Group #41006174 — subgroups

Subgroups — #41006174

Order 26973.

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

normal subgroups only

This group has 198 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	#22582571	C₃	1	2	yes	maximal, complemented
2	#22582559	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582454	C₃	1	2	yes	maximal, complemented
4	#22582451	C₃	1	1	yes	maximal, characteristic, complemented
5	#22582447	C₃	1	2	yes	maximal, complemented
6	#22582445	C₃	1	1	yes	maximal, characteristic, complemented
7	#22582444	C₃	1	2	yes	maximal, complemented
8	#22582442	C₃	1	1	yes	maximal, characteristic
9	C₃₇×#6893	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃×#11758608	C₃²	1	2	yes	—
11	C₃×#11758606	—	3	1	no	—
12	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	1	yes	characteristic
13	#15688505	—	3	2	no	—
14	#15688504	—	3	2	no	—
15	#15688504	—	3	2	no	—
16	C₃²⋊#66524	C₃²	1	1	yes	characteristic
17	C₃²⋊#66524	C₃²	1	1	yes	characteristic
18	C₃²⋊#66524	C₃²	1	2	yes	—
19	C₃²⋊#66524	C₃²	1	2	yes	—
20	#15688502	—	3	2	no	—
21	C₃×3₋¹⁺²×C₃₇	C₃²	1	2	yes	—
22	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
23	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
24	C₃₇×C₃²⋊C₉	—	3	2	no	—
25	C₃₇×C₃²⋊C₉	—	3	1	no	—
26	C₃³×C₃₇	—	3	1	no	—
27	C₃³×C₃₇	C₃³	1	1	yes	characteristic
28	C₃²×C₃₇⋊C₃	—	3	1	no	—
29	3₋¹⁺²×C₃₇	—	9	2	no	—
30	H(3,𝔽₃)×C₃₇	—	9	1	no	—
31	C₃×C₉×C₃₇	—	3	2	no	—
32	C₃×C₉×C₃₇	—	9	1	no	—
33	C₃×C₉×C₃₇	H(3,𝔽₃)	1	2	yes	—
34	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
35	C₃²⋊(C₃₇⋊C₃)	—	9	1	no	—
36	#11758608	—	9	2	no	—
37	C₃×#66524	—	3	1	no	—
38	C₃×#66524	—	3	1	no	—
39	C₃×#66524	—	3	2	no	—
40	C₃×#66524	—	3	2	no	—
41	C₃×#66524	—	3	2	no	—
42	#11758606	—	9	3	no	—
43	#10624677	—	37	1	no	maximal, Sylow
44	C₃²×C₃₇	—	3	2	no	—
45	C₃²×C₃₇	—	3	1	no	—
46	C₃²×C₃₇	—	3	1	no	—
47	C₃²×C₃₇	—	9	2	no	—
48	C₃²×C₃₇	—	9	1	no	—
49	C₃²×C₃₇	—	9	1	no	—
50	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
51	C₃×C₃₇⋊C₃	—	9	1	no	—
52	C₃×C₃₇⋊C₃	—	9	2	no	—
53	C₃×C₃₇⋊C₃	—	9	1	no	—
54	C₉×C₃₇	—	3	1	no	—
55	C₉×C₃₇	—	3	2	no	—
56	C₉×C₃₇	—	9	2	no	—
57	#66524	—	9	2	no	—
58	#66524	—	9	2	no	—
59	#66524	—	9	2	no	—
60	#66524	—	9	1	no	—
61	#66524	—	9	1	no	—
62	#6894	—	37	2	no	—
63	#6893	C₃₇⋊C₃	1	1	yes	characteristic, complemented
64	C₉⋊H(3,𝔽₃)	—	37	1	no	—
65	#6860	—	37	2	no	—
66	#6859	—	37	1	no	—
67	#6858	—	37	2	no	—
68	#6857	—	37	2	no	—
69	#6857	—	37	1	no	—
70	#6856	—	37	1	no	—
71	C₃×C₃₇	—	9	1	no	—
72	C₃×C₃₇	—	27	1	no	—
73	C₃×C₃₇	#6909	1	1	yes	characteristic
74	C₃×C₃₇	#6913	1	2	yes	—
75	C₃×C₃₇	#6913	1	1	yes	characteristic
76	C₃₇⋊C₃	—	27	1	no	—
77	C₃×3₋¹⁺²	—	37	2	no	—
78	C₃×3₋¹⁺²	—	111	1	no	—
79	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	2	yes	—
80	C₃×H(3,𝔽₃)	—	37	1	no	—
81	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
82	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
83	C₉⋊C₉	—	111	2	no	—
84	C₉⋊C₉	—	111	2	no	—
85	C₉⋊C₉	—	111	2	no	—
86	C₃²⋊C₉	—	3	2	no	—
87	C₃²⋊C₉	—	3	1	no	—
88	C₃²⋊C₉	—	37	2	no	—
89	C₃²⋊C₉	—	37	1	no	—
90	C₃²⋊C₉	—	37	2	no	—
91	C₃²⋊C₉	—	37	1	no	—
92	C₃²⋊C₉	—	111	2	no	—
93	C₃₇	#10624677	1	1	yes	characteristic, complemented, Sylow
94	C₃³	—	3	1	no	—
95	C₃³	—	111	1	no	—
96	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
97	3₋¹⁺²	—	9	2	no	—
98	3₋¹⁺²	—	333	3	no	—
99	3₋¹⁺²	—	333	2	no	—
100	H(3,𝔽₃)	—	9	1	no	—
101	H(3,𝔽₃)	—	333	1	no	—
102	C₃×C₉	—	3	2	no	—
103	C₃×C₉	—	9	1	no	—
104	C₃×C₉	—	111	1	no	—
105	C₃×C₉	—	111	1	no	—
106	C₃×C₉	—	111	2	no	—
107	C₃×C₉	—	111	2	no	—
108	C₃×C₉	—	111	2	no	—
109	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
110	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
111	C₃²	—	3	1	no	—
112	C₃²	—	3	2	no	—
113	C₃²	—	3	1	no	—
114	C₃²	—	9	1	no	—
115	C₃²	—	9	2	no	—
116	C₃²	—	9	1	no	—
117	C₃²	—	333	1	no	—
118	C₃²	—	333	2	no	—
119	C₃²	—	333	1	no	—
120	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
121	C₉	—	3	1	no	—
122	C₉	—	3	2	no	—
123	C₉	—	9	2	no	—
124	C₉	—	333	1	no	—
125	C₉	—	333	1	no	—
126	C₉	—	333	2	no	—
127	C₉	—	333	2	no	—
128	C₉	—	333	2	no	—
129	C₃	—	9	1	no	—
130	C₃	—	27	1	no	—
131	C₃	—	999	1	no	—
132	C₃	#22582638	1	1	yes	characteristic
133	C₃	#22582655	1	2	yes	—
134	C₃	#22582656	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22582571

