Group #41007418 — subgroups

Subgroups — #41007418

Order 26973.

200 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 46 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	#22582661	C₃	1	2	yes	maximal, complemented
2	#22582648	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582523	C₃	1	2	yes	maximal, complemented
4	#22582513	C₃	1	2	yes	maximal, complemented
5	#22582513	C₃	1	2	yes	maximal, complemented
6	#22582507	C₃	1	1	yes	maximal, characteristic, complemented
7	#22582503	C₃	1	1	yes	maximal, characteristic, complemented
8	#22582502	C₃	1	1	yes	maximal, characteristic, complemented
9	C₃₇×#6900	C₃	1	1	yes	maximal, characteristic, complemented
10	#15688531	—	3	2	no	—
11	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
12	#15688521	C₃²	1	2	yes	complemented
13	#15688521	C₃²	1	2	yes	complemented
14	#15688519	—	3	2	no	—
15	#15688519	C₃²	1	1	yes	characteristic
16	#15688519	C₃²	1	1	yes	characteristic
17	#15688519	C₃²	1	1	yes	characteristic
18	#15688519	C₃²	1	2	yes	complemented
19	#15688516	—	3	2	no	—
20	#15688513	—	3	2	no	—
21	C₃×3₋¹⁺²×C₃₇	—	3	1	no	—
22	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
23	C₃₇×C₉⋊C₉	—	3	2	no	—
24	C₃₇×C₉⋊C₉	C₃²	1	2	yes	complemented
25	C₉²×C₃₇	C₃²	1	1	yes	characteristic, complemented
26	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
27	C₃²×C₃₇⋊C₃	—	9	1	no	—
28	3₋¹⁺²×C₃₇	—	9	1	no	—
29	3₋¹⁺²×C₃₇	—	9	3	no	—
30	C₃×C₉×C₃₇	—	3	1	no	—
31	C₃×C₉×C₃₇	—	3	2	no	—
32	C₃×C₉×C₃₇	H(3,𝔽₃)	1	2	yes	—
33	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
34	C₃²⋊(C₃₇⋊C₃)	—	3	2	no	—
35	C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
36	C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
37	C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
38	C₃²⋊(C₃₇⋊C₃)	—	9	2	no	—
39	#11758608	—	3	2	no	—
40	#11758608	—	3	2	no	—
41	#11758606	—	9	2	no	—
42	#11758605	—	9	2	no	—
43	C₉×C₃₇⋊C₃	—	9	2	no	—
44	#10624851	—	37	1	no	maximal, Sylow
45	C₃²×C₃₇	—	3	1	no	—
46	C₃²×C₃₇	—	9	1	no	—
47	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
48	C₃×C₃₇⋊C₃	—	9	2	no	—
49	C₃×C₃₇⋊C₃	—	9	1	no	—
50	C₃×C₃₇⋊C₃	—	9	1	no	—
51	C₃×C₃₇⋊C₃	—	9	1	no	—
52	C₃×C₃₇⋊C₃	—	27	3	no	—
53	C₉×C₃₇	—	3	3	no	—
54	C₉×C₃₇	—	3	2	no	—
55	C₉×C₃₇	—	3	1	no	—
56	C₉×C₃₇	—	9	2	no	—
57	#66524	—	9	2	no	—
58	#66524	—	9	2	no	—
59	#6913	—	37	2	no	—
60	#6911	—	37	1	no	—
61	#6900	C₃₇⋊C₃	1	1	yes	characteristic, complemented
62	#6883	—	37	2	no	—
63	#6881	—	37	2	no	—
64	#6881	—	37	2	no	—
65	#6879	—	37	1	no	—
66	#6878	—	37	1	no	—
67	#6878	—	37	1	no	—
68	C₃×C₃₇	—	3	1	no	—
69	C₃×C₃₇	—	9	1	no	—
70	C₃×C₃₇	#6911	1	1	yes	characteristic
71	C₃₇⋊C₃	—	27	2	no	—
72	C₃₇⋊C₃	—	27	1	no	—
73	C₃₇⋊C₃	—	27	1	no	—
74	C₃₇⋊C₃	—	27	1	no	—
75	#737	—	111	2	no	—
76	C₃×3₋¹⁺²	—	3	1	no	—
77	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
78	C₃×H(3,𝔽₃)	—	111	1	no	—
79	#733	—	37	2	no	—
80	#733	—	37	2	no	—
81	#732	—	37	2	no	—
82	#732	—	37	1	no	—
83	#732	—	37	1	no	—
84	#732	—	37	1	no	—
85	#732	—	111	2	no	—
86	#731	—	111	2	no	—
87	AΣL(1,𝔽₂₇)	—	111	2	no	—
88	C₉⋊C₉	—	3	2	no	—
89	C₉⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	complemented
90	C₉²	C₃×C₃₇⋊C₃	1	1	yes	characteristic, complemented
91	C₃₇	#10624851	1	1	yes	characteristic, complemented, Sylow
92	C₃³	—	333	1	no	—
93	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
94	3₋¹⁺²	—	9	1	no	—
95	3₋¹⁺²	—	9	3	no	—
96	3₋¹⁺²	—	333	2	no	—
97	3₋¹⁺²	—	333	2	no	—
98	3₋¹⁺²	—	111	2	no	—
99	3₋¹⁺²	—	111	2	no	—
100	H(3,𝔽₃)	—	333	2	no	—
101	H(3,𝔽₃)	—	111	2	no	—
102	H(3,𝔽₃)	—	111	1	no	—
103	H(3,𝔽₃)	—	111	1	no	—
104	H(3,𝔽₃)	—	111	1	no	—
105	C₃×C₉	—	3	1	no	—
106	C₃×C₉	—	3	2	no	—
107	C₃×C₉	—	333	2	no	—
108	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
109	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
110	C₃²	—	3	1	no	—
111	C₃²	—	9	1	no	—
112	C₃²	—	333	2	no	—
113	C₃²	—	333	1	no	—
114	C₃²	—	333	1	no	—
115	C₃²	—	333	1	no	—
116	C₃²	—	999	3	no	—
117	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
118	C₉	—	3	3	no	—
119	C₉	—	3	2	no	—
120	C₉	—	3	1	no	—
121	C₉	—	9	2	no	—
122	C₉	—	333	2	no	—
123	C₉	—	333	2	no	—
124	C₃	—	3	1	no	—
125	C₃	—	9	1	no	—
126	C₃	—	999	1	no	—
127	C₃	—	999	1	no	—
128	C₃	—	999	1	no	—
129	C₃	—	999	2	no	—
130	C₃	#22582648	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22582661

