Group #41008228 — subgroups

Subgroups — #41008228

Order 26973.

948 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

normal subgroups only

This group has 948 conjugacy classes of subgroups — too many to draw. Only its 286 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	#22582679	C₃	1	3	yes	maximal, complemented
2	#22582661	C₃	1	18	yes	maximal, complemented
3	#22582652	C₃	1	2	yes	maximal, complemented
4	#22582648	C₃	1	1	yes	maximal, characteristic, complemented
5	#22582638	C₃	1	3	yes	maximal, complemented
6	#22582633	C₃	1	9	yes	maximal, complemented
7	C₃×#15688516	C₃	1	3	yes	maximal, complemented
8	C₃×C₃₇×#737	C₃	1	1	yes	maximal, characteristic, complemented
9	#15688532	—	3	18	no	—
10	#15688531	C₃²	1	27	yes	complemented
11	C₃×#11758608	C₃²	1	6	yes	—
12	3₋¹⁺²×C₃₇⋊C₃	—	3	9	no	—
13	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	3	yes	—
14	#15688521	C₃²	1	18	yes	complemented
15	#15688519	C₃²	1	9	yes	—
16	#15688516	C₃²	1	9	yes	complemented
17	#15688516	C₃²	1	18	yes	complemented
18	#15688513	C₃²	1	27	yes	complemented
19	C₃₇×#737	—	3	9	no	—
20	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
21	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
22	C₃×3₋¹⁺²×C₃₇	C₃²	1	6	yes	complemented
23	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
24	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
25	C₃²×C₉×C₃₇	C₃²	1	3	yes	complemented
26	C₃³×C₃₇	C₃³	1	3	yes	—
27	C₃³×C₃₇	C₃³	1	1	yes	characteristic
28	C₃²×C₃₇⋊C₃	—	3	3	no	—
29	3₋¹⁺²×C₃₇	—	3	3	no	—
30	3₋¹⁺²×C₃₇	—	3	18	no	—
31	3₋¹⁺²×C₃₇	H(3,𝔽₃)	1	9	yes	—
32	H(3,𝔽₃)×C₃₇	—	3	3	no	—
33	C₃×C₉×C₃₇	—	3	3	no	—
34	C₃×C₉×C₃₇	—	3	9	no	—
35	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
36	C₃×C₉×C₃₇	C₃³	1	2	yes	—
37	C₃×C₉×C₃₇	C₃³	1	3	yes	—
38	C₃×C₉×C₃₇	C₃³	1	3	yes	—
39	C₃²⋊(C₃₇⋊C₃)	—	3	6	no	—
40	C₃²⋊(C₃₇⋊C₃)	C₃³	1	9	yes	—
41	#11758608	—	3	6	no	—
42	#11758608	—	3	6	no	—
43	#11758608	C₃³	1	18	yes	—
44	C₃×#66524	—	3	6	no	—
45	#11758606	—	3	27	no	—
46	#11758605	—	3	27	no	—
47	C₉×C₃₇⋊C₃	—	3	27	no	—
48	#10624935	—	37	1	no	maximal, Sylow
49	C₃²×C₃₇	—	3	3	no	—
50	C₃²×C₃₇	—	9	1	no	—
51	C₃²×C₃₇	—	9	3	no	—
52	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
53	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
54	C₃×C₃₇⋊C₃	—	3	9	no	—
55	C₃×C₃₇⋊C₃	—	27	3	no	—
56	C₉×C₃₇	—	3	1	no	—
57	C₉×C₃₇	—	3	2	no	—
58	C₉×C₃₇	—	3	3	no	—
59	C₉×C₃₇	C₃×H(3,𝔽₃)	1	9	yes	—
60	#66524	—	3	18	no	—
61	3₋¹⁺⁴	—	37	3	no	—
62	C₃×#737	C₃₇⋊C₃	1	1	yes	characteristic, complemented
63	#6913	—	37	18	no	—
64	#6912	—	37	2	no	—
65	#6911	—	37	1	no	—
66	#6909	—	37	3	no	—
67	#6908	—	37	9	no	—
68	C₃×#731	—	37	3	no	—
69	C₃×C₃₇	—	3	3	no	—
70	C₃×C₃₇	—	3	1	no	—
71	C₃×C₃₇	—	9	3	no	—
72	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
73	C₃₇⋊C₃	—	9	9	no	—
74	#737	—	3	9	no	—
75	#737	—	37	27	no	—
76	#737	—	111	18	no	—
77	C₃×3₋¹⁺²	—	37	6	no	—
78	C₃×3₋¹⁺²	—	111	9	no	—
79	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
80	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
81	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	6	yes	complemented
82	C₃×H(3,𝔽₃)	—	37	3	no	—
83	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
84	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
85	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	complemented
86	#733	—	37	18	no	—
87	#732	—	37	9	no	—
88	#731	—	37	18	no	—
89	#731	—	37	9	no	—
90	AΣL(1,𝔽₂₇)	—	37	27	no	—
91	C₃₇	#10624935	1	1	yes	characteristic, complemented, Sylow
92	C₃³	—	111	3	no	—
93	C₃³	C₃²×C₃₇⋊C₃	1	3	yes	—
94	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
95	3₋¹⁺²	—	3	3	no	—
96	3₋¹⁺²	—	3	18	no	—
97	3₋¹⁺²	—	37	18	no	—
98	3₋¹⁺²	—	111	6	no	—
99	3₋¹⁺²	—	111	6	no	—
100	3₋¹⁺²	—	111	27	no	—
101	3₋¹⁺²	—	111	27	no	—
102	3₋¹⁺²	C₃²⋊(C₃₇⋊C₃)	1	9	yes	—
103	H(3,𝔽₃)	—	3	3	no	—
104	H(3,𝔽₃)	—	37	9	no	—
105	H(3,𝔽₃)	—	111	6	no	—
106	C₃×C₉	—	3	3	no	—
107	C₃×C₉	—	3	9	no	—
108	C₃×C₉	—	111	6	no	—
109	C₃×C₉	—	111	27	no	—
110	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
111	C₃×C₉	C₃²×C₃₇⋊C₃	1	2	yes	—
112	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
113	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
114	C₃²	—	3	3	no	—
115	C₃²	—	9	1	no	—
116	C₃²	—	9	3	no	—
117	C₃²	—	111	9	no	—
118	C₃²	—	999	3	no	—
119	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
120	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
121	C₉	—	3	1	no	—
122	C₉	—	3	2	no	—
123	C₉	—	3	3	no	—
124	C₉	—	111	18	no	—
125	C₉	C₃×C₃²⋊(C₃₇⋊C₃)	1	9	yes	—
126	C₃	—	3	1	no	—
127	C₃	—	3	3	no	—
128	C₃	—	9	3	no	—
129	C₃	—	333	9	no	—
130	C₃	C₃²×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22582679

