Group #41008332 — subgroups

Subgroups — #41008332

Order 26973.

170 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 60 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×#15688543	C₃	1	2	yes	maximal
2	C₃×#15688538	C₃	1	1	yes	maximal, characteristic
3	C₃×#15688509	C₃	1	1	yes	maximal, characteristic
4	C₃²×#66526	—	3	1	no	—
5	#15688543	—	3	2	no	—
6	#15688543	—	3	2	no	—
7	#15688543	C₉	1	6	yes	complemented
8	C₃×#11758603	—	3	2	no	—
9	#15688538	—	3	1	no	—
10	#15688538	—	3	1	no	—
11	#15688538	C₉	1	3	yes	—
12	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
13	#15688509	—	3	1	no	—
14	#15688509	—	3	2	no	—
15	C₃×#11758601	C₉	1	2	yes	complemented
16	C₃×C₂₇×C₃₇	C₉	1	1	yes	characteristic, complemented
17	C₃²×C₃₇⋊C₃	3₋¹⁺²	1	1	yes	characteristic
18	C₃×#66526	—	3	3	no	—
19	C₃×#66526	—	9	1	no	—
20	C₃×#66526	—	9	1	no	—
21	C₃×#66526	—	9	1	no	—
22	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
23	C₃×#66524	3₋¹⁺²	1	2	yes	—
24	C₉×C₃₇⋊C₃	C₃×C₉	1	2	yes	—
25	C₉×C₃₇⋊C₃	C₃×C₉	1	1	yes	characteristic
26	C₉×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
27	C₉×C₃₇⋊C₃	3₋¹⁺²	1	2	yes	—
28	C₉×C₃₇⋊C₃	3₋¹⁺²	1	2	yes	—
29	C₉×C₃₇⋊C₃	3₋¹⁺²	1	1	yes	characteristic
30	#11758603	—	3	6	no	—
31	C₂₇×C₃₇	—	3	1	no	—
32	#11758601	—	3	2	no	—
33	#10624472	—	37	1	no	maximal, Sylow
34	C₃²×C₃₇	C₉⋊C₉	1	1	yes	characteristic
35	C₃×C₃₇⋊C₃	—	3	1	no	—
36	C₃×C₃₇⋊C₃	—	3	1	no	—
37	C₃×C₃₇⋊C₃	—	3	1	no	—
38	C₃×C₃₇⋊C₃	C₃²⋊C₉	1	1	yes	characteristic
39	C₃×C₃₇⋊C₃	C₉⋊C₉	1	2	yes	—
40	#66526	—	9	3	no	—
41	C₉×C₃₇	C₉²	1	1	yes	characteristic
42	C₉×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
43	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
44	#66524	C₃²⋊C₉	1	2	yes	—
45	#66524	C₉⋊C₉	1	2	yes	—
46	#66524	C₉⋊C₉	1	2	yes	—
47	C₃×C₂₇⋊C₃	—	37	1	no	—
48	C₃×C₂₇⋊C₃	—	37	2	no	—
49	C₃×C₉⋊C₉	—	37	1	no	—
50	C₃×C₃₇	#6855	1	1	yes	characteristic
51	C₃×C₃₇	#6875	1	1	yes	characteristic
52	C₃×C₃₇	#6875	1	1	yes	characteristic
53	C₃×C₃₇	#6875	1	1	yes	characteristic
54	C₃₇⋊C₃	—	3	1	no	—
55	C₃₇⋊C₃	—	3	2	no	—
56	C₃²×C₉	—	37	1	no	—
57	C₃²×C₉	—	111	1	no	—
58	C₂₇⋊C₃	—	37	6	no	—
59	C₂₇⋊C₃	—	111	1	no	—
60	C₂₇⋊C₃	—	111	2	no	—
61	C₂₇⋊C₃	—	111	2	no	—
62	C₂₇⋊C₃	—	111	2	no	—
63	C₃×C₂₇	—	37	2	no	—
64	C₃×C₂₇	—	111	2	no	—
65	C₃×C₂₇	#66526	1	1	yes	characteristic, complemented
66	C₉⋊C₉	—	37	3	no	—
67	C₉⋊C₉	—	111	1	no	—
68	C₉⋊C₉	—	111	1	no	—
69	C₃₇	#10624472	1	1	yes	characteristic, complemented, Sylow
70	C₃³	—	37	1	no	—
71	C₃×C₉	—	37	2	no	—
72	C₃×C₉	—	37	1	no	—
73	C₃×C₉	—	37	2	no	—
74	C₃×C₉	—	37	1	no	—
75	C₃×C₉	—	37	2	no	—
76	C₃×C₉	—	37	1	no	—
77	C₃×C₉	—	37	2	no	—
78	C₃×C₉	—	333	1	no	—
79	C₃×C₉	—	333	1	no	—
80	C₃×C₉	—	333	1	no	—
81	C₃×C₉	—	111	3	no	—
82	C₃×C₉	C₃×#66526	1	1	yes	characteristic
83	C₂₇	—	3	1	no	—
84	C₂₇	—	111	6	no	—
85	C₂₇	—	111	2	no	—
86	C₃²	—	37	1	no	—
87	C₃²	—	37	2	no	—
88	C₃²	—	111	1	no	—
89	C₃²	—	111	1	no	—
90	C₃²	—	111	1	no	—
91	C₃²	#15688538	1	1	yes	characteristic
92	C₉	—	37	2	no	—
93	C₉	—	37	2	no	—
94	C₉	—	37	2	no	—
95	C₉	—	333	3	no	—
96	C₉	C₉×#66526	1	1	yes	characteristic
97	C₉	C₃²⋊#66526	1	1	yes	characteristic
98	C₉	#15688537	1	1	yes	characteristic
99	C₃	—	111	1	no	—
100	C₃	—	111	2	no	—
101	C₃	(C₃×#66526)⋊C₉	1	1	yes	characteristic
102	C₃	#22582708	1	1	yes	characteristic
103	C₃	#22582710	1	1	yes	characteristic
104	C₃	#22582712	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688543

