Group #41008331 — subgroups

Subgroups — #41008331

Order 26973.

170 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 57 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	(C₃₇⋊C₃)⋊(C₂₇⋊C₃)	C₃	1	1	yes	maximal, characteristic
2	(C₃₇⋊C₃)⋊(C₂₇⋊C₃)	C₃	1	1	yes	maximal, characteristic
3	(C₃₇⋊C₃)⋊(C₂₇⋊C₃)	C₃	1	1	yes	maximal, characteristic
4	C₉⋊C₉×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
5	#15688543	—	3	1	no	—
6	#15688543	—	3	1	no	—
7	#15688543	C₉	1	3	yes	complemented
8	#15688542	—	3	1	no	—
9	#15688542	—	3	1	no	—
10	#15688542	C₉	1	3	yes	complemented
11	#15688541	—	3	1	no	—
12	#15688541	—	3	1	no	—
13	#15688541	—	3	1	no	—
14	(C₃₇⋊C₃)⋊C₂₇	—	3	1	no	—
15	(C₃₇⋊C₃)⋊C₂₇	—	3	1	no	—
16	(C₃₇⋊C₃)⋊C₂₇	C₉	1	3	yes	complemented
17	C₃×C₉×C₃₇⋊C₃	—	3	1	no	—
18	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
19	#15688506	C₉	1	1	yes	characteristic
20	#15688505	C₉	1	1	yes	characteristic
21	#15688504	—	3	1	no	—
22	#15688504	—	3	1	no	—
23	C₃₇×C₉⋊C₉	C₉	1	1	yes	characteristic
24	C₃²×C₃₇⋊C₃	3₋¹⁺²	1	1	yes	characteristic
25	C₃×C₉×C₃₇	—	3	1	no	—
26	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
27	C₃×#66524	—	3	1	no	—
28	C₃×#66524	—	3	1	no	—
29	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
30	C₃×#66524	3₋¹⁺²	1	1	yes	characteristic
31	C₉×C₃₇⋊C₃	—	9	3	no	—
32	C₉×C₃₇⋊C₃	C₃×C₉	1	1	yes	characteristic
33	C₉×C₃₇⋊C₃	C₃×C₉	1	1	yes	characteristic
34	C₉×C₃₇⋊C₃	C₃×C₉	1	1	yes	characteristic
35	C₉×C₃₇⋊C₃	3₋¹⁺²	1	1	yes	characteristic
36	C₉×C₃₇⋊C₃	3₋¹⁺²	1	1	yes	characteristic
37	C₉×C₃₇⋊C₃	3₋¹⁺²	1	1	yes	characteristic
38	C₉×C₃₇⋊C₃	3₋¹⁺²	1	1	yes	characteristic
39	C₉×C₃₇⋊C₃	3₋¹⁺²	1	1	yes	characteristic
40	C₉×C₃₇⋊C₃	3₋¹⁺²	1	1	yes	characteristic
41	#11758603	—	3	3	no	—
42	#11758603	—	3	3	no	—
43	#11758603	—	3	3	no	—
44	#10624472	—	37	1	no	maximal, Sylow
45	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
46	C₃×C₃₇⋊C₃	—	3	3	no	—
47	C₃×C₃₇⋊C₃	C₉⋊C₉	1	1	yes	characteristic
48	C₃×C₃₇⋊C₃	C₉⋊C₉	1	1	yes	characteristic
49	C₃×C₃₇⋊C₃	C₉⋊C₉	1	1	yes	characteristic
50	C₉×C₃₇	—	9	1	no	—
51	C₉×C₃₇	C₉²	1	1	yes	characteristic
52	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
