Group #41007439 — subgroups

Subgroups — #41007439

Order 26973.

188 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 43 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	#22582662	C₃	1	2	yes	maximal, complemented
2	#22582649	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582605	C₃	1	1	yes	maximal, characteristic, complemented
4	#22582523	C₃	1	2	yes	maximal, complemented
5	#22582519	C₃	1	2	yes	maximal, complemented
6	#22582514	C₃	1	1	yes	maximal, characteristic, complemented
7	#22582509	C₃	1	1	yes	maximal, characteristic, complemented
8	#22582505	C₃	1	2	yes	maximal, complemented
9	C₃₇×#6882	C₃	1	1	yes	maximal, characteristic, complemented
10	#15688532	—	3	2	no	—
11	C₃×#11758605	—	3	1	no	—
12	C₃×#11758605	C₃²	1	1	yes	characteristic
13	3₋¹⁺²×C₃₇⋊C₃	—	3	1	no	—
14	#15688521	C₃²	1	2	yes	—
15	#15688521	C₃²	1	2	yes	—
16	#15688520	—	3	2	no	—
17	#15688519	C₃²	1	2	yes	—
18	#15688517	—	3	2	no	—
19	#15688514	—	3	2	no	—
20	#15688506	C₃²	1	1	yes	characteristic
21	#15688504	—	3	2	no	—
22	C₉×#66524	C₃²	1	1	yes	characteristic
23	C₃₇×#733	C₃²	1	1	yes	characteristic
24	C₃₇×#733	C₃²	1	2	yes	—
25	C₃₇×C₉⋊C₉	C₃²	1	1	yes	characteristic
26	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
27	3₋¹⁺²×C₃₇	—	3	2	no	—
28	3₋¹⁺²×C₃₇	—	3	1	no	—
29	C₃×C₉×C₃₇	—	3	1	no	—
30	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
31	C₃²⋊(C₃₇⋊C₃)	—	3	2	no	—
32	#11758608	—	3	2	no	—
33	#11758608	—	3	2	no	—
34	C₃×#66524	—	3	1	no	—
35	C₃×#66524	—	3	1	no	—
36	C₃×#66524	H(3,𝔽₃)	1	2	yes	—
37	#11758606	—	9	2	no	—
38	#11758606	—	9	1	no	—
39	#11758605	—	9	1	no	—
40	#11758605	—	9	3	no	—
41	#11758605	—	9	2	no	—
42	#11758605	—	9	1	no	—
43	C₉×C₃₇⋊C₃	—	9	2	no	—
44	C₉×C₃₇⋊C₃	—	9	1	no	—
45	#10624852	—	37	1	no	maximal, Sylow
46	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
47	C₃×C₃₇⋊C₃	—	3	1	no	—
48	C₃×C₃₇⋊C₃	—	9	1	no	—
49	C₃×C₃₇⋊C₃	—	9	2	no	—
50	C₉×C₃₇	—	3	1	no	—
51	C₉×C₃₇	—	9	1	no	—
52	C₉×C₃₇	—	9	2	no	—
53	C₉×C₃₇	—	9	1	no	—
54	#66524	—	3	3	no	—
55	#66524	—	3	2	no	—
56	#66524	—	9	1	no	—
57	#66524	—	9	2	no	—
58	#66524	—	9	2	no	—
59	#6913	—	37	2	no	—
60	#6912	—	37	1	no	—
61	#6900	—	37	1	no	—
62	#6883	—	37	2	no	—
63	#6882	—	37	2	no	—
64	#6882	C₃₇⋊C₃	1	1	yes	characteristic, complemented
65	#6881	—	37	1	no	—
66	#6880	—	37	1	no	—
67	#6878	—	37	2	no	—
68	C₃×C₃₇	—	3	1	no	—
69	C₃×C₃₇	#6911	1	1	yes	characteristic
70	C₃₇⋊C₃	—	9	1	no	—
71	C₃₇⋊C₃	—	27	2	no	—
72	#737	—	111	2	no	—
73	C₃×3₋¹⁺²	—	37	1	no	—
74	C₃×3₋¹⁺²	—	111	1	no	—
75	C₃×3₋¹⁺²	—	111	1	no	—
76	#733	—	37	2	no	—
77	#733	—	37	2	no	—
78	#733	—	111	2	no	—
79	#733	C₃×C₃₇⋊C₃	1	1	yes	characteristic
80	#733	C₃×C₃₇⋊C₃	1	2	yes	—
81	#732	—	37	2	no	—
82	#731	—	111	2	no	—
83	AΣL(1,𝔽₂₇)	—	111	2	no	—
84	C₉⋊C₉	—	37	1	no	—
85	C₉⋊C₉	—	111	2	no	—
86	C₉⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
87	C₉²	—	37	1	no	—
88	C₃₇	#10624852	1	1	yes	characteristic, complemented, Sylow
89	C₃³	—	37	1	no	—
90	3₋¹⁺²	—	3	2	no	—
91	3₋¹⁺²	—	3	1	no	—
92	3₋¹⁺²	—	333	1	no	—
93	3₋¹⁺²	—	333	1	no	—
94	3₋¹⁺²	—	333	1	no	—
95	3₋¹⁺²	—	333	3	no	—
96	3₋¹⁺²	—	333	2	no	—
97	3₋¹⁺²	—	333	2	no	—
98	3₋¹⁺²	—	111	2	no	—
99	3₋¹⁺²	—	111	2	no	—
100	H(3,𝔽₃)	—	111	2	no	—
101	C₃×C₉	—	3	1	no	—
102	C₃×C₉	—	37	2	no	—
103	C₃×C₉	—	333	1	no	—
104	C₃×C₉	—	333	2	no	—
105	C₃×C₉	—	111	1	no	—
106	C₃×C₉	—	111	1	no	—
107	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
108	C₃²	—	333	1	no	—
109	C₃²	—	333	2	no	—
110	C₃²	—	111	1	no	—
111	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
112	C₉	—	3	1	no	—
113	C₉	—	9	1	no	—
114	C₉	—	9	1	no	—
115	C₉	—	9	2	no	—
116	C₉	—	333	1	no	—
117	C₉	—	333	2	no	—
118	C₉	—	333	2	no	—
119	C₉	—	111	3	no	—
120	C₉	—	111	2	no	—
121	C₃	—	3	1	no	—
122	C₃	—	333	1	no	—
123	C₃	—	999	2	no	—
124	C₃	#22582645	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22582662

