Group #41007664 — subgroups

Subgroups — #41007664

Order 26973.

716 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 274 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×#11758608	C₃	1	1	yes	maximal, characteristic
2	#22582601	C₃	1	9	yes	maximal, complemented
3	#22582593	C₃	1	18	yes	maximal, complemented
4	#22582576	C₃	1	3	yes	maximal, complemented
5	#22582575	C₃	1	3	yes	maximal, complemented
6	#22582554	C₃	1	2	yes	maximal
7	#22582535	C₃	1	3	yes	maximal, complemented
8	C₃₇×#6891	C₃	1	1	yes	maximal, characteristic
9	C₃×#11758608	—	3	9	no	—
10	C₃×#11758608	C₃²	1	6	yes	—
11	C₃×#11758608	C₃²	1	3	yes	—
12	C₃×#11758608	C₃²	1	6	yes	—
13	C₃²×#66524	C₃²	1	1	yes	characteristic
14	C₃²×#66524	C₃²	1	1	yes	characteristic
15	C₃²×#66524	C₃²	1	1	yes	characteristic
16	#15688506	C₃²	1	9	yes	complemented
17	#15688505	C₃²	1	9	yes	complemented
18	#15688504	C₃²	1	18	yes	complemented
19	#15688504	C₃²	1	18	yes	complemented
20	#15688504	C₃²	1	18	yes	complemented
21	C₃²⋊#66524	C₃²	1	6	yes	—
22	C₃²⋊#66524	C₃²	1	12	yes	—
23	C₉×#66524	C₃²	1	9	yes	complemented
24	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
25	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
26	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
27	C₃₇×C₃²⋊C₉	C₃²	1	6	yes	—
28	C₃³×C₃₇	—	3	3	no	—
29	C₃³×C₃₇	—	9	2	no	—
30	C₃³×C₃₇	—	9	1	no	—
31	C₃³×C₃₇	C₃³	1	2	yes	—
32	C₃³×C₃₇	C₃³	1	1	yes	characteristic
33	C₃³×C₃₇	C₃³	1	1	yes	characteristic
34	3₋¹⁺²×C₃₇	—	9	3	no	—
35	C₃×C₉×C₃₇	C₃³	1	9	yes	—
36	#11758608	—	3	18	no	—
37	#11758608	—	3	9	no	—
38	#11758608	—	9	6	no	—
39	C₃×#66524	—	3	3	no	—
40	C₃×#66524	—	3	3	no	—
41	C₃×#66524	3₋¹⁺²	1	9	yes	complemented
42	C₃×#66524	C₃³	1	18	yes	—
43	C₃×#66524	C₃³	1	3	yes	—
44	C₃×#66524	C₃³	1	3	yes	—
45	C₃×#66524	C₃³	1	3	yes	—
46	#10624884	—	37	1	no	maximal, Sylow
47	C₃²×C₃₇	—	3	2	no	—
48	C₃²×C₃₇	—	3	1	no	—
49	C₃²×C₃₇	—	3	2	no	—
50	C₃²×C₃₇	—	3	1	no	—
51	C₃²×C₃₇	—	3	2	no	—
52	C₃²×C₃₇	—	3	1	no	—
53	C₃²×C₃₇	—	3	4	no	—
54	C₃²×C₃₇	—	3	2	no	—
55	C₃²×C₃₇	—	9	9	no	—
56	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
57	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
58	C₉×C₃₇	—	3	9	no	—
59	#66524	—	3	3	no	—
60	#66524	—	3	3	no	—
