Group #41006517 — subgroups

Subgroups — #41006517

Order 26973.

572 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 67 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, complemented
2	C₃×#15688519	C₃	1	2	yes	maximal, complemented
3	C₃×#15688518	C₃	1	6	yes	maximal, complemented
4	(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	C₃	1	3	yes	maximal, complemented
5	C₃×C₃₇×#732	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	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	9	no	—
11	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	6	yes	—
12	C₃²×#66524	C₃²	1	2	yes	—
13	#15688519	—	3	6	no	—
14	#15688518	—	3	18	no	—
15	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	3	yes	—
16	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
17	C₃₇×#732	—	3	3	no	—
18	C₃³×C₃₇	—	3	3	no	—
19	C₃³×C₃₇	C₃³	1	1	yes	characteristic
20	C₃²×C₃₇⋊C₃	—	3	2	no	—
21	C₃²×C₃₇⋊C₃	—	3	1	no	—
22	C₃²×C₃₇⋊C₃	—	3	6	no	—
23	C₃²×C₃₇⋊C₃	—	9	3	no	—
24	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	3	yes	—
25	H(3,𝔽₃)×C₃₇	—	3	3	no	—
26	H(3,𝔽₃)×C₃₇	—	3	6	no	—
27	C₃×C₉×C₃₇	—	3	2	no	—
28	C₃×C₉×C₃₇	—	3	1	no	—
29	C₃×C₉×C₃₇	H(3,𝔽₃)	1	3	yes	complemented
30	C₃²⋊(C₃₇⋊C₃)	—	3	12	no	—
31	C₃²⋊(C₃₇⋊C₃)	—	3	6	no	—
32	C₃²⋊(C₃₇⋊C₃)	—	9	27	no	—
33	C₃×#66524	—	3	2	no	—
34	C₃×#66524	—	3	2	no	—
35	C₃×#66524	—	3	2	no	—
36	C₃×#66524	H(3,𝔽₃)	1	6	yes	complemented
37	#11758606	—	3	27	no	—
38	#10624737	—	37	1	no	maximal, Sylow
39	C₃²×C₃₇	—	3	3	no	—
40	C₃²×C₃₇	—	9	3	no	—
41	C₃²×C₃₇	—	9	9	no	—
42	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	2	yes	—
43	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
44	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
45	C₃×C₃₇⋊C₃	—	3	3	no	—
46	C₃×C₃₇⋊C₃	—	3	9	no	—
47	C₃×C₃₇⋊C₃	—	9	6	no	—
48	C₃×C₃₇⋊C₃	—	9	3	no	—
49	C₃×C₃₇⋊C₃	—	9	6	no	—
50	C₃×C₃₇⋊C₃	—	9	18	no	—
51	C₉×C₃₇	—	3	3	no	—
52	#66524	—	3	6	no	—
53	C₃²×3₋¹⁺²	—	37	1	no	—
54	C₃×#732	—	37	6	no	—
55	C₃×#732	—	37	2	no	—
56	C₃×#732	C₃₇⋊C₃	1	1	yes	characteristic, complemented
57	C₃²⋊H(3,𝔽₃)	—	37	3	no	—
58	C₃×C₃₇	—	3	2	no	—
59	C₃×C₃₇	—	3	1	no	—
60	C₃×C₃₇	—	27	3	no	—
61	C₃×C₃₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
62	C₃×C₃₇	#6911	1	3	yes	—
63	C₃₇⋊C₃	—	9	3	no	—
64	C₃₇⋊C₃	—	27	6	no	—
65	C₃⁴	—	37	1	no	—
66	C₃×3₋¹⁺²	—	111	6	no	—
67	C₃×3₋¹⁺²	—	111	3	no	—
68	C₃×3₋¹⁺²	—	111	3	no	—
69	C₃×H(3,𝔽₃)	—	37	6	no	—
70	C₃×H(3,𝔽₃)	—	111	9	no	—
71	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	3	yes	—
72	C₃²×C₉	—	37	2	no	—
73	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
74	#732	—	3	3	no	—
75	#732	—	111	18	no	—
76	#732	—	111	6	no	—
77	C₃₇	#10624737	1	1	yes	characteristic, complemented, Sylow
78	C₃³	—	3	3	no	—
79	C₃³	—	37	3	no	—
80	C₃³	—	333	3	no	—
81	C₃³	—	111	6	no	—
82	C₃³	—	111	2	no	—
83	C₃³	—	111	1	no	—
84	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
85	3₋¹⁺²	—	111	27	no	—
86	H(3,𝔽₃)	—	3	6	no	—
87	H(3,𝔽₃)	—	3	3	no	—
88	H(3,𝔽₃)	—	333	27	no	—
89	H(3,𝔽₃)	—	111	12	no	—
90	H(3,𝔽₃)	—	111	6	no	—
91	C₃×C₉	—	3	2	no	—
92	C₃×C₉	—	3	1	no	—
93	C₃×C₉	—	37	6	no	—
94	C₃×C₉	—	111	2	no	—
95	C₃×C₉	—	111	2	no	—
96	C₃×C₉	—	111	2	no	—
97	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	3	yes	complemented
98	C₃²	—	3	3	no	—
99	C₃²	—	9	3	no	—
100	C₃²	—	9	9	no	—
101	C₃²	—	333	6	no	—
102	C₃²	—	333	3	no	—
103	C₃²	—	333	6	no	—
104	C₃²	—	333	18	no	—
105	C₃²	—	111	3	no	—
106	C₃²	—	111	9	no	—
107	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	2	yes	—
108	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
109	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
110	C₉	—	3	3	no	—
111	C₉	—	111	6	no	—
112	C₃	—	3	2	no	—
113	C₃	—	3	1	no	—
114	C₃	—	27	3	no	—
115	C₃	—	333	3	no	—
116	C₃	—	999	6	no	—
117	C₃	(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	1	1	yes	characteristic
118	C₃	#22582646	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758606

