₃	—	3	3	no	—
21	C₃²×C₃₇⋊C₃	—	3	9	no	—
22	C₃²×C₃₇⋊C₃	—	9	9	no	—
23	C₃×C₉×C₃₇	—	9	1	no	—
24	C₃²⋊(C₃₇⋊C₃)	—	3	9	no	—
25	C₃²⋊(C₃₇⋊C₃)	—	3	9	no	—
26	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
27	C₃²⋊(C₃₇⋊C₃)	—	3	6	no	—
28	C₃×#66524	—	9	2	no	—
29	#10624505	—	37	1	no	maximal, Sylow
30	C₃²×C₃₇	—	3	3	no	—
31	C₃²×C₃₇	—	3	1	no	—
32	C₃²×C₃₇	—	3	1	no	—
33	C₃²×C₃₇	—	3	1	no	—
34	C₃²×C₃₇	—	9	1	no	—
35	C₃²×C₃₇	—	9	6	no	—
36	C₃²×C₃₇	—	9	3	no	—
37	C₃²×C₃₇	—	9	1	no	—
38	C₃²×C₃₇	—	9	1	no	—
39	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
40	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
41	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	2	yes	—
42	C₃×C₃₇⋊C₃	—	3	9	no	—
43	C₃×C₃₇⋊C₃	—	9	9	no	—
44	C₃×C₃₇⋊C₃	—	9	27	no	—
45	C₉×C₃₇	—	27	1	no	—
46	#66524	—	27	2	no	—
47	C₃²×H(3,𝔽₃)	—	37	1	no	—
48	#6866	—	37	2	no	—
49	#6866	C₃₇⋊C₃	1	1	yes	characteristic, complemented
50	C₃×C₃₇	—	3	2	no	—
51	C₃×C₃₇	—	3	1	no	—
52	C₃×C₃₇	—	3	3	no	—
53	C₃×C₃₇	—	3	1	no	—
54	C₃×C₃₇	—	9	1	no	—
55	C₃×C₃₇	—	9	1	no	—
56	C₃×C₃₇	#6866	1	1	yes	characteristic
57	C₃₇⋊C₃	—	9	9	no	—
58	C₃⁴	—	37	3	no	—
59	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
60	C₃×H(3,𝔽₃)	—	111	1	no	—
61	C₃×H(3,𝔽₃)	—	111	1	no	—
62	C₃×H(3,𝔽₃)	—	111	1	no	—
63	C₃×H(3,𝔽₃)	—	111	3	no	—
64	C₃×H(3,𝔽₃)	—	111	6	no	—
65	C₃²⋊C₉	—	3	1	no	—
66	C₃²⋊C₉	—	111	2	no	—
67	C₃₇	#10624505	1	1	yes	characteristic, complemented, Sylow
68	C₃³	—	3	1	no	—
69	C₃³	—	3	2	no	—
70	C₃³	—	9	3	no	—
71	C₃³	—	333	9	no	—
72	C₃³	—	111	9	no	—
73	C₃³	—	111	3	no	—
74	C₃³	C₉×C₃₇⋊C₃	1	1	yes	characteristic
75	C₃³	#11758605	1	1	yes	characteristic
76	C₃³	#11758606	1	1	yes	characteristic
77	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
78	H(3,𝔽₃)	—	111	9	no	—
79	H(3,𝔽₃)	—	111	9	no	—
80	H(3,𝔽₃)	—	111	3	no	—
81	H(3,𝔽₃)	—	111	6	no	—
82	C₃×C₉	—	9	1	no	—
83	C₃×C₉	—	333	2	no	—
84	C₃²	—	3	3	no	—
85	C₃²	—	3	1	no	—
86	C₃²	—	3	1	no	—
87	C₃²	—	3	1	no	—
88	C₃²	—	9	1	no	—
89	C₃²	—	9	3	no	—
90	C₃²	—	9	6	no	—
91	C₃²	—	9	1	no	—
92	C₃²	—	9	1	no	—
93	C₃²	—	333	27	no	—
94	C₃²	—	333	9	no	—
95	C₃²	—	111	9	no	—
96	C₃²	#15688502	1	1	yes	characteristic
97	C₃²	#15688512	1	1	yes	characteristic
98	C₃²	#15688514	1	2	yes	—
99	C₉	—	27	1	no	—
100	C₉	—	999	2	no	—
101	C₃	—	3	1	no	—
102	C₃	—	3	2	no	—
103	C₃	—	3	1	no	—
104	C₃	—	3	3	no	—
105	C₃	—	9	1	no	—
106	C₃	—	9	1	no	—
107	C₃	—	333	9	no	—
108	C₃	#22582467	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₃²⋊(C₃₇⋊C₃)

