Group #41008593 — subgroups

Subgroups — #41008593

Order 26973.

224 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 66 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	#22582732	C₃	1	3	yes	maximal, complemented
2	C₃×(C₃₇⋊C₃)⋊C₂₇	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582702	C₃	1	3	yes	maximal, complemented
4	#22582700	C₃	1	3	yes	maximal, complemented
5	#22582695	C₃	1	2	yes	maximal, complemented
6	#22582634	C₃	1	1	yes	maximal, characteristic
7	#15688543	—	3	3	no	—
8	#15688542	—	3	3	no	—
9	#15688541	C₃²	1	6	yes	—
10	(C₃₇⋊C₃)⋊C₂₇	—	3	1	no	—
11	(C₃₇⋊C₃)⋊C₂₇	—	3	1	no	—
12	(C₃₇⋊C₃)⋊C₂₇	—	3	1	no	—
13	(C₃₇⋊C₃)⋊C₂₇	—	3	3	no	—
14	C₃×#11758603	C₃²	1	1	yes	characteristic
15	C₃×#11758603	C₃²	1	1	yes	characteristic
16	C₃×#11758603	C₃²	1	1	yes	characteristic
17	#15688532	C₃²	1	3	yes	—
18	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
19	#15688521	C₉	1	1	yes	characteristic
20	#15688519	C₉	1	1	yes	characteristic
21	#15688517	C₉	1	3	yes	—
22	#15688514	C₉	1	3	yes	—
23	C₃₇×#733	C₉	1	1	yes	characteristic
24	C₃²×C₃₇⋊C₃	C₃×C₉	1	1	yes	characteristic
25	3₋¹⁺²×C₃₇	C₃×C₉	1	3	yes	—
26	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
27	C₃²⋊(C₃₇⋊C₃)	C₃×C₉	1	3	yes	—
28	#11758608	C₃×C₉	1	3	yes	—
29	C₃×#66524	C₃×C₉	1	1	yes	characteristic
30	C₃×#66524	C₃³	1	1	yes	characteristic
31	#11758606	—	3	3	no	—
32	#11758605	—	3	3	no	—
33	C₉×C₃₇⋊C₃	—	3	1	no	—
34	C₉×C₃₇⋊C₃	—	3	3	no	—
35	C₉×C₃₇⋊C₃	—	3	1	no	—
36	C₉×C₃₇⋊C₃	—	3	1	no	—
37	#11758603	—	3	1	no	—
38	#11758603	—	3	1	no	—
39	#11758603	—	3	6	no	—
40	#11758603	H(3,𝔽₃)	1	1	yes	characteristic
41	#11758603	H(3,𝔽₃)	1	1	yes	characteristic
42	#11758603	H(3,𝔽₃)	1	1	yes	characteristic
43	#10624724	—	37	1	no	maximal, Sylow
44	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
45	C₃×C₃₇⋊C₃	—	3	1	no	—
46	C₃×C₃₇⋊C₃	—	3	3	no	—
47	C₃×C₃₇⋊C₃	—	3	3	no	—
48	C₉×C₃₇	—	3	1	no	—
49	C₉×C₃₇	—	3	3	no	—
50	#66524	—	3	1	no	—
51	#66524	—	3	3	no	—
52	#66524	C₃×H(3,𝔽₃)	1	1	yes	characteristic
53	#66524	#737	1	1	yes	characteristic
54	#66524	#737	1	1	yes	characteristic
55	#6908	—	37	1	no	—
56	#6903	—	37	3	no	—
57	C₃²×C₂₇	—	37	1	no	—
58	#6873	—	37	3	no	—
59	#6872	—	37	3	no	—
60	#6869	—	37	2	no	—
61	C₃×C₃₇	—	3	1	no	—
62	C₃×C₃₇	C₉×H(3,𝔽₃)	1	1	yes	characteristic
63	C₃₇⋊C₃	—	3	3	no	—
64	C₃₇⋊C₃	—	9	3	no	—
65	#737	—	37	3	no	—
66	C₃²×C₉	—	37	1	no	—
67	#733	—	37	1	no	—
68	#733	#66526	1	1	yes	characteristic
69	#732	—	37	1	no	—
70	#731	—	37	3	no	—
71	AΣL(1,𝔽₂₇)	—	37	3	no	—
72	C₂₇⋊C₃	—	37	6	no	—
73	C₂₇⋊C₃	—	111	3	no	—
74	C₂₇⋊C₃	—	111	3	no	—
75	C₃×C₂₇	—	37	1	no	—
76	C₃×C₂₇	—	37	1	no	—
77	C₃×C₂₇	—	37	1	no	—
78	C₃×C₂₇	—	111	1	no	—
79	C₃×C₂₇	—	111	1	no	—
80	C₃×C₂₇	—	111	1	no	—
81	C₃×C₂₇	—	111	3	no	—
82	C₃₇	#10624724	1	1	yes	characteristic, complemented, Sylow
83	C₃³	—	37	1	no	—
84	3₋¹⁺²	—	37	3	no	—
85	3₋¹⁺²	—	111	3	no	—
86	3₋¹⁺²	—	111	3	no	—
87	3₋¹⁺²	C₃×#66526	1	3	yes	—
88	H(3,𝔽₃)	—	37	3	no	—
89	C₃×C₉	—	37	1	no	—
90	C₃×C₉	—	37	1	no	—
91	C₃×C₉	—	111	1	no	—
92	C₃×C₉	—	111	3	no	—
93	C₃×C₉	—	111	1	no	—
94	C₃×C₉	—	111	1	no	—
95	C₃×C₉	C₃×#66526	1	1	yes	characteristic
96	C₂₇	—	37	1	no	—
97	C₂₇	—	37	1	no	—
98	C₂₇	—	37	1	no	—
99	C₂₇	—	111	1	no	—
100	C₂₇	—	111	1	no	—
101	C₂₇	—	111	6	no	—
102	C₃²	—	111	3	no	—
103	C₃²	—	111	3	no	—
104	C₃²	—	111	1	no	—
105	C₃²	C₃²×#66526	1	1	yes	characteristic
106	C₉	—	3	1	no	—
107	C₉	—	3	3	no	—
108	C₉	—	37	1	no	—
109	C₉	—	37	1	no	—
110	C₉	—	37	1	no	—
111	C₉	—	111	1	no	—
112	C₉	—	111	3	no	—
113	C₃	—	3	1	no	—
114	C₃	—	333	3	no	—
115	C₃	—	111	3	no	—
116	C₃	H(3,𝔽₃)×#66526	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22582732