C₃

yes

maximal, complemented

#22582661

C₃

yes

maximal, complemented

#22582652

C₃

yes

maximal, complemented

#22582648

C₃

yes

maximal, characteristic, complemented

#22582638

C₃

yes

maximal, complemented

#22582633

C₃

yes

maximal, complemented

C₃×#15688516

C₃

yes

maximal, complemented

C₃×C₃₇×#737

C₃

yes

maximal, characteristic, complemented

#15688532

—

#15688531

C₃²

yes

complemented

C₃×#11758608

C₃²

yes

—

3₋¹⁺²×C₃₇⋊C₃

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

—

#15688521

C₃²

yes

complemented

#15688519

C₃²

yes

—

#15688516

C₃²

yes

complemented

#15688516

C₃²

yes

complemented

#15688513

C₃²

yes

complemented

C₃₇×#737

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₇

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₇

C₃²

yes

complemented

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

characteristic

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃²×C₉×C₃₇

C₃²

yes

complemented

C₃³×C₃₇

C₃³

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

H(3,𝔽₃)

yes

—

H(3,𝔽₃)×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃³

yes

characteristic

C₃×C₉×C₃₇

C₃³

yes

—

C₃×C₉×C₃₇

C₃³

yes

—

C₃×C₉×C₃₇

C₃³

yes

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

C₃³

yes

—

#11758608

—

#11758608

—

#11758608

C₃³

yes

—

C₃×#66524

—

#11758606

—

#11758605

—

C₉×C₃₇⋊C₃

—

#10624935

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₇

C₃⁴

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

C₃×H(3,𝔽₃)

yes

—

#66524

—

3₋¹⁺⁴

—

C₃×#737

C₃₇⋊C₃

yes

characteristic, complemented

—

—

—

—

—

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃²×H(3,𝔽₃)

yes

characteristic

C₃₇⋊C₃

—

#737

—

#737

—

#737

—

111

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

complemented

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×C₃₇⋊C₃

yes

characteristic

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₃²×C₉

C₃×C₃₇⋊C₃

yes

—

—

—

—

AΣL(1,𝔽₂₇)

