Group #41008592 — subgroups

Subgroups — #41008592

Order 26973.

224 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 66 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	#22582732	C₃	1	2	yes	maximal, complemented
2	#22582731	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃²×#11758603	C₃	1	1	yes	maximal, characteristic, complemented
4	#22582702	C₃	1	2	yes	maximal, complemented
5	#22582701	C₃	1	1	yes	maximal, characteristic, complemented
6	#22582700	C₃	1	2	yes	maximal, complemented
7	#22582699	C₃	1	1	yes	maximal, characteristic, complemented
8	#22582694	C₃	1	2	yes	maximal, complemented
9	#22582629	C₃	1	1	yes	maximal, characteristic
10	#15688543	—	3	2	no	—
11	#15688542	—	3	2	no	—
12	#15688541	—	3	2	no	—
13	#15688541	C₃²	1	2	yes	—
14	#15688541	C₃²	1	2	yes	—
15	#15688541	C₃²	1	2	yes	—
16	(C₃₇⋊C₃)⋊C₂₇	—	3	2	no	—
17	C₃×#11758603	—	3	1	no	—
18	C₃×#11758603	—	3	2	no	—
19	C₃×#11758603	—	3	1	no	—
20	C₃×#11758603	C₃²	1	1	yes	characteristic
21	C₃×#11758603	C₃²	1	1	yes	characteristic
22	C₃×#11758603	C₃²	1	1	yes	characteristic
23	#15688532	C₃²	1	2	yes	—
24	#15688530	C₃²	1	1	yes	characteristic
25	C₃²×#66524	C₃²	1	1	yes	characteristic
26	#15688520	C₉	1	2	yes	—
27	#15688518	C₉	1	1	yes	characteristic
28	#15688517	C₉	1	2	yes	—
29	#15688515	C₉	1	1	yes	characteristic
30	#15688513	C₉	1	2	yes	—
31	C₃₇×AΣL(1,𝔽₂₇)	C₉	1	1	yes	characteristic
32	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
33	3₋¹⁺²×C₃₇	C₃×C₉	1	2	yes	—
34	H(3,𝔽₃)×C₃₇	C₃×C₉	1	1	yes	characteristic
35	C₃²⋊(C₃₇⋊C₃)	C₃×C₉	1	2	yes	—
36	#11758608	—	3	2	no	—
37	#11758608	C₃×C₉	1	2	yes	—
38	#11758608	C₃×C₉	1	2	yes	—
39	C₃×#66524	—	3	1	no	—
40	C₃×#66524	—	3	1	no	—
41	C₃×#66524	—	3	2	no	—
42	C₃×#66524	C₃×C₉	1	1	yes	characteristic
43	C₃×#66524	C₃×C₉	1	1	yes	characteristic
44	C₃×#66524	C₃³	1	1	yes	characteristic
45	#11758606	—	3	2	no	—
46	#11758605	—	3	2	no	—
47	C₉×C₃₇⋊C₃	—	3	2	no	—
48	#11758603	—	3	1	no	—
49	#11758603	—	3	1	no	—
50	#11758603	—	3	2	no	—
51	#11758603	—	3	2	no	—
52	#11758603	—	3	2	no	—
53	#11758603	H(3,𝔽₃)	1	1	yes	characteristic
54	#11758603	H(3,𝔽₃)	1	2	yes	—
55	#10624724	—	37	1	no	maximal, Sylow
56	C₃²×C₃₇	—	3	3	no	—
57	C₃²×C₃₇	—	3	1	no	—
58	C₃²×C₃₇	—	3	1	no	—
59	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
60	C₃×C₃₇⋊C₃	—	3	2	no	—
61	C₉×C₃₇	—	3	2	no	—
62	#66524	—	3	2	no	—
63	#66524	—	3	1	no	—
64	#66524	—	3	1	no	—
65	#66524	—	3	2	no	—
66	#66524	C₃×H(3,𝔽₃)	1	1	yes	characteristic
67	#66524	#737	1	2	yes	—
68	#6908	—	37	1	no	—
69	#6903	—	37	1	no	—
70	#6903	—	37	2	no	—
71	C₃²×C₂₇	—	37	1	no	—
72	#6873	—	37	1	no	—
73	#6873	—	37	2	no	—
74	#6872	—	37	1	no	—
75	#6872	—	37	2	no	—
76	#6869	—	37	2	no	—
77	C₃×C₃₇	—	3	3	no	—
78	C₃×C₃₇	—	3	1	no	—
79	C₃×C₃₇	—	9	1	no	—
80	C₃×C₃₇	C₉×H(3,𝔽₃)	1	1	yes	characteristic
81	C₃₇⋊C₃	—	9	2	no	—
82	#737	—	37	2	no	—
83	#737	—	37	1	no	—
84	C₃²×C₉	—	37	1	no	—
85	#733	—	37	2	no	—
86	#732	—	37	1	no	—
87	#731	—	37	2	no	—
88	#731	—	37	1	no	—
89	AΣL(1,𝔽₂₇)	—	37	2	no	—
90	AΣL(1,𝔽₂₇)	#66526	1	1	yes	characteristic
91	C₂₇⋊C₃	—	37	2	no	—
92	C₂₇⋊C₃	—	37	2	no	—
93	C₂₇⋊C₃	—	37	2	no	—
94	C₂₇⋊C₃	—	111	2	no	—
95	C₂₇⋊C₃	—	111	2	no	—
96	C₂₇⋊C₃	—	111	2	no	—
97	C₃×C₂₇	—	37	1	no	—
98	C₃×C₂₇	—	37	1	no	—
99	C₃×C₂₇	—	37	1	no	—
100	C₃×C₂₇	—	111	1	no	—
101	C₃×C₂₇	—	111	2	no	—
102	C₃×C₂₇	—	111	1	no	—
103	C₃×C₂₇	—	111	2	no	—
104	C₃₇	#10624724	1	1	yes	characteristic, complemented, Sylow
105	C₃³	C₃×#66526	1	1	yes	characteristic
106	3₋¹⁺²	—	37	2	no	—
107	3₋¹⁺²	—	37	2	no	—
108	3₋¹⁺²	—	111	2	no	—
109	3₋¹⁺²	—	111	2	no	—
110	3₋¹⁺²	—	111	2	no	—
111	3₋¹⁺²	C₃×#66526	1	2	yes	—
112	H(3,𝔽₃)	—	37	2	no	—
113	H(3,𝔽₃)	C₃×#66526	1	1	yes	characteristic
114	C₃×C₉	—	37	1	no	—
115	C₃×C₉	—	37	1	no	—
116	C₃×C₉	—	37	1	no	—
117	C₃×C₉	—	111	2	no	—
118	C₃×C₉	—	111	1	no	—
119	C₃×C₉	—	111	1	no	—
120	C₃×C₉	—	111	2	no	—
121	C₂₇	—	37	1	no	—
122	C₂₇	—	37	2	no	—
123	C₂₇	—	111	1	no	—
124	C₂₇	—	111	1	no	—
125	C₂₇	—	111	2	no	—
126	C₂₇	—	111	2	no	—
127	C₂₇	—	111	2	no	—
128	C₃²	—	3	3	no	—
129	C₃²	—	3	1	no	—
130	C₃²	—	3	1	no	—
131	C₃²	—	111	2	no	—
132	C₃²	C₃²×#66526	1	1	yes	characteristic
133	C₉	—	3	2	no	—
134	C₉	—	37	1	no	—
135	C₉	—	37	2	no	—
136	C₉	—	111	1	no	—
137	C₉	—	111	1	no	—
138	C₉	—	111	2	no	—
139	C₉	—	111	2	no	—
140	C₃	—	3	1	no	—
141	C₃	—	3	3	no	—
142	C₃	—	9	1	no	—
143	C₃	—	333	2	no	—
144	C₃	H(3,𝔽₃)×#66526	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22582732

