Group #41008284 — subgroups

Subgroups — #41008284

Order 26973.

132 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 30 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×#15688541	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃²⋊#11758603	C₃	1	2	yes	maximal, complemented
3	#22582539	C₃	1	1	yes	maximal, characteristic
4	#15688541	—	3	3	no	—
5	C₃×#11758603	—	3	1	no	—
6	C₃×#11758603	—	3	2	no	—
7	C₃×C₃²⋊(C₃₇⋊C₃)	C₉	1	1	yes	characteristic
8	C₃²×#66524	C₃²	1	1	yes	characteristic
9	C₃×C₉×C₃₇⋊C₃	—	3	1	no	—
10	C₃²⋊#66524	C₉	1	1	yes	characteristic
11	#15688502	—	3	2	no	—
12	C₃₇×C₃²⋊C₉	C₉	1	1	yes	characteristic
13	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
14	C₃²×C₃₇⋊C₃	—	3	1	no	—
15	C₃×C₉×C₃₇	—	3	1	no	—
16	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
17	C₃×#66524	—	3	1	no	—
18	C₃×#66524	—	3	1	no	—
19	C₃×#66524	—	3	1	no	—
20	C₃×#66524	—	3	1	no	—
21	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
22	C₃×#66524	3₋¹⁺²	1	2	yes	—
23	C₉×C₃₇⋊C₃	—	9	1	no	—
24	C₉×C₃₇⋊C₃	—	9	1	no	—
25	C₉×C₃₇⋊C₃	—	9	1	no	—
26	#11758603	—	3	3	no	—
27	#11758603	—	9	2	no	—
28	#10624453	—	37	1	no	maximal, Sylow
29	C₃²×C₃₇	—	3	1	no	—
30	C₃²×C₃₇	—	3	1	no	—
31	C₃²×C₃₇	—	3	1	no	—
32	C₃²×C₃₇	—	3	1	no	—
33	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
34	C₃×C₃₇⋊C₃	—	3	3	no	—
35	C₃×C₃₇⋊C₃	—	9	1	no	—
36	C₃×C₃₇⋊C₃	—	9	1	no	—
37	C₃×C₃₇⋊C₃	—	9	1	no	—
38	C₉×C₃₇	—	9	1	no	—
39	#66524	—	3	2	no	—
40	#66524	—	9	1	no	—
41	#66524	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
42	#66524	#731	1	1	yes	characteristic
43	#66524	#732	1	1	yes	characteristic
44	C₃×C₂₇⋊C₃	—	37	1	no	—
45	C₉×H(3,𝔽₃)	—	37	1	no	—
46	C₃²⋊C₂₇	—	37	2	no	—
47	C₃×C₃₇	—	9	1	no	—
48	C₃×C₃₇	#6869	1	1	yes	characteristic
49	C₃×C₃₇	H(3,𝔽₃)⋊C₉	1	1	yes	characteristic
50	C₃×C₃₇	#6872	1	1	yes	characteristic
51	C₃×C₃₇	#6873	1	1	yes	characteristic
52	C₃₇⋊C₃	—	9	3	no	—
53	C₃×H(3,𝔽₃)	—	37	1	no	—
54	C₃²×C₉	—	37	1	no	—
55	C₃²×C₉	—	111	1	no	—
56	C₂₇⋊C₃	—	111	3	no	—
57	C₃×C₂₇	—	111	2	no	—
58	C₃×C₂₇	—	111	1	no	—
59	C₃²⋊C₉	—	37	1	no	—
60	C₃²⋊C₉	—	111	2	no	—
61	C₃²⋊C₉	#66526	1	1	yes	characteristic
62	C₃₇	#10624453	1	1	yes	characteristic, complemented, Sylow
63	C₃³	—	111	1	no	—
64	C₃³	C₃×#66526	1	1	yes	characteristic
65	H(3,𝔽₃)	—	111	3	no	—
66	C₃×C₉	—	3	1	no	—
67	C₃×C₉	—	37	1	no	—
68	C₃×C₉	—	37	2	no	—
69	C₃×C₉	—	333	1	no	—
70	C₃×C₉	—	333	1	no	—
71	C₃×C₉	—	333	1	no	—
72	C₃×C₉	—	111	1	no	—
73	C₃×C₉	—	111	1	no	—
74	C₃×C₉	—	111	1	no	—
75	C₃×C₉	—	111	1	no	—
76	C₂₇	—	333	2	no	—
77	C₂₇	—	111	3	no	—
78	C₃²	—	3	1	no	—
79	C₃²	—	3	1	no	—
80	C₃²	—	3	1	no	—
81	C₃²	—	3	1	no	—
82	C₃²	—	333	1	no	—
83	C₃²	—	333	1	no	—
84	C₃²	—	333	1	no	—
85	C₃²	—	111	3	no	—
86	C₃²	C₃²⋊#66526	1	1	yes	characteristic
87	C₉	—	9	1	no	—
88	C₉	—	37	1	no	—
89	C₉	—	37	1	no	—
90	C₉	—	37	1	no	—
91	C₉	—	333	1	no	—
92	C₉	—	111	2	no	—
93	C₃	—	9	1	no	—
94	C₃	—	333	3	no	—
95	C₃	#22582695	1	1	yes	characteristic
96	C₃	H(3,𝔽₃)⋊#66526	1	1	yes	characteristic
97	C₃	#22582700	1	1	yes	characteristic
98	C₃	#22582702	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688541

