Group #41006874 — subgroups

Subgroups — #41006874

Order 26973.

228 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 64 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	C₃×#15688516	C₃	1	1	yes	maximal, characteristic
2	#22582571	C₃	1	2	yes	maximal, complemented
3	#22582558	C₃	1	1	yes	maximal, characteristic
4	#22582480	C₃	1	4	yes	maximal, complemented
5	#22582477	C₃	1	2	yes	maximal
6	#22582470	C₃	1	2	yes	maximal, complemented
7	C₃×C₃₇×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃×#11758608	C₃²	1	2	yes	—
9	C₃×#11758605	—	3	1	no	—
10	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	1	yes	characteristic
11	#15688516	—	3	3	no	—
12	#15688505	—	3	2	no	—
13	#15688504	—	3	2	no	—
14	#15688504	—	3	2	no	—
15	C₃²⋊#66524	C₃²	1	2	yes	—
16	C₃²⋊#66524	C₃²	1	4	yes	—
17	#15688502	—	3	2	no	—
18	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
19	C₃²×C₉×C₃₇	C₃²	1	2	yes	—
20	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
21	C₃₇×C₉⋊C₉	—	3	2	no	—
22	C₃₇×C₉⋊C₉	—	3	1	no	—
23	C₃³×C₃₇	C₃³	1	1	yes	characteristic
24	C₃²×C₃₇⋊C₃	—	3	1	no	—
25	C₃×C₉×C₃₇	—	3	1	no	—
26	C₃×C₉×C₃₇	—	3	2	no	—
27	C₃×C₉×C₃₇	—	3	2	no	—
28	C₃×C₉×C₃₇	—	9	1	no	—
29	C₃×C₉×C₃₇	—	9	2	no	—
30	C₃×C₉×C₃₇	H(3,𝔽₃)	1	2	yes	—
31	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
32	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	complemented
33	C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
34	C₃²⋊(C₃₇⋊C₃)	—	3	2	no	—
35	#11758608	—	3	2	no	—
36	#11758608	—	3	4	no	—
37	C₃×#66524	—	3	2	no	—
38	C₃×#66524	—	3	4	no	—
39	C₃×#66524	—	3	2	no	—
40	#11758605	—	9	3	no	—
41	#10624789	—	37	1	no	maximal, Sylow
42	C₃²×C₃₇	—	3	2	no	—
43	C₃²×C₃₇	—	3	1	no	—
44	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
45	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
46	C₃²×C₃₇	#737	1	2	yes	—
47	C₃×C₃₇⋊C₃	—	9	2	no	—
48	C₃×C₃₇⋊C₃	—	9	1	no	—
49	C₃×C₃₇⋊C₃	—	9	1	no	—
50	C₉×C₃₇	—	3	3	no	—
51	C₉×C₃₇	—	3	1	no	—
52	C₉×C₃₇	—	3	2	no	—
53	C₉×C₃₇	—	9	2	no	—
54	#66524	—	9	4	no	—
55	#66524	—	9	2	no	—
56	#66524	—	9	2	no	—
57	C₃×#731	—	37	1	no	—
58	#6894	—	37	2	no	—
59	C₉⋊H(3,𝔽₃)	—	37	1	no	—
60	C₃×C₉⋊C₉	C₃₇⋊C₃	1	1	yes	characteristic, complemented
61	3₋¹⁺²⋊C₉	—	37	4	no	—
62	H(3,𝔽₃)⋊C₉	—	37	2	no	—
63	#6867	—	37	2	no	—
64	C₃×C₃₇	—	3	1	no	—
65	C₃×C₃₇	—	3	2	no	—
66	C₃×C₃₇	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
67	C₃×C₃₇	#6910	1	1	yes	characteristic
68	C₃×C₃₇	#6913	1	2	yes	—
69	C₃₇⋊C₃	—	27	1	no	—
70	C₃×3₋¹⁺²	—	37	2	no	—
71	C₃×3₋¹⁺²	—	111	1	no	—
72	C₃×H(3,𝔽₃)	—	37	1	no	—
73	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
74	C₃²×C₉	C₃×C₃₇⋊C₃	1	2	yes	—
75	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
76	#731	—	111	3	no	—
77	C₉⋊C₉	—	3	2	no	—
78	C₉⋊C₉	—	3	1	no	—
79	C₉⋊C₉	—	111	2	no	—
80	C₉⋊C₉	—	111	2	no	—
81	C₉⋊C₉	—	111	2	no	—
82	C₃²⋊C₉	—	37	4	no	—
83	C₃²⋊C₉	—	37	2	no	—
84	C₃²⋊C₉	—	111	2	no	—
85	C₃₇	#10624789	1	1	yes	characteristic, complemented, Sylow
86	C₃³	—	111	1	no	—
87	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
88	3₋¹⁺²	—	333	3	no	—
89	3₋¹⁺²	—	111	2	no	—
90	3₋¹⁺²	—	111	4	no	—
91	H(3,𝔽₃)	—	111	1	no	—
92	H(3,𝔽₃)	—	111	2	no	—
93	C₃×C₉	—	3	1	no	—
94	C₃×C₉	—	3	2	no	—
95	C₃×C₉	—	3	2	no	—
96	C₃×C₉	—	9	1	no	—
97	C₃×C₉	—	9	2	no	—
98	C₃×C₉	—	111	2	no	—
99	C₃×C₉	—	111	4	no	—
100	C₃×C₉	—	111	2	no	—
101	C₃×C₉	#11758605	1	3	yes	complemented
102	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
103	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
104	C₃²	—	3	1	no	—
105	C₃²	—	3	2	no	—
106	C₃²	—	333	1	no	—
107	C₃²	—	333	1	no	—
108	C₃²	—	333	2	no	—
109	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
110	C₃²	C₃×#11758605	1	1	yes	characteristic
111	C₃²	#15688531	1	2	yes	—
112	C₉	—	3	3	no	—
113	C₉	—	3	1	no	—
114	C₉	—	3	2	no	—
115	C₉	—	9	2	no	—
116	C₉	—	333	2	no	—
117	C₉	—	333	2	no	—
118	C₉	—	333	4	no	—
119	C₃	—	3	1	no	—
120	C₃	—	3	2	no	—
121	C₃	—	999	1	no	—
122	C₃	#22582558	1	1	yes	characteristic
123	C₃	#22582643	1	1	yes	characteristic
124	C₃	#22582661	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688516

