Group #41005302 — subgroups

Subgroups — #41005302

Order 26973.

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

normal subgroups only

This group has 256 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	H(3,𝔽₃)×#66524	C₃	1	1	yes	maximal, characteristic, complemented
2	#22582539	C₃	1	2	yes	maximal, complemented
3	#22582535	C₃	1	1	yes	maximal, characteristic, complemented
4	#22582457	C₃	1	2	yes	maximal, complemented
5	#22582456	C₃	1	1	yes	maximal, characteristic, complemented
6	#22582453	C₃	1	2	yes	maximal, complemented
7	#22582443	C₃	1	1	yes	maximal, characteristic, complemented
8	#22582442	C₃	1	2	yes	maximal, complemented
9	C₃₇×#6859	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	2	yes	—
11	C₃²×#66524	—	3	1	no	—
12	C₃²×#66524	C₃²	1	1	yes	characteristic
13	C₃×C₉×C₃₇⋊C₃	—	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	2	yes	—
18	C₃²⋊#66524	C₃²	1	1	yes	characteristic
19	C₃²⋊#66524	C₃²	1	1	yes	characteristic
20	#15688502	—	3	4	no	—
21	C₉×#66524	—	3	1	no	—
22	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
23	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
24	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
25	C₃³×C₃₇	—	3	1	no	—
26	C₃³×C₃₇	C₃³	1	1	yes	characteristic
27	C₃²×C₃₇⋊C₃	—	3	2	no	—
28	H(3,𝔽₃)×C₃₇	—	3	3	no	—
29	C₃×C₉×C₃₇	—	3	1	no	—
30	C₃×C₉×C₃₇	—	3	2	no	—
31	C₃²⋊(C₃₇⋊C₃)	—	3	6	no	—
32	C₃×#66524	—	3	3	no	—
33	C₃×#66524	—	3	2	no	—
34	C₃×#66524	—	3	2	no	—
35	C₃×#66524	—	3	1	no	—
36	C₃×#66524	—	3	1	no	—
37	C₃×#66524	—	9	3	no	—
38	C₃×#66524	H(3,𝔽₃)	1	2	yes	—
39	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
40	C₉×C₃₇⋊C₃	—	9	6	no	—
41	#10624585	—	37	1	no	maximal, Sylow
42	C₃²×C₃₇	—	3	1	no	—
43	C₃²×C₃₇	—	3	2	no	—
44	C₃²×C₃₇	—	3	1	no	—
45	C₃²×C₃₇	—	3	3	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	C₃×C₃₇⋊C₃	—	3	6	no	—
50	C₃×C₃₇⋊C₃	—	9	2	no	—
51	C₃×C₃₇⋊C₃	—	9	2	no	—
52	C₃×C₃₇⋊C₃	—	9	2	no	—
53	C₉×C₃₇	—	9	2	no	—
54	C₉×C₃₇	—	9	1	no	—
55	#66524	—	3	2	no	—
56	#66524	—	9	2	no	—
57	#66524	—	9	1	no	—
58	#66524	—	9	1	no	—
59	#66524	—	9	2	no	—
60	#66524	#732	1	3	yes	—
61	C₉×H(3,𝔽₃)	—	37	1	no	—
62	C₉×H(3,𝔽₃)	—	37	2	no	—
63	#6887	—	37	1	no	—
64	#6861	—	37	1	no	—
65	#6861	—	37	2	no	—
66	#6859	—	37	2	no	—
67	#6859	C₃₇⋊C₃	1	1	yes	characteristic, complemented
68	#6856	—	37	2	no	—
69	#6856	—	37	1	no	—
70	C₃×C₃₇	—	9	1	no	—
71	C₃×C₃₇	—	9	3	no	—
72	C₃×C₃₇	C₃×#732	1	1	yes	characteristic
73	C₃×C₃₇	#6908	1	1	yes	characteristic
74	C₃×C₃₇	#6908	1	2	yes	—
75	C₃₇⋊C₃	—	9	6	no	—
76	C₃×H(3,𝔽₃)	—	37	2	no	—
77	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
78	C₃²×C₉	—	37	1	no	—
79	C₃²×C₉	—	111	2	no	—
80	C₃²×C₉	—	111	1	no	—
81	C₉⋊C₉	—	111	2	no	—
82	C₃²⋊C₉	—	37	2	no	—
83	C₃²⋊C₉	—	37	2	no	—
84	C₃²⋊C₉	—	37	1	no	—
85	C₃²⋊C₉	—	37	1	no	—
86	C₃²⋊C₉	—	111	4	no	—
87	C₃²⋊C₉	—	111	2	no	—
88	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
89	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	—
90	C₉²	—	111	1	no	—
91	C₃₇	#10624585	1	1	yes	characteristic, complemented, Sylow
92	C₃³	—	3	1	no	—
93	C₃³	—	111	2	no	—
94	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
95	H(3,𝔽₃)	—	3	3	no	—
96	H(3,𝔽₃)	—	111	6	no	—
97	C₃×C₉	—	3	1	no	—
98	C₃×C₉	—	3	2	no	—
99	C₃×C₉	—	37	1	no	—
100	C₃×C₉	—	37	2	no	—
101	C₃×C₉	—	333	6	no	—
102	C₃×C₉	—	333	3	no	—
103	C₃×C₉	—	111	3	no	—
104	C₃×C₉	—	111	2	no	—
105	C₃×C₉	—	111	2	no	—
106	C₃×C₉	—	111	1	no	—
107	C₃×C₉	—	111	1	no	—
108	C₃²	—	3	1	no	—
109	C₃²	—	3	1	no	—
110	C₃²	—	3	3	no	—
111	C₃²	—	3	2	no	—
112	C₃²	—	9	1	no	—
113	C₃²	—	9	2	no	—
114	C₃²	—	333	2	no	—
115	C₃²	—	333	2	no	—
116	C₃²	—	333	2	no	—
117	C₃²	—	111	6	no	—
118	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
119	C₉	—	9	2	no	—
120	C₉	—	9	1	no	—
121	C₉	—	37	3	no	—
122	C₉	—	333	2	no	—
123	C₉	—	333	1	no	—
124	C₉	—	333	1	no	—
125	C₉	—	333	2	no	—
126	C₉	—	111	2	no	—
127	C₃	—	9	1	no	—
128	C₃	—	9	3	no	—
129	C₃	—	333	6	no	—
130	C₃	#732×C₃₇⋊C₃	1	1	yes	characteristic
131	C₃	#22582629	1	1	yes	characteristic
132	C₃	#22582632	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

