Group #41006654 — subgroups

Subgroups — #41006654

Order 26973.

426 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 70 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	#22582647	C₃	1	3	yes	maximal, complemented
2	#22582639	C₃	1	6	yes	maximal, complemented
3	C₃×#15688521	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×#15688519	C₃	1	2	yes	maximal, complemented
5	C₃×C₃₇×#737	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃×#11758608	C₃²	1	3	yes	—
7	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	6	yes	—
8	#15688521	—	3	1	no	—
9	#15688521	—	3	2	no	—
10	#15688519	—	3	2	no	—
11	#15688519	—	3	4	no	—
12	#15688518	—	3	9	no	—
13	#15688514	—	3	18	no	—
14	C₃₇×#737	—	3	6	no	—
15	C₃₇×#737	—	3	3	no	—
16	C₃×3₋¹⁺²×C₃₇	—	3	2	no	—
17	C₃×3₋¹⁺²×C₃₇	C₃²	1	2	yes	—
18	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
19	C₃²×C₉×C₃₇	—	3	1	no	—
20	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
21	C₃³×C₃₇	—	3	1	no	—
22	C₃³×C₃₇	C₃³	1	1	yes	characteristic
23	C₃²×C₃₇⋊C₃	—	3	6	no	—
24	3₋¹⁺²×C₃₇	—	3	18	no	—
25	3₋¹⁺²×C₃₇	—	3	2	no	—
26	3₋¹⁺²×C₃₇	—	3	2	no	—
27	3₋¹⁺²×C₃₇	H(3,𝔽₃)	1	6	yes	—
28	H(3,𝔽₃)×C₃₇	—	3	2	no	—
29	H(3,𝔽₃)×C₃₇	H(3,𝔽₃)	1	3	yes	—
30	C₃×C₉×C₃₇	—	3	9	no	—
31	C₃×C₉×C₃₇	—	3	2	no	—
32	C₃×C₉×C₃₇	—	3	1	no	—
33	C₃×C₉×C₃₇	—	3	2	no	—
34	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
35	C₃×C₉×C₃₇	H(3,𝔽₃)	1	2	yes	—
36	C₃²⋊(C₃₇⋊C₃)	—	3	6	no	—
37	C₃²⋊(C₃₇⋊C₃)	—	3	6	no	—
38	C₃²⋊(C₃₇⋊C₃)	—	3	6	no	—
39	#11758608	—	3	6	no	—
40	#11758608	—	3	3	no	—
41	C₃×#66524	—	3	3	no	—
42	#10624757	—	37	1	no	maximal, Sylow
43	C₃²×C₃₇	—	3	3	no	—
44	C₃²×C₃₇	—	9	1	no	—
45	C₃²×C₃₇	—	9	3	no	—
46	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
47	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
48	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	2	yes	—
49	C₃×C₃₇⋊C₃	—	9	6	no	—
50	C₃×C₃₇⋊C₃	—	27	6	no	—
51	C₉×C₃₇	—	3	1	no	—
52	C₉×C₃₇	—	3	6	no	—
53	C₉×C₃₇	—	3	2	no	—
54	#66524	—	9	3	no	—
55	C₃×#737	C₃₇⋊C₃	1	1	yes	characteristic, complemented
56	#6911	—	37	3	no	—
57	#6909	—	37	6	no	—
58	C₃×#733	—	37	1	no	—
59	C₃×#732	—	37	2	no	—
60	C₃×C₃₇	—	3	2	no	—
61	C₃×C₃₇	—	3	1	no	—
62	C₃×C₃₇	—	3	1	no	—
63	C₃×C₃₇	—	9	3	no	—
64	C₃×C₃₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
65	C₃₇⋊C₃	—	27	6	no	—
66	#737	—	3	6	no	—
67	#737	—	3	3	no	—
68	C₃×3₋¹⁺²	—	3	2	no	—
69	C₃×3₋¹⁺²	—	37	3	no	—
70	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	2	yes	—
71	C₃×H(3,𝔽₃)	—	37	6	no	—
72	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
73	C₃²×C₉	—	3	1	no	—
74	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
75	#733	—	111	1	no	—
76	#733	—	111	2	no	—
77	#732	—	111	9	no	—
78	#732	—	111	2	no	—
79	#732	—	111	4	no	—
80	AΣL(1,𝔽₂₇)	—	111	18	no	—
81	C₃₇	#10624757	1	1	yes	characteristic, complemented, Sylow
82	C₃³	—	3	1	no	—
83	C₃³	—	111	6	no	—
84	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
85	3₋¹⁺²	—	3	18	no	—
86	3₋¹⁺²	—	3	2	no	—
87	3₋¹⁺²	—	3	2	no	—
88	3₋¹⁺²	—	111	6	no	—
89	3₋¹⁺²	—	111	3	no	—
90	3₋¹⁺²	C₃²⋊(C₃₇⋊C₃)	1	6	yes	—
91	H(3,𝔽₃)	—	3	2	no	—
92	H(3,𝔽₃)	—	111	6	no	—
93	H(3,𝔽₃)	—	111	6	no	—
94	H(3,𝔽₃)	—	111	6	no	—
95	H(3,𝔽₃)	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
96	C₃×C₉	—	3	9	no	—
97	C₃×C₉	—	3	2	no	—
98	C₃×C₉	—	3	1	no	—
99	C₃×C₉	—	3	2	no	—
100	C₃×C₉	—	111	3	no	—
101	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
102	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
103	C₃²	—	3	3	no	—
104	C₃²	—	9	1	no	—
105	C₃²	—	9	3	no	—
106	C₃²	—	333	6	no	—
107	C₃²	—	999	6	no	—
108	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
109	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
110	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
111	C₉	—	3	1	no	—
112	C₉	—	3	6	no	—
113	C₉	—	3	2	no	—
114	C₉	—	333	3	no	—
115	C₃	—	3	1	no	—
116	C₃	—	3	2	no	—
117	C₃	—	3	1	no	—
118	C₃	—	9	3	no	—
119	C₃	—	999	6	no	—
120	C₃	#22582550	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22582647

