Group #41007378 — subgroups

Subgroups — #41007378

Order 26973.

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

normal subgroups only

This group has 188 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	#22582633	C₃	1	2	yes	maximal, complemented
2	#22582538	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582524	C₃	1	1	yes	maximal, characteristic, complemented
4	#22582521	C₃	1	2	yes	maximal, complemented
5	#22582519	C₃	1	1	yes	maximal, characteristic, complemented
6	#22582516	C₃	1	2	yes	maximal, complemented
7	#22582514	C₃	1	1	yes	maximal, characteristic, complemented
8	#22582511	C₃	1	2	yes	maximal, complemented
9	C₃₇×#6908	C₃	1	1	yes	maximal, characteristic, complemented
10	#15688531	—	3	2	no	—
11	#15688521	C₃²	1	2	yes	—
12	#15688521	C₃²	1	2	yes	—
13	#15688519	C₃²	1	2	yes	—
14	#15688516	—	3	2	no	—
15	#15688513	—	3	2	no	—
16	#15688506	C₃²	1	2	yes	—
17	#15688506	C₃²	1	1	yes	characteristic
18	#15688505	—	3	1	no	—
19	C₃²⋊#66524	—	3	1	no	—
20	C₉×#66524	—	3	1	no	—
21	C₃₇×#737	—	3	1	no	—
22	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
23	C₃₇×#733	C₃²	1	1	yes	characteristic
24	C₃₇×#733	C₃²	1	1	yes	characteristic
25	C₃₇×#732	C₃²	1	1	yes	characteristic
26	C₃₇×#731	—	3	1	no	—
27	C₃₇×AΣL(1,𝔽₂₇)	—	3	1	no	—
28	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
29	3₋¹⁺²×C₃₇	—	3	1	no	—
30	3₋¹⁺²×C₃₇	—	3	1	no	—
31	3₋¹⁺²×C₃₇	—	9	2	no	—
32	H(3,𝔽₃)×C₃₇	—	3	1	no	—
33	C₃×C₉×C₃₇	—	3	2	no	—
34	C₃×C₉×C₃₇	—	3	1	no	—
35	C₃×C₉×C₃₇	—	9	1	no	—
36	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
37	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
38	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
39	C₃²⋊(C₃₇⋊C₃)	—	3	2	no	—
40	#11758608	—	3	2	no	—
41	#11758608	—	3	2	no	—
42	C₃×#66524	—	3	2	no	—
43	C₃×#66524	—	3	1	no	—
44	#11758606	—	9	2	no	—
45	#11758605	—	9	2	no	—
46	C₉×C₃₇⋊C₃	—	9	2	no	—
47	#10624845	—	37	1	no	maximal, Sylow
48	C₃²×C₃₇	—	3	1	no	—
49	C₃²×C₃₇	—	9	1	no	—
50	C₃²×C₃₇	—	9	1	no	—
51	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
52	C₃×C₃₇⋊C₃	—	9	2	no	—
53	C₉×C₃₇	—	3	1	no	—
54	C₉×C₃₇	—	3	1	no	—
55	C₉×C₃₇	—	9	1	no	—
56	C₉×C₃₇	—	9	1	no	—