C₃

yes

maximal, characteristic, complemented

C₃×#15688519

C₃

yes

maximal, complemented

C₃×#15688518

C₃

yes

maximal, complemented

(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)

C₃

yes

maximal, complemented

C₃×C₃₇×#732

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

C₃²

yes

—

—

—

C₃×C₃²⋊(C₃₇⋊C₃)

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

#15688519

—

#15688518

—

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×#732

—

C₃³×C₃₇

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

H(3,𝔽₃)

yes

—

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

complemented

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

—

—

—

yes

—

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

#66524

—

C₃²×3₋¹⁺²

—

C₃×#732

—

C₃×#732

—

C₃×#732

C₃₇⋊C₃

yes

characteristic, complemented

C₃²⋊H(3,𝔽₃)

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃²⋊H(3,𝔽₃)

yes

characteristic

C₃×C₃₇

#6911

yes

—

C₃₇⋊C₃

—

C₃₇⋊C₃

—

C₃⁴

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

111

—

C₃×H(3,𝔽₃)

C₃×C₃₇⋊C₃

yes

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

#732

—

#732

—

111

—

#732

—

111

—

C₃₇

#10624737

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

333

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

111

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

333

—

H(3,𝔽₃)

—

111

—

H(3,𝔽₃)

—

111

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

complemented

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

—

333

—

102

C₃²

—

333

—

103

C₃²

—

333

—

104

C₃²

—

333

—

105

C₃²

—

111

—

106

C₃²

—

111

—

107

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

—

108

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

109

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

110

C₉

—

111

C₉

—

111

—

112

C₃

—

113

C₃

—

114

C₃

—

115

C₃

—

333

—

116

C₃

—

999

—

117

C₃

(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)

yes

characteristic

118

C₃

#22582646

yes

—

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃×C₃₇×#732	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
hypercenter	C₃×#732	C₃₇⋊C₃
hypocenter	C₃₇	#10624737
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#732

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃×C₃₇×#732

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#732

N	H ≅ G/N	type
1. C₃²×#11758606	C₃	semidirect
2. C₃×#15688519	C₃	semidirect
3. C₃×#15688518	C₃	semidirect
4. (C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	C₃	semidirect
5. C₃×C₃₇×#732	C₃	semidirect
29. C₃×C₉×C₃₇	H(3,𝔽₃)	semidirect
36. C₃×#66524	H(3,𝔽₃)	semidirect
56. C₃×#732	C₃₇⋊C₃	semidirect
77. C₃₇	#10624737	semidirect
97. C₃×C₉	C₃²⋊(C₃₇⋊C₃)	semidirect

H ≅ G/N

type

1. C₃²×#11758606

C₃

semidirect

2. C₃×#15688519

C₃

semidirect

3. C₃×#15688518

C₃

semidirect

4. (C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)

C₃

semidirect

5. C₃×C₃₇×#732

C₃

semidirect

29. C₃×C₉×C₃₇

H(3,𝔽₃)

semidirect

36. C₃×#66524

H(3,𝔽₃)

semidirect

56. C₃×#732

C₃₇⋊C₃

semidirect

77. C₃₇

#10624737

semidirect

97. C₃×C₉

C₃²⋊(C₃₇⋊C₃)

semidirect

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

subgroup	index
C₃²×#11758606	3
C₃×#15688519	3
C₃×#15688518	3
(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	3
C₃×C₃₇×#732	3
#10624737	37

subgroup

index

C₃²×#11758606

C₃×#15688519

C₃×#15688518

(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)

C₃×C₃₇×#732

#10624737

Subgroups — #41006517

Conjugacy classes of subgroups

Complements

Maximal subgroups