Group #41008274 — subgroups

Subgroups — H(3,𝔽₃)⋊#11758603 #41008274

Order 26973.

144 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 42 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×#11758603	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃²⋊#11758603	C₃	1	2	yes	maximal, complemented
3	H(3,𝔽₃)×#66524	C₃	1	1	yes	maximal, characteristic
4	C₃×#11758603	—	3	1	no	—
5	C₃×#11758603	—	3	3	no	—
6	C₃×#11758603	—	3	2	no	—
7	C₃²×#66524	—	3	1	no	—
8	C₃²×#66524	C₃²	1	1	yes	characteristic
9	C₃²⋊#66524	—	3	2	no	—
10	C₃²⋊#66524	C₉	1	2	yes	—
11	C₃×H(3,𝔽₃)×C₃₇	C₉	1	1	yes	characteristic
12	C₃³×C₃₇	—	3	1	no	—
13	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
14	H(3,𝔽₃)×C₃₇	—	3	2	no	—
15	H(3,𝔽₃)×C₃₇	C₂₇	1	3	yes	complemented
16	C₃×#66524	—	3	1	no	—
17	C₃×#66524	—	3	1	no	—
18	C₃×#66524	—	3	1	no	—
19	C₃×#66524	—	3	2	no	—
20	C₃×#66524	—	9	1	no	—
21	C₃×#66524	—	9	1	no	—
22	C₃×#66524	—	9	1	no	—
23	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
24	C₃×#66524	3₋¹⁺²	1	2	yes	—
25	#11758603	—	3	3	no	—
26	#11758603	—	9	2	no	—
27	H(3,𝔽₃)⋊C₂₇	—	37	1	no	maximal, Sylow
28	C₃²×C₃₇	—	3	1	no	—
29	C₃²×C₃₇	—	3	1	no	—
30	C₃²×C₃₇	—	3	1	no	—
31	C₃²×C₃₇	—	3	3	no	—
32	C₃²×C₃₇	—	9	1	no	—
33	C₃²×C₃₇	—	9	1	no	—
34	C₃²×C₃₇	—	9	1	no	—
35	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
36	C₃²×C₃₇	C₃×C₂₇	1	1	yes	characteristic
37	C₃²×C₃₇	C₂₇⋊C₃	1	2	yes	—
38	#66524	—	3	2	no	—
39	#66524	—	9	2	no	—
40	#66524	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
41	#66524	#731	1	1	yes	characteristic
42	#66524	#732	1	1	yes	characteristic
43	C₃²×C₂₇	—	37	1	no	—
44	C₉×H(3,𝔽₃)	—	37	1	no	—
45	C₃²⋊C₂₇	—	37	2	no	—
46	C₃×C₃₇	—	3	1	no	—
47	C₃×C₃₇	—	3	2	no	—
48	C₃×C₃₇	—	9	3	no	—
49	C₃×C₃₇	C₃²⋊C₂₇	1	1	yes	characteristic
50	C₃×C₃₇	H(3,𝔽₃)⋊C₉	1	1	yes	characteristic
51	C₃×C₃₇	#6872	1	1	yes	characteristic
52	C₃×C₃₇	#6873	1	1	yes	characteristic
53	C₃×H(3,𝔽₃)	#66526	1	1	yes	characteristic
54	C₃²×C₉	—	37	1	no	—
55	C₃²×C₉	—	111	1	no	—
56	C₃×C₂₇	—	111	2	no	—
57	C₃×C₂₇	—	111	1	no	—
58	C₃×C₂₇	—	111	3	no	—
59	C₃²⋊C₉	—	37	2	no	—
60	C₃²⋊C₉	—	111	2	no	—
61	C₃₇	H(3,𝔽₃)⋊C₂₇	1	1	yes	characteristic, complemented, Sylow
62	C₃³	—	3	1	no	—
63	C₃³	C₃×#66526	1	1	yes	characteristic
64	H(3,𝔽₃)	—	3	2	no	—
65	H(3,𝔽₃)	#11758603	1	3	yes	complemented
66	C₃×C₉	—	37	1	no	—
67	C₃×C₉	—	37	2	no	—
68	C₃×C₉	—	333	1	no	—
69	C₃×C₉	—	333	1	no	—
70	C₃×C₉	—	333	1	no	—
71	C₃×C₉	—	111	1	no	—
72	C₃×C₉	—	111	1	no	—
73	C₃×C₉	—	111	1	no	—
74	C₃×C₉	—	111	2	no	—
75	C₂₇	—	333	2	no	—
76	C₂₇	—	111	3	no	—
77	C₃²	—	3	1	no	—
78	C₃²	—	3	1	no	—
79	C₃²	—	3	1	no	—
80	C₃²	—	3	3	no	—
81	C₃²	—	9	1	no	—
82	C₃²	—	9	1	no	—
83	C₃²	—	9	1	no	—
84	C₃²	C₃²⋊#66526	1	1	yes	characteristic
85	C₃²	C₃×#11758603	1	1	yes	characteristic
86	C₃²	#15688541	1	2	yes	—
87	C₉	—	37	1	no	—
88	C₉	—	37	1	no	—
89	C₉	—	37	1	no	—
90	C₉	—	333	2	no	—
91	C₉	—	111	2	no	—
92	C₃	—	3	1	no	—
93	C₃	—	3	2	no	—
94	C₃	—	9	3	no	—
95	C₃	C₃²⋊#11758603	1	1	yes	characteristic
96	C₃	H(3,𝔽₃)⋊#66526	1	1	yes	characteristic
97	C₃	#22582699	1	1	yes	characteristic
98	C₃	#22582701	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758603

