Group #41004739 — subgroups

Subgroups — #41004739

Order 26973.

316 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 46 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×C₉×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
2	C₃²⋊#11758601	C₃	1	2	yes	maximal, complemented
3	C₃₇×C₃²⋊C₂₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃³×C₃₇⋊C₃	C₉	1	1	yes	characteristic
5	C₃²×#66524	C₉	1	2	yes	—
6	C₃×C₉×C₃₇⋊C₃	—	3	1	no	—
7	C₃×C₉×C₃₇⋊C₃	—	3	1	no	—
8	C₃×C₉×C₃₇⋊C₃	—	3	1	no	—
9	C₃×C₉×C₃₇⋊C₃	—	3	3	no	—
10	C₃×C₉×C₃₇⋊C₃	—	3	6	no	—
11	C₃×#11758601	—	3	2	no	—
12	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
13	C₃×C₂₇×C₃₇	—	3	1	no	—
14	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
15	C₃²×C₃₇⋊C₃	—	3	1	no	—
16	C₃²×C₃₇⋊C₃	—	3	1	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	C₂₇	1	3	yes	complemented
21	C₃×C₉×C₃₇	—	3	1	no	—
22	C₃×C₉×C₃₇	—	3	2	no	—
23	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
24	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
25	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
26	C₃×#66524	—	3	2	no	—
27	C₃×#66524	—	3	6	no	—
28	C₉×C₃₇⋊C₃	—	3	9	no	—
29	C₉×C₃₇⋊C₃	—	3	9	no	—
30	C₉×C₃₇⋊C₃	—	3	3	no	—
31	C₉×C₃₇⋊C₃	—	3	6	no	—
32	C₂₇×C₃₇	—	9	1	no	—
33	#11758601	—	9	2	no	—
34	#10624444	—	37	1	no	maximal, Sylow
35	C₃²×C₃₇	—	3	1	no	—
36	C₃²×C₃₇	—	3	2	no	—
37	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
38	C₃²×C₃₇	C₃×C₂₇	1	1	yes	characteristic
39	C₃²×C₃₇	C₂₇⋊C₃	1	1	yes	characteristic
40	C₃²×C₃₇	C₂₇⋊C₃	1	1	yes	characteristic
41	C₃×C₃₇⋊C₃	—	3	9	no	—
42	C₃×C₃₇⋊C₃	—	3	9	no	—
43	C₃×C₃₇⋊C₃	—	3	3	no	—
44	C₃×C₃₇⋊C₃	—	3	3	no	—
45	C₃×C₃₇⋊C₃	—	3	6	no	—
46	C₃×C₃₇⋊C₃	—	3	9	no	—
47	C₉×C₃₇	—	3	1	no	—
48	C₉×C₃₇	—	3	1	no	—
49	C₉×C₃₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
50	C₉×C₃₇	AΣL(1,𝔽₂₇)	1	2	yes	—
51	#66524	—	3	6	no	—
52	C₃³×C₉	—	37	1	no	—
53	C₃²⋊C₂₇	—	37	2	no	—
54	C₃²⋊C₂₇	C₃₇⋊C₃	1	1	yes	characteristic, complemented
55	C₃×C₃₇	—	3	1	no	—
56	C₃×C₃₇	—	3	1	no	—
57	C₃×C₃₇	—	3	1	no	—
58	C₃×C₃₇	C₃²⋊C₂₇	1	1	yes	characteristic
59	C₃×C₃₇	#6866	1	1	yes	characteristic
60	C₃×C₃₇	#6869	1	2	yes	—
61	C₃₇⋊C₃	—	3	9	no	—
62	C₃⁴	—	37	1	no	—
63	C₃²×C₉	—	37	2	no	—
64	C₃²×C₉	—	111	3	no	—
65	C₃²×C₉	—	111	6	no	—
66	C₃²×C₉	—	111	1	no	—
67	C₃²×C₉	—	111	1	no	—
68	C₃²×C₉	—	111	1	no	—
69	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
70	C₃×C₂₇	—	3	1	no	—
71	C₃×C₂₇	—	111	2	no	—
72	C₃₇	#10624444	1	1	yes	characteristic, complemented, Sylow
73	C₃³	—	37	3	no	—
74	C₃³	—	111	3	no	—
75	C₃³	—	111	6	no	—
76	C₃³	—	111	1	no	—
77	C₃³	—	111	1	no	—
78	C₃³	—	111	1	no	—
79	C₃³	C₉×C₃₇⋊C₃	1	1	yes	characteristic
80	C₃×C₉	—	3	1	no	—
81	C₃×C₉	—	3	2	no	—
82	C₃×C₉	—	111	9	no	—
83	C₃×C₉	—	111	9	no	—
84	C₃×C₉	—	111	2	no	—
85	C₃×C₉	—	111	3	no	—
86	C₃×C₉	—	111	6	no	—
87	C₃×C₉	—	111	6	no	—
88	C₃×C₉	#11758605	1	1	yes	characteristic
89	C₃×C₉	#11758606	1	1	yes	characteristic
90	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
91	C₂₇	—	9	1	no	—
92	C₂₇	—	333	2	no	—
93	C₃²	—	3	1	no	—
94	C₃²	—	3	2	no	—
95	C₃²	—	111	9	no	—
96	C₃²	—	111	9	no	—
97	C₃²	—	111	9	no	—
98	C₃²	—	111	3	no	—
99	C₃²	—	111	3	no	—
100	C₃²	—	111	6	no	—
101	C₃²	#15688502	1	1	yes	characteristic
102	C₃²	C₂₇×C₃₇⋊C₃	1	1	yes	characteristic
103	C₃²	#15688509	1	1	yes	characteristic
104	C₃²	#15688510	1	1	yes	characteristic
105	C₉	—	3	1	no	—
106	C₉	—	3	1	no	—
107	C₉	—	111	6	no	—
108	C₉	#15688512	1	1	yes	characteristic
109	C₉	#15688514	1	2	yes	—
110	C₃	—	3	1	no	—
111	C₃	—	3	1	no	—
112	C₃	—	3	1	no	—
113	C₃	—	111	9	no	—
114	C₃	#22582465	1	1	yes	characteristic
115	C₃	#22582467	1	1	yes	characteristic
116	C₃	#22582473	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₉×C₃₇⋊C₃

