Group #41005309 — subgroups

Subgroups — #41005309

Order 26973.

220 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 52 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	3₋¹⁺²×#66524	C₃	1	2	yes	maximal, complemented
2	H(3,𝔽₃)×#66524	C₃	1	1	yes	maximal, characteristic
3	#22582535	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃₇⋊#6862	C₃	1	1	yes	maximal, characteristic, complemented
5	#22582457	C₃	1	1	yes	maximal, characteristic, complemented
6	#22582451	C₃	1	2	yes	maximal
7	#22582447	C₃	1	2	yes	maximal, complemented
8	#22582447	C₃	1	2	yes	maximal, complemented
9	C₃₇×#6857	C₃	1	1	yes	maximal, characteristic
10	C₃²×#66524	—	3	1	no	—
11	C₃²×#66524	C₃²	1	1	yes	characteristic
12	#15688506	—	3	2	no	—
13	#15688505	—	3	2	no	—
14	#15688504	—	3	2	no	—
15	C₃²⋊#66524	—	3	2	no	—
16	C₃²⋊#66524	C₃²	1	2	yes	—
17	C₃²⋊#66524	C₃²	1	1	yes	characteristic
18	C₃²⋊#66524	C₃²	1	1	yes	characteristic
19	C₃²⋊#66524	C₃²	1	2	yes	—
20	C₃²⋊#66524	C₃²	1	2	yes	—
21	C₉×#66524	—	3	1	no	—
22	C₉×#66524	—	3	2	no	—
23	C₃×3₋¹⁺²×C₃₇	C₃²	1	2	yes	—
24	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
25	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
26	C₃³×C₃₇	—	3	1	no	—
27	C₃³×C₃₇	C₃³	1	1	yes	characteristic
28	3₋¹⁺²×C₃₇	—	3	6	no	—
29	H(3,𝔽₃)×C₃₇	—	3	3	no	—
30	C₃×C₉×C₃₇	—	3	2	no	—
31	C₃×C₉×C₃₇	—	3	1	no	—
32	C₃×#66524	—	3	3	no	—
33	C₃×#66524	—	3	2	no	—
34	C₃×#66524	—	3	2	no	—
35	C₃×#66524	—	3	2	no	—
36	C₃×#66524	—	3	1	no	—
37	C₃×#66524	—	3	1	no	—
38	C₃×#66524	—	9	3	no	—
39	C₃×#66524	H(3,𝔽₃)	1	2	yes	—
40	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
41	#10624587	—	37	1	no	maximal, Sylow
42	C₃²×C₃₇	—	3	1	no	—
43	C₃²×C₃₇	—	3	1	no	—
44	C₃²×C₃₇	—	3	3	no	—
45	C₃²×C₃₇	—	3	2	no	—
46	C₃²×C₃₇	—	9	1	no	—
47	C₃²×C₃₇	—	9	2	no	—
48	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
49	C₉×C₃₇	—	3	6	no	—
50	C₉×C₃₇	—	9	1	no	—
51	#66524	—	3	2	no	—
52	#66524	—	9	1	no	—
53	#66524	—	9	1	no	—
54	#66524	—	9	2	no	—
55	#66524	—	9	2	no	—
56	#66524	—	9	2	no	—
57	#66524	#731	1	3	yes	—
58	C₉×3₋¹⁺²	—	37	2	no	—
59	C₉×H(3,𝔽₃)	—	37	1	no	—
60	#6887	—	37	1	no	—
61	#6862	—	37	1	no	—
62	#6861	—	37	1	no	—
63	#6859	—	37	2	no	—
64	#6858	—	37	2	no	—
65	#6858	—	37	2	no	—
66	#6857	C₃₇⋊C₃	1	1	yes	characteristic
67	C₃×C₃₇	—	9	1	no	—
68	C₃×C₃₇	—	9	3	no	—
69	C₃×C₃₇	C₃×#731	1	1	yes	characteristic
70	C₃×C₃₇	#6908	1	2	yes	—
71	C₃×C₃₇	#6908	1	1	yes	characteristic
72	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	2	yes	—
73	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
74	C₃²×C₉	—	37	1	no	—
75	C₃²×C₉	—	111	1	no	—
76	C₉⋊C₉	—	111	2	no	—
77	C₉⋊C₉	—	111	2	no	—
78	C₉⋊C₉	—	111	2	no	—
79	C₃²⋊C₉	—	37	2	no	—
80	C₃²⋊C₉	—	37	1	no	—
81	C₃²⋊C₉	—	37	1	no	—
82	C₃²⋊C₉	—	37	2	no	—
83	C₃²⋊C₉	—	37	2	no	—
84	C₃²⋊C₉	—	111	2	no	—
85	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
86	C₉²	—	111	1	no	—
87	C₉²	—	111	2	no	—
88	C₃₇	#10624587	1	1	yes	characteristic, complemented, Sylow
89	C₃³	—	3	1	no	—
90	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
91	3₋¹⁺²	—	3	6	no	—
92	H(3,𝔽₃)	—	3	3	no	—
93	C₃×C₉	—	3	2	no	—
94	C₃×C₉	—	3	1	no	—
95	C₃×C₉	—	37	1	no	—
96	C₃×C₉	—	37	2	no	—
97	C₃×C₉	—	333	3	no	—
98	C₃×C₉	—	111	3	no	—
99	C₃×C₉	—	111	2	no	—
100	C₃×C₉	—	111	2	no	—
101	C₃×C₉	—	111	2	no	—
102	C₃×C₉	—	111	1	no	—
103	C₃×C₉	—	111	1	no	—
104	C₃²	—	3	1	no	—
105	C₃²	—	3	1	no	—
106	C₃²	—	3	3	no	—
107	C₃²	—	3	2	no	—
108	C₃²	—	9	1	no	—
109	C₃²	—	9	2	no	—
110	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
111	C₉	—	3	6	no	—
112	C₉	—	9	1	no	—
113	C₉	—	37	3	no	—
114	C₉	—	333	1	no	—
115	C₉	—	333	1	no	—
116	C₉	—	333	2	no	—
117	C₉	—	333	2	no	—
118	C₉	—	333	2	no	—
119	C₉	—	111	2	no	—
120	C₃	—	9	1	no	—
121	C₃	—	9	3	no	—
122	C₃	#731×C₃₇⋊C₃	1	1	yes	characteristic
123	C₃	#22582629	1	2	yes	—
124	C₃	#22582629	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