C₃

yes

maximal

C₃×#15688538

C₃

yes

maximal, characteristic

C₃×#15688509

C₃

yes

maximal, characteristic

C₃²×#66526

—

#15688543

—

#15688543

—

#15688543

C₉

yes

—

—

—

C₉

yes

—

C₃×C₉×C₃₇⋊C₃

C₃²

yes

—

—

C₉

yes

complemented

C₃×C₂₇×C₃₇

C₉

yes

characteristic, complemented

C₃²×C₃₇⋊C₃

3₋¹⁺²

yes

—

—

—

—

C₃×C₉×C₃₇

C₃×C₉

yes

characteristic

C₃×#66524

3₋¹⁺²

yes

—

C₉×C₃₇⋊C₃

C₃×C₉

yes

—

C₉×C₃₇⋊C₃

C₃×C₉

yes

characteristic

C₉×C₃₇⋊C₃

H(3,𝔽₃)

yes

characteristic

C₉×C₃₇⋊C₃

3₋¹⁺²

yes

—

C₉×C₃₇⋊C₃

3₋¹⁺²

yes

—

C₉×C₃₇⋊C₃

3₋¹⁺²

yes

characteristic

#11758603

—

C₂₇×C₃₇

—

#11758601

—

#10624472

—

maximal, Sylow

C₃²×C₃₇

C₉⋊C₉

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

C₃²⋊C₉

yes

characteristic

C₃×C₃₇⋊C₃

C₉⋊C₉

yes

—

#66526

—

C₉×C₃₇

C₉²

yes

characteristic

C₉×C₃₇

C₃²⋊C₉

yes

characteristic

C₉×C₃₇

C₉⋊C₉

yes

characteristic

#66524

C₃²⋊C₉

yes

—

#66524

C₉⋊C₉

yes

—

#66524

C₉⋊C₉

yes

—

C₃×C₂₇⋊C₃

—

C₃×C₂₇⋊C₃

—

C₃×C₉⋊C₉

—

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃₇⋊C₃

—

C₃₇⋊C₃

—

C₃²×C₉

—

C₃²×C₉

—

111

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

—

111

—

C₃×C₂₇

—

C₃×C₂₇

—

111

—

C₃×C₂₇

#66526

yes

characteristic, complemented

C₉⋊C₉

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₃₇

#10624472

yes

characteristic, complemented, Sylow

C₃³

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

333

—

C₃×C₉

—

333

—

C₃×C₉

—

333

—

C₃×C₉

—

111

—

C₃×C₉

C₃×#66526

yes

characteristic

C₂₇

—

C₂₇

—

111

—

C₂₇

—

111

—

C₃²

—

C₃²

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

#15688538

yes

characteristic

C₉

—

C₉

—

C₉

—

C₉

—

333

—

C₉

yes

C₉

yes

C₉

yes

C₃

—

111

—

100

C₃

—

111

—

101

C₃

(C₃×#66526)⋊C₉

yes

characteristic

102

C₃

#22582708

yes

characteristic

103

C₃

#22582710

yes

characteristic

104

C₃

#22582712

yes

characteristic

	subgroup	G/H
center	C₃²	#15688538
commutator	C₉×C₃₇	C₉²
Frattini	C₃×C₉	C₃×#66526
Fitting	C₃×C₂₇×C₃₇	C₉
socle	C₃²×C₃₇	C₉⋊C₉
prime radical	C₃²×C₃₇⋊C₃	3₋¹⁺²
cosocle	C₃×C₉×C₃₇⋊C₃	C₃²
hypercenter	C₃×C₂₇	#66526
hypocenter	C₃₇	#10624472
LCS	G ▷ C₉×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃×C₉ ◁ C₃×C₂₇

subgroup

G/H

center

C₃²

#15688538

commutator

C₉×C₃₇

C₉²

Frattini

C₃×C₉

C₃×#66526

Fitting

C₃×C₂₇×C₃₇

C₉

socle

C₃²×C₃₇

C₉⋊C₉

prime radical

C₃²×C₃₇⋊C₃

3₋¹⁺²

cosocle

C₃×C₉×C₃₇⋊C₃

C₃²

C₃×C₂₇

C₃₇

G ▷ C₉×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃×C₉ ◁ C₃×C₂₇

N	H ≅ G/N	type
7. #15688543	C₉	semidirect
15. C₃×#11758601	C₉	semidirect
16. C₃×C₂₇×C₃₇	C₉	semidirect
65. C₃×C₂₇	#66526	semidirect
69. C₃₇	#10624472	semidirect

H ≅ G/N

type

7. #15688543

C₉

semidirect

15. C₃×#11758601

C₉

semidirect

16. C₃×C₂₇×C₃₇

C₉

semidirect

65. C₃×C₂₇

#66526

semidirect

69. C₃₇

#10624472

semidirect

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

subgroup	index
C₃×#15688543	3
C₃×#15688538	3
C₃×#15688509	3
#10624472	37

subgroup

index

Subgroups — #41008332

Conjugacy classes of subgroups

Complements

Maximal subgroups