Group #41007365 — subgroups

Subgroups — C₃₇⋊C₃×#6882 #41007365

Order 26973.

200 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 49 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	#733×C₃₇⋊C₃	C₃	1	2	yes	maximal
2	#733×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
3	C₉⋊C₉×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
4	#22582520	C₃	1	1	yes	maximal, characteristic, complemented
5	#22582519	C₃	1	1	yes	maximal, characteristic, complemented
6	#22582518	C₃	1	2	yes	maximal, complemented
7	#22582517	C₃	1	2	yes	maximal, complemented
8	#22582516	C₃	1	2	yes	maximal, complemented
9	C₃₇×#6882	C₃	1	1	yes	maximal, characteristic, complemented
10	3₋¹⁺²×C₃₇⋊C₃	—	3	2	no	—
11	3₋¹⁺²×C₃₇⋊C₃	—	3	1	no	—
12	C₃×C₉×C₃₇⋊C₃	—	3	1	no	—
13	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
14	#15688521	C₃²	1	2	yes	—
15	#15688521	C₃²	1	2	yes	—
16	#15688521	C₃²	1	2	yes	—
17	#15688520	—	3	2	no	—
18	#15688520	—	3	4	no	—
19	#15688506	C₃²	1	1	yes	characteristic
20	#15688505	C₃²	1	1	yes	characteristic
21	#15688504	—	3	2	no	—
22	C₃₇×#733	C₃²	1	2	yes	—
23	C₃₇×#733	C₃²	1	1	yes	characteristic
24	C₃₇×C₉⋊C₉	C₃²	1	1	yes	characteristic
25	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
26	3₋¹⁺²×C₃₇	—	3	2	no	—
27	3₋¹⁺²×C₃₇	—	3	1	no	—
28	C₃×C₉×C₃₇	—	3	1	no	—
29	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
30	#11758608	—	3	2	no	—
31	#11758608	—	3	2	no	—
32	#11758608	—	3	2	no	—
33	C₃×#66524	—	3	1	no	—
34	C₃×#66524	—	3	1	no	—
35	C₃×#66524	H(3,𝔽₃)	1	2	yes	—
36	#11758606	—	9	1	no	—
37	#11758606	—	9	2	no	—
38	#11758605	—	9	2	no	—
39	#11758605	—	9	1	no	—
40	C₉×C₃₇⋊C₃	—	3	1	no	—
41	C₉×C₃₇⋊C₃	—	3	1	no	—
42	C₉×C₃₇⋊C₃	—	3	1	no	—
43	C₉×C₃₇⋊C₃	—	9	3	no	—
44	C₉×C₃₇⋊C₃	—	9	2	no	—
45	C₉×C₃₇⋊C₃	—	9	1	no	—
46	C₃×#6882	—	37	1	no	maximal, Sylow
47	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
48	C₃×C₃₇⋊C₃	—	3	3	no	—
49	C₃×C₃₇⋊C₃	#732	1	1	yes	characteristic
50	C₃×C₃₇⋊C₃	#732	1	1	yes	characteristic
51	C₃×C₃₇⋊C₃	#732	1	1	yes	characteristic
52	C₉×C₃₇	—	3	1	no	—
53	C₉×C₃₇	—	9	1	no	—
54	C₉×C₃₇	—	9	2	no	—
55	C₉×C₃₇	—	9	1	no	—
56	#66524	—	3	2	no	—
57	#66524	—	9	1	no	—
58	#66524	—	9	1	no	—
59	#66524	—	9	2	no	—
60	#66524	—	9	2	no	—
61	#66524	—	9	2	no	—
62	C₃×#733	—	37	1	no	—
63	C₃×#733	—	37	2	no	—
64	C₃×C₉⋊C₉	—	37	1	no	—
65	#6882	—	37	2	no	—
66	#6882	—	37	2	no	—
67	#6882	—	37	1	no	—
68	#6882	—	37	1	no	—
69	#6882	—	37	2	no	—
70	#6882	C₃₇⋊C₃	1	1	yes	characteristic, direct product factor
71	C₃×C₃₇	—	3	1	no	—
72	C₃×C₃₇	C₃×#732	1	1	yes	characteristic
73	C₃₇⋊C₃	—	3	1	no	—
74	C₃₇⋊C₃	—	3	1	no	—
75	C₃₇⋊C₃	#6882	1	3	yes	direct product factor
76	C₃×3₋¹⁺²	—	111	2	no	—
77	C₃×3₋¹⁺²	—	111	1	no	—
78	C₃²×C₉	—	37	1	no	—
79	C₃²×C₉	—	111	1	no	—
80	#733	—	37	2	no	—
81	#733	—	37	2	no	—
82	#733	—	37	2	no	—
83	#733	—	111	4	no	—
84	#733	—	111	2	no	—
85	#733	C₃×C₃₇⋊C₃	1	2	yes	—
86	#733	C₃×C₃₇⋊C₃	1	1	yes	characteristic
87	C₉⋊C₉	—	37	1	no	—
88	C₉⋊C₉	—	37	1	no	—
89	C₉⋊C₉	—	111	2	no	—
90	C₉⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
91	C₃₇	C₃×#6882	1	1	yes	characteristic, complemented, Sylow
92	C₃³	—	37	1	no	—
93	3₋¹⁺²	—	3	2	no	—
94	3₋¹⁺²	—	3	1	no	—
95	3₋¹⁺²	—	333	2	no	—
96	3₋¹⁺²	—	333	1	no	—
97	3₋¹⁺²	—	333	1	no	—
98	3₋¹⁺²	—	333	2	no	—
99	3₋¹⁺²	—	111	2	no	—
100	3₋¹⁺²	—	111	2	no	—
101	3₋¹⁺²	—	111	2	no	—
102	C₃×C₉	—	3	1	no	—
103	C₃×C₉	—	37	2	no	—
104	C₃×C₉	—	333	2	no	—
105	C₃×C₉	—	333	1	no	—
106	C₃×C₉	—	333	3	no	—
107	C₃×C₉	—	111	1	no	—
108	C₃×C₉	—	111	1	no	—
109	C₃×C₉	—	111	1	no	—
110	C₃×C₉	—	111	1	no	—
111	C₃×C₉	—	111	1	no	—
112	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
113	C₃²	—	37	1	no	—
114	C₃²	—	37	1	no	—
115	C₃²	—	37	1	no	—
116	C₃²	—	111	3	no	—
117	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
118	C₉	—	3	1	no	—
119	C₉	—	9	1	no	—
120	C₉	—	9	2	no	—
121	C₉	—	9	1	no	—
122	C₉	—	333	1	no	—
123	C₉	—	333	1	no	—
124	C₉	—	333	2	no	—
125	C₉	—	333	2	no	—
126	C₉	—	333	2	no	—
127	C₉	—	111	2	no	—
128	C₃	—	3	1	no	—
129	C₃	—	37	3	no	—
130	C₃	—	111	1	no	—
131	C₃	—	111	1	no	—
132	C₃	#732×C₃₇⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#733×C₃₇⋊C₃