C₃

yes

maximal, complemented

C₃×(C₃₇⋊C₃)⋊C₂₇

C₃

yes

maximal, characteristic, complemented

#22582702

C₃

yes

maximal, complemented

#22582700

C₃

yes

maximal, complemented

#22582695

C₃

yes

maximal, complemented

#22582634

C₃

yes

maximal, characteristic

#15688543

—

#15688542

—

#15688541

C₃²

yes

—

(C₃₇⋊C₃)⋊C₂₇

—

(C₃₇⋊C₃)⋊C₂₇

—

(C₃₇⋊C₃)⋊C₂₇

—

(C₃₇⋊C₃)⋊C₂₇

—

C₃×#11758603

C₃²

yes

characteristic

C₃×#11758603

C₃²

yes

characteristic

C₃×#11758603

C₃²

yes

characteristic

#15688532

C₃²

yes

—

C₃×C₉×C₃₇⋊C₃

C₃²

yes

characteristic

#15688521

C₉

yes

characteristic

#15688519

C₉

yes

characteristic

#15688517

C₉

yes

—

#15688514

C₉

yes

—

C₃₇×#733

C₉

yes

characteristic

C₃²×C₃₇⋊C₃

C₃×C₉

yes

characteristic

3₋¹⁺²×C₃₇

C₃×C₉

yes

—

C₃×C₉×C₃₇

C₃×C₉

yes

characteristic

C₃²⋊(C₃₇⋊C₃)

