Group #41007336 — subgroups

Subgroups — #41007336

Order 26973.

226 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 46 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	#22582634	C₃	1	3	yes	maximal, complemented
2	C₉²×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582510	C₃	1	1	yes	maximal, characteristic, complemented
4	#22582507	C₃	1	1	yes	maximal, characteristic, complemented
5	#22582505	C₃	1	3	yes	maximal, complemented
6	#22582504	C₃	1	3	yes	maximal, complemented
7	C₃₇×#6880	C₃	1	1	yes	maximal, characteristic, complemented
8	#15688532	—	3	3	no	—
9	C₃×C₉×C₃₇⋊C₃	—	3	1	no	—
10	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
11	#15688521	C₃²	1	3	yes	—
12	#15688519	C₃²	1	3	yes	—
13	#15688517	—	3	3	no	—
14	#15688514	—	3	3	no	—
15	C₉×#66524	—	3	1	no	—
16	C₉×#66524	—	3	1	no	—
17	C₉×#66524	C₃²	1	1	yes	characteristic
18	C₉×#66524	C₃²	1	1	yes	characteristic
19	C₃₇×#733	C₃²	1	3	yes	—
20	C₉²×C₃₇	C₃²	1	1	yes	characteristic
21	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
22	3₋¹⁺²×C₃₇	—	3	3	no	—
23	C₃×C₉×C₃₇	—	3	1	no	—
24	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
25	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
26	#11758608	—	3	3	no	—
27	C₃×#66524	—	3	1	no	—
28	C₃×#66524	—	3	1	no	—
29	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
30	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
31	#11758606	—	9	3	no	—
32	#11758605	—	9	3	no	—
33	C₉×C₃₇⋊C₃	—	3	9	no	—
34	C₉×C₃₇⋊C₃	—	3	1	no	—
35	C₉×C₃₇⋊C₃	—	3	1	no	—
36	C₉×C₃₇⋊C₃	—	3	1	no	—
37	C₉×C₃₇⋊C₃	—	9	3	no	—
38	#10624837	—	37	1	no	maximal, Sylow
39	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
40	C₃×C₃₇⋊C₃	—	3	1	no	—
41	C₃×C₃₇⋊C₃	—	3	3	no	—
42	C₃×C₃₇⋊C₃	—	9	3	no	—
43	C₉×C₃₇	—	3	3	no	—
44	C₉×C₃₇	—	3	1	no	—
45	C₉×C₃₇	—	9	3	no	—
46	#66524	—	3	3	no	—
47	#66524	—	3	3	no	—
48	#66524	—	3	1	no	—
49	#66524	—	9	3	no	—
50	#66524	#732	1	1	yes	characteristic
51	#66524	#732	1	1	yes	characteristic
52	#66524	#732	1	1	yes	characteristic
53	#6908	—	37	3	no	—
54	C₃×C₉²	—	37	1	no	—
55	#6880	—	37	1	no	—
56	#6880	C₃₇⋊C₃	1	1	yes	characteristic, complemented
57	#6879	—	37	1	no	—
58	#6878	—	37	3	no	—
59	#6878	—	37	3	no	—
60	C₃×C₃₇	—	3	1	no	—
61	C₃×C₃₇	C₃×#732	1	1	yes	characteristic
62	C₃₇⋊C₃	—	3	3	no	—
63	C₃₇⋊C₃	—	27	3	no	—
64	#737	—	111	3	no	—
65	C₃²×C₉	—	37	1	no	—
66	C₃²×C₉	—	111	1	no	—
67	#733	—	37	3	no	—
68	#733	C₃×C₃₇⋊C₃	1	3	yes	—
69	#732	—	37	3	no	—
70	#731	—	111	3	no	—
71	AΣL(1,𝔽₂₇)	—	111	3	no	—
72	C₉²	—	37	1	no	—
73	C₉²	—	37	1	no	—
74	C₉²	—	111	1	no	—
75	C₉²	—	111	1	no	—
76	C₉²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
77	C₃₇	#10624837	1	1	yes	characteristic, complemented, Sylow
78	C₃³	—	37	1	no	—
79	3₋¹⁺²	—	3	3	no	—
80	3₋¹⁺²	—	333	3	no	—
81	3₋¹⁺²	—	333	3	no	—
82	3₋¹⁺²	—	111	3	no	—
83	H(3,𝔽₃)	—	111	3	no	—
84	C₃×C₉	—	3	1	no	—
85	C₃×C₉	—	37	1	no	—
86	C₃×C₉	—	37	1	no	—
87	C₃×C₉	—	333	3	no	—
88	C₃×C₉	—	111	1	no	—
89	C₃×C₉	—	111	1	no	—
90	C₃×C₉	—	111	1	no	—
91	C₃×C₉	—	111	9	no	—
92	C₃×C₉	—	111	1	no	—
93	C₃×C₉	—	111	1	no	—
94	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
95	C₃²	—	333	3	no	—
96	C₃²	—	111	3	no	—
97	C₃²	—	111	1	no	—
98	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
99	C₉	—	3	3	no	—
100	C₉	—	3	1	no	—
101	C₉	—	9	3	no	—
102	C₉	—	37	1	no	—
103	C₉	—	37	1	no	—
104	C₉	—	37	1	no	—
105	C₉	—	333	3	no	—
106	C₉	—	111	3	no	—
107	C₉	—	111	3	no	—
108	C₉	—	111	1	no	—
109	C₃	—	3	1	no	—
110	C₃	—	111	3	no	—
111	C₃	—	999	3	no	—
112	C₃	#732×C₃₇⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22582634