57	C₉×C₃₇	#732	1	2	yes	—
58	C₉×C₃₇	#732	1	1	yes	characteristic
59	#66524	—	9	2	no	—
60	#66524	—	9	1	no	—
61	#66524	—	9	2	no	—
62	#66524	—	9	2	no	—
63	#6908	—	37	2	no	—
64	#6908	C₃₇⋊C₃	1	1	yes	characteristic, complemented
65	#6887	—	37	1	no	—
66	#6883	—	37	2	no	—
67	#6883	—	37	1	no	—
68	#6882	—	37	2	no	—
69	#6882	—	37	1	no	—
70	#6881	—	37	1	no	—
71	#6881	—	37	2	no	—
72	C₃×C₃₇	—	3	1	no	—
73	C₃×C₃₇	—	9	1	no	—
74	C₃×C₃₇	—	27	1	no	—
75	C₃×C₃₇	C₃×#732	1	1	yes	characteristic
76	C₃₇⋊C₃	—	27	2	no	—
77	#737	—	3	1	no	—
78	#737	—	111	2	no	—
79	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
80	#733	—	37	2	no	—
81	#733	—	37	2	no	—
82	#733	C₃×C₃₇⋊C₃	1	1	yes	characteristic
83	#733	C₃×C₃₇⋊C₃	1	1	yes	characteristic
84	#732	—	37	2	no	—
85	#732	C₃×C₃₇⋊C₃	1	1	yes	characteristic
86	#731	—	3	1	no	—
87	#731	—	111	2	no	—
88	AΣL(1,𝔽₂₇)	—	3	1	no	—
89	AΣL(1,𝔽₂₇)	—	111	2	no	—
90	C₉⋊C₉	—	37	2	no	—
91	C₉⋊C₉	—	37	1	no	—
92	C₉⋊C₉	—	111	1	no	—
93	C₃²⋊C₉	—	111	1	no	—
94	C₉²	—	111	1	no	—
95	C₃₇	#10624845	1	1	yes	characteristic, complemented, Sylow
96	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
97	3₋¹⁺²	—	3	1	no	—
98	3₋¹⁺²	—	3	1	no	—
99	3₋¹⁺²	—	9	2	no	—
100	3₋¹⁺²	—	333	2	no	—
101	3₋¹⁺²	—	333	2	no	—
102	3₋¹⁺²	—	111	2	no	—
103	3₋¹⁺²	—	111	2	no	—
104	H(3,𝔽₃)	—	3	1	no	—
105	H(3,𝔽₃)	—	111	2	no	—
106	C₃×C₉	—	3	2	no	—
107	C₃×C₉	—	3	1	no	—
108	C₃×C₉	—	9	1	no	—
109	C₃×C₉	—	333	2	no	—
110	C₃×C₉	—	111	2	no	—
111	C₃×C₉	—	111	1	no	—
112	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
113	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
114	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
115	C₃²	—	3	1	no	—
116	C₃²	—	9	1	no	—
117	C₃²	—	9	1	no	—
118	C₃²	—	333	2	no	—
119	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
120	C₉	—	3	1	no	—
121	C₉	—	3	1	no	—
122	C₉	—	9	1	no	—
123	C₉	—	9	1	no	—
124	C₉	—	333	2	no	—
125	C₉	—	333	1	no	—
126	C₉	—	333	2	no	—
127	C₉	—	333	2	no	—
128	C₉	#15688518	1	2	yes	—
129	C₉	#15688518	1	1	yes	characteristic
130	C₃	—	3	1	no	—
131	C₃	—	9	1	no	—
132	C₃	—	27	1	no	—
133	C₃	—	999	2	no	—
134	C₃	C₃×#15688518	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22582633