61	#66524	—	3	18	no	—
62	#66524	C₃×3₋¹⁺²	1	9	yes	—
63	C₃²×3₋¹⁺²	—	37	1	no	—
64	#6900	—	37	9	no	—
65	#6899	—	37	18	no	—
66	#6895	—	37	3	no	—
67	#6895	—	37	3	no	—
68	#6891	—	37	2	no	—
69	#6891	C₃₇⋊C₃	1	1	yes	characteristic
70	#6887	—	37	3	no	—
71	C₃×C₃₇	—	3	3	no	—
72	C₃×C₃₇	—	9	1	no	—
73	C₃×C₃₇	—	9	2	no	—
74	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
75	C₃×C₃₇	3₋¹⁺⁴	1	2	yes	—
76	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
77	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
78	C₃×3₋¹⁺²	—	37	6	no	—
79	C₃×3₋¹⁺²	—	37	6	no	—
80	C₃×3₋¹⁺²	—	37	3	no	—
81	C₃×3₋¹⁺²	—	111	9	no	—
82	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	3	yes	—
83	C₃²×C₉	—	37	1	no	—
84	C₃²×C₉	—	37	1	no	—
85	C₃²×C₉	—	37	1	no	—
86	C₉⋊C₉	—	37	18	no	—
87	C₉⋊C₉	—	37	18	no	—
88	C₉⋊C₉	—	37	18	no	—
89	C₉⋊C₉	—	37	9	no	—
90	C₉⋊C₉	—	37	9	no	—
91	C₃²⋊C₉	—	37	6	no	—
92	C₃²⋊C₉	—	37	12	no	—
93	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
94	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	6	yes	—
95	C₉²	—	37	9	no	—
96	C₃₇	#10624884	1	1	yes	characteristic, complemented, Sylow
97	C₃³	—	3	3	no	—
98	C₃³	—	9	2	no	—
99	C₃³	—	9	1	no	—
100	C₃³	C₃²×C₃₇⋊C₃	1	2	yes	—
101	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
102	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
103	3₋¹⁺²	—	9	3	no	—
104	3₋¹⁺²	—	333	6	no	—
105	3₋¹⁺²	—	111	18	no	—
106	3₋¹⁺²	—	111	9	no	—
107	C₃×C₉	—	37	9	no	—
108	C₃×C₉	—	37	3	no	—
109	C₃×C₉	—	37	18	no	—
110	C₃×C₉	—	37	3	no	—
111	C₃×C₉	—	37	3	no	—
112	C₃×C₉	—	111	3	no	—
113	C₃×C₉	—	111	3	no	—
114	C₃×C₉	C₃²×C₃₇⋊C₃	1	9	yes	—
115	C₃²	—	3	2	no	—
116	C₃²	—	3	1	no	—
117	C₃²	—	3	2	no	—
118	C₃²	—	3	1	no	—
119	C₃²	—	3	2	no	—
120	C₃²	—	3	4	no	—
121	C₃²	—	3	1	no	—
122	C₃²	—	3	2	no	—
123	C₃²	—	9	9	no	—
124	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	3	yes	—
125	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
126	C₉	—	3	9	no	—
127	C₉	—	37	9	no	—
128	C₉	—	111	3	no	—
129	C₉	—	111	3	no	—
130	C₉	—	111	18	no	—
131	C₃	—	3	3	no	—
132	C₃	—	9	1	no	—
133	C₃	—	9	2	no	—
134	C₃	C₃×3₋¹⁺²×C₃₇⋊C₃	1	1	yes	characteristic
135	C₃	#22582679	1	2	yes	—
136	C₃	#22582680	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758608