C₃

yes

maximal, complemented

C₉²×C₃₇⋊C₃

C₃

yes

maximal, characteristic, complemented

#22582510

C₃

yes

maximal, characteristic, complemented

#22582507

C₃

yes

maximal, characteristic, complemented

#22582505

C₃

yes

maximal, complemented

#22582504

C₃

yes

maximal, complemented

C₃₇×#6880

C₃

yes

maximal, characteristic, complemented

#15688532

—

C₃×C₉×C₃₇⋊C₃

—

C₃×C₉×C₃₇⋊C₃

C₃²

yes

characteristic

#15688521

C₃²

yes

—

#15688519

C₃²

yes

—

—

—

—

—

C₃²

yes

characteristic

C₉×#66524

C₃²

yes

characteristic

C₃₇×#733

C₃²

yes

—

C₉²×C₃₇

C₃²

yes

characteristic

C₃²×C₃₇⋊C₃

H(3,𝔽₃)

yes

characteristic

3₋¹⁺²×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃³

yes

characteristic

C₃²⋊(C₃₇⋊C₃)

—

—

—

—

yes

yes

—

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

#10624837

—

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

—

—

—

—

—

yes

yes

yes

—

C₃×C₉²

—

#6880

—

#6880

C₃₇⋊C₃

yes

characteristic, complemented

#6879

—

#6878

—

#6878

—

C₃×C₃₇

—

C₃×C₃₇

C₃×#732

yes

characteristic

C₃₇⋊C₃

—

C₃₇⋊C₃

—

#737

—

111

—

C₃²×C₉

—

C₃²×C₉

—

111

—

#733

—

#733

C₃×C₃₇⋊C₃

yes

—

#732

—

#731

—

111

—

AΣL(1,𝔽₂₇)

—

111

—

C₉²

—

C₉²

—

C₉²

—

111

—

C₉²

—

111

—

C₉²

C₃×C₃₇⋊C₃

yes

characteristic

C₃₇

#10624837

yes

characteristic, complemented, Sylow

C₃³

—

3₋¹⁺²

—

3₋¹⁺²

—

333

—

3₋¹⁺²

—

333

—

3₋¹⁺²

—

111

—

H(3,𝔽₃)

—

111

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

333

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

C₃²×C₃₇⋊C₃

yes

characteristic

C₃²

—

333

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

C₉

—

100

C₉

—

101

C₉

—

102

C₉

—

103

C₉

—

104

C₉

—

105

C₉

—

333

—

106

C₉

—

111

—

107

C₉

—

111

—

108

C₉

—

111

—

109

C₃

—

110

C₃

—

111

—

111

C₃

—

999

—

112

C₃

#732×C₃₇⋊C₃

yes

characteristic

	subgroup	G/H
center	C₃	#732×C₃₇⋊C₃
commutator, cosocle	C₃×C₉×C₃₇	C₃³
Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6880	C₃
socle	C₃×C₃₇	C₃×#732
hypercenter	#6880	C₃₇⋊C₃
hypocenter	C₃₇	#10624837
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉ ◁ #6880

subgroup

G/H

center

C₃

#732×C₃₇⋊C₃

commutator, cosocle

C₃×C₉×C₃₇

C₃³

Frattini

C₃×C₉

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6880

C₃

C₃×C₃₇

C₃₇⋊C₃

C₃₇

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

UCS

C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉ ◁ #6880

N	H ≅ G/N	type
1. #22582634	C₃	semidirect
2. C₉²×C₃₇⋊C₃	C₃	semidirect
3. #22582510	C₃	semidirect
4. #22582507	C₃	semidirect
5. #22582505	C₃	semidirect
6. #22582504	C₃	semidirect
7. C₃₇×#6880	C₃	semidirect
56. #6880	C₃₇⋊C₃	semidirect
77. C₃₇	#10624837	semidirect

H ≅ G/N

type

1. #22582634

C₃

semidirect

2. C₉²×C₃₇⋊C₃

C₃

semidirect

3. #22582510

C₃

semidirect

4. #22582507

C₃

semidirect

5. #22582505

C₃

semidirect

6. #22582504

C₃

semidirect

7. C₃₇×#6880

C₃

semidirect

56. #6880

C₃₇⋊C₃

semidirect

77. C₃₇

#10624837

semidirect

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

subgroup	index
#22582634	3
C₉²×C₃₇⋊C₃	3
#22582510	3
#22582507	3
#22582505	3
#22582504	3
C₃₇×#6880	3
#10624837	37

subgroup

index

C₉²×C₃₇⋊C₃

C₃₇×#6880

Subgroups — #41007336

Conjugacy classes of subgroups

Complements

Maximal subgroups