Group #41008276 — subgroups

Subgroups — 3₋¹⁺²⋊#11758603 #41008276

Order 26973.

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

normal subgroups only

This group has 126 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
2	C₃²⋊#11758603	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊#11758603	C₃	1	1	yes	maximal, characteristic
4	3₋¹⁺²×#66524	C₃	1	1	yes	maximal, characteristic
5	C₃×#11758603	—	3	1	no	—
6	C₃×#11758603	—	3	3	no	—
7	C₃×#11758603	—	3	1	no	—
8	C₃×#11758603	—	3	1	no	—
9	C₃²×#66524	C₃²	1	1	yes	characteristic
10	#15688506	—	3	1	no	—
11	#15688505	—	3	1	no	—
12	C₃²⋊#66524	C₉	1	1	yes	characteristic
13	C₃²⋊#66524	C₉	1	1	yes	characteristic
14	C₉×#66524	—	3	1	no	—
15	C₃×3₋¹⁺²×C₃₇	C₉	1	1	yes	characteristic
16	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
17	3₋¹⁺²×C₃₇	—	3	2	no	—
18	3₋¹⁺²×C₃₇	C₂₇	1	3	yes	complemented
19	C₃×C₉×C₃₇	—	3	1	no	—
20	C₃×#66524	—	3	1	no	—
21	C₃×#66524	—	3	1	no	—
22	C₃×#66524	—	3	1	no	—
23	C₃×#66524	—	3	1	no	—
24	C₃×#66524	—	3	1	no	—
25	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
26	C₃×#66524	3₋¹⁺²	1	1	yes	characteristic
27	C₃×#66524	3₋¹⁺²	1	1	yes	characteristic
28	#11758603	—	3	3	no	—
29	#11758603	—	9	1	no	—
30	#11758603	—	9	1	no	—
31	3₋¹⁺²⋊C₂₇	—	37	1	no	maximal, Sylow
32	C₃²×C₃₇	—	3	1	no	—
33	C₃²×C₃₇	—	3	1	no	—
34	C₃²×C₃₇	—	3	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	C₉×C₃₇	—	3	3	no	—
39	#66524	—	3	1	no	—
40	#66524	—	3	1	no	—
41	#66524	—	9	1	no	—
42	#66524	—	9	1	no	—
43	#66524	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
44	#66524	#731	1	1	yes	characteristic
45	#66524	#733	1	1	yes	characteristic
46	C₃²×C₂₇	—	37	1	no	—
47	C₉×3₋¹⁺²	—	37	1	no	—
48	C₃²⋊C₂₇	—	37	1	no	—
49	C₃²⋊C₂₇	—	37	1	no	—
50	C₃×C₃₇	—	3	1	no	—
51	C₃×C₃₇	—	3	2	no	—
52	C₃×C₃₇	C₃²⋊C₂₇	1	1	yes	characteristic
53	C₃×C₃₇	3₋¹⁺²⋊C₉	1	1	yes	characteristic
54	C₃×C₃₇	#6872	1	1	yes	characteristic
55	C₃×C₃₇	#6873	1	1	yes	characteristic
56	C₃×3₋¹⁺²	#66526	1	1	yes	characteristic
57	C₃²×C₉	—	37	1	no	—
58	C₃×C₂₇	—	111	1	no	—
59	C₃×C₂₇	—	111	1	no	—
60	C₃×C₂₇	—	111	1	no	—
61	C₃×C₂₇	—	111	3	no	—
62	C₉⋊C₉	—	111	1	no	—
63	C₉⋊C₉	—	111	1	no	—
64	C₃²⋊C₉	—	37	1	no	—
65	C₃²⋊C₉	—	37	1	no	—
66	C₉²	—	111	1	no	—
67	C₃₇	3₋¹⁺²⋊C₂₇	1	1	yes	characteristic, complemented, Sylow
68	C₃³	C₃×#66526	1	1	yes	characteristic
69	3₋¹⁺²	—	3	2	no	—
70	3₋¹⁺²	#11758603	1	3	yes	complemented
71	C₃×C₉	—	3	1	no	—
72	C₃×C₉	—	37	1	no	—
73	C₃×C₉	—	37	1	no	—
74	C₃×C₉	—	37	1	no	—
75	C₃×C₉	—	111	1	no	—
76	C₃×C₉	—	111	1	no	—
77	C₃×C₉	—	111	1	no	—
78	C₃×C₉	—	111	1	no	—
79	C₃×C₉	—	111	1	no	—
80	C₂₇	—	333	1	no	—
81	C₂₇	—	333	1	no	—
82	C₂₇	—	111	3	no	—
83	C₃²	—	3	1	no	—
84	C₃²	—	3	1	no	—
85	C₃²	—	3	1	no	—
86	C₃²	C₃²⋊#66526	1	1	yes	characteristic
87	C₃²	C₃×#11758603	1	1	yes	characteristic
88	C₃²	#15688541	1	2	yes	—
89	C₉	—	3	3	no	—
90	C₉	—	37	1	no	—
91	C₉	—	37	1	no	—
92	C₉	—	37	1	no	—
93	C₉	—	333	1	no	—
94	C₉	—	333	1	no	—
95	C₉	—	111	1	no	—
96	C₉	—	111	1	no	—
97	C₃	—	3	1	no	—
98	C₃	—	3	2	no	—
99	C₃	C₃²⋊#11758603	1	1	yes	characteristic
100	C₃	3₋¹⁺²⋊#66526	1	1	yes	characteristic
101	C₃	#22582700	1	1	yes	characteristic
102	C₃	#22582702	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758603