C₃

yes

maximal

#733×C₃₇⋊C₃

C₃

yes

maximal, characteristic

C₉⋊C₉×C₃₇⋊C₃

C₃

yes

maximal, characteristic

#22582520

C₃

yes

maximal, characteristic, complemented

#22582519

C₃

yes

maximal, characteristic, complemented

#22582518

C₃

yes

maximal, complemented

#22582517

C₃

yes

maximal, complemented

#22582516

C₃

yes

maximal, complemented

C₃₇×#6882

C₃

yes

maximal, characteristic, complemented

3₋¹⁺²×C₃₇⋊C₃

—

3₋¹⁺²×C₃₇⋊C₃

—

C₃×C₉×C₃₇⋊C₃

—

C₃×C₉×C₃₇⋊C₃

C₃²

yes

characteristic

#15688521

C₃²

yes

—

#15688521

C₃²

yes

—

#15688521

C₃²

yes

—

#15688520

—

#15688520

—

#15688506

C₃²

yes

characteristic

#15688505

C₃²

yes

characteristic

#15688504

—

C₃₇×#733

C₃²

yes

—

C₃₇×#733

C₃²

yes

characteristic

C₃₇×C₉⋊C₉

C₃²

yes

characteristic

C₃²×C₃₇⋊C₃

H(3,𝔽₃)

