Group #41008407 — subgroups

Subgroups — #41008407

Order 26973.

142 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 27 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×#15688538	C₃	1	1	yes	maximal, characteristic, complemented
2	#22582693	C₃	1	2	yes	maximal, complemented
3	#732×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
4	C₃²×#66526	—	3	1	no	—
5	#15688538	—	3	3	no	—
6	C₃²⋊#66526	—	3	2	no	—
7	H(3,𝔽₃)×C₃₇⋊C₃	—	3	1	no	—
8	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
9	#15688519	C₉	1	2	yes	complemented
10	#15688518	—	3	2	no	—
11	C₃₇×#732	C₉	1	1	yes	characteristic, complemented
12	C₃²×C₃₇⋊C₃	—	9	1	no	—
13	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
14	C₃×#66526	—	3	3	no	—
15	C₃×#66526	—	9	2	no	—
16	C₃×#66526	—	9	1	no	—
17	C₃×#66526	—	9	2	no	—
18	H(3,𝔽₃)×C₃₇	—	3	1	no	—
19	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
20	C₃²⋊(C₃₇⋊C₃)	—	3	2	no	—
21	C₃²⋊(C₃₇⋊C₃)	—	9	1	no	—
22	C₃²⋊(C₃₇⋊C₃)	—	9	1	no	—
23	C₃×#66524	3₋¹⁺²	1	2	yes	—
24	C₉×C₃₇⋊C₃	—	3	1	no	—
25	C₉×C₃₇⋊C₃	—	3	1	no	—
26	C₉×C₃₇⋊C₃	—	3	1	no	—
27	#10624527	—	37	1	no	maximal, Sylow
28	C₃²×C₃₇	—	9	1	no	—
29	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
30	C₃×C₃₇⋊C₃	—	3	2	no	—
31	C₃×C₃₇⋊C₃	—	3	1	no	—
32	C₃×C₃₇⋊C₃	—	9	2	no	—
33	C₃×C₃₇⋊C₃	—	27	2	no	—
34	C₃×C₃₇⋊C₃	—	27	1	no	—
35	C₃×C₃₇⋊C₃	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
36	C₃×C₃₇⋊C₃	#731	1	1	yes	characteristic
37	C₃×C₃₇⋊C₃	#732	1	1	yes	characteristic
38	#66526	—	9	3	no	—
39	#66526	—	27	2	no	—
40	C₉×C₃₇	—	3	1	no	—
41	#66524	—	3	2	no	—
42	C₃×#732	—	37	1	no	—
43	C₃×C₉⋊C₉	—	37	1	no	—
44	#6868	—	37	2	no	—
45	C₃×C₃₇	—	3	1	no	—
46	C₃×C₃₇	—	27	1	no	—
47	C₃×C₃₇	H(3,𝔽₃)⋊C₉	1	1	yes	characteristic
48	C₃₇⋊C₃	—	3	1	no	—
49	C₃₇⋊C₃	—	3	1	no	—
50	C₃₇⋊C₃	—	27	2	no	—
51	C₃₇⋊C₃	#6881	1	2	yes	—
52	C₃₇⋊C₃	#6883	1	1	yes	characteristic
53	C₃×H(3,𝔽₃)	—	111	1	no	—
54	C₃²×C₉	—	37	1	no	—
55	C₃²×C₉	—	111	1	no	—
56	#732	—	37	2	no	—
57	#732	—	111	2	no	—
58	#732	#66526	1	1	yes	characteristic, complemented
59	C₉⋊C₉	—	111	3	no	—
60	C₃²⋊C₉	—	111	2	no	—
61	C₃₇	#10624527	1	1	yes	characteristic, complemented, Sylow
62	C₃³	—	37	1	no	—
63	C₃³	—	333	1	no	—
64	H(3,𝔽₃)	—	3	1	no	—
65	H(3,𝔽₃)	—	333	1	no	—
66	H(3,𝔽₃)	—	333	1	no	—
67	H(3,𝔽₃)	—	111	2	no	—
68	C₃×C₉	—	37	2	no	—
69	C₃×C₉	—	333	2	no	—
70	C₃×C₉	—	333	1	no	—
71	C₃×C₉	—	333	2	no	—
72	C₃×C₉	—	111	1	no	—
73	C₃×C₉	—	111	1	no	—
74	C₃×C₉	—	111	1	no	—
75	C₃×C₉	—	111	3	no	—
76	C₃×C₉	C₃×#66526	1	1	yes	characteristic
77	C₃²	—	9	1	no	—
78	C₃²	—	37	1	no	—
79	C₃²	—	37	1	no	—
80	C₃²	—	37	1	no	—
81	C₃²	—	333	2	no	—
82	C₃²	—	111	2	no	—
83	C₃²	—	111	1	no	—
84	C₃²	—	999	2	no	—
85	C₃²	—	999	1	no	—
86	C₃²	C₃²⋊#66526	1	1	yes	characteristic
87	C₉	—	3	1	no	—
88	C₉	—	333	3	no	—
89	C₉	—	111	2	no	—
90	C₉	—	999	2	no	—
91	C₃	—	3	1	no	—
92	C₃	—	27	1	no	—
93	C₃	—	37	2	no	—
94	C₃	—	37	1	no	—
95	C₃	—	111	1	no	—
96	C₃	—	111	1	no	—
97	C₃	—	999	2	no	—
98	C₃	H(3,𝔽₃)⋊#66526	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688538

