Group #41008373 — subgroups

Subgroups — #41008373

Order 26973.

350 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 24 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	#22582691	C₃	1	3	yes	maximal, complemented
2	C₃×H(3,𝔽₃)×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊#66526	—	3	3	no	—
4	C₃³×C₃₇⋊C₃	—	3	1	no	—
5	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
6	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
7	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
8	C₃×C₃²⋊(C₃₇⋊C₃)	C₉	1	2	yes	complemented
9	H(3,𝔽₃)×C₃₇⋊C₃	—	3	9	no	—
10	C₃×H(3,𝔽₃)×C₃₇	C₉	1	1	yes	characteristic, complemented
11	C₃³×C₃₇	—	3	1	no	—
12	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
13	C₃²×C₃₇⋊C₃	—	3	9	no	—
14	C₃²×C₃₇⋊C₃	—	3	2	no	—
15	C₃²×C₃₇⋊C₃	—	3	3	no	—
16	C₃²×C₃₇⋊C₃	—	9	9	no	—
17	C₃²×C₃₇⋊C₃	—	9	3	no	—
18	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
19	C₃²×C₃₇⋊C₃	3₋¹⁺²	1	1	yes	characteristic
20	C₃²×C₃₇⋊C₃	3₋¹⁺²	1	1	yes	characteristic
21	C₃×#66526	—	9	3	no	—
22	H(3,𝔽₃)×C₃₇	—	3	3	no	—
23	C₃²⋊(C₃₇⋊C₃)	—	3	9	no	—
24	C₃²⋊(C₃₇⋊C₃)	—	3	9	no	—
25	C₃²⋊(C₃₇⋊C₃)	—	3	6	no	—
26	#10624508	—	37	1	no	maximal, Sylow
27	C₃²×C₃₇	—	3	1	no	—
28	C₃²×C₃₇	—	3	3	no	—
29	C₃²×C₃₇	—	9	1	no	—
30	C₃²×C₃₇	—	9	3	no	—
31	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
32	C₃×C₃₇⋊C₃	—	3	1	no	—
33	C₃×C₃₇⋊C₃	—	3	3	no	—
34	C₃×C₃₇⋊C₃	—	3	1	no	—
35	C₃×C₃₇⋊C₃	—	3	6	no	—
36	C₃×C₃₇⋊C₃	—	9	1	no	—
37	C₃×C₃₇⋊C₃	—	9	9	no	—
38	C₃×C₃₇⋊C₃	—	9	27	no	—
39	C₃×C₃₇⋊C₃	—	9	6	no	—
40	C₃×C₃₇⋊C₃	—	9	1	no	—
41	C₃×C₃₇⋊C₃	AΣL(1,𝔽₂₇)	1	3	yes	—
42	#66526	—	27	3	no	—
43	C₃²×H(3,𝔽₃)	—	37	1	no	—
44	#6866	—	37	3	no	—
45	C₃×C₃₇	—	3	1	no	—
46	C₃×C₃₇	—	9	1	no	—
47	C₃×C₃₇	—	9	3	no	—
48	C₃×C₃₇	#6866	1	1	yes	characteristic
49	C₃₇⋊C₃	—	3	3	no	—
50	C₃₇⋊C₃	—	3	3	no	—
51	C₃₇⋊C₃	—	9	1	no	—
52	C₃₇⋊C₃	—	9	6	no	—
53	C₃⁴	—	37	1	no	—
54	C₃⁴	—	111	1	no	—
55	C₃×H(3,𝔽₃)	—	37	2	no	—
56	C₃×H(3,𝔽₃)	—	111	9	no	—
57	C₃×H(3,𝔽₃)	—	111	1	no	—
58	C₃×H(3,𝔽₃)	—	111	1	no	—
59	C₃×H(3,𝔽₃)	#66526	1	1	yes	characteristic, complemented
60	C₃²⋊C₉	—	111	3	no	—
61	C₃₇	#10624508	1	1	yes	characteristic, complemented, Sylow
62	C₃³	—	3	1	no	—
63	C₃³	—	37	1	no	—
64	C₃³	—	37	1	no	—
65	C₃³	—	37	1	no	—
66	C₃³	—	333	9	no	—
67	C₃³	—	333	3	no	—
68	C₃³	—	111	3	no	—
69	C₃³	—	111	9	no	—
70	C₃³	—	111	2	no	—
71	C₃³	C₃×#66526	1	1	yes	characteristic
72	H(3,𝔽₃)	—	3	3	no	—
73	H(3,𝔽₃)	—	111	9	no	—
74	H(3,𝔽₃)	—	111	6	no	—
75	H(3,𝔽₃)	—	111	9	no	—
76	C₃×C₉	—	333	3	no	—
77	C₃²	—	3	1	no	—
78	C₃²	—	3	3	no	—
79	C₃²	—	9	1	no	—
80	C₃²	—	9	3	no	—
81	C₃²	—	37	3	no	—
82	C₃²	—	333	1	no	—
83	C₃²	—	333	6	no	—
84	C₃²	—	333	1	no	—
85	C₃²	—	333	27	no	—
86	C₃²	—	333	9	no	—
87	C₃²	—	111	3	no	—
88	C₃²	—	111	1	no	—
89	C₃²	—	111	6	no	—
90	C₃²	—	111	1	no	—
91	C₃²	C₃²⋊#66526	1	1	yes	characteristic
92	C₉	—	999	3	no	—
93	C₃	—	3	1	no	—
94	C₃	—	9	3	no	—
95	C₃	—	9	1	no	—
96	C₃	—	333	6	no	—
97	C₃	—	333	1	no	—
98	C₃	—	111	3	no	—
99	C₃	—	111	3	no	—
100	C₃	#22582691	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22582691

