Group #41007660 — subgroups

Subgroups — #41007660

Order 26973.

716 conjugacy classes of subgroups; only the 292 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 292 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×#11758606	C₃	1	1	yes	maximal, characteristic
2	#22582604	C₃	1	6	yes	maximal, complemented
3	#22582603	C₃	1	3	yes	maximal, complemented
4	#22582600	C₃	1	12	yes	maximal, complemented
5	#22582595	C₃	1	6	yes	maximal, complemented
6	#22582575	C₃	1	6	yes	maximal, complemented
7	#22582552	C₃	1	3	yes	maximal
8	#22582538	C₃	1	2	yes	maximal, complemented
9	C₃₇×#6887	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
11	C₃×#11758606	—	3	9	no	—
12	C₃×#11758606	C₃²	1	2	yes	—
13	C₃×#11758606	C₃²	1	4	yes	—
14	C₃×#11758606	C₃²	1	1	yes	characteristic
15	C₃×#11758606	C₃²	1	2	yes	—
16	C₃×#11758606	C₃²	1	9	yes	—
17	C₃²×#66524	C₃²	1	2	yes	—
18	#15688506	C₃²	1	6	yes	complemented
19	#15688506	C₃²	1	18	yes	complemented
20	#15688505	C₃²	1	6	yes	complemented
21	#15688504	C₃²	1	36	yes	complemented
22	C₃²⋊#66524	C₃²	1	6	yes	—
23	#15688502	C₃²	1	18	yes	—
24	C₉×#66524	C₃²	1	6	yes	complemented
25	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
26	C₃₇×C₉⋊C₉	C₃²	1	6	yes	complemented
27	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
28	C₉²×C₃₇	C₃²	1	3	yes	complemented
29	C₃³×C₃₇	C₃³	1	1	yes	characteristic
30	C₃²×C₃₇⋊C₃	—	3	3	no	—
31	C₃²×C₃₇⋊C₃	—	9	3	no	—
32	C₃²×C₃₇⋊C₃	C₃³	1	2	yes	—
33	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
34	C₃×C₉×C₃₇	3₋¹⁺²	1	9	yes	complemented
35	C₃×C₉×C₃₇	C₃³	1	9	yes	—
36	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
37	C₃×C₉×C₃₇	C₃³	1	2	yes	—
38	C₃×#66524	—	3	6	no	—
39	C₃×#66524	C₃³	1	18	yes	—
40	C₃×#66524	C₃³	1	4	yes	—
41	C₃×#66524	C₃³	1	2	yes	—
42	#11758606	—	3	6	no	—
43	#11758606	—	3	12	no	—
44	#11758606	—	3	3	no	—
45	#11758606	—	3	6	no	—
46	#11758606	—	9	9	no	—
47	#10624884	—	37	1	no	maximal, Sylow
48	C₃²×C₃₇	—	3	3	no	—
49	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
50	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
51	C₃×C₃₇⋊C₃	—	3	2	no	—
52	C₃×C₃₇⋊C₃	—	3	1	no	—
53	C₃×C₃₇⋊C₃	—	3	3	no	—
54	C₃×C₃₇⋊C₃	—	3	6	no	—
55	C₃×C₃₇⋊C₃	—	9	9	no	—
56	C₉×C₃₇	—	3	9	no	—