C₃×C₉

yes

—

#11758608

C₃×C₉

yes

—

C₃×#66524

C₃×C₉

yes

characteristic

C₃×#66524

C₃³

yes

characteristic

#11758606

—

#11758605

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

—

—

—

yes

yes

yes

—

C₃²×C₃₇

C₃²×C₉

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

—

#66524

—

#66524

—

#66524

C₃×H(3,𝔽₃)

yes

yes

yes

—

—

C₃²×C₂₇

—

#6873

—

#6872

—

#6869

—

C₃×C₃₇

—

C₃×C₃₇

C₉×H(3,𝔽₃)

yes

characteristic

C₃₇⋊C₃

—

C₃₇⋊C₃

—

#737

—

C₃²×C₉

—

—

yes

—

—

AΣL(1,𝔽₂₇)

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

—

111

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

111

—

C₃×C₂₇

—

111

—

C₃×C₂₇

—

111

—

C₃×C₂₇

—

111

—

C₃₇

#10624724

yes

characteristic, complemented, Sylow

C₃³

—

3₋¹⁺²

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

3₋¹⁺²

C₃×#66526

yes

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

C₃×#66526

yes

characteristic

C₂₇

—

C₂₇

—

C₂₇

—

C₂₇

—

111

—

100

C₂₇

—

111

—

101

C₂₇

—

111

—

102

C₃²

—

111

—

103

C₃²

—

111

—

104

C₃²

—

111

—

105

C₃²

C₃²×#66526

yes

characteristic

106

C₉

—

107

C₉

—

108

C₉

—

109

C₉

—

110

C₉

—

111

C₉

—

111

—

112

C₉

—

111

—

113

C₃

—

114

C₃

—

333

—

115

C₃

—

111

—

116

C₃

H(3,𝔽₃)×#66526

yes

characteristic

	subgroup	G/H
center	C₃	H(3,𝔽₃)×#66526
commutator	C₃²×C₃₇	C₃²×C₉
Frattini	C₃²	C₃²×#66526
Fitting	C₃₇×#733	C₉
socle	C₃×C₃₇	C₉×H(3,𝔽₃)
prime radical	#22582634	C₃
cosocle	C₃×#66524	C₃³
hypercenter	#733	#66526
hypocenter	C₃₇	#10624724
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ #733

subgroup

G/H

center

C₃

H(3,𝔽₃)×#66526

commutator

C₃²×C₃₇

C₃²×C₉

Frattini

C₃²

C₃²×#66526

Fitting

C₃₇×#733

C₉

socle

C₃×C₃₇

C₉×H(3,𝔽₃)

C₃

C₃³

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃ ◁ C₃² ◁ #733

N	H ≅ G/N	type
1. #22582732	C₃	semidirect
2. C₃×(C₃₇⋊C₃)⋊C₂₇	C₃	semidirect
3. #22582702	C₃	semidirect
4. #22582700	C₃	semidirect
5. #22582695	C₃	semidirect
82. C₃₇	#10624724	semidirect

H ≅ G/N

type

1. #22582732

C₃

semidirect

2. C₃×(C₃₇⋊C₃)⋊C₂₇

C₃

semidirect

3. #22582702

C₃

semidirect

4. #22582700

C₃

semidirect

5. #22582695

C₃

semidirect

82. C₃₇

#10624724

semidirect

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

subgroup	index
#22582732	3
C₃×(C₃₇⋊C₃)⋊C₂₇	3
#22582702	3
#22582700	3
#22582695	3
#22582634	3
#10624724	37

subgroup

index

C₃×(C₃₇⋊C₃)⋊C₂₇

Subgroups — #41008593

Conjugacy classes of subgroups

Complements

Maximal subgroups