C₃

yes

maximal, complemented

#22582559

C₃

yes

maximal, characteristic, complemented

#22582454

C₃

yes

maximal, complemented

#22582451

C₃

yes

maximal, characteristic, complemented

#22582447

C₃

yes

maximal, complemented

#22582445

C₃

yes

maximal, characteristic, complemented

#22582444

C₃

yes

maximal, complemented

#22582442

C₃

yes

maximal, characteristic

C₃₇×#6893

C₃

yes

maximal, characteristic, complemented

C₃×#11758608

C₃²

yes

—

C₃×#11758606

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

—

—

—

C₃²

yes

characteristic

C₃²⋊#66524

C₃²

yes

characteristic

C₃²⋊#66524

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

#15688502

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

characteristic

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×C₃²⋊C₉

—

C₃₇×C₃²⋊C₉

—

C₃³×C₃₇

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

3₋¹⁺²×C₃₇

—

H(3,𝔽₃)×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

—

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

characteristic

C₃²⋊(C₃₇⋊C₃)

—

—

—

—

—

—

—

—

—

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

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

—

—

—

—

—

—

C₃₇⋊C₃

yes

characteristic, complemented

C₉⋊H(3,𝔽₃)

—

—

—

—

—

—

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

#6909

yes

characteristic

C₃×C₃₇

#6913

yes

—

C₃×C₃₇

#6913

yes

characteristic

C₃₇⋊C₃

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×C₃₇⋊C₃

yes

characteristic

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃₇

#10624677

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

111

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

333

—

3₋¹⁺²

—

333

—

100

H(3,𝔽₃)

—

101

H(3,𝔽₃)

—

333

—

102

C₃×C₉

—

103

C₃×C₉

—

104

C₃×C₉

—

111

—

105

C₃×C₉

—

111

—

106

C₃×C₉

—

111

—

107

C₃×C₉

—

111

—

108

C₃×C₉

—

111

—

109

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

—

110

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

333

—

118

C₃²

—

333

—

119

C₃²

—

333

—

120

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

121

C₉

—

122

C₉

—

123

C₉

—

124

C₉

—

333

—

125

C₉

—

333

—

126

C₉

—

333

—

127

C₉

—

333

—

128

C₉

—

333

—

129

C₃

—

130

C₃

—

131

C₃

—

999

—

132

C₃

#22582638

yes

characteristic

133

C₃

#22582655

yes

—

134

C₃

#22582656

yes

characteristic

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6893	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	#22582442	C₃
hypercenter	#6893	C₃₇⋊C₃
hypocenter	C₃₇	#10624677
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²×C₉ ◁ #6893

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6893

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

C₃

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃²×C₉ ◁ #6893

N	H ≅ G/N	type
1. #22582571	C₃	semidirect
2. #22582559	C₃	semidirect
3. #22582454	C₃	semidirect
4. #22582451	C₃	semidirect
5. #22582447	C₃	semidirect
6. #22582445	C₃	semidirect
7. #22582444	C₃	semidirect
9. C₃₇×#6893	C₃	semidirect
63. #6893	C₃₇⋊C₃	semidirect
93. C₃₇	#10624677	semidirect

H ≅ G/N

type

1. #22582571

C₃

semidirect

2. #22582559

C₃

semidirect

3. #22582454

C₃

semidirect

4. #22582451

C₃

semidirect

5. #22582447

C₃

semidirect

6. #22582445

C₃

semidirect

7. #22582444

C₃

semidirect

9. C₃₇×#6893

C₃

semidirect

63. #6893

C₃₇⋊C₃

semidirect

93. C₃₇

#10624677

semidirect

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

subgroup	index
#22582571	3
#22582559	3
#22582454	3
#22582451	3
#22582447	3
#22582445	3
#22582444	3
#22582442	3
C₃₇×#6893	3
#10624677	37

subgroup

index

C₃₇×#6893

Subgroups — #41006174

Conjugacy classes of subgroups

Complements

Maximal subgroups