C₃

yes

maximal, characteristic, complemented

C₃²⋊#11758603

C₃

yes

maximal, complemented

H(3,𝔽₃)×#66524

C₃

yes

maximal, characteristic

C₃×#11758603

—

C₃×#11758603

—

C₃×#11758603

—

C₃²×#66524

—

C₃²×#66524

C₃²

yes

characteristic

C₃²⋊#66524

—

C₃²⋊#66524

C₉

yes

—

C₃×H(3,𝔽₃)×C₃₇

C₉

yes

characteristic

C₃³×C₃₇

—

C₃³×C₃₇

C₃×C₉

yes

characteristic

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×C₃₇

C₂₇

yes

—

—

—

—

—

—

—

yes

3₋¹⁺²

yes

—

#11758603

—

#11758603

—

H(3,𝔽₃)⋊C₂₇

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃²⋊C₉

yes

characteristic

C₃²×C₃₇

C₃×C₂₇

yes

characteristic

C₃²×C₃₇

C₂₇⋊C₃

yes

—

#66524

—

#66524

—

#66524

AΣL(1,𝔽₂₇)

yes

yes

yes

C₃²×C₂₇

—

C₉×H(3,𝔽₃)

—

C₃²⋊C₂₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃²⋊C₂₇

yes

characteristic

C₃×C₃₇

H(3,𝔽₃)⋊C₉

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×H(3,𝔽₃)

yes

C₃²×C₉

—

C₃²×C₉

—

111

—

C₃×C₂₇

—

111

—

C₃×C₂₇

—

111

—

C₃×C₂₇

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃₇

H(3,𝔽₃)⋊C₂₇

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

yes

—

yes

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

333

—

C₃×C₉

—

333

—

C₃×C₉

—

333

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₂₇

—

333

—

C₂₇

—

111

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

yes

C₃²

yes

C₃²

yes

—

C₉

—

C₉

—

C₉

—

C₉

—

333

—

C₉

—

111

—

C₃

—

C₃

—

C₃

—

C₃

yes

C₃

yes

C₃

yes

C₃

yes

	subgroup	G/H
center	C₃²	C₃²⋊#66526
commutator	C₃²×C₃₇	C₃×C₂₇
Frattini	C₃³	C₃×#66526
Fitting, prime radical	C₃×H(3,𝔽₃)×C₃₇	C₉
socle	C₃²×C₃₇	C₃²⋊C₉
cosocle	C₃²×#66524	C₃²
hypercenter	C₃×H(3,𝔽₃)	#66526
hypocenter	C₃₇	H(3,𝔽₃)⋊C₂₇
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×H(3,𝔽₃)

subgroup

G/H

center

C₃²

C₃²⋊#66526

commutator

C₃²×C₃₇

C₃×C₂₇

Frattini

C₃³

C₃×#66526

Fitting, prime radical

C₃×H(3,𝔽₃)×C₃₇

C₉

socle

C₃²×C₃₇

C₃²⋊C₉

cosocle

C₃²×#66524

C₃²

hypercenter

C₃×H(3,𝔽₃)

#66526

hypocenter

C₃₇

H(3,𝔽₃)⋊C₂₇

LCS

G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ C₃×H(3,𝔽₃)

N	H ≅ G/N	type
1. C₃²×#11758603	C₃	semidirect
2. C₃²⋊#11758603	C₃	semidirect
15. H(3,𝔽₃)×C₃₇	C₂₇	semidirect
61. C₃₇	H(3,𝔽₃)⋊C₂₇	semidirect
65. H(3,𝔽₃)	#11758603	semidirect

H ≅ G/N

type

1. C₃²×#11758603

C₃

semidirect

2. C₃²⋊#11758603

C₃

semidirect

15. H(3,𝔽₃)×C₃₇

C₂₇

semidirect

61. C₃₇

H(3,𝔽₃)⋊C₂₇

semidirect

65. H(3,𝔽₃)

#11758603

semidirect

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

subgroup	index
C₃²×#11758603	3
C₃²⋊#11758603	3
H(3,𝔽₃)×#66524	3
H(3,𝔽₃)⋊C₂₇	37

subgroup

index

C₃²×#11758603

C₃²⋊#11758603

H(3,𝔽₃)×#66524

H(3,𝔽₃)⋊C₂₇

Subgroups — H(3,𝔽₃)⋊#11758603 #41008274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽3)⋊#11758603 #41008274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽₃)⋊#11758603 #41008274