Group #41006392 — subgroups

Subgroups — #733×#66524 #41006392

Order 26973.

274 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 100 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	3₋¹⁺²×#66524	C₃	1	3	yes	maximal
2	C₃×C₉×#66524	C₃	1	1	yes	maximal, characteristic
3	3₋¹⁺²⋊#66524	C₃	1	6	yes	maximal
4	#22582472	C₃	1	2	yes	maximal
5	C₃×C₃₇×#733	C₃	1	1	yes	maximal, characteristic
6	C₃²×#66524	C₃²	1	2	yes	—
7	C₃²×#66524	C₃²	1	1	yes	characteristic
8	#15688506	—	3	3	no	—
9	#15688505	—	3	3	no	—
10	C₃²⋊#66524	C₃²	1	6	yes	—
11	C₉×#66524	—	3	3	no	—
12	C₉×#66524	—	3	1	no	—
13	C₉×#66524	—	3	1	no	—
14	C₉×#66524	—	3	1	no	—
15	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
16	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
17	C₃₇×#733	C₉	1	9	yes	complemented
18	C₃³×C₃₇	C₃³	1	1	yes	characteristic
19	3₋¹⁺²×C₃₇	—	3	6	no	—
20	3₋¹⁺²×C₃₇	C₃×C₉	1	9	yes	—
21	C₃×C₉×C₃₇	—	3	1	no	—
22	C₃×C₉×C₃₇	—	3	3	no	—
23	C₃×C₉×C₃₇	—	3	2	no	—
24	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
25	C₃×#66524	—	3	2	no	—
26	C₃×#66524	—	3	3	no	—
27	C₃×#66524	—	3	6	no	—
28	C₃×#66524	—	3	6	no	—
29	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
30	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
31	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
32	C₉×#733	—	37	1	no	maximal, Sylow
33	C₃²×C₃₇	—	3	1	no	—
34	C₃²×C₃₇	—	3	2	no	—
35	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
36	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
37	C₃²×C₃₇	#737	1	2	yes	—
38	C₉×C₃₇	—	3	3	no	—
39	C₉×C₃₇	—	3	9	no	—
40	#66524	—	3	1	no	—
41	#66524	—	3	1	no	—
42	#66524	—	3	6	no	—
43	#66524	—	9	6	no	—
44	#66524	#733	1	3	yes	direct product factor
45	C₃×#733	C₃₇⋊C₃	1	1	yes	characteristic
46	C₉×3₋¹⁺²	—	37	3	no	—
47	C₃×C₉²	—	37	1	no	—
48	3₋¹⁺²⋊C₉	—	37	6	no	—
49	#6868	—	37	2	no	—
50	C₃×C₃₇	—	3	1	no	—
51	C₃×C₃₇	—	3	2	no	—
52	C₃×C₃₇	C₉×H(3,𝔽₃)	1	1	yes	characteristic
53	C₃×C₃₇	C₃×#733	1	1	yes	characteristic
54	C₃×C₃₇	#6908	1	2	yes	—
55	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	3	yes	—
56	C₃²×C₉	—	37	1	no	—
57	C₃²×C₉	—	37	2	no	—
58	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
59	#733	#66524	1	9	yes	direct product factor
60	C₉⋊C₉	—	111	3	no	—
61	C₉⋊C₉	—	111	3	no	—
62	C₃²⋊C₉	—	37	6	no	—
63	C₉²	—	111	1	no	—
64	C₉²	—	111	1	no	—
65	C₉²	—	111	1	no	—
66	C₉²	—	111	3	no	—
67	C₃₇	C₉×#733	1	1	yes	characteristic, complemented, Sylow
68	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
69	3₋¹⁺²	—	3	6	no	—
70	3₋¹⁺²	C₃×#66524	1	9	yes	—
71	C₃×C₉	—	3	1	no	—
72	C₃×C₉	—	3	3	no	—
73	C₃×C₉	—	3	2	no	—
74	C₃×C₉	—	37	1	no	—
75	C₃×C₉	—	37	1	no	—
76	C₃×C₉	—	37	1	no	—
77	C₃×C₉	—	111	3	no	—
78	C₃×C₉	—	111	6	no	—
79	C₃×C₉	—	111	2	no	—
80	C₃×C₉	—	111	6	no	—
81	C₃×C₉	C₃×#66524	1	3	yes	—
82	C₃²	—	3	1	no	—
83	C₃²	—	3	2	no	—
84	C₃²	C₃²×#66524	1	1	yes	characteristic
85	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
86	C₃²	#15688530	1	2	yes	—
87	C₉	—	3	3	no	—
88	C₉	—	3	9	no	—
89	C₉	—	37	3	no	—
90	C₉	—	333	6	no	—
91	C₉	—	111	1	no	—
92	C₉	—	111	1	no	—
93	C₉	—	111	6	no	—
94	C₃	—	3	1	no	—
95	C₃	—	3	2	no	—
96	C₃	H(3,𝔽₃)×#66524	1	1	yes	characteristic
97	C₃	#733×C₃₇⋊C₃	1	1	yes	characteristic
98	C₃	#22582634	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