C₃

yes

maximal, characteristic

C₃²⋊#11758601

C₃

yes

maximal, complemented

C₃₇×C₃²⋊C₂₇

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

C₉

yes

characteristic

C₃²×#66524

C₉

yes

—

C₃×C₉×C₃₇⋊C₃

—

C₃×C₉×C₃₇⋊C₃

—

C₃×C₉×C₃₇⋊C₃

—

C₃×C₉×C₃₇⋊C₃

—

C₃×C₉×C₃₇⋊C₃

—

C₃×#11758601

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃×C₂₇×C₃₇

—

C₃³×C₃₇

C₃×C₉

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

C₂₇

yes

complemented

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

characteristic

C₃×C₉×C₃₇

3₋¹⁺²

yes

characteristic

C₃×C₉×C₃₇

3₋¹⁺²

yes

characteristic

C₃×#66524

—

C₃×#66524

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₂₇×C₃₇

—

#11758601

—

#10624444

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃²⋊C₉

yes

characteristic

C₃²×C₃₇

C₃×C₂₇

yes

characteristic

C₃²×C₃₇

C₂₇⋊C₃

yes

characteristic

C₃²×C₃₇

C₂₇⋊C₃

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

AΣL(1,𝔽₂₇)

yes

characteristic

C₉×C₃₇

AΣL(1,𝔽₂₇)

yes

—

#66524

—

C₃³×C₉

—

C₃²⋊C₂₇

—

C₃²⋊C₂₇

C₃₇⋊C₃

yes

characteristic, complemented

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃²⋊C₂₇

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

—

C₃₇⋊C₃

—

C₃⁴

—

C₃²×C₉

—

C₃²×C₉

—

111

—

C₃²×C₉

—

111

—

C₃²×C₉

—

111

—

C₃²×C₉

—

111

—

C₃²×C₉

—

111

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₃×C₂₇

—

C₃×C₂₇

—

111

—

C₃₇

#10624444

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

C₉×C₃₇⋊C₃

yes

characteristic

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

yes

C₃×C₉

yes

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

C₂₇

—

C₂₇

—

333

—

C₃²

—

C₃²

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

—

111

—

100

C₃²

—

111

—

101

C₃²

#15688502

yes

characteristic

102

C₃²

C₂₇×C₃₇⋊C₃

yes

characteristic

103

C₃²

#15688509

yes

characteristic

104

C₃²

#15688510

yes

characteristic

105

C₉

—

106

C₉

—

107

C₉

—

111

—

108

C₉

#15688512

yes

characteristic

109

C₉

#15688514

yes

—

110

C₃

—

111

C₃

—

112

C₃

—

113

C₃

—

111

—

114

C₃

#22582465

yes

characteristic

115

C₃

#22582467

yes

characteristic

116

C₃

#22582473

yes

—

	subgroup	G/H
center	C₃×C₉	C₃²⋊(C₃₇⋊C₃)
commutator	C₃²×C₃₇	C₃×C₂₇
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃₇×C₃²⋊C₂₇	C₃
socle	C₃²×C₃₇	C₃²⋊C₉
prime radical	C₃³×C₃₇⋊C₃	C₉
cosocle	C₃²×C₉×C₃₇	C₃²
hypercenter	C₃²⋊C₂₇	C₃₇⋊C₃
hypocenter	C₃₇	#10624444
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ C₃²⋊C₂₇

subgroup

G/H

center

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

commutator

C₃²×C₃₇

C₃×C₂₇

Frattini

C₃²×C₉

C₃×C₃₇⋊C₃

Fitting

C₃₇×C₃²⋊C₂₇

C₃

socle

C₃²×C₃₇

C₃²⋊C₉

prime radical

C₃³×C₃₇⋊C₃

C₉

cosocle

C₃²×C₉×C₃₇

C₃²

hypercenter

C₃²⋊C₂₇

C₃₇⋊C₃

hypocenter

C₃₇

#10624444

LCS

G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ C₃²⋊C₂₇

N	H ≅ G/N	type
2. C₃²⋊#11758601	C₃	semidirect
3. C₃₇×C₃²⋊C₂₇	C₃	semidirect
20. C₃²×C₃₇⋊C₃	C₂₇	semidirect
54. C₃²⋊C₂₇	C₃₇⋊C₃	semidirect
72. C₃₇	#10624444	semidirect

H ≅ G/N

type

2. C₃²⋊#11758601

C₃

semidirect

3. C₃₇×C₃²⋊C₂₇

C₃

semidirect

20. C₃²×C₃₇⋊C₃

C₂₇

semidirect

54. C₃²⋊C₂₇

C₃₇⋊C₃

semidirect

72. C₃₇

#10624444

semidirect

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

subgroup	index
C₃²×C₉×C₃₇⋊C₃	3
C₃²⋊#11758601	3
C₃₇×C₃²⋊C₂₇	3
#10624444	37

subgroup

index

C₃²×C₉×C₃₇⋊C₃

C₃²⋊#11758601

C₃₇×C₃²⋊C₂₇

#10624444

Subgroups — #41004739

Conjugacy classes of subgroups

Complements

Maximal subgroups