Group #41008321 — subgroups

Subgroups — #41008321

Order 26973.

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

normal subgroups only

This group has 164 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×#15688541	C₃	1	2	yes	maximal
2	C₃²×#11758603	C₃	1	1	yes	maximal, characteristic
3	#22582535	C₃	1	1	yes	maximal, characteristic
4	#15688541	—	3	2	no	—
5	#15688541	—	3	2	no	—
6	#15688541	C₉	1	6	yes	complemented
7	C₃×#11758603	—	3	1	no	—
8	C₃×#11758603	—	3	2	no	—
9	C₃×#11758603	—	3	1	no	—
10	C₃×#11758603	—	3	1	no	—
11	C₃×#11758603	C₉	1	3	yes	complemented
12	C₃²×#66524	C₃²	1	1	yes	characteristic
13	#15688504	—	3	2	no	—
14	C₃²⋊#66524	C₉	1	2	yes	—
15	C₉×#66524	—	3	1	no	—
16	C₃₇×C₃²⋊C₉	C₉	1	1	yes	characteristic
17	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
18	C₃×C₉×C₃₇	—	3	1	no	—
19	C₃×#66524	—	3	2	no	—
20	C₃×#66524	C₃×C₉	1	1	yes	characteristic
21	C₃×#66524	C₃×C₉	1	2	yes	—
22	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
23	C₃×#66524	3₋¹⁺²	1	1	yes	characteristic
24	C₃×#66524	3₋¹⁺²	1	1	yes	characteristic
25	C₃×#66524	3₋¹⁺²	1	2	yes	—
26	C₃×#66524	3₋¹⁺²	1	2	yes	—
27	C₃×#66524	3₋¹⁺²	1	2	yes	—
28	#11758603	—	3	3	no	—
29	#11758603	—	3	6	no	—
30	#10624469	—	37	1	no	maximal, Sylow
31	C₃²×C₃₇	—	3	1	no	—
32	C₃²×C₃₇	—	3	2	no	—
33	C₃²×C₃₇	C₉²	1	1	yes	characteristic
34	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
35	C₃²×C₃₇	C₉⋊C₉	1	1	yes	characteristic
36	C₃²×C₃₇	C₉⋊C₉	1	1	yes	characteristic
37	C₉×C₃₇	—	9	1	no	—
38	#66524	—	9	2	no	—
39	#66524	C₃²⋊C₉	1	1	yes	characteristic
40	#66524	C₃²⋊C₉	1	2	yes	—
41	#66524	C₉⋊C₉	1	2	yes	—
42	#66524	C₉⋊C₉	1	2	yes	—
43	#66524	C₉⋊C₉	1	2	yes	—
44	C₃×C₂₇⋊C₃	—	37	2	no	—
45	C₃²×C₂₇	—	37	1	no	—
46	#6887	—	37	1	no	—
47	C₃×C₃₇	—	3	1	no	—
48	C₃×C₃₇	—	3	1	no	—
49	C₃×C₃₇	—	3	1	no	—
50	C₃×C₃₇	#6855	1	1	yes	characteristic
51	C₃×C₃₇	#6872	1	1	yes	characteristic
52	C₃×C₃₇	#6872	1	2	yes	—
53	C₃²×C₉	—	37	1	no	—
54	C₂₇⋊C₃	—	37	6	no	—
55	C₂₇⋊C₃	—	111	2	no	—
56	C₂₇⋊C₃	—	111	2	no	—
57	C₃×C₂₇	—	37	3	no	—
58	C₃×C₂₇	—	111	1	no	—
59	C₃×C₂₇	—	111	1	no	—
60	C₃×C₂₇	—	111	1	no	—
61	C₃×C₂₇	—	111	2	no	—
62	C₉⋊C₉	—	111	2	no	—
63	C₃²⋊C₉	—	37	2	no	—
64	C₃²⋊C₉	#66526	1	1	yes	characteristic
65	C₉²	—	111	1	no	—
66	C₃₇	#10624469	1	1	yes	characteristic, complemented, Sylow
67	C₃³	C₃×#66526	1	1	yes	characteristic
68	C₃×C₉	—	3	1	no	—
69	C₃×C₉	—	37	1	no	—
70	C₃×C₉	—	37	2	no	—
71	C₃×C₉	—	37	1	no	—
72	C₃×C₉	—	37	2	no	—
73	C₃×C₉	—	37	1	no	—
74	C₃×C₉	—	37	2	no	—
75	C₃×C₉	—	37	1	no	—
76	C₃×C₉	—	37	2	no	—
77	C₃×C₉	—	111	2	no	—
78	C₂₇	—	111	3	no	—
79	C₂₇	—	111	6	no	—
80	C₃²	—	3	1	no	—
81	C₃²	—	3	2	no	—
82	C₃²	C₉×#66526	1	1	yes	characteristic
83	C₃²	C₃²⋊#66526	1	1	yes	characteristic
84	C₃²	#15688537	1	1	yes	characteristic
85	C₃²	#15688538	1	1	yes	characteristic
86	C₉	—	9	1	no	—
87	C₉	—	37	1	no	—
88	C₉	—	37	2	no	—
89	C₉	—	37	2	no	—
90	C₉	—	37	2	no	—
91	C₉	—	37	2	no	—
92	C₉	—	333	2	no	—
93	C₃	—	3	1	no	—
94	C₃	—	3	1	no	—
95	C₃	—	3	1	no	—
96	C₃	(C₃×#66526)⋊C₉	1	1	yes	characteristic
97	C₃	#22582699	1	1	yes	characteristic
98	C₃	#22582700	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688541