C₃

yes

maximal, complemented

#22582731

C₃

yes

maximal, characteristic, complemented

C₃²×#11758603

C₃

yes

maximal, characteristic, complemented

#22582702

C₃

yes

maximal, complemented

#22582701

C₃

yes

maximal, characteristic, complemented

#22582700

C₃

yes

maximal, complemented

#22582699

C₃

yes

maximal, characteristic, complemented

#22582694

C₃

yes

maximal, complemented

#22582629

C₃

yes

maximal, characteristic

—

—

—

C₃²

yes

—

#15688541

C₃²

yes

—

#15688541

C₃²

yes

—

(C₃₇⋊C₃)⋊C₂₇

—

—

—

—

C₃²

yes

characteristic

C₃×#11758603

C₃²

yes

characteristic

C₃×#11758603

C₃²

yes

characteristic

#15688532

C₃²

yes

—

#15688530

C₃²

yes

characteristic

C₃²×#66524

C₃²

yes

characteristic

#15688520

C₉

yes

—

#15688518

C₉

yes

characteristic

#15688517

C₉

yes

—

#15688515

C₉

yes

characteristic

#15688513

C₉

yes

—

C₃₇×AΣL(1,𝔽₂₇)

C₉

yes

characteristic

C₃³×C₃₇

C₃×C₉

yes

characteristic

3₋¹⁺²×C₃₇

C₃×C₉

yes

—

H(3,𝔽₃)×C₃₇

C₃×C₉

yes

characteristic

C₃²⋊(C₃₇⋊C₃)