53	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
54	#66524	—	9	1	no	—
55	#66524	—	9	1	no	—
56	#66524	C₃²⋊C₉	1	1	yes	characteristic
57	#66524	C₃²⋊C₉	1	1	yes	characteristic
58	#66524	C₃²⋊C₉	1	1	yes	characteristic
59	#66524	C₉⋊C₉	1	1	yes	characteristic
60	#66524	C₉⋊C₉	1	1	yes	characteristic
61	#66524	C₉⋊C₉	1	1	yes	characteristic
62	C₃×C₂₇⋊C₃	—	37	1	no	—
63	C₃×C₂₇⋊C₃	—	37	1	no	—
64	C₃×C₂₇⋊C₃	—	37	1	no	—
65	C₃×C₉⋊C₉	—	37	1	no	—
66	C₃×C₃₇	—	3	1	no	—
67	C₃×C₃₇	#6855	1	1	yes	characteristic
68	C₃₇⋊C₃	—	3	1	no	—
69	C₃₇⋊C₃	—	3	1	no	—
70	C₃₇⋊C₃	#6875	1	3	yes	—
71	C₃²×C₉	—	37	1	no	—
72	C₃²×C₉	—	111	1	no	—
73	C₂₇⋊C₃	—	37	3	no	—
74	C₂₇⋊C₃	—	37	3	no	—
75	C₂₇⋊C₃	—	111	1	no	—
76	C₂₇⋊C₃	—	111	1	no	—
77	C₂₇⋊C₃	—	111	1	no	—
78	C₂₇⋊C₃	—	111	1	no	—
79	C₂₇⋊C₃	—	111	1	no	—
80	C₂₇⋊C₃	—	111	1	no	—
81	C₂₇⋊C₃	—	111	1	no	—
82	C₃×C₂₇	—	37	3	no	—
83	C₃×C₂₇	—	111	1	no	—
84	C₃×C₂₇	—	111	1	no	—
85	C₉⋊C₉	—	37	1	no	—
86	C₉⋊C₉	—	37	1	no	—
87	C₉⋊C₉	—	111	1	no	—
88	C₉⋊C₉	—	111	1	no	—
89	C₉⋊C₉	#66526	1	1	yes	characteristic
90	C₃₇	#10624472	1	1	yes	characteristic, complemented, Sylow
91	C₃³	—	37	1	no	—
92	C₃×C₉	—	3	1	no	—
93	C₃×C₉	—	37	1	no	—
94	C₃×C₉	—	37	1	no	—
95	C₃×C₉	—	37	1	no	—
96	C₃×C₉	—	37	1	no	—
97	C₃×C₉	—	37	1	no	—
98	C₃×C₉	—	37	1	no	—
99	C₃×C₉	—	37	1	no	—
100	C₃×C₉	—	37	1	no	—
101	C₃×C₉	—	37	1	no	—
102	C₃×C₉	—	37	1	no	—
103	C₃×C₉	—	37	1	no	—
104	C₃×C₉	—	333	3	no	—
105	C₃×C₉	—	111	1	no	—
106	C₃×C₉	—	111	1	no	—
107	C₃×C₉	C₃×#66526	1	1	yes	characteristic
108	C₂₇	—	111	3	no	—
109	C₂₇	—	111	3	no	—
110	C₂₇	—	111	3	no	—
111	C₃²	—	37	1	no	—
112	C₃²	—	37	1	no	—
113	C₃²	—	37	1	no	—
114	C₃²	—	111	3	no	—
115	C₃²	C₃²⋊#66526	1	1	yes	characteristic
116	C₉	—	9	1	no	—
117	C₉	—	37	1	no	—
118	C₉	—	37	1	no	—
119	C₉	—	37	1	no	—
120	C₉	—	37	1	no	—
121	C₉	—	37	1	no	—
122	C₉	—	37	1	no	—
123	C₉	—	333	1	no	—
124	C₉	—	333	1	no	—
125	C₉	C₉×#66526	1	1	yes	characteristic
126	C₉	#15688537	1	1	yes	characteristic
127	C₉	#15688538	1	1	yes	characteristic
128	C₃	—	3	1	no	—
129	C₃	—	37	3	no	—
130	C₃	—	111	1	no	—
131	C₃	—	111	1	no	—
132	C₃	(C₃×#66526)⋊C₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