C₃

yes

maximal, characteristic

C₃²⋊#11758603

C₃

yes

maximal, characteristic

C₃²⋊#11758603

C₃

yes

maximal, characteristic

3₋¹⁺²×#66524

C₃

yes

maximal, characteristic

—

—

—

—

C₃²×#66524

C₃²

yes

—

—

C₉

yes

characteristic

C₃²⋊#66524

C₉

yes

characteristic

C₉×#66524

—

C₃×3₋¹⁺²×C₃₇

C₉

yes

characteristic

C₃³×C₃₇

C₃×C₉

yes

characteristic

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

C₂₇

yes

complemented

C₃×C₉×C₃₇

—

—

—

—

—

—

yes

3₋¹⁺²

yes

characteristic

C₃×#66524

3₋¹⁺²

yes

—

—

—

3₋¹⁺²⋊C₂₇

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃²⋊C₉

yes

characteristic

C₃²×C₃₇

C₃×C₂₇

yes

characteristic

C₃²×C₃₇

C₂₇⋊C₃

yes

—

C₉×C₃₇

—

—

—

—

—

AΣL(1,𝔽₂₇)

yes

yes

yes

C₃²×C₂₇

—

C₉×3₋¹⁺²

—

C₃²⋊C₂₇

—

C₃²⋊C₂₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃²⋊C₂₇

yes

characteristic

C₃×C₃₇

3₋¹⁺²⋊C₉

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×3₋¹⁺²

yes

C₃²×C₉

—

C₃×C₂₇

—

111

—

C₃×C₂₇

—

111

—

C₃×C₂₇

—

111

—

C₃×C₂₇

—

111

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₉²

—

111

—

C₃₇

3₋¹⁺²⋊C₂₇

yes

characteristic, complemented, Sylow

C₃³

C₃×#66526

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

#11758603

yes

complemented

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₂₇

—

333

—

C₂₇

—

333

—

C₂₇

—

111

—

C₃²

—

C₃²

—

C₃²

—

C₃²

yes

C₃²

yes

C₃²

yes

—

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

333

—

C₉

—

333

—

C₉

—

111

—

C₉

—

111

—

C₃

—

C₃

—

C₃

C₃²⋊#11758603

yes

characteristic

100

C₃

3₋¹⁺²⋊#66526

yes

characteristic

101

C₃

#22582700

yes

characteristic

102

C₃

#22582702

yes

characteristic

	subgroup	G/H
center	C₃²	C₃²⋊#66526
commutator	C₃²×C₃₇	C₃×C₂₇
Frattini	C₃³	C₃×#66526
Fitting	C₃×3₋¹⁺²×C₃₇	C₉
socle	C₃²×C₃₇	C₃²⋊C₉
prime radical	C₃³×C₃₇	C₃×C₉
cosocle	C₃²×#66524	C₃²
hypercenter	C₃×3₋¹⁺²	#66526
hypocenter	C₃₇	3₋¹⁺²⋊C₂₇
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×3₋¹⁺²

subgroup

G/H

center

C₃²

C₃²⋊#66526

commutator

C₃²×C₃₇

C₃×C₂₇

Frattini

C₃³

C₃×#66526

Fitting

C₃×3₋¹⁺²×C₃₇

C₉

socle

C₃²×C₃₇

C₃²⋊C₉

prime radical

C₃³×C₃₇

C₃×C₉

cosocle

C₃²×#66524

C₃²

hypercenter

C₃×3₋¹⁺²

#66526

hypocenter

C₃₇

3₋¹⁺²⋊C₂₇

LCS

G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ C₃×3₋¹⁺²

N	H ≅ G/N	type
18. 3₋¹⁺²×C₃₇	C₂₇	semidirect
67. C₃₇	3₋¹⁺²⋊C₂₇	semidirect
70. 3₋¹⁺²	#11758603	semidirect

H ≅ G/N

type

18. 3₋¹⁺²×C₃₇

C₂₇

semidirect

67. C₃₇

3₋¹⁺²⋊C₂₇

semidirect

70. 3₋¹⁺²

#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
3₋¹⁺²×#66524	3
3₋¹⁺²⋊C₂₇	37

subgroup

index

C₃²×#11758603

C₃²⋊#11758603

3₋¹⁺²×#66524

3₋¹⁺²⋊C₂₇

Subgroups — 3₋¹⁺²⋊#11758603 #41008276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3−1+2⋊#11758603 #41008276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²⋊#11758603 #41008276