Group #41005224 — subgroups

Subgroups — #41005224

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×#11758606	C₃	1	1	yes	maximal, characteristic
2	C₃×#15688502	C₃	1	3	yes	maximal
3	#22582441	C₃	1	6	yes	maximal, complemented
4	#22582441	C₃	1	2	yes	maximal, complemented
5	C₃₇×#6855	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
7	C₃×#11758606	—	3	3	no	—
8	C₃×#11758606	—	3	3	no	—
9	C₃×#11758606	—	3	6	no	—
10	C₃²×#66524	C₃²	1	6	yes	—
11	C₃²×#66524	C₃²	1	2	yes	—
12	#15688502	—	3	9	no	—
13	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
14	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
15	C₃³×C₃₇	C₃³	1	1	yes	characteristic
16	C₃²×C₃₇⋊C₃	—	3	2	no	—
17	C₃²×C₃₇⋊C₃	—	3	1	no	—
18	C₃²×C₃₇⋊C₃	—	3	3	no	—
19	C₃²×C₃₇⋊C₃	—	3	6	no	—
20	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	3	yes	—
21	C₃×C₉×C₃₇	—	3	1	no	—
22	C₃×C₉×C₃₇	—	3	2	no	—
23	C₃×C₉×C₃₇	—	3	3	no	—
24	C₃×C₉×C₃₇	—	3	3	no	—
25	C₃×C₉×C₃₇	—	3	6	no	—
26	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
27	C₃×#66524	—	3	2	no	—
28	C₃×#66524	—	3	2	no	—
29	C₃×#66524	—	3	2	no	—
30	C₃×#66524	—	3	6	no	—
31	C₃×#66524	—	3	6	no	—
32	C₃×#66524	—	3	12	no	—
33	C₃×#66524	3₋¹⁺²	1	6	yes	—
34	#11758606	—	3	27	no	—
35	#10624570	—	37	1	no	maximal, Sylow
36	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
37	C₃²×C₃₇	C₃×3₋¹⁺²	1	2	yes	—
38	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
39	C₃²×C₃₇	#737	1	3	yes	—
40	C₃²×C₃₇	#737	1	6	yes	—
41	C₃×C₃₇⋊C₃	—	3	3	no	—
42	C₃×C₃₇⋊C₃	—	3	9	no	—
43	C₃×C₃₇⋊C₃	—	9	1	no	—
44	C₃×C₃₇⋊C₃	—	9	6	no	—
45	C₃×C₃₇⋊C₃	—	9	2	no	—
46	C₉×C₃₇	—	3	3	no	—
47	C₉×C₃₇	—	9	3	no	—
48	#66524	—	3	6	no	—
49	#66524	—	9	6	no	—
50	C₃²×3₋¹⁺²	—	37	1	no	—
51	C₃×C₃²⋊C₉	—	37	3	no	—
52	#6855	—	37	6	no	—
53	#6855	—	37	2	no	—
54	#6855	C₃₇⋊C₃	1	1	yes	characteristic, complemented
55	C₃×C₃₇	#6891	1	1	yes	characteristic
56	C₃×C₃₇	#6893	1	3	yes	—
57	C₃×C₃₇	#6899	1	1	yes	characteristic
58	C₃×C₃₇	#6899	1	6	yes	—
59	C₃×C₃₇	#6899	1	2	yes	—
60	C₃₇⋊C₃	—	9	3	no	—
61	C₃⁴	—	37	1	no	—
62	C₃×3₋¹⁺²	—	111	6	no	—
63	C₃×3₋¹⁺²	—	111	3	no	—
64	C₃×3₋¹⁺²	—	111	3	no	—
65	C₃²×C₉	—	37	2	no	—
66	C₃²×C₉	—	37	6	no	—
67	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	—
68	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
69	C₃²⋊C₉	—	111	9	no	—
70	C₃₇	#10624570	1	1	yes	characteristic, complemented, Sylow
71	C₃³	—	37	3	no	—
72	C₃³	—	111	2	no	—
73	C₃³	—	111	1	no	—
74	C₃³	—	111	3	no	—
75	C₃³	—	111	6	no	—
76	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
77	3₋¹⁺²	—	111	27	no	—
78	C₃×C₉	—	3	1	no	—
79	C₃×C₉	—	3	2	no	—
80	C₃×C₉	—	3	3	no	—
81	C₃×C₉	—	3	3	no	—
82	C₃×C₉	—	3	6	no	—
83	C₃×C₉	—	37	6	no	—
84	C₃×C₉	—	111	6	no	—
85	C₃×C₉	—	111	6	no	—
86	C₃×C₉	—	111	12	no	—
87	C₃×C₉	—	111	2	no	—
88	C₃×C₉	—	111	2	no	—
89	C₃×C₉	—	111	2	no	—
90	C₃×C₉	#11758606	1	3	yes	—
91	C₃²	—	333	1	no	—
92	C₃²	—	333	6	no	—
93	C₃²	—	333	2	no	—
94	C₃²	—	111	3	no	—
95	C₃²	—	111	9	no	—
96	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
97	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	2	yes	—
98	C₃²	C₃×#11758606	1	1	yes	characteristic
99	C₃²	#15688531	1	3	yes	—
100	C₃²	#15688532	1	6	yes	—
101	C₉	—	3	3	no	—
102	C₉	—	9	3	no	—
103	C₉	—	333	6	no	—
104	C₉	—	111	6	no	—
105	C₃	—	333	3	no	—
106	C₃	#22582552	1	1	yes	characteristic
107	C₃	#22582563	1	3	yes	—
108	C₃	#22582595	1	1	yes	characteristic
109	C₃	#22582598	1	6	yes	—
110	C₃	#22582600	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758606

