Group #41008052 — subgroups

Subgroups — #41008052

Order 26973.

740 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 250 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	#22582563	C₃	1	9	yes	maximal, complemented
2	#22582557	C₃	1	18	yes	maximal, complemented
3	#22582555	C₃	1	9	yes	maximal, complemented
4	#22582554	C₃	1	2	yes	maximal, complemented
5	#22582550	C₃	1	1	yes	maximal, characteristic
6	C₃₇×#6891	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃×#11758608	C₃²	1	6	yes	—
8	C₃×#11758605	C₃²	1	9	yes	complemented
9	C₃×#11758606	C₃²	1	9	yes	complemented
10	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	3	yes	—
11	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	6	yes	—
12	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	3	yes	—
13	C₃×C₉×C₃₇⋊C₃	C₃²	1	9	yes	complemented
14	C₃²⋊#66524	C₃²	1	6	yes	—
15	C₃²⋊#66524	C₃²	1	6	yes	—
16	C₃²⋊#66524	C₃²	1	6	yes	—
17	#15688502	C₃²	1	54	yes	complemented
18	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
19	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
20	C₃₇×C₃²⋊C₉	C₃²	1	6	yes	—
21	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
22	C₃³×C₃₇	—	9	1	no	—
23	C₃³×C₃₇	—	9	2	no	—
24	C₃³×C₃₇	—	9	1	no	—
25	C₃³×C₃₇	C₃³	1	1	yes	characteristic
26	C₃³×C₃₇	C₃³	1	2	yes	—
27	C₃³×C₃₇	C₃³	1	1	yes	characteristic
28	C₃²×C₃₇⋊C₃	C₃³	1	9	yes	—
29	3₋¹⁺²×C₃₇	—	9	3	no	—
30	C₃×C₉×C₃₇	C₃³	1	9	yes	—
31	C₃²⋊(C₃₇⋊C₃)	—	9	3	no	—
32	C₃²⋊(C₃₇⋊C₃)	—	9	3	no	—
33	C₃²⋊(C₃₇⋊C₃)	—	9	6	no	—
34	#11758608	—	9	6	no	—
35	C₃×#66524	C₃³	1	18	yes	—
36	#11758606	—	9	9	no	—
37	#11758605	—	9	9	no	—
38	C₉×C₃₇⋊C₃	—	9	9	no	—
39	#10624910	—	37	1	no	maximal, Sylow
40	C₃²×C₃₇	—	3	2	no	—
41	C₃²×C₃₇	—	3	1	no	—
42	C₃²×C₃₇	—	3	1	no	—
43	C₃²×C₃₇	—	3	2	no	—
44	C₃²×C₃₇	—	3	1	no	—
45	C₃²×C₃₇	—	3	2	no	—
46	C₃²×C₃₇	—	3	1	no	—
47	C₃²×C₃₇	—	3	2	no	—
48	C₃²×C₃₇	—	3	2	no	—
49	C₃²×C₃₇	—	3	2	no	—
50	C₃²×C₃₇	—	9	9	no	—
51	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
52	C₃×C₃₇⋊C₃	—	3	9	no	—
53	C₃×C₃₇⋊C₃	—	3	9	no	—
54	C₃×C₃₇⋊C₃	—	3	18	no	—
55	C₉×C₃₇	—	3	9	no	—
56	#66524	—	3	18	no	—
57	#6893	—	37	9	no	—
58	C₉⋊H(3,𝔽₃)	—	37	9	no	—
59	C₉⋊H(3,𝔽₃)	—	37	18	no	—
60	#6891	—	37	2	no	—
61	#6891	C₃₇⋊C₃	1	1	yes	characteristic, complemented
62	C₃²⋊H(3,𝔽₃)	—	37	1	no	—
63	C₃×C₃₇	—	9	2	no	—
64	C₃×C₃₇	—	9	1	no	—
65	C₃×C₃₇	—	9	1	no	—
66	C₃×C₃₇	H(5,𝔽₃)	1	1	yes	characteristic
67	C₃×C₃₇	3₋¹⁺⁴	1	2	yes	—
68	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
69	C₃₇⋊C₃	—	9	9	no	—
70	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
71	C₃×3₋¹⁺²	—	37	6	no	—
72	C₃×3₋¹⁺²	—	37	9	no	—
73	C₃×3₋¹⁺²	—	37	9	no	—
74	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	3	yes	—
75	C₃×H(3,𝔽₃)	—	37	3	no	—
76	C₃×H(3,𝔽₃)	—	37	6	no	—
77	C₃×H(3,𝔽₃)	—	37	3	no	—
78	C₃²×C₉	—	37	9	no	—
79	C₃²⋊C₉	—	37	6	no	—
80	C₃²⋊C₉	—	37	6	no	—
81	C₃²⋊C₉	—	37	6	no	—
82	C₃²⋊C₉	—	37	54	no	—
83	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	6	yes	—
84	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
85	C₃₇	#10624910	1	1	yes	characteristic, complemented, Sylow
86	C₃³	—	9	1	no	—
87	C₃³	—	9	1	no	—
88	C₃³	—	9	2	no	—
89	C₃³	—	37	9	no	—
90	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
91	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
92	C₃³	C₃²×C₃₇⋊C₃	1	2	yes	—
93	3₋¹⁺²	—	9	3	no	—
94	3₋¹⁺²	—	333	9	no	—
95	3₋¹⁺²	—	333	9	no	—
96	3₋¹⁺²	—	333	6	no	—
97	H(3,𝔽₃)	—	333	3	no	—
98	H(3,𝔽₃)	—	333	3	no	—
99	H(3,𝔽₃)	—	333	6	no	—
100	C₃×C₉	—	37	18	no	—
101	C₃×C₉	—	333	9	no	—
102	C₃×C₉	C₃²×C₃₇⋊C₃	1	9	yes	—
103	C₃²	—	3	1	no	—
104	C₃²	—	3	1	no	—
105	C₃²	—	3	2	no	—
106	C₃²	—	3	1	no	—
107	C₃²	—	3	2	no	—
108	C₃²	—	3	1	no	—
109	C₃²	—	3	2	no	—
110	C₃²	—	3	2	no	—
111	C₃²	—	3	2	no	—
112	C₃²	—	3	2	no	—
113	C₃²	—	9	9	no	—
114	C₃²	—	111	9	no	—
115	C₃²	—	111	9	no	—
116	C₃²	—	111	18	no	—
117	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
118	C₉	—	3	9	no	—
119	C₉	—	111	18	no	—
120	C₃	—	9	1	no	—
121	C₃	—	9	1	no	—
122	C₃	—	9	2	no	—
123	C₃	—	333	9	no	—
124	C₃	C₃₇⋊H(5,𝔽₃)	1	1	yes	characteristic
125	C₃	#22582679	1	1	yes	characteristic
126	C₃	#22582679	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22582563