3₋¹⁺²×#66524

C₃

yes

maximal, complemented

H(3,𝔽₃)×#66524

C₃

yes

maximal, characteristic

#22582535

C₃

yes

maximal, characteristic, complemented

C₃₇⋊#6862

C₃

yes

maximal, characteristic, complemented

#22582457

C₃

yes

maximal, characteristic, complemented

#22582451

C₃

yes

maximal

#22582447

C₃

yes

maximal, complemented

#22582447

C₃

yes

maximal, complemented

C₃₇×#6857

C₃

yes

maximal, characteristic

C₃²×#66524

—

C₃²×#66524

C₃²

yes

—

—

—

—

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

characteristic

C₃²⋊#66524

C₃²

yes

characteristic

C₃²⋊#66524

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

C₉×#66524

—

C₉×#66524

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

characteristic

C₃₇×C₃²⋊C₉

C₃²

yes

characteristic

C₃³×C₃₇

—

C₃³×C₃₇

C₃³

yes

characteristic

3₋¹⁺²×C₃₇

—

H(3,𝔽₃)×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

—

—

—

—

—

—

—

yes

—

yes

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₃₇

—

C₉×C₃₇

—

—

—

—

—

—

—

yes

—

C₉×3₋¹⁺²

—

C₉×H(3,𝔽₃)

—

—

—

—

—

—

—

C₃₇⋊C₃

yes

characteristic

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃×#731

yes

characteristic

C₃×C₃₇

#6908

yes

—

C₃×C₃₇

#6908

yes

characteristic

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

—

C₃×H(3,𝔽₃)

C₃×C₃₇⋊C₃

yes

characteristic

C₃²×C₉

—

C₃²×C₉

—

111

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₉²

—

111

—

C₉²

—

111

—

C₃₇

#10624587

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

333

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

100

C₃×C₉

—

111

—

101

C₃×C₉

—

111

—

102

C₃×C₉

—

111

—

103

C₃×C₉

—

111

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

111

C₉

—

112

C₉

—

113

C₉

—

114

C₉

—

333

—

115

C₉

—

333

—

116

C₉

—

333

—

117

C₉

—

333

—

118

C₉

—

333

—

119

C₉

—

111

—

120

C₃

—

121

C₃

—

122

C₃

#731×C₃₇⋊C₃

yes

characteristic

123

C₃

#22582629

yes

—

124

C₃

#22582629

yes

characteristic

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6857	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃×H(3,𝔽₃)×C₃₇	C₃²
hypercenter	#6857	C₃₇⋊C₃
hypocenter	C₃₇	#10624587
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ #6857

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6857

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃×H(3,𝔽₃)×C₃₇

C₃²

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ #6857

N	H ≅ G/N	type
1. 3₋¹⁺²×#66524	C₃	semidirect
3. #22582535	C₃	semidirect
4. C₃₇⋊#6862	C₃	semidirect
5. #22582457	C₃	semidirect
7. #22582447	C₃	semidirect
8. #22582447	C₃	semidirect
88. C₃₇	#10624587	semidirect

H ≅ G/N

type

1. 3₋¹⁺²×#66524

C₃

semidirect

3. #22582535

C₃

semidirect

4. C₃₇⋊#6862

C₃

semidirect

5. #22582457

C₃

semidirect

7. #22582447

C₃

semidirect

8. #22582447

C₃

semidirect

88. C₃₇

#10624587

semidirect

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

subgroup	index
3₋¹⁺²×#66524	3
H(3,𝔽₃)×#66524	3
#22582535	3
C₃₇⋊#6862	3
#22582457	3
#22582451	3
#22582447	3
C₃₇×#6857	3
#10624587	37

subgroup

index

3₋¹⁺²×#66524

C₃₇⋊#6862

C₃₇×#6857

Subgroups — #41005309

Conjugacy classes of subgroups

Complements

Maximal subgroups