57	C₉×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
58	C₉×C₃₇	C₃×3₋¹⁺²	1	6	yes	—
59	#66524	—	3	4	no	—
60	#66524	—	3	2	no	—
61	#66524	—	3	18	no	—
62	C₃²×3₋¹⁺²	—	37	1	no	—
63	#6900	—	37	6	no	—
64	#6900	—	37	3	no	—
65	#6899	—	37	12	no	—
66	#6899	—	37	6	no	—
67	#6895	—	37	6	no	—
68	#6891	—	37	3	no	—
69	#6887	—	37	2	no	—
70	#6887	C₃₇⋊C₃	1	1	yes	characteristic, complemented
71	C₃×C₃₇	—	3	3	no	—
72	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
73	C₃×C₃₇	3₋¹⁺⁴	1	3	yes	—
74	C₃₇⋊C₃	—	9	1	no	—
75	C₃₇⋊C₃	—	9	2	no	—
76	C₃⁴	—	37	1	no	—
77	C₃×3₋¹⁺²	—	37	9	no	—
78	C₃×3₋¹⁺²	—	37	4	no	—
79	C₃×3₋¹⁺²	—	37	2	no	—
80	C₃×3₋¹⁺²	—	37	2	no	—
81	C₃×3₋¹⁺²	—	37	1	no	—
82	C₃×3₋¹⁺²	—	111	9	no	—
83	C₃²×C₉	—	37	2	no	—
84	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
85	C₉⋊C₉	—	37	36	no	—
86	C₉⋊C₉	—	37	6	no	—
87	C₉⋊C₉	—	37	6	no	—
88	C₉⋊C₉	—	37	18	no	—
89	C₉⋊C₉	C₃×C₃₇⋊C₃	1	6	yes	complemented
90	C₃²⋊C₉	—	37	6	no	—
91	C₃²⋊C₉	—	37	18	no	—
92	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
93	C₉²	—	37	6	no	—
94	C₉²	C₃×C₃₇⋊C₃	1	3	yes	complemented
95	C₃₇	#10624884	1	1	yes	characteristic, complemented, Sylow
96	C₃³	—	37	2	no	—
97	C₃³	—	37	1	no	—
98	C₃³	—	333	3	no	—
99	C₃³	—	111	3	no	—
100	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
101	3₋¹⁺²	—	333	9	no	—
102	3₋¹⁺²	—	111	12	no	—
103	3₋¹⁺²	—	111	6	no	—
104	3₋¹⁺²	—	111	6	no	—
105	3₋¹⁺²	—	111	3	no	—
106	C₃×C₉	—	37	18	no	—
107	C₃×C₉	—	37	4	no	—
108	C₃×C₉	—	37	2	no	—
109	C₃×C₉	—	111	6	no	—
110	C₃×C₉	#11758606	1	9	yes	complemented
111	C₃×C₉	C₃²×C₃₇⋊C₃	1	9	yes	—
112	C₃×C₉	C₃²×C₃₇⋊C₃	1	2	yes	—
113	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
114	C₃²	—	3	3	no	—
115	C₃²	—	333	9	no	—
116	C₃²	—	111	2	no	—
117	C₃²	—	111	1	no	—
118	C₃²	—	111	3	no	—
119	C₃²	—	111	6	no	—
120	C₃²	C₃×#11758606	1	3	yes	—
121	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
122	C₉	—	3	9	no	—
123	C₉	—	111	4	no	—
124	C₉	—	111	2	no	—
125	C₉	—	111	18	no	—
126	C₉	C₃×#11758606	1	6	yes	—
127	C₉	C₃×#11758606	1	3	yes	—
128	C₃	—	3	3	no	—
129	C₃	—	333	1	no	—
130	C₃	—	333	2	no	—
131	C₃	C₃²×#11758606	1	1	yes	characteristic
132	C₃	#22582677	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758606