C₃

yes

maximal, complemented

#22582538

C₃

yes

maximal, characteristic, complemented

#22582524

C₃

yes

maximal, characteristic, complemented

#22582521

C₃

yes

maximal, complemented

#22582519

C₃

yes

maximal, characteristic, complemented

#22582516

C₃

yes

maximal, complemented

#22582514

C₃

yes

maximal, characteristic, complemented

#22582511

C₃

yes

maximal, complemented

C₃₇×#6908

C₃

yes

maximal, characteristic, complemented

#15688531

—

#15688521

C₃²

yes

—

#15688521

C₃²

yes

—

#15688519

C₃²

yes

—

#15688516

—

#15688513

—

#15688506

C₃²

yes

—

#15688506

C₃²

yes

—

—

—

C₃₇×#737

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×#733

C₃²

yes

characteristic

C₃₇×#733

C₃²

yes

characteristic

C₃₇×#732

C₃²

yes

characteristic

C₃₇×#731

—

C₃₇×AΣL(1,𝔽₂₇)

—

C₃³×C₃₇

H(3,𝔽₃)

yes

characteristic

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

H(3,𝔽₃)×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

yes

C₃×C₉×C₃₇

yes

C₃×C₉×C₃₇

C₃³

yes

characteristic

C₃²⋊(C₃₇⋊C₃)

—

—

—

—

—

—

—

C₉×C₃₇⋊C₃

—

#10624845

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

#732

yes

—

C₉×C₃₇

yes

—

—

—

—

—

C₃₇⋊C₃

yes

characteristic, complemented

—

—

—

—

—

—

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃×#732

yes

characteristic

C₃₇⋊C₃

—

#737

—

#737

—

111

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

—

—

C₃×C₃₇⋊C₃

yes

characteristic

#733

C₃×C₃₇⋊C₃

yes

characteristic

#732

—

#732

C₃×C₃₇⋊C₃

yes

characteristic

#731

—

#731

—

111

—

AΣL(1,𝔽₂₇)

—

AΣL(1,𝔽₂₇)

—

111

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₉²

—

111

—

C₃₇

#10624845

yes

characteristic, complemented, Sylow

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

100

3₋¹⁺²

—

333

—

101

3₋¹⁺²

—

333

—

102

3₋¹⁺²

—

111

—

103

3₋¹⁺²

—

111

—

104

H(3,𝔽₃)

—

105

H(3,𝔽₃)

—

111

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

—

333

—

110

C₃×C₉

—

111

—

111

C₃×C₉

—

111

—

112

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

113

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

114

C₃×C₉

C₃²×C₃₇⋊C₃

yes

characteristic

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

—

333

—

119

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

120

C₉

—

121

C₉

—

122

C₉

—

123

C₉

—

124

C₉

—

333

—

125

C₉

—

333

—

126

C₉

—

333

—

127

C₉

—

333

—

128

C₉

#15688518

yes

—

129

C₉

#15688518

yes

characteristic

130

C₃

—

131

C₃

—

132

C₃

—

133

C₃

—

999

—

134

C₃

C₃×#15688518

yes

characteristic

	subgroup	G/H
center	C₉	#15688518
commutator, cosocle	C₃×C₉×C₃₇	C₃³
Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6908	C₃
socle	C₃×C₃₇	C₃×#732
hypercenter	#6908	C₃₇⋊C₃
hypocenter	C₃₇	#10624845
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₉ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ #6908

subgroup

G/H

center

C₉

#15688518

commutator, cosocle

C₃×C₉×C₃₇

C₃³

Frattini

C₃×C₉

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6908

C₃

C₃×C₃₇

C₃₇⋊C₃

C₃₇

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

UCS

C₁ ◁ C₉ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ #6908

N	H ≅ G/N	type
1. #22582633	C₃	semidirect
2. #22582538	C₃	semidirect
3. #22582524	C₃	semidirect
4. #22582521	C₃	semidirect
5. #22582519	C₃	semidirect
6. #22582516	C₃	semidirect
7. #22582514	C₃	semidirect
8. #22582511	C₃	semidirect
9. C₃₇×#6908	C₃	semidirect
64. #6908	C₃₇⋊C₃	semidirect
95. C₃₇	#10624845	semidirect

H ≅ G/N

type

1. #22582633

C₃

semidirect

2. #22582538

C₃

semidirect

3. #22582524

C₃

semidirect

4. #22582521

C₃

semidirect

5. #22582519

C₃

semidirect

6. #22582516

C₃

semidirect

7. #22582514

C₃

semidirect

8. #22582511

C₃

semidirect

9. C₃₇×#6908

C₃

semidirect

64. #6908

C₃₇⋊C₃

semidirect

95. C₃₇

#10624845

semidirect

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

subgroup	index
#22582633	3
#22582538	3
#22582524	3
#22582521	3
#22582519	3
#22582516	3
#22582514	3
#22582511	3
C₃₇×#6908	3
#10624845	37

subgroup

index

C₃₇×#6908

Subgroups — #41007378

Conjugacy classes of subgroups

Complements

Maximal subgroups