C₃

yes

maximal, characteristic

#22582571

C₃

yes

maximal, complemented

#22582558

C₃

yes

maximal, characteristic

#22582480

C₃

yes

maximal, complemented

#22582477

C₃

yes

maximal

#22582470

C₃

yes

maximal, complemented

C₃×C₃₇×C₉⋊C₉

C₃

yes

maximal, characteristic, complemented

C₃×#11758608

C₃²

yes

—

C₃×#11758605

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

—

—

—

—

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

#15688502

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃²×C₉×C₃₇

C₃²

yes

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×C₉⋊C₉

—

C₃₇×C₉⋊C₉

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

—

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

characteristic

C₃×C₉×C₃₇

3₋¹⁺²

yes

complemented

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

—

—

—

—

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₃₇

#737

yes

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

—

—

—

—

—

C₉⋊H(3,𝔽₃)

—

C₃×C₉⋊C₉

C₃₇⋊C₃

yes

characteristic, complemented

3₋¹⁺²⋊C₉

—

H(3,𝔽₃)⋊C₉

—

#6867

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₉⋊H(3,𝔽₃)

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

—

C₃₇⋊C₃

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₃²×C₉

C₃×C₃₇⋊C₃

yes

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

#731

—

111

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃₇

#10624789

yes

characteristic, complemented, Sylow

C₃³

—

111

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

333

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

H(3,𝔽₃)

—

111

—

H(3,𝔽₃)

—

111

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

100

C₃×C₉

—

111

—

101

C₃×C₉

#11758605

yes

complemented

102

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

—

103

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

104

C₃²

—

105

C₃²

—

106

C₃²

—

333

—

107

C₃²

—

333

—

108

C₃²

—

333

—

109

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

110

C₃²

C₃×#11758605

yes

characteristic

111

C₃²

#15688531

yes

—

112

C₉

—

113

C₉

—

114

C₉

—

115

C₉

—

116

C₉

—

333

—

117

C₉

—

333

—

118

C₉

—

333

—

119

C₃

—

120

C₃

—

121

C₃

—

999

—

122

C₃

#22582558

yes

characteristic

123

C₃

#22582643

yes

characteristic

124

C₃

#22582661

yes

—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃×C₃₇×C₉⋊C₉	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²
hypercenter	C₃×C₉⋊C₉	C₃₇⋊C₃
hypocenter	C₃₇	#10624789
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²×C₉ ◁ C₃×C₉⋊C₉

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃×C₃₇×C₉⋊C₉

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

hypercenter

C₃×C₉⋊C₉

C₃₇⋊C₃

hypocenter

C₃₇

#10624789

LCS

G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃²×C₉ ◁ C₃×C₉⋊C₉

N	H ≅ G/N	type
2. #22582571	C₃	semidirect
4. #22582480	C₃	semidirect
6. #22582470	C₃	semidirect
7. C₃×C₃₇×C₉⋊C₉	C₃	semidirect
32. C₃×C₉×C₃₇	3₋¹⁺²	semidirect
60. C₃×C₉⋊C₉	C₃₇⋊C₃	semidirect
85. C₃₇	#10624789	semidirect
101. C₃×C₉	#11758605	semidirect

H ≅ G/N

type

2. #22582571

C₃

semidirect

4. #22582480

C₃

semidirect

6. #22582470

C₃

semidirect

7. C₃×C₃₇×C₉⋊C₉

C₃

semidirect

32. C₃×C₉×C₃₇

3₋¹⁺²

semidirect

60. C₃×C₉⋊C₉

C₃₇⋊C₃

semidirect

85. C₃₇

#10624789

semidirect

101. C₃×C₉

#11758605

semidirect

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

subgroup	index
C₃×#15688516	3
#22582571	3
#22582558	3
#22582480	3
#22582477	3
#22582470	3
C₃×C₃₇×C₉⋊C₉	3
#10624789	37

subgroup

index

C₃×C₃₇×C₉⋊C₉

Subgroups — #41006874

Conjugacy classes of subgroups

Complements

Maximal subgroups