C₃×C₉

yes

—

#11758608

—

#11758608

C₃×C₉

yes

—

#11758608

C₃×C₉

yes

—

—

—

—

C₃×C₉

yes

characteristic

C₃×#66524

C₃×C₉

yes

characteristic

C₃×#66524

C₃³

yes

characteristic

#11758606

—

#11758605

—

C₉×C₃₇⋊C₃

—

—

—

—

—

—

yes

yes

—

#10624724

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃²×C₉

yes

characteristic

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

—

—

—

—

C₃×H(3,𝔽₃)

yes

characteristic

#66524

#737

yes

—

#6908

—

#6903

—

#6903

—

C₃²×C₂₇

—

—

—

—

—

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₉×H(3,𝔽₃)

yes

characteristic

C₃₇⋊C₃

—

#737

—

#737

—

C₃²×C₉

—

—

—

—

—

AΣL(1,𝔽₂₇)

—

AΣL(1,𝔽₂₇)

#66526

yes

characteristic

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

—

111

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

100

C₃×C₂₇

—

111

—

101

C₃×C₂₇

—

111

—

102

C₃×C₂₇

—

111

—

103

C₃×C₂₇

—

111

—

104

C₃₇

#10624724

yes

characteristic, complemented, Sylow

105

C₃³

C₃×#66526

yes

characteristic

106

3₋¹⁺²

—

107

3₋¹⁺²

—

108

3₋¹⁺²

—

111

—

109

3₋¹⁺²

—

111

—

110

3₋¹⁺²

—

111

—

111

3₋¹⁺²

C₃×#66526

yes

—

112

H(3,𝔽₃)

—

113

H(3,𝔽₃)

C₃×#66526

yes

characteristic

114

C₃×C₉

—

115

C₃×C₉

—

116

C₃×C₉

—

117

C₃×C₉

—

111

—

118

C₃×C₉

—

111

—

119

C₃×C₉

—

111

—

120

C₃×C₉

—

111

—

121

C₂₇

—

122

C₂₇

—

123

C₂₇

—

111

—

124

C₂₇

—

111

—

125

C₂₇

—

111

—

126

C₂₇

—

111

—

127

C₂₇

—

111

—

128

C₃²

—

129

C₃²

—

130

C₃²

—

131

C₃²

—

111

—

132

C₃²

C₃²×#66526

yes

characteristic

133

C₉

—

134

C₉

—

135

C₉

—

136

C₉

—

111

—

137

C₉

—

111

—

138

C₉

—

111

—

139

C₉

—

111

—

140

C₃

—

141

C₃

—

142

C₃

—

143

C₃

—

333

—

144

C₃

H(3,𝔽₃)×#66526

yes

characteristic

	subgroup	G/H
center	C₃	H(3,𝔽₃)×#66526
commutator	C₃²×C₃₇	C₃²×C₉
Frattini	C₃²	C₃²×#66526
Fitting	C₃₇×AΣL(1,𝔽₂₇)	C₉
socle	C₃×C₃₇	C₉×H(3,𝔽₃)
prime radical	#22582629	C₃
cosocle	C₃×#66524	C₃³
hypercenter	AΣL(1,𝔽₂₇)	#66526
hypocenter	C₃₇	#10624724
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ AΣL(1,𝔽₂₇)

subgroup

G/H

center

C₃

H(3,𝔽₃)×#66526

commutator

C₃²×C₃₇

C₃²×C₉

Frattini

C₃²

C₃²×#66526

Fitting

C₃₇×AΣL(1,𝔽₂₇)

C₉

socle

C₃×C₃₇

C₉×H(3,𝔽₃)

C₃

C₃³

AΣL(1,𝔽₂₇)

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃ ◁ C₃² ◁ AΣL(1,𝔽₂₇)

N	H ≅ G/N	type
1. #22582732	C₃	semidirect
2. #22582731	C₃	semidirect
3. C₃²×#11758603	C₃	semidirect
4. #22582702	C₃	semidirect
5. #22582701	C₃	semidirect
6. #22582700	C₃	semidirect
7. #22582699	C₃	semidirect
8. #22582694	C₃	semidirect
104. C₃₇	#10624724	semidirect

H ≅ G/N

type

1. #22582732

C₃

semidirect

2. #22582731

C₃

semidirect

3. C₃²×#11758603

C₃

semidirect

4. #22582702

C₃

semidirect

5. #22582701

C₃

semidirect

6. #22582700

C₃

semidirect

7. #22582699

C₃

semidirect

8. #22582694

C₃

semidirect

104. C₃₇

#10624724

semidirect

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

subgroup	index
#22582732	3
#22582731	3
C₃²×#11758603	3
#22582702	3
#22582701	3
#22582700	3
#22582699	3
#22582694	3
#22582629	3
#10624724	37

subgroup

index

Subgroups — #41008592

Conjugacy classes of subgroups

Complements

Maximal subgroups