—

C₃₇

#10624935

yes

characteristic, complemented, Sylow

C₃³

—

111

—

C₃³

C₃²×C₃₇⋊C₃

yes

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

100

3₋¹⁺²

—

111

—

101

3₋¹⁺²

—

111

—

102

3₋¹⁺²

C₃²⋊(C₃₇⋊C₃)

yes

—

103

H(3,𝔽₃)

—

104

H(3,𝔽₃)

—

105

H(3,𝔽₃)

—

111

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

111

—

109

C₃×C₉

—

111

—

110

C₃×C₉

C₃²×C₃₇⋊C₃

yes

characteristic

111

C₃×C₉

C₃²×C₃₇⋊C₃

yes

—

112

C₃×C₉

C₃²×C₃₇⋊C₃

yes

—

113

C₃×C₉

C₃²×C₃₇⋊C₃

yes

—

114

C₃²

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

111

—

118

C₃²

—

999

—

119

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

—

120

C₃²

C₃³×C₃₇⋊C₃

yes

characteristic

121

C₉

—

122

C₉

—

123

C₉

—

124

C₉

—

111

—

125

C₉

C₃×C₃²⋊(C₃₇⋊C₃)

yes

—

126

C₃

—

127

C₃

—

128

C₃

—

129

C₃

—

333

—

130

C₃

C₃²×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

	subgroup	G/H
center	C₃	C₃²×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃²×C₃₇	C₃⁴
Frattini	C₃²	C₃³×C₃₇⋊C₃
Fitting	C₃×C₃₇×#737	C₃
socle	C₃×C₃₇	C₃²×H(3,𝔽₃)
hypercenter	C₃×#737	C₃₇⋊C₃
hypocenter	C₃₇	#10624935
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃×3₋¹⁺² ◁ C₃×#737

subgroup

G/H

center

C₃

C₃²×C₃²⋊(C₃₇⋊C₃)

commutator, cosocle

C₃²×C₃₇

C₃⁴

Frattini

C₃²

C₃³×C₃₇⋊C₃

Fitting

C₃×C₃₇×#737

C₃

socle

C₃×C₃₇

C₃²×H(3,𝔽₃)

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃ ◁ C₃×3₋¹⁺² ◁ C₃×#737

N	H ≅ G/N	type
1. #22582679	C₃	semidirect
2. #22582661	C₃	semidirect
3. #22582652	C₃	semidirect
4. #22582648	C₃	semidirect
5. #22582638	C₃	semidirect
6. #22582633	C₃	semidirect
7. C₃×#15688516	C₃	semidirect
8. C₃×C₃₇×#737	C₃	semidirect
10. #15688531	C₃²	semidirect
14. #15688521	C₃²	semidirect
16. #15688516	C₃²	semidirect
17. #15688516	C₃²	semidirect
18. #15688513	C₃²	semidirect
22. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
25. C₃²×C₉×C₃₇	C₃²	semidirect
62. C₃×#737	C₃₇⋊C₃	semidirect
81. C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	semidirect
85. C₃²×C₉	C₃×C₃₇⋊C₃	semidirect
91. C₃₇	#10624935	semidirect

H ≅ G/N

type

1. #22582679

C₃

semidirect

2. #22582661

C₃

semidirect

3. #22582652

C₃

semidirect

4. #22582648

C₃

semidirect

5. #22582638

C₃

semidirect

6. #22582633

C₃

semidirect

7. C₃×#15688516

C₃

semidirect

8. C₃×C₃₇×#737

C₃

semidirect

10. #15688531

C₃²

semidirect

14. #15688521

C₃²

semidirect

16. #15688516

C₃²

semidirect

17. #15688516

C₃²

semidirect

18. #15688513

C₃²

semidirect

22. C₃×3₋¹⁺²×C₃₇

C₃²

semidirect

25. C₃²×C₉×C₃₇

C₃²

semidirect

62. C₃×#737

C₃₇⋊C₃

semidirect

81. C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

semidirect

85. C₃²×C₉

C₃×C₃₇⋊C₃

semidirect

91. C₃₇

#10624935

semidirect

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

subgroup	index
#22582679	3
#22582661	3
#22582652	3
#22582648	3
#22582638	3
#22582633	3
C₃×#15688516	3
C₃×C₃₇×#737	3
#10624935	37

subgroup

index

C₃×C₃₇×#737

Subgroups — #41008228

Conjugacy classes of subgroups

Complements

Maximal subgroups