C₃

yes

maximal, complemented

#22582649

C₃

yes

maximal, characteristic, complemented

#22582605

C₃

yes

maximal, characteristic, complemented

#22582523

C₃

yes

maximal, complemented

#22582519

C₃

yes

maximal, complemented

#22582514

C₃

yes

maximal, characteristic, complemented

#22582509

C₃

yes

maximal, characteristic, complemented

#22582505

C₃

yes

maximal, complemented

C₃₇×#6882

C₃

yes

maximal, characteristic, complemented

#15688532

—

C₃×#11758605

—

C₃×#11758605

C₃²

yes

characteristic

3₋¹⁺²×C₃₇⋊C₃

—

#15688521

C₃²

yes

—

#15688521

C₃²

yes

—

#15688520

—

#15688519

C₃²

yes

—

#15688517

—

#15688514

—

#15688506

C₃²

yes

characteristic

#15688504

—

C₉×#66524

C₃²

yes

characteristic

C₃₇×#733

C₃²

yes

characteristic

C₃₇×#733

C₃²

yes

—

C₃₇×C₉⋊C₉

C₃²

yes

characteristic

C₃²×C₃₇⋊C₃

H(3,𝔽₃)

yes

characteristic

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃³

yes

characteristic

C₃²⋊(C₃₇⋊C₃)