H(3,𝔽₃)×#66524

C₃

yes

maximal, characteristic, complemented

#22582539

C₃

yes

maximal, complemented

#22582535

C₃

yes

maximal, characteristic, complemented

#22582457

C₃

yes

maximal, complemented

#22582456

C₃

yes

maximal, characteristic, complemented

#22582453

C₃

yes

maximal, complemented

#22582443

C₃

yes

maximal, characteristic, complemented

#22582442

C₃

yes

maximal, complemented

C₃₇×#6859

C₃

yes

maximal, characteristic, complemented

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

—

C₃²×#66524

—

C₃²×#66524

C₃²

yes

characteristic

C₃×C₉×C₃₇⋊C₃

—

#15688504

—

C₃²⋊#66524

—

C₃²⋊#66524

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

characteristic

C₃²⋊#66524

C₃²

yes

characteristic

#15688502

—

C₉×#66524

—

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

characteristic

C₃₇×C₃²⋊C₉

C₃²

yes

characteristic

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃³×C₃₇

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

H(3,𝔽₃)×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃²⋊(C₃₇⋊C₃)

—

—

—

—

—

—

—

yes

—

C₃×#66524

H(3,𝔽₃)

yes

characteristic

C₉×C₃₇⋊C₃

—

#10624585

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

—

—

—

—

—

—

yes

—

C₉×H(3,𝔽₃)

—

C₉×H(3,𝔽₃)

—

—

—

—

—

C₃₇⋊C₃

yes

characteristic, complemented

#6856

—

#6856

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

—

C₃₇⋊C₃

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×C₃₇⋊C₃

yes

characteristic

C₃²×C₉

—

C₃²×C₉

—

111

—

C₃²×C₉

—

111

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₉²

—

111

—

C₃₇

#10624585

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

111

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

H(3,𝔽₃)

—

H(3,𝔽₃)

—

111

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

100

C₃×C₉

—

101

C₃×C₉

—

333

—

102

C₃×C₉

—

333

—

103

C₃×C₉

—

111

—

104

C₃×C₉

—

111

—

105

C₃×C₉

—

111

—

106

C₃×C₉

—

111

—

107

C₃×C₉

—

111

—

108

C₃²

—

109

C₃²

—

110

C₃²

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

333

—

115

C₃²

—

333

—

116

C₃²

—

333

—

117

C₃²

—

111

—

118

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

119

C₉

—

120

C₉

—

121

C₉

—

122

C₉

—

333

—

123

C₉

—

333

—

124

C₉

—

333

—

125

C₉

—

333

—

126

C₉

—

111

—

127

C₃

—

128

C₃

—

129

C₃

—

333

—

130

C₃

#732×C₃₇⋊C₃

yes

characteristic

131

C₃

#22582629

yes

characteristic

132

C₃

#22582632

yes

—

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6859	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
hypercenter	#6859	C₃₇⋊C₃
hypocenter	C₃₇	#10624585
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ #6859

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6859

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ #6859

N	H ≅ G/N	type
1. H(3,𝔽₃)×#66524	C₃	semidirect
2. #22582539	C₃	semidirect
3. #22582535	C₃	semidirect
4. #22582457	C₃	semidirect
5. #22582456	C₃	semidirect
6. #22582453	C₃	semidirect
7. #22582443	C₃	semidirect
8. #22582442	C₃	semidirect
9. C₃₇×#6859	C₃	semidirect
67. #6859	C₃₇⋊C₃	semidirect
91. C₃₇	#10624585	semidirect

H ≅ G/N

type

1. H(3,𝔽₃)×#66524

C₃

semidirect

2. #22582539

C₃

semidirect

3. #22582535

C₃

semidirect

4. #22582457

C₃

semidirect

5. #22582456

C₃

semidirect

6. #22582453

C₃

semidirect

7. #22582443

C₃

semidirect

8. #22582442

C₃

semidirect

9. C₃₇×#6859

C₃

semidirect

67. #6859

C₃₇⋊C₃

semidirect

91. C₃₇

#10624585

semidirect

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

subgroup	index
H(3,𝔽₃)×#66524	3
#22582539	3
#22582535	3
#22582457	3
#22582456	3
#22582453	3
#22582443	3
#22582442	3
C₃₇×#6859	3
#10624585	37

subgroup

index

C₃₇×#6859

Subgroups — #41005302

Conjugacy classes of subgroups

Complements

Maximal subgroups