C₃

yes

maximal, characteristic, complemented

C₃²⋊#11758603

C₃

yes

maximal, complemented

#22582539

C₃

yes

maximal, characteristic

#15688541

—

C₃×#11758603

—

C₃×#11758603

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₉

yes

characteristic

C₃²×#66524

C₃²

yes

characteristic

C₃×C₉×C₃₇⋊C₃

—

C₃²⋊#66524

C₉

yes

characteristic

#15688502

—

C₃₇×C₃²⋊C₉

C₉

yes

characteristic

C₃³×C₃₇

C₃×C₉

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃×C₉×C₃₇

—

C₃²⋊(C₃₇⋊C₃)

—

—

—

—

—

yes

3₋¹⁺²

yes

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

—

—

—

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

—

C₉×C₃₇

—

#66524

—

#66524

—

#66524

AΣL(1,𝔽₂₇)

yes

yes

yes

C₃×C₂₇⋊C₃

—

C₉×H(3,𝔽₃)

—

C₃²⋊C₂₇

—

C₃×C₃₇

—

C₃×C₃₇

#6869

yes

characteristic

C₃×C₃₇

H(3,𝔽₃)⋊C₉

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃₇⋊C₃

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

111

—

C₂₇⋊C₃

—

111

—

C₃×C₂₇

—

111

—

C₃×C₂₇

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

#66526

yes

characteristic

C₃₇

#10624453

yes

characteristic, complemented, Sylow

C₃³

—

111

—

C₃³

C₃×#66526

yes

characteristic

H(3,𝔽₃)

—

111

—

C₃×C₉

—

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

—

333

—

C₃²

—

333

—

C₃²

—

333

—

C₃²

—

111

—

C₃²

C₃²⋊#66526

yes

characteristic

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

333

—

C₉

—

111

—

C₃

—

C₃

—

333

—

C₃

yes

C₃

yes

C₃

yes

C₃

yes

	subgroup	G/H
center	C₃²	C₃²⋊#66526
commutator	C₃³×C₃₇	C₃×C₉
Frattini	C₃³	C₃×#66526
Fitting	C₃₇×C₃²⋊C₉	C₉
socle	C₃²×C₃₇	C₃²⋊C₉
prime radical	C₃×C₃²⋊(C₃₇⋊C₃)	C₉
cosocle	C₃²×#66524	C₃²
hypercenter	C₃²⋊C₉	#66526
hypocenter	C₃₇	#10624453
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²⋊C₉

subgroup

G/H

center

C₃²

C₃²⋊#66526

commutator

C₃³×C₃₇

C₃×C₉

Frattini

C₃³

C₃×#66526

Fitting

C₃₇×C₃²⋊C₉

C₉

socle

C₃²×C₃₇

C₃²⋊C₉

prime radical

C₃×C₃²⋊(C₃₇⋊C₃)

C₉

cosocle

C₃²×#66524

C₃²

C₃²⋊C₉

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ C₃²⋊C₉

N	H ≅ G/N	type
1. C₃×#15688541	C₃	semidirect
2. C₃²⋊#11758603	C₃	semidirect
62. C₃₇	#10624453	semidirect

H ≅ G/N

type

1. C₃×#15688541

C₃

semidirect

2. C₃²⋊#11758603

C₃

semidirect

62. C₃₇

#10624453

semidirect

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

subgroup	index
C₃×#15688541	3
C₃²⋊#11758603	3
#22582539	3
#10624453	37

subgroup

index

Subgroups — #41008284

Conjugacy classes of subgroups

Complements

Maximal subgroups