C₃

yes

maximal, characteristic

C₃×#15688502

C₃

yes

maximal

#22582441

C₃

yes

maximal, complemented

#22582441

C₃

yes

maximal, complemented

C₃₇×#6855

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

C₃²

yes

—

—

—

C₃²×#66524

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

#15688502

—

C₃²×C₉×C₃₇

C₃²

yes

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

H(3,𝔽₃)

yes

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

3₋¹⁺²

yes

—

—

—

—

—

—

—

3₋¹⁺²

yes

—

#11758606

—

#10624570

—

maximal, Sylow

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₃²×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₃₇

#737

yes

—

C₃²×C₃₇

#737

yes

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

—

#66524

—

#66524

—

C₃²×3₋¹⁺²

—

C₃×C₃²⋊C₉

—

#6855

—

#6855

—

#6855

C₃₇⋊C₃

yes

characteristic, complemented

C₃×C₃₇

#6891

yes

characteristic

C₃×C₃₇

#6893

yes

—

C₃×C₃₇

#6899

yes

characteristic

C₃×C₃₇

#6899

yes

—

C₃×C₃₇

#6899

yes

—

C₃₇⋊C₃

—

C₃⁴

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₃²⋊C₉

—

111

—

C₃₇

#10624570

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

111

—

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

—

111

—

C₃×C₉

—

111

—

C₃×C₉

#11758606

yes

—

C₃²

—

333

—

C₃²

—

333

—

C₃²

—

333

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

C₃²

3₋¹⁺²×C₃₇⋊C₃

yes

—

C₃²

C₃×#11758606

yes

characteristic

C₃²

#15688531

yes

—

100

C₃²

#15688532

yes

—

101

C₉

—

102

C₉

—

103

C₉

—

333

—

104

C₉

—

111

—

105

C₃

—

333

—

106

C₃

#22582552

yes

characteristic

107

C₃

#22582563

yes

—

108

C₃

#22582595

yes

characteristic

109

C₃

#22582598

yes

—

110

C₃

#22582600

yes

—

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₃₇⋊C₃
commutator, socle, cosocle	C₃³×C₃₇	C₃³
Fitting	C₃₇×#6855	C₃
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	#6855	C₃₇⋊C₃
hypocenter	C₃₇	#10624570
LCS	G ▷ C₃³×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ #6855

subgroup

G/H

center, Frattini

C₃³

C₃²×C₃₇⋊C₃

commutator, socle, cosocle

C₃³×C₃₇

C₃³

Fitting

C₃₇×#6855

C₃

prime radical

C₃³×C₃₇⋊C₃

C₃²

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃₇

C₁ ◁ C₃³ ◁ #6855

N	H ≅ G/N	type
3. #22582441	C₃	semidirect
4. #22582441	C₃	semidirect
5. C₃₇×#6855	C₃	semidirect
54. #6855	C₃₇⋊C₃	semidirect
70. C₃₇	#10624570	semidirect

H ≅ G/N

type

3. #22582441

C₃

semidirect

4. #22582441

C₃

semidirect

5. C₃₇×#6855

C₃

semidirect

54. #6855

C₃₇⋊C₃

semidirect

70. C₃₇

#10624570

semidirect

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

subgroup	index
C₃²×#11758606	3
C₃×#15688502	3
#22582441	3
C₃₇×#6855	3
#10624570	37

subgroup

index

C₃₇×#6855

Subgroups — #41005224

Conjugacy classes of subgroups

Complements

Maximal subgroups