C₃

yes

maximal, complemented

#22582648

C₃

yes

maximal, characteristic, complemented

#22582523

C₃

yes

maximal, complemented

#22582513

C₃

yes

maximal, complemented

#22582513

C₃

yes

maximal, complemented

#22582507

C₃

yes

maximal, characteristic, complemented

#22582503

C₃

yes

maximal, characteristic, complemented

#22582502

C₃

yes

maximal, characteristic, complemented

C₃₇×#6900

C₃

yes

maximal, characteristic, complemented

#15688531

—

C₃×C₃²⋊(C₃₇⋊C₃)

—

#15688521

C₃²

yes

complemented

#15688521

C₃²

yes

complemented

#15688519

—

#15688519

C₃²

yes

characteristic

#15688519

C₃²

yes

characteristic

#15688519

C₃²

yes

characteristic

#15688519

C₃²

yes

complemented

#15688516

—

#15688513

—

C₃×3₋¹⁺²×C₃₇

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

characteristic

C₃₇×C₉⋊C₉

—

C₃₇×C₉⋊C₉

C₃²

yes

complemented

C₉²×C₃₇

C₃²

yes

characteristic, complemented

C₃³×C₃₇

H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇⋊C₃

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

—

C₃×C₉×C₃₇

C₃³

yes

characteristic

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

—

—

—

—

C₉×C₃₇⋊C₃

—

#10624851

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

—

—

—

—

C₃₇⋊C₃

yes

characteristic, complemented

—

—

—

—

—

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

#6911

yes

characteristic

C₃₇⋊C₃

—

C₃₇⋊C₃

—

C₃₇⋊C₃

—

C₃₇⋊C₃

—

#737

—

111

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

characteristic

C₃×H(3,𝔽₃)

—

111

—

—

—

—

—

—

—

—

111

—

#731

—

111

—

AΣL(1,𝔽₂₇)