yes

characteristic

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃³

yes

—

—

—

—

—

yes

—

—

—

—

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₃×#6882

—

maximal, Sylow

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

yes

C₃×C₃₇⋊C₃

yes

C₃×C₃₇⋊C₃

yes

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

—

—

—

—

—

—

—

—

C₃×C₉⋊C₉

—

—

—

—

—

—

C₃₇⋊C₃

yes

characteristic, direct product factor

C₃×C₃₇

—

C₃×C₃₇

C₃×#732

yes

characteristic

C₃₇⋊C₃

—

C₃₇⋊C₃

—

C₃₇⋊C₃

#6882

yes

direct product factor

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃²×C₉

—

C₃²×C₉

—

111

—

—

—

—

—

111

—

#733

—

111

—

#733

C₃×C₃₇⋊C₃

yes

—

#733

C₃×C₃₇⋊C₃

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

111

—

C₉⋊C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₃₇

C₃×#6882

yes

characteristic, complemented, Sylow

C₃³

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

333

—

3₋¹⁺²

—

333

—

3₋¹⁺²

—

333

—

3₋¹⁺²

—

333

—

3₋¹⁺²

—

111

—

100

3₋¹⁺²

—

111

—

101

3₋¹⁺²

—

111

—

102

C₃×C₉

—

103

C₃×C₉

—

104

C₃×C₉

—

333

—

105

C₃×C₉

—

333

—

106

C₃×C₉

—

333

—

107

C₃×C₉

—

111

—

108

C₃×C₉

—

111

—

109

C₃×C₉

—

111

—

110

C₃×C₉

—

111

—

111

C₃×C₉

—

111

—

112

C₃×C₉

C₃²×C₃₇⋊C₃

yes

characteristic

113

C₃²

—

114

C₃²

—

115

C₃²

—

116

C₃²

—

111

—

117

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

118

C₉

—

119

C₉

—

120

C₉

—

121

C₉

—

122

C₉

—

333

—

123

C₉

—

333

—

124

C₉

—

333

—

125

C₉

—

333

—

126

C₉

—

333

—

127

C₉

—

111

—

128

C₃

—

129

C₃

—

130

C₃

—

111

—

131

C₃

—

111

—

132

C₃

#732×C₃₇⋊C₃

yes

characteristic

	subgroup	G/H
center	C₃	#732×C₃₇⋊C₃
commutator, cosocle	C₃×C₉×C₃₇	C₃³
Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6882	C₃
socle	C₃×C₃₇	C₃×#732
prime radical	C₃²×C₃₇⋊C₃	H(3,𝔽₃)
hypercenter	#6882	C₃₇⋊C₃
hypocenter	C₃₇	C₃×#6882
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉ ◁ #6882

subgroup

G/H

center

C₃

#732×C₃₇⋊C₃

commutator, cosocle

C₃×C₉×C₃₇

C₃³

Frattini

C₃×C₉

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6882

C₃

C₃×C₃₇

C₃²×C₃₇⋊C₃

C₃₇⋊C₃

C₃₇

G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉ ◁ #6882

N	H ≅ G/N	type
4. #22582520	C₃	semidirect
5. #22582519	C₃	semidirect
6. #22582518	C₃	semidirect
7. #22582517	C₃	semidirect
8. #22582516	C₃	semidirect
9. C₃₇×#6882	C₃	semidirect
70. #6882	C₃₇⋊C₃	direct (N × H)
75. C₃₇⋊C₃	#6882	direct (N × H)
91. C₃₇	C₃×#6882	semidirect

H ≅ G/N

type

4. #22582520

C₃

semidirect

5. #22582519

C₃

semidirect

6. #22582518

C₃

semidirect

7. #22582517

C₃

semidirect

8. #22582516

C₃

semidirect

9. C₃₇×#6882

C₃

semidirect

70. #6882

C₃₇⋊C₃

direct (N × H)

75. C₃₇⋊C₃

#6882

direct (N × H)

91. C₃₇

C₃×#6882

semidirect

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

subgroup	index
#733×C₃₇⋊C₃	3
C₉⋊C₉×C₃₇⋊C₃	3
#22582520	3
#22582519	3
#22582518	3
#22582517	3
#22582516	3
C₃₇×#6882	3
C₃×#6882	37

subgroup

index

#733×C₃₇⋊C₃

C₉⋊C₉×C₃₇⋊C₃

C₃₇×#6882

Subgroups — C₃₇⋊C₃×#6882 #41007365

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C37⋊C3×#6882 #41007365

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇⋊C₃×#6882 #41007365