C₃

yes

maximal, complemented

C₃×H(3,𝔽₃)×C₃₇⋊C₃

C₃

yes

maximal, characteristic

C₃²⋊#66526

—

C₃³×C₃₇⋊C₃

—

C₃³×C₃₇⋊C₃

C₃²

yes

characteristic

C₃×C₃²⋊(C₃₇⋊C₃)

—

C₃×C₃²⋊(C₃₇⋊C₃)

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₉

yes

complemented

H(3,𝔽₃)×C₃₇⋊C₃

—

C₃×H(3,𝔽₃)×C₃₇

C₉

yes

characteristic, complemented

C₃³×C₃₇

—

C₃³×C₃₇

C₃×C₉

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇⋊C₃

3₋¹⁺²

yes

characteristic

C₃²×C₃₇⋊C₃

3₋¹⁺²

yes

characteristic

C₃×#66526

—

H(3,𝔽₃)×C₃₇

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

#10624508

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

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

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

AΣL(1,𝔽₂₇)

yes

—

#66526

—

C₃²×H(3,𝔽₃)

—

#6866

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

#6866

yes

characteristic

C₃₇⋊C₃

—

C₃₇⋊C₃

—

C₃₇⋊C₃

—

C₃₇⋊C₃

—

C₃⁴

—

C₃⁴

—

111

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

111

—

C₃×H(3,𝔽₃)

—

111

—

C₃×H(3,𝔽₃)

—

111

—

C₃×H(3,𝔽₃)

#66526

yes

characteristic, complemented

C₃²⋊C₉

—

111

—

C₃₇

#10624508

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

333

—

C₃³

—

333

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

yes

—

—

111

—

H(3,𝔽₃)

—

111

—

H(3,𝔽₃)

—

111

—

C₃×C₉

—

333

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

333

—

C₃²

—

333

—

C₃²

—

333

—

C₃²

—

333

—

C₃²

—

333

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

C₃²⋊#66526

yes

characteristic

C₉

—

999

—

C₃

—

C₃

—

C₃

—

C₃

—

333

—

C₃

—

333

—

C₃

—

111

—

C₃

—

111

—

100

C₃

#22582691

yes

characteristic

	subgroup	G/H
center	C₃	#22582691
commutator	C₃³×C₃₇	C₃×C₉
Frattini	C₃³	C₃×#66526
Fitting	C₃×H(3,𝔽₃)×C₃₇	C₉
socle	C₃×C₃₇	#6866
prime radical	C₃×H(3,𝔽₃)×C₃₇⋊C₃	C₃
cosocle	C₃³×C₃₇⋊C₃	C₃²
hypercenter	C₃×H(3,𝔽₃)	#66526
hypocenter	C₃₇	#10624508
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃³ ◁ C₃×H(3,𝔽₃)

subgroup

G/H

center

C₃

#22582691

commutator

C₃³×C₃₇

C₃×C₉

Frattini

C₃³

C₃×#66526

Fitting

C₃×H(3,𝔽₃)×C₃₇

C₉

socle

C₃×C₃₇

#6866

prime radical

C₃×H(3,𝔽₃)×C₃₇⋊C₃

C₃

cosocle

C₃³×C₃₇⋊C₃

C₃²

C₃×H(3,𝔽₃)

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃ ◁ C₃² ◁ C₃³ ◁ C₃×H(3,𝔽₃)

N	H ≅ G/N	type
1. #22582691	C₃	semidirect
8. C₃×C₃²⋊(C₃₇⋊C₃)	C₉	semidirect
10. C₃×H(3,𝔽₃)×C₃₇	C₉	semidirect
59. C₃×H(3,𝔽₃)	#66526	semidirect
61. C₃₇	#10624508	semidirect

H ≅ G/N

type

1. #22582691

C₃

semidirect

8. C₃×C₃²⋊(C₃₇⋊C₃)

C₉

semidirect

10. C₃×H(3,𝔽₃)×C₃₇

C₉

semidirect

59. C₃×H(3,𝔽₃)

#66526

semidirect

61. C₃₇

#10624508

semidirect

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

subgroup	index
#22582691	3
C₃×H(3,𝔽₃)×C₃₇⋊C₃	3
#10624508	37

subgroup

index

#22582691

C₃×H(3,𝔽₃)×C₃₇⋊C₃

#10624508

Subgroups — #41008373

Conjugacy classes of subgroups

Complements

Maximal subgroups