C₃

yes

maximal, characteristic

#22582604

C₃

yes

maximal, complemented

#22582603

C₃

yes

maximal, complemented

#22582600

C₃

yes

maximal, complemented

#22582595

C₃

yes

maximal, complemented

#22582575

C₃

yes

maximal, complemented

#22582552

C₃

yes

maximal

#22582538

C₃

yes

maximal, complemented

C₃₇×#6887

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

C₃²

yes

characteristic

C₃×#11758606

—

C₃×#11758606

C₃²

yes

—

C₃×#11758606

C₃²

yes

—

C₃×#11758606

C₃²

yes

characteristic

C₃×#11758606

C₃²

yes

—

C₃×#11758606

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

#15688506

C₃²

yes

complemented

#15688506

C₃²

yes

complemented

#15688505

C₃²

yes

complemented

#15688504

C₃²

yes

complemented

C₃²⋊#66524

C₃²

yes

—

#15688502

C₃²

yes

—

C₉×#66524

C₃²

yes

complemented

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×C₉⋊C₉

C₃²

yes

complemented

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₉²×C₃₇

C₃²

yes

complemented

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

C₃³

yes

—

C₃²×C₃₇⋊C₃

C₃³

yes

characteristic

C₃×C₉×C₃₇

3₋¹⁺²

yes

complemented

C₃×C₉×C₃₇

C₃³

yes

—

C₃×C₉×C₃₇

C₃³

yes

characteristic

C₃×C₉×C₃₇

C₃³

yes

—

C₃×#66524

—

C₃×#66524

C₃³

yes

—

C₃×#66524

C₃³

yes

—

C₃×#66524

C₃³

yes

—

—

—

—

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₃²×C₃₇

C₃⁴

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

C₃×3₋¹⁺²

yes

—

C₉×C₃₇

C₃×3₋¹⁺²

yes

—

#66524

—

#66524

—

#66524

—

C₃²×3₋¹⁺²

—

—

—

—

—

—

—

—

C₃₇⋊C₃

yes

characteristic, complemented

C₃×C₃₇

—

C₃×C₃₇

C₃²×3₋¹⁺²

yes

characteristic

C₃×C₃₇

3₋¹⁺⁴

yes

—

C₃₇⋊C₃

—

C₃₇⋊C₃

—

C₃⁴

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

C₃×C₃₇⋊C₃

yes

complemented

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₉²

—

C₉²

C₃×C₃₇⋊C₃

yes

complemented

C₃₇

#10624884

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

333

—

C₃³

—

111

—

100

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

101

3₋¹⁺²

—

333

—

102

3₋¹⁺²

—

111

—

103

3₋¹⁺²

—

111

—

104

3₋¹⁺²

—

111

—

105

3₋¹⁺²

—

111

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

—

111

—

110

C₃×C₉

#11758606

yes

complemented

111

C₃×C₉

C₃²×C₃₇⋊C₃

yes

—

112

C₃×C₉

C₃²×C₃₇⋊C₃

yes

—

113

C₃×C₉

C₃²×C₃₇⋊C₃

yes

characteristic

114

C₃²

—

115

C₃²

—

333

—

116

C₃²

—

111

—

117

C₃²

—

111

—

118

C₃²

—

111

—

119

C₃²

—

111

—

120

C₃²

C₃×#11758606

yes

—

121

C₃²

C₃³×C₃₇⋊C₃

yes

characteristic

122

C₉

—

123

C₉

—

111

—

124

C₉

—

111

—

125

C₉

—

111

—

126

C₉

C₃×#11758606

yes

—

127

C₉

C₃×#11758606

yes

—

128

C₃

—

129

C₃

—

333

—

130

C₃

—

333

—

131

C₃

C₃²×#11758606

yes

characteristic

132

C₃

#22582677

yes

—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting	C₃₇×#6887	C₃
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	#6887	C₃₇⋊C₃
hypocenter	C₃₇	#10624884
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ #6887

subgroup

G/H

center, Frattini

C₃²

C₃³×C₃₇⋊C₃

commutator, socle, cosocle

C₃²×C₃₇

C₃⁴

Fitting

C₃₇×#6887

C₃

prime radical

C₃³×C₃₇⋊C₃

C₃²

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃₇

C₁ ◁ C₃² ◁ #6887

N	H ≅ G/N	type
2. #22582604	C₃	semidirect
3. #22582603	C₃	semidirect
4. #22582600	C₃	semidirect
5. #22582595	C₃	semidirect
6. #22582575	C₃	semidirect
8. #22582538	C₃	semidirect
9. C₃₇×#6887	C₃	semidirect
18. #15688506	C₃²	semidirect
19. #15688506	C₃²	semidirect
20. #15688505	C₃²	semidirect
21. #15688504	C₃²	semidirect
24. C₉×#66524	C₃²	semidirect
26. C₃₇×C₉⋊C₉	C₃²	semidirect
28. C₉²×C₃₇	C₃²	semidirect
34. C₃×C₉×C₃₇	3₋¹⁺²	semidirect
70. #6887	C₃₇⋊C₃	semidirect
89. C₉⋊C₉	C₃×C₃₇⋊C₃	semidirect
94. C₉²	C₃×C₃₇⋊C₃	semidirect
95. C₃₇	#10624884	semidirect
110. C₃×C₉	#11758606	semidirect

H ≅ G/N

type

2. #22582604

C₃

semidirect

3. #22582603

C₃

semidirect

4. #22582600

C₃

semidirect

5. #22582595

C₃

semidirect

6. #22582575

C₃

semidirect

8. #22582538

C₃

semidirect

9. C₃₇×#6887

C₃

semidirect

18. #15688506

C₃²

semidirect

19. #15688506

C₃²

semidirect

20. #15688505

C₃²

semidirect

21. #15688504

C₃²

semidirect

24. C₉×#66524

C₃²

semidirect

26. C₃₇×C₉⋊C₉

C₃²

semidirect

28. C₉²×C₃₇

C₃²

semidirect

34. C₃×C₉×C₃₇

3₋¹⁺²

semidirect

70. #6887

C₃₇⋊C₃

semidirect

89. C₉⋊C₉

C₃×C₃₇⋊C₃

semidirect

94. C₉²

C₃×C₃₇⋊C₃

semidirect

95. C₃₇

#10624884

semidirect

110. C₃×C₉

#11758606

semidirect

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

subgroup	index
C₃²×#11758606	3
#22582604	3
#22582603	3
#22582600	3
#22582595	3
#22582575	3
#22582552	3
#22582538	3
C₃₇×#6887	3
#10624884	37

subgroup

index

C₃₇×#6887

Subgroups — #41007660

Conjugacy classes of subgroups

Complements

Maximal subgroups