C₃

yes

maximal, characteristic

#22582601

C₃

yes

maximal, complemented

#22582593

C₃

yes

maximal, complemented

#22582576

C₃

yes

maximal, complemented

#22582575

C₃

yes

maximal, complemented

#22582554

C₃

yes

maximal

#22582535

C₃

yes

maximal, complemented

C₃₇×#6891

C₃

yes

maximal, characteristic

C₃×#11758608

—

C₃×#11758608

C₃²

yes

—

C₃×#11758608

C₃²

yes

—

C₃×#11758608

C₃²

yes

—

C₃²×#66524

C₃²

yes

characteristic

C₃²×#66524

C₃²

yes

characteristic

C₃²×#66524

C₃²

yes

characteristic

#15688506

C₃²

yes

complemented

#15688505

C₃²

yes

complemented

#15688504

C₃²

yes

complemented

#15688504

C₃²

yes

complemented

#15688504

C₃²

yes

complemented

C₃²⋊#66524

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

C₉×#66524

C₃²

yes

complemented

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

C₃³

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃³×C₃₇

C₃³

yes

characteristic

3₋¹⁺²×C₃₇

—

C₃×C₉×C₃₇

C₃³

yes

—

—

—

—

—

—

3₋¹⁺²

yes

complemented

C₃×#66524

C₃³

yes

—

C₃×#66524

C₃³

yes

—

C₃×#66524

C₃³

yes

—

C₃×#66524

C₃³

yes

—

#10624884

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₃²×C₃₇

C₃⁴

yes

characteristic

C₉×C₃₇

—

—

—

—

C₃×3₋¹⁺²

yes

—

C₃²×3₋¹⁺²

—

—

—

—

—

—

C₃₇⋊C₃

yes

characteristic

#6887

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃²×3₋¹⁺²

yes

characteristic

C₃×C₃₇

3₋¹⁺⁴

yes

—

C₃×C₃₇

3₋¹⁺⁴

yes

characteristic

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₉²

—

C₃₇

#10624884

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

100

C₃³

C₃²×C₃₇⋊C₃

yes

—

101

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

102

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

103

3₋¹⁺²

—

104

3₋¹⁺²

—

333

—

105

3₋¹⁺²

—

111

—

106

3₋¹⁺²

—

111

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

—

110

C₃×C₉

—

111

C₃×C₉

—

112

C₃×C₉

—

111

—

113

C₃×C₉

—

111

—

114

C₃×C₉

C₃²×C₃₇⋊C₃

yes

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

3₋¹⁺²×C₃₇⋊C₃

yes

—

125

C₃²

C₃³×C₃₇⋊C₃

yes

characteristic

126

C₉

—

127

C₉

—

128

C₉

—

111

—

129

C₉

—

111

—

130

C₉

—

111

—

131

C₃

—

132

C₃

—

133

C₃

—

134

C₃

C₃×3₋¹⁺²×C₃₇⋊C₃

yes

characteristic

135

C₃

#22582679

yes

—

136

C₃

#22582680

yes

characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting	C₃₇×#6891	C₃
prime radical	C₃⁴×C₃₇	C₃²
hypercenter	#6891	C₃₇⋊C₃
hypocenter	C₃₇	#10624884
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ #6891

subgroup

G/H

center, Frattini

C₃²

C₃³×C₃₇⋊C₃

commutator, socle, cosocle

C₃²×C₃₇

C₃⁴

Fitting

C₃₇×#6891

C₃

prime radical

C₃⁴×C₃₇

C₃²

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃₇

C₁ ◁ C₃² ◁ #6891

N	H ≅ G/N	type
2. #22582601	C₃	semidirect
3. #22582593	C₃	semidirect
4. #22582576	C₃	semidirect
5. #22582575	C₃	semidirect
7. #22582535	C₃	semidirect
16. #15688506	C₃²	semidirect
17. #15688505	C₃²	semidirect
18. #15688504	C₃²	semidirect
19. #15688504	C₃²	semidirect
20. #15688504	C₃²	semidirect
23. C₉×#66524	C₃²	semidirect
41. C₃×#66524	3₋¹⁺²	semidirect
96. C₃₇	#10624884	semidirect

H ≅ G/N

type

2. #22582601

C₃

semidirect

3. #22582593

C₃

semidirect

4. #22582576

C₃

semidirect

5. #22582575

C₃

semidirect

7. #22582535

C₃

semidirect

16. #15688506

C₃²

semidirect

17. #15688505

C₃²

semidirect

18. #15688504

C₃²

semidirect

19. #15688504

C₃²

semidirect

20. #15688504

C₃²

semidirect

23. C₉×#66524

C₃²

semidirect

41. C₃×#66524

3₋¹⁺²

semidirect

96. C₃₇

#10624884

semidirect

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

subgroup	index
C₃²×#11758608	3
#22582601	3
#22582593	3
#22582576	3
#22582575	3
#22582554	3
#22582535	3
C₃₇×#6891	3
#10624884	37

subgroup

index

C₃₇×#6891

Subgroups — #41007664

Conjugacy classes of subgroups

Complements

Maximal subgroups