—

—

—

—

—

yes

—

—

—

—

—

—

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

#10624852

—

maximal, Sylow

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

—

—

—

—

—

—

—

—

—

—

C₃₇⋊C₃

yes

characteristic, complemented

#6881

—

#6880

—

#6878

—

C₃×C₃₇

—

C₃×C₃₇

#6911

yes

characteristic

C₃₇⋊C₃

—

C₃₇⋊C₃

—

#737

—

111

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

#733

—

#733

—

#733

—

111

—

#733

C₃×C₃₇⋊C₃

yes

characteristic

#733

C₃×C₃₇⋊C₃

yes

—

#732

—

#731

—

111

—

AΣL(1,𝔽₂₇)

—

111

—

C₉⋊C₉

—

C₉⋊C₉

—

111

—

C₉⋊C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₉²

—

C₃₇

#10624852

yes

characteristic, complemented, Sylow

C₃³

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

333

—

3₋¹⁺²

—

333

—

3₋¹⁺²

—

333

—

3₋¹⁺²

—

333

—

3₋¹⁺²

—

333

—

3₋¹⁺²

—

333

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

100

H(3,𝔽₃)

—

111

—

101

C₃×C₉

—

102

C₃×C₉

—

103

C₃×C₉

—

333

—

104

C₃×C₉

—

333

—

105

C₃×C₉

—

111

—

106

C₃×C₉

—

111

—

107

C₃×C₉

C₃²×C₃₇⋊C₃

yes

characteristic

108

C₃²

—

333

—

109

C₃²

—

333

—

110

C₃²

—

111

—

111

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

112

C₉

—

113

C₉

—

114

C₉

—

115

C₉

—

116

C₉

—

333

—

117

C₉

—

333

—

118

C₉

—

333

—

119

C₉

—

111

—

120

C₉

—

111

—

121

C₃

—

122

C₃

—

333

—

123

C₃

—

999

—

124

C₃

#22582645

yes

characteristic

	subgroup	G/H
center	C₃	#22582645
commutator, cosocle	C₃×C₉×C₃₇	C₃³
Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6882	C₃
socle	C₃×C₃₇	#6911
hypercenter	#6882	C₃₇⋊C₃
hypocenter	C₃₇	#10624852
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉ ◁ #6882

subgroup

G/H

center

C₃

#22582645

commutator, cosocle

C₃×C₉×C₃₇

C₃³

Frattini

C₃×C₉

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6882

C₃

C₃×C₃₇

C₃₇⋊C₃

C₃₇

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

UCS

C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉ ◁ #6882

N	H ≅ G/N	type
1. #22582662	C₃	semidirect
2. #22582649	C₃	semidirect
3. #22582605	C₃	semidirect
4. #22582523	C₃	semidirect
5. #22582519	C₃	semidirect
6. #22582514	C₃	semidirect
7. #22582509	C₃	semidirect
8. #22582505	C₃	semidirect
9. C₃₇×#6882	C₃	semidirect
64. #6882	C₃₇⋊C₃	semidirect
88. C₃₇	#10624852	semidirect

H ≅ G/N

type

1. #22582662

C₃

semidirect

2. #22582649

C₃

semidirect

3. #22582605

C₃

semidirect

4. #22582523

C₃

semidirect

5. #22582519

C₃

semidirect

6. #22582514

C₃

semidirect

7. #22582509

C₃

semidirect

8. #22582505

C₃

semidirect

9. C₃₇×#6882

C₃

semidirect

64. #6882

C₃₇⋊C₃

semidirect

88. C₃₇

#10624852

semidirect

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

subgroup	index
#22582662	3
#22582649	3
#22582605	3
#22582523	3
#22582519	3
#22582514	3
#22582509	3
#22582505	3
C₃₇×#6882	3
#10624852	37

subgroup

index

C₃₇×#6882

Subgroups — #41007439

Conjugacy classes of subgroups

Complements

Maximal subgroups