C₃

yes

maximal

C₃²×#11758603

C₃

yes

maximal, characteristic

#22582535

C₃

yes

maximal, characteristic

#15688541

—

#15688541

—

#15688541

C₉

yes

—

—

—

—

C₉

yes

complemented

C₃²×#66524

C₃²

yes

characteristic

#15688504

—

C₃²⋊#66524

C₉

yes

—

C₉×#66524

—

C₃₇×C₃²⋊C₉

C₉

yes

characteristic

C₃³×C₃₇

C₃×C₉

yes

characteristic

C₃×C₉×C₃₇

—

C₃×#66524

—

C₃×#66524

C₃×C₉

yes

characteristic

C₃×#66524

C₃×C₉

yes

—

yes

3₋¹⁺²

yes

characteristic

C₃×#66524

3₋¹⁺²

yes

characteristic

C₃×#66524

3₋¹⁺²

yes

—

C₃×#66524

3₋¹⁺²

yes

—

C₃×#66524

3₋¹⁺²

yes

—

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₉²

yes

characteristic

C₃²×C₃₇

C₃²⋊C₉

yes

characteristic

C₃²×C₃₇

C₉⋊C₉

yes

characteristic

C₃²×C₃₇

C₉⋊C₉

yes

characteristic

C₉×C₃₇

—

#66524

—

#66524

C₃²⋊C₉

yes

characteristic

#66524

C₃²⋊C₉

yes

—

#66524

C₉⋊C₉

yes

—

#66524

C₉⋊C₉

yes

—

#66524

C₉⋊C₉

yes

—

C₃×C₂₇⋊C₃

—

C₃²×C₂₇

—

#6887

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

—

C₃²×C₉

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

—

111

—

C₃×C₂₇

—

C₃×C₂₇

—

111

—

C₃×C₂₇

—

111

—

C₃×C₂₇

—

111

—

C₃×C₂₇

—

111

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

#66526

yes

characteristic

C₉²

—

111

—

C₃₇

#10624469

yes

characteristic, complemented, Sylow

C₃³

C₃×#66526

yes

characteristic

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

C₂₇

—

111

—

C₂₇

—

111

—

C₃²

—

C₃²

—

C₃²

yes

C₃²

yes

C₃²

yes

C₃²

yes

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

333

—

C₃

—

C₃

—

C₃

—

C₃

yes

C₃

yes

C₃

yes

—

	subgroup	G/H
center	C₃²	C₃²⋊#66526
commutator	C₃²×C₃₇	C₉²
Frattini	C₃³	C₃×#66526
Fitting	C₃₇×C₃²⋊C₉	C₉
socle	C₃²×C₃₇	C₃²⋊C₉
prime radical	C₃³×C₃₇	C₃×C₉
cosocle	C₃²×#66524	C₃²
hypercenter	C₃²⋊C₉	#66526
hypocenter	C₃₇	#10624469
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²⋊C₉

subgroup

G/H

center

C₃²

C₃²⋊#66526

commutator

C₃²×C₃₇

C₉²

Frattini

C₃³

C₃×#66526

Fitting

C₃₇×C₃²⋊C₉

C₉

socle

C₃²×C₃₇

C₃²⋊C₉

prime radical

C₃³×C₃₇

C₃×C₉

cosocle

C₃²×#66524

C₃²

C₃²⋊C₉

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ C₃²⋊C₉

N	H ≅ G/N	type
6. #15688541	C₉	semidirect
11. C₃×#11758603	C₉	semidirect
66. C₃₇	#10624469	semidirect

H ≅ G/N

type

6. #15688541

C₉

semidirect

11. C₃×#11758603

C₉

semidirect

66. C₃₇

#10624469

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
#22582535	3
#10624469	37

subgroup

index

Subgroups — #41008321

Conjugacy classes of subgroups

Complements

Maximal subgroups