C₃

yes

maximal, characteristic

#22582468

C₃

yes

maximal, complemented

C₃₇×#6866

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

C₉

yes

complemented

C₃×C₃²⋊(C₃₇⋊C₃)

—

C₃×C₃²⋊(C₃₇⋊C₃)

—

C₃×C₃²⋊(C₃₇⋊C₃)

—

C₃×C₃²⋊(C₃₇⋊C₃)

—

C₃×C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊#66524

—

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃₇×C₃²⋊C₉

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

C₃×C₉

yes

characteristic

C₃³×C₃₇

H(3,𝔽₃)

yes

characteristic

C₃³×C₃₇

3₋¹⁺²

yes

characteristic

C₃³×C₃₇

3₋¹⁺²

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃×C₉×C₃₇

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

C₃×#66524

—

#10624505

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃²⋊C₉

yes

characteristic

C₃²×C₃₇

AΣL(1,𝔽₂₇)

yes

characteristic

C₃²×C₃₇

AΣL(1,𝔽₂₇)

yes

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

#66524

—

C₃²×H(3,𝔽₃)

—

#6866

—

#6866

C₃₇⋊C₃

yes

characteristic, complemented

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

#6866

yes

characteristic

C₃₇⋊C₃

—

C₃⁴

—

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃×H(3,𝔽₃)

—

111

—

C₃×H(3,𝔽₃)

—

111

—

C₃×H(3,𝔽₃)

—

111

—

C₃×H(3,𝔽₃)

—

111

—

C₃×H(3,𝔽₃)

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃₇

#10624505

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

—

333

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

C₉×C₃₇⋊C₃

yes

C₃³

yes

C₃³

yes

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

H(3,𝔽₃)

—

111

—

H(3,𝔽₃)

—

111

—

H(3,𝔽₃)

—

111

—

H(3,𝔽₃)

—

111

—

C₃×C₉

—

C₃×C₉

—

333

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

333

—

C₃²

—

333

—

C₃²

—

111

—

C₃²

yes

C₃²

yes

C₃²

yes

—

C₉

—

100

C₉

—

999

—

101

C₃

—

102

C₃

—

103

C₃

—

104

C₃

—

105

C₃

—

106

C₃

—

107

C₃

—

333

—

108

C₃

#22582467

yes

characteristic

	subgroup	G/H
center	C₃	#22582467
commutator	C₃³×C₃₇	C₃×C₉
Frattini	C₃⁴	C₃×C₃₇⋊C₃
Fitting	C₃₇×#6866	C₃
socle	C₃×C₃₇	#6866
prime radical	C₃²×C₃²⋊(C₃₇⋊C₃)	C₃
cosocle	C₃⁴×C₃₇	C₃²
hypercenter	#6866	C₃₇⋊C₃
hypocenter	C₃₇	#10624505
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃⁴ ◁ #6866

subgroup

G/H

center

C₃

#22582467

commutator

C₃³×C₃₇

C₃×C₉

Frattini

C₃⁴

C₃×C₃₇⋊C₃

Fitting

C₃₇×#6866

C₃

socle

C₃×C₃₇

#6866

prime radical

C₃²×C₃²⋊(C₃₇⋊C₃)

C₃

cosocle

C₃⁴×C₃₇

C₃²

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃ ◁ C₃³ ◁ C₃⁴ ◁ #6866

N	H ≅ G/N	type
2. #22582468	C₃	semidirect
3. C₃₇×#6866	C₃	semidirect
4. C₃³×C₃₇⋊C₃	C₉	semidirect
49. #6866	C₃₇⋊C₃	semidirect
67. C₃₇	#10624505	semidirect

H ≅ G/N

type

2. #22582468

C₃

semidirect

3. C₃₇×#6866

C₃

semidirect

4. C₃³×C₃₇⋊C₃

C₉

semidirect

49. #6866

C₃₇⋊C₃

semidirect

67. C₃₇

#10624505

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
#22582468	3
C₃₇×#6866	3
#10624505	37

subgroup

index

C₃²×C₃²⋊(C₃₇⋊C₃)

#22582468

C₃₇×#6866

#10624505

Subgroups — #41004954

Conjugacy classes of subgroups

Complements

Maximal subgroups