—

111

—

C₉⋊C₉

—

C₉⋊C₉

C₃×C₃₇⋊C₃

yes

complemented

C₉²

C₃×C₃₇⋊C₃

yes

characteristic, complemented

C₃₇

#10624851

yes

characteristic, complemented, Sylow

C₃³

—

333

—

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

333

—

3₋¹⁺²

—

333

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

100

H(3,𝔽₃)

—

333

—

101

H(3,𝔽₃)

—

111

—

102

H(3,𝔽₃)

—

111

—

103

H(3,𝔽₃)

—

111

—

104

H(3,𝔽₃)

—

111

—

105

C₃×C₉

—

106

C₃×C₉

—

107

C₃×C₉

—

333

—

108

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

—

109

C₃×C₉

C₃²×C₃₇⋊C₃

yes

characteristic

110

C₃²

—

111

C₃²

—

112

C₃²

—

333

—

113

C₃²

—

333

—

114

C₃²

—

333

—

115

C₃²

—

333

—

116

C₃²

—

999

—

117

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

118

C₉

—

119

C₉

—

120

C₉

—

121

C₉

—

122

C₉

—

333

—

123

C₉

—

333

—

124

C₃

—

125

C₃

—

126

C₃

—

999

—

127

C₃

—

999

—

128

C₃

—

999

—

129

C₃

—

999

—

130

C₃

#22582648

yes

characteristic

	subgroup	G/H
center	C₃	#22582648
commutator, cosocle	C₃×C₉×C₃₇	C₃³
Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6900	C₃
socle	C₃×C₃₇	#6911
hypercenter	#6900	C₃₇⋊C₃
hypocenter	C₃₇	#10624851
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×3₋¹⁺² ◁ #6900

subgroup

G/H

center

C₃

#22582648

commutator, cosocle

C₃×C₉×C₃₇

C₃³

Frattini

C₃×C₉

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6900

C₃

C₃×C₃₇

C₃₇⋊C₃

C₃₇

G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃ ◁ C₃² ◁ C₃×3₋¹⁺² ◁ #6900

N	H ≅ G/N	type
1. #22582661	C₃	semidirect
2. #22582648	C₃	semidirect
3. #22582523	C₃	semidirect
4. #22582513	C₃	semidirect
5. #22582513	C₃	semidirect
6. #22582507	C₃	semidirect
7. #22582503	C₃	semidirect
8. #22582502	C₃	semidirect
9. C₃₇×#6900	C₃	semidirect
12. #15688521	C₃²	semidirect
13. #15688521	C₃²	semidirect
18. #15688519	C₃²	semidirect
24. C₃₇×C₉⋊C₉	C₃²	semidirect
25. C₉²×C₃₇	C₃²	semidirect
61. #6900	C₃₇⋊C₃	semidirect
89. C₉⋊C₉	C₃×C₃₇⋊C₃	semidirect
90. C₉²	C₃×C₃₇⋊C₃	semidirect
91. C₃₇	#10624851	semidirect

H ≅ G/N

type

1. #22582661

C₃

semidirect

2. #22582648

C₃

semidirect

3. #22582523

C₃

semidirect

4. #22582513

C₃

semidirect

5. #22582513

C₃

semidirect

6. #22582507

C₃

semidirect

7. #22582503

C₃

semidirect

8. #22582502

C₃

semidirect

9. C₃₇×#6900

C₃

semidirect

12. #15688521

C₃²

semidirect

13. #15688521

C₃²

semidirect

18. #15688519

C₃²

semidirect

24. C₃₇×C₉⋊C₉

C₃²

semidirect

25. C₉²×C₃₇

C₃²

semidirect

61. #6900

C₃₇⋊C₃

semidirect

89. C₉⋊C₉

C₃×C₃₇⋊C₃

semidirect

90. C₉²

C₃×C₃₇⋊C₃

semidirect

91. C₃₇

#10624851

semidirect

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

subgroup	index
#22582661	3
#22582648	3
#22582523	3
#22582513	3
#22582507	3
#22582503	3
#22582502	3
C₃₇×#6900	3
#10624851	37

subgroup

index

C₃₇×#6900

Subgroups — #41007418

Conjugacy classes of subgroups

Complements

Maximal subgroups