Group #41005221 — subgroups

Subgroups — #41005221

Order 26973.

428 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 103 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×#11758608	C₃	1	1	yes	maximal, characteristic
2	C₃×C₃²⋊#66524	C₃	1	1	yes	maximal, characteristic
3	C₃×C₃²⋊#66524	C₃	1	1	yes	maximal, characteristic
4	#22582441	C₃	1	9	yes	maximal, complemented
5	C₃×C₃₇×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
6	C₃×#11758608	—	3	3	no	—
7	C₃×#11758608	—	3	9	no	—
8	C₃²×#66524	C₃²	1	3	yes	—
9	C₃²×#66524	C₃²	1	3	yes	—
10	C₃²×#66524	C₃²	1	3	yes	—
11	C₃²⋊#66524	—	3	3	no	—
12	C₃²⋊#66524	—	3	3	no	—
13	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
14	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
15	C₃₇×C₃²⋊C₉	—	3	3	no	—
16	C₃³×C₃₇	—	3	3	no	—
17	C₃³×C₃₇	—	3	3	no	—
18	C₃³×C₃₇	—	3	3	no	—
19	C₃³×C₃₇	—	3	3	no	—
20	C₃³×C₃₇	H(3,𝔽₃)	1	3	yes	—
21	C₃³×C₃₇	C₃³	1	1	yes	characteristic
22	C₃×C₉×C₃₇	—	3	3	no	—
23	C₃×C₉×C₃₇	—	3	9	no	—
24	#11758608	—	3	27	no	—
25	C₃×#66524	—	3	3	no	—
26	C₃×#66524	—	3	6	no	—
27	C₃×#66524	—	3	3	no	—
28	C₃×#66524	—	3	3	no	—
29	C₃×#66524	—	3	9	no	—
30	C₃×#66524	—	3	9	no	—
31	C₃×#66524	3₋¹⁺²	1	9	yes	—
32	#10624570	—	37	1	no	maximal, Sylow
33	C₃²×C₃₇	—	3	3	no	—
34	C₃²×C₃₇	—	3	3	no	—
35	C₃²×C₃₇	—	3	3	no	—
36	C₃²×C₃₇	—	3	3	no	—
37	C₃²×C₃₇	—	9	9	no	—
38	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
39	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
40	C₃²×C₃₇	#737	1	3	yes	—
41	C₃²×C₃₇	#737	1	3	yes	—
42	C₃²×C₃₇	#737	1	3	yes	—
43	C₉×C₃₇	—	9	3	no	—
44	#66524	—	3	9	no	—
45	#66524	—	9	3	no	—
46	#66524	—	9	3	no	—
47	C₃²×3₋¹⁺²	—	37	1	no	—
48	C₃×C₃²⋊C₉	—	37	1	no	—
49	C₃×C₃²⋊C₉	—	37	1	no	—
50	C₃×C₃²⋊C₉	C₃₇⋊C₃	1	1	yes	characteristic
51	#6855	—	37	9	no	—
52	C₃×C₃₇	—	9	3	no	—
53	C₃×C₃₇	#6891	1	1	yes	characteristic
54	C₃×C₃₇	#6893	1	1	yes	characteristic
55	C₃×C₃₇	#6893	1	1	yes	characteristic
56	C₃×C₃₇	#6893	1	1	yes	characteristic
57	C₃×C₃₇	#6899	1	9	yes	—
58	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
59	C₃×3₋¹⁺²	—	111	9	no	—
60	C₃×3₋¹⁺²	—	111	3	no	—
61	C₃²×C₉	—	37	3	no	—
62	C₃²×C₉	—	37	3	no	—
63	C₃²×C₉	—	37	3	no	—
64	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	—
65	C₃²⋊C₉	—	3	3	no	—
66	C₃²⋊C₉	—	111	3	no	—
67	C₃²⋊C₉	—	111	3	no	—
68	C₃₇	#10624570	1	1	yes	characteristic, complemented, Sylow
69	C₃³	—	3	3	no	—
70	C₃³	—	3	3	no	—
71	C₃³	—	3	3	no	—
72	C₃³	—	3	3	no	—
73	C₃³	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
74	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
75	3₋¹⁺²	—	111	27	no	—
76	C₃×C₉	—	3	3	no	—
77	C₃×C₉	—	3	9	no	—
78	C₃×C₉	—	37	9	no	—
79	C₃×C₉	—	111	3	no	—
80	C₃×C₉	—	111	3	no	—
81	C₃×C₉	—	111	9	no	—
82	C₃×C₉	—	111	9	no	—
83	C₃×C₉	—	111	3	no	—
84	C₃×C₉	—	111	6	no	—
85	C₃²	—	3	3	no	—
86	C₃²	—	3	3	no	—
87	C₃²	—	3	3	no	—
88	C₃²	—	3	3	no	—
89	C₃²	—	9	9	no	—
90	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
91	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	3	yes	—
92	C₃²	#15688530	1	3	yes	—
93	C₃²	#15688531	1	3	yes	—
94	C₃²	#15688532	1	3	yes	—
95	C₉	—	9	3	no	—
96	C₉	—	333	3	no	—
97	C₉	—	333	3	no	—
98	C₉	—	111	9	no	—
99	C₃	—	9	3	no	—
100	C₃	#22582551	1	1	yes	characteristic
101	C₃	#22582562	1	1	yes	characteristic
102	C₃	#22582564	1	1	yes	characteristic
103	C₃	#22582565	1	1	yes	characteristic
104	C₃	#22582593	1	9	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758608