C₃

yes

maximal, characteristic, complemented

#22582693

C₃

yes

maximal, complemented

#732×C₃₇⋊C₃

C₃

yes

maximal, characteristic

C₃²×#66526

—

#15688538

—

C₃²⋊#66526

—

H(3,𝔽₃)×C₃₇⋊C₃

—

C₃×C₉×C₃₇⋊C₃

C₃²

yes

characteristic

#15688519

C₉

yes

complemented

#15688518

—

C₃₇×#732

C₉

yes

characteristic, complemented

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

yes

—

—

—

—

H(3,𝔽₃)×C₃₇

—

C₃×C₉×C₃₇

C₃×C₉

yes

characteristic

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

C₃×#66524

3₋¹⁺²

yes

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

#10624527

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

C₃²⋊C₉

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

yes

C₃×C₃₇⋊C₃

yes

—

—

C₉×C₃₇

—

#66524

—

C₃×#732

—

C₃×C₉⋊C₉

—

#6868

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

H(3,𝔽₃)⋊C₉

yes

characteristic

C₃₇⋊C₃

—

C₃₇⋊C₃

—

C₃₇⋊C₃

—

C₃₇⋊C₃

#6881

yes

—

C₃₇⋊C₃

#6883

yes

characteristic

C₃×H(3,𝔽₃)

—

111

—

C₃²×C₉

—

C₃²×C₉

—

111

—

#732

—

#732

—

111

—

#732

#66526

yes

characteristic, complemented

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃₇

#10624527

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

333

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

333

—

H(3,𝔽₃)

—

333

—

H(3,𝔽₃)

—

111

—

C₃×C₉

—

C₃×C₉

—

333

—

C₃×C₉

—

333

—

C₃×C₉

—

333

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

C₃×#66526

yes

characteristic

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

333

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

—

999

—

C₃²

—

999

—

C₃²

C₃²⋊#66526

yes

characteristic

C₉

—

C₉

—

333

—

C₉

—

111

—

C₉

—

999

—

C₃

—

C₃

—

C₃

—

C₃

—

C₃

—

111

—

C₃

—

111

—

C₃

—

999

—

C₃

H(3,𝔽₃)⋊#66526

yes

characteristic

	subgroup	G/H
center	C₃	H(3,𝔽₃)⋊#66526
commutator	C₃×C₉×C₃₇	C₃×C₉
Frattini	C₃×C₉	C₃×#66526
Fitting	C₃₇×#732	C₉
socle	C₃×C₃₇	H(3,𝔽₃)⋊C₉
prime radical	#732×C₃₇⋊C₃	C₃
cosocle	C₃×C₉×C₃₇⋊C₃	C₃²
hypercenter	#732	#66526
hypocenter	C₃₇	#10624527
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉ ◁ #732

subgroup

G/H

center

C₃

H(3,𝔽₃)⋊#66526

commutator

C₃×C₉×C₃₇

C₃×C₉

Frattini

C₃×C₉

C₃×#66526

Fitting

C₃₇×#732

C₉

socle

C₃×C₃₇

H(3,𝔽₃)⋊C₉

prime radical

#732×C₃₇⋊C₃

C₃

cosocle

C₃×C₉×C₃₇⋊C₃

C₃²

C₃₇

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

UCS

C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉ ◁ #732

N	H ≅ G/N	type
1. C₃×#15688538	C₃	semidirect
2. #22582693	C₃	semidirect
9. #15688519	C₉	semidirect
11. C₃₇×#732	C₉	semidirect
58. #732	#66526	semidirect
61. C₃₇	#10624527	semidirect

H ≅ G/N

type

1. C₃×#15688538

C₃

semidirect

2. #22582693

C₃

semidirect

9. #15688519

C₉

semidirect

11. C₃₇×#732

C₉

semidirect

58. #732

#66526

semidirect

61. C₃₇

#10624527

semidirect

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

subgroup	index
C₃×#15688538	3
#22582693	3
#732×C₃₇⋊C₃	3
#10624527	37

subgroup

index

C₃×#15688538

#22582693

#732×C₃₇⋊C₃

#10624527

Subgroups — #41008407

Conjugacy classes of subgroups

Complements

Maximal subgroups