C₃

yes

maximal, complemented

#22582557

C₃

yes

maximal, complemented

#22582555

C₃

yes

maximal, complemented

#22582554

C₃

yes

maximal, complemented

#22582550

C₃

yes

maximal, characteristic

C₃₇×#6891

C₃

yes

maximal, characteristic, complemented

C₃×#11758608

C₃²

yes

—

C₃×#11758605

C₃²

yes

complemented

C₃×#11758606

C₃²

yes

complemented

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

—

C₃×C₉×C₃₇⋊C₃

C₃²

yes

complemented

C₃²⋊#66524

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

#15688502

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

characteristic

C₃³×C₃₇

C₃³

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

C₃³

yes

—

3₋¹⁺²×C₃₇

—

C₃×C₉×C₃₇

C₃³

yes

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

#11758608

—

C₃×#66524

C₃³

yes

—

#11758606

—

#11758605

—

C₉×C₃₇⋊C₃

—

#10624910

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃⁴

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

#66524

—

#6893

—

C₉⋊H(3,𝔽₃)

—

C₉⋊H(3,𝔽₃)

—

#6891

—

#6891

C₃₇⋊C₃

yes

characteristic, complemented

C₃²⋊H(3,𝔽₃)

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

H(5,𝔽₃)

yes

characteristic

C₃×C₃₇

3₋¹⁺⁴

yes

—

C₃×C₃₇

3₋¹⁺⁴

yes

characteristic

C₃₇⋊C₃

—

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₃₇

#10624910

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

C₃³

C₃²×C₃₇⋊C₃

yes

—

3₋¹⁺²

—

3₋¹⁺²

—

333

—

3₋¹⁺²

—

333

—

3₋¹⁺²

—

333

—

H(3,𝔽₃)

—

333

—

H(3,𝔽₃)

—

333

—

H(3,𝔽₃)

—

333

—

100

C₃×C₉

—

101

C₃×C₉

—

333

—

102

C₃×C₉

C₃²×C₃₇⋊C₃

yes

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

111

—

115

C₃²

—

111

—

116

C₃²

—

111

—

117

C₃²

C₃³×C₃₇⋊C₃

yes

characteristic

118

C₉

—

119

C₉

—

111

—

120

C₃

—

121

C₃

—

122

C₃

—

123

C₃

—

333

—

124

C₃

C₃₇⋊H(5,𝔽₃)

yes

characteristic

125

C₃

#22582679

yes

characteristic

126

C₃

#22582679

yes

—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting	C₃₇×#6891	C₃
prime radical	#22582550	C₃
hypercenter	#6891	C₃₇⋊C₃
hypocenter	C₃₇	#10624910
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

C₃

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃₇

C₁ ◁ C₃² ◁ #6891

N	H ≅ G/N	type
1. #22582563	C₃	semidirect
2. #22582557	C₃	semidirect
3. #22582555	C₃	semidirect
4. #22582554	C₃	semidirect
6. C₃₇×#6891	C₃	semidirect
8. C₃×#11758605	C₃²	semidirect
9. C₃×#11758606	C₃²	semidirect
13. C₃×C₉×C₃₇⋊C₃	C₃²	semidirect
17. #15688502	C₃²	semidirect
61. #6891	C₃₇⋊C₃	semidirect
85. C₃₇	#10624910	semidirect

H ≅ G/N

type

1. #22582563

C₃

semidirect

2. #22582557

C₃

semidirect

3. #22582555

C₃

semidirect

4. #22582554

C₃

semidirect

6. C₃₇×#6891

C₃

semidirect

8. C₃×#11758605

C₃²

semidirect

9. C₃×#11758606

C₃²

semidirect

13. C₃×C₉×C₃₇⋊C₃

C₃²

semidirect

17. #15688502

C₃²

semidirect

61. #6891

C₃₇⋊C₃

semidirect

85. C₃₇

#10624910

semidirect

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

subgroup	index
#22582563	3
#22582557	3
#22582555	3
#22582554	3
#22582550	3
C₃₇×#6891	3
#10624910	37

subgroup

index

C₃₇×#6891

Subgroups — #41008052

Conjugacy classes of subgroups

Complements

Maximal subgroups