C₃

yes

maximal, characteristic

C₃×C₃²⋊#66524

C₃

yes

maximal, characteristic

C₃×C₃²⋊#66524

C₃

yes

maximal, characteristic

#22582441

C₃

yes

maximal, complemented

C₃×C₃₇×C₃²⋊C₉

C₃

yes

maximal, characteristic

C₃×#11758608

—

C₃×#11758608

—

C₃²×#66524

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

C₃²⋊#66524

—

C₃²⋊#66524

—

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃²×C₉×C₃₇

C₃²

yes

—

C₃₇×C₃²⋊C₉

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

H(3,𝔽₃)

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

—

—

—

—

—

—

—

3₋¹⁺²

yes

—

#10624570

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₃²×C₃₇

#737

yes

—

C₃²×C₃₇

#737

yes

—

C₃²×C₃₇

#737

yes

—

C₉×C₃₇

—

#66524

—

#66524

—

#66524

—

C₃²×3₋¹⁺²

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

C₃₇⋊C₃

yes

characteristic

#6855

—

C₃×C₃₇

—

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

—

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃₇

#10624570

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

111

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

C₃²

3₋¹⁺²×C₃₇⋊C₃

yes

—

C₃²

#15688530

yes

—

C₃²

#15688531

yes

—

C₃²

#15688532

yes

—

C₉

—

C₉

—

333

—

C₉

—

333

—

C₉

—

111

—

C₃

—

100

C₃

#22582551

yes

characteristic

101

C₃

#22582562

yes

characteristic

102

C₃

#22582564

yes

characteristic

103

C₃

#22582565

yes

characteristic

104

C₃

#22582593

yes

—

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₃₇⋊C₃
commutator, socle, cosocle	C₃³×C₃₇	C₃³
Fitting	C₃×C₃₇×C₃²⋊C₉	C₃
prime radical	C₃⁴×C₃₇	C₃²
hypercenter	C₃×C₃²⋊C₉	C₃₇⋊C₃
hypocenter	C₃₇	#10624570
LCS	G ▷ C₃³×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ C₃×C₃²⋊C₉

subgroup

G/H

center, Frattini

C₃³

C₃²×C₃₇⋊C₃

commutator, socle, cosocle

C₃³×C₃₇

C₃³

Fitting

C₃×C₃₇×C₃²⋊C₉

C₃

prime radical

C₃⁴×C₃₇

C₃²

hypercenter

C₃×C₃²⋊C₉

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃₇

C₁ ◁ C₃³ ◁ C₃×C₃²⋊C₉

N	H ≅ G/N	type
4. #22582441	C₃	semidirect
68. C₃₇	#10624570	semidirect

H ≅ G/N

type

4. #22582441

C₃

semidirect

68. C₃₇

#10624570

semidirect

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

subgroup	index
C₃²×#11758608	3
C₃×C₃²⋊#66524	3
#22582441	3
C₃×C₃₇×C₃²⋊C₉	3
#10624570	37

subgroup

index

C₃²×#11758608

C₃×C₃²⋊#66524

#22582441

C₃×C₃₇×C₃²⋊C₉

#10624570

Subgroups — #41005221

Conjugacy classes of subgroups

Complements

Maximal subgroups