3₋¹⁺²×#66524

C₃

yes

maximal

C₃×C₉×#66524

C₃

yes

maximal, characteristic

3₋¹⁺²⋊#66524

C₃

yes

maximal

#22582472

C₃

yes

maximal

C₃×C₃₇×#733

C₃

yes

maximal, characteristic

C₃²×#66524

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

—

C₃²

yes

—

—

—

—

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×#733

C₉

yes

complemented

C₃³×C₃₇

C₃³

yes

characteristic

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

C₃×C₉

yes

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃×C₉

yes

—

—

—

—

—

yes

yes

yes

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃²×C₉

yes

characteristic

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

#737

yes

—

C₉×C₃₇

—

C₉×C₃₇

—

—

—

—

—

yes

direct product factor

C₃×#733

C₃₇⋊C₃

yes

characteristic

C₉×3₋¹⁺²

—

C₃×C₉²

—

3₋¹⁺²⋊C₉

—

#6868

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₉×H(3,𝔽₃)

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

#733

#66524

yes

direct product factor

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

C₉²

—

111

—

C₉²

—

111

—

C₉²

—

111

—

C₉²

—

111

—

C₃₇

C₉×#733

yes

characteristic, complemented, Sylow

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

C₃×#66524

yes

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

C₃×#66524

yes

—

C₃²

—

C₃²

—

C₃²

C₃²×#66524

yes

characteristic

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

C₃²

#15688530

yes

—

C₉

—

C₉

—

C₉

—

C₉

—

333

—

C₉

—

111

—

C₉

—

111

—

C₉

—

111

—

C₃

—

C₃

—

C₃

H(3,𝔽₃)×#66524

yes

characteristic

C₃

#733×C₃₇⋊C₃

yes

characteristic

C₃

#22582634

yes

—

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator	C₃²×C₃₇	C₃²×C₉
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃×C₃₇×#733	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical, cosocle	C₃³×C₃₇	C₃³
hypercenter	C₃×#733	C₃₇⋊C₃
hypocenter	C₃₇	C₉×#733
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#733

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

commutator

C₃²×C₃₇

C₃²×C₉

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃×C₃₇×#733

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical, cosocle

C₃³×C₃₇

C₃³

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#733

N	H ≅ G/N	type
17. C₃₇×#733	C₉	semidirect
44. #66524	#733	direct (N × H)
59. #733	#66524	direct (N × H)
67. C₃₇	C₉×#733	semidirect

H ≅ G/N

type

17. C₃₇×#733

C₉

semidirect

44. #66524

#733

direct (N × H)

59. #733

#66524

direct (N × H)

67. C₃₇

C₉×#733

semidirect

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

subgroup	index
3₋¹⁺²×#66524	3
C₃×C₉×#66524	3
3₋¹⁺²⋊#66524	3
#22582472	3
C₃×C₃₇×#733	3
C₉×#733	37

subgroup

index

3₋¹⁺²×#66524

C₃×C₉×#66524

3₋¹⁺²⋊#66524

#22582472

C₃×C₃₇×#733

C₉×#733

Subgroups — #733×#66524 #41006392

Conjugacy classes of subgroups

Complements

Maximal subgroups