C₃

yes

maximal, complemented

#22582639

C₃

yes

maximal, complemented

C₃×#15688521

C₃

yes

maximal, characteristic, complemented

C₃×#15688519

C₃

yes

maximal, complemented

C₃×C₃₇×#737

C₃

yes

maximal, characteristic, complemented

C₃×#11758608

C₃²

yes

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

—

—

—

—

—

—

—

C₃₇×#737

—

C₃₇×#737

—

C₃×3₋¹⁺²×C₃₇

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

characteristic

C₃²×C₉×C₃₇

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃³×C₃₇

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

H(3,𝔽₃)

yes

—

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×C₃₇

H(3,𝔽₃)

yes

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

characteristic

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

—

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

#66524

—

C₃×#737

C₃₇⋊C₃

yes

characteristic, complemented

—

—

—

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃²⋊H(3,𝔽₃)

yes

characteristic

C₃₇⋊C₃

—

#737

—

#737

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×C₃₇⋊C₃

yes

characteristic

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

#733

—

111

—

#733

—

111

—

#732

—

111

—

#732

—

111

—

#732

—

111

—

AΣL(1,𝔽₂₇)

—

111

—

C₃₇

#10624757

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

111

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

3₋¹⁺²

C₃²⋊(C₃₇⋊C₃)

yes

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

111

—

H(3,𝔽₃)

—

111

—

H(3,𝔽₃)

—

111

—

H(3,𝔽₃)

C₃²⋊(C₃₇⋊C₃)

yes

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

100

C₃×C₉

—

111

—

101

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

102

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

333

—

107

C₃²

—

999

—

108

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

109

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

110

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

—

111

C₉

—

112

C₉

—

113

C₉

—

114

C₉

—

333

—

115

C₃

—

116

C₃

—

117

C₃

—

118

C₃

—

119

C₃

—

999

—

120

C₃

#22582550

yes

characteristic

	subgroup	G/H
center	C₃	#22582550
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃×C₃₇×#737	C₃
socle	C₃×C₃₇	C₃²⋊H(3,𝔽₃)
hypercenter	C₃×#737	C₃₇⋊C₃
hypocenter	C₃₇	#10624757
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×#737

subgroup

G/H

center

C₃

#22582550

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃×C₃₇×#737

C₃

socle

C₃×C₃₇

C₃²⋊H(3,𝔽₃)

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃ ◁ C₃³ ◁ C₃×#737

N	H ≅ G/N	type
1. #22582647	C₃	semidirect
2. #22582639	C₃	semidirect
3. C₃×#15688521	C₃	semidirect
4. C₃×#15688519	C₃	semidirect
5. C₃×C₃₇×#737	C₃	semidirect
55. C₃×#737	C₃₇⋊C₃	semidirect
81. C₃₇	#10624757	semidirect

H ≅ G/N

type

1. #22582647

C₃

semidirect

2. #22582639

C₃

semidirect

3. C₃×#15688521

C₃

semidirect

4. C₃×#15688519

C₃

semidirect

5. C₃×C₃₇×#737

C₃

semidirect

55. C₃×#737

C₃₇⋊C₃

semidirect

81. C₃₇

#10624757

semidirect

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

subgroup	index
#22582647	3
#22582639	3
C₃×#15688521	3
C₃×#15688519	3
C₃×C₃₇×#737	3
#10624757	37

subgroup

index

C₃×C₃₇×#737

Subgroups — #41006654

Conjugacy classes of subgroups

Complements

Maximal subgroups