(C₃₇⋊C₃)⋊(C₂₇⋊C₃)

C₃

yes

maximal, characteristic

(C₃₇⋊C₃)⋊(C₂₇⋊C₃)

C₃

yes

maximal, characteristic

(C₃₇⋊C₃)⋊(C₂₇⋊C₃)

C₃

yes

maximal, characteristic

C₉⋊C₉×C₃₇⋊C₃

C₃

yes

maximal, characteristic

#15688543

—

#15688543

—

#15688543

C₉

yes

—

—

C₉

yes

—

—

—

(C₃₇⋊C₃)⋊C₂₇

—

(C₃₇⋊C₃)⋊C₂₇

—

(C₃₇⋊C₃)⋊C₂₇

C₉

yes

complemented

C₃×C₉×C₃₇⋊C₃

—

C₃×C₉×C₃₇⋊C₃

C₃²

yes

characteristic

#15688506

C₉

yes

characteristic

#15688505

C₉

yes

characteristic

#15688504

—

#15688504

—

C₃₇×C₉⋊C₉

C₉

yes

characteristic

C₃²×C₃₇⋊C₃

3₋¹⁺²

yes

characteristic

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃×C₉

yes

—

—

yes

3₋¹⁺²

yes

characteristic

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

C₃×C₉

yes

characteristic

C₉×C₃₇⋊C₃

C₃×C₉

yes

characteristic

C₉×C₃₇⋊C₃

C₃×C₉

yes

characteristic

C₉×C₃₇⋊C₃

3₋¹⁺²

yes

characteristic

C₉×C₃₇⋊C₃

3₋¹⁺²

yes

characteristic

C₉×C₃₇⋊C₃

3₋¹⁺²

yes

characteristic

C₉×C₃₇⋊C₃

3₋¹⁺²

yes

characteristic

C₉×C₃₇⋊C₃

3₋¹⁺²

yes

characteristic

C₉×C₃₇⋊C₃

3₋¹⁺²

yes

—

—

—

—

C₃²×C₃₇

C₃²⋊C₉

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

C₉⋊C₉

yes

characteristic

C₃×C₃₇⋊C₃

C₉⋊C₉

yes

characteristic

C₃×C₃₇⋊C₃

C₉⋊C₉

yes

characteristic

C₉×C₃₇

—

C₉×C₃₇

C₉²

yes

characteristic

C₉×C₃₇

C₉⋊C₉

yes

characteristic

C₉×C₃₇

C₉⋊C₉

yes

—

—

C₃²⋊C₉

yes

characteristic

#66524

C₃²⋊C₉

yes

characteristic

#66524

C₃²⋊C₉

yes

characteristic

#66524

C₉⋊C₉

yes

characteristic

#66524

C₉⋊C₉

yes

characteristic

#66524

C₉⋊C₉

yes

characteristic

C₃×C₂₇⋊C₃

—

C₃×C₂₇⋊C₃

—

C₃×C₂₇⋊C₃

—

C₃×C₉⋊C₉

—

C₃×C₃₇

—

C₃×C₃₇

#6855

yes

characteristic

C₃₇⋊C₃

—

C₃₇⋊C₃

—

C₃₇⋊C₃

#6875

yes

—

C₃²×C₉

—

C₃²×C₉

—

111

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

—

111

—

C₃×C₂₇

—

C₃×C₂₇

—

111

—

C₃×C₂₇

—

111

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₉⋊C₉

#66526

yes

characteristic

C₃₇

#10624472

yes

characteristic, complemented, Sylow

C₃³

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

100

C₃×C₉

—

101

C₃×C₉

—

102

C₃×C₉

—

103

C₃×C₉

—

104

C₃×C₉

—

333

—

105

C₃×C₉

—

111

—

106

C₃×C₉

—

111

—

107

C₃×C₉

C₃×#66526

yes

characteristic

108

C₂₇

—

111

—

109

C₂₇

—

111

—

110

C₂₇

—

111

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

111

—

115

C₃²

C₃²⋊#66526

yes

characteristic

116

C₉

—

117

C₉

—

118

C₉

—

119

C₉

—

120

C₉

—

121

C₉

—

122

C₉

—

123

C₉

—

333

—

124

C₉

—

333

—

125

C₉

C₉×#66526

yes

characteristic

126

C₉

#15688537

yes

characteristic

127

C₉

#15688538

yes

characteristic

128

C₃

—

129

C₃

—

130

C₃

—

111

—

131

C₃

—

111

—

132

C₃

(C₃×#66526)⋊C₉

yes

characteristic

	subgroup	G/H
center	C₃	(C₃×#66526)⋊C₉
commutator	C₉×C₃₇	C₉²
Frattini	C₃×C₉	C₃×#66526
Fitting	C₃₇×C₉⋊C₉	C₉
socle	C₃×C₃₇	#6855
prime radical	C₃²×C₃₇⋊C₃	3₋¹⁺²
cosocle	C₃×C₉×C₃₇⋊C₃	C₃²
hypercenter	C₉⋊C₉	#66526
hypocenter	C₃₇	#10624472
LCS	G ▷ C₉×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃×C₉ ◁ C₉⋊C₉

subgroup

G/H

center

C₃

(C₃×#66526)⋊C₉

commutator

C₉×C₃₇

C₉²

Frattini

C₃×C₉

C₃×#66526

Fitting

C₃₇×C₉⋊C₉

C₉

socle

C₃×C₃₇

#6855

prime radical

C₃²×C₃₇⋊C₃

3₋¹⁺²

cosocle

C₃×C₉×C₃₇⋊C₃

C₃²

C₉⋊C₉

C₃₇

G ▷ C₉×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃ ◁ C₃×C₉ ◁ C₉⋊C₉

N	H ≅ G/N	type
7. #15688543	C₉	semidirect
10. #15688542	C₉	semidirect
16. (C₃₇⋊C₃)⋊C₂₇	C₉	semidirect
90. C₃₇	#10624472	semidirect

H ≅ G/N

type

7. #15688543

C₉

semidirect

10. #15688542

C₉

semidirect

16. (C₃₇⋊C₃)⋊C₂₇

C₉

semidirect

90. C₃₇

#10624472

semidirect

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

subgroup	index
(C₃₇⋊C₃)⋊(C₂₇⋊C₃)	3
C₉⋊C₉×C₃₇⋊C₃	3
#10624472	37

subgroup

index

(C₃₇⋊C₃)⋊(C₂₇⋊C₃)

C₉⋊C₉×C₃₇⋊C₃

#10624472

Subgroups — #41008331

Conjugacy classes of subgroups

Complements

Maximal subgroups