Group #41006668 — subgroups

Subgroups — #41006668

Order 26973.

456 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 70 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	#22582647	C₃	1	4	yes	maximal, complemented
2	#22582637	C₃	1	4	yes	maximal, complemented
3	C₃×#15688518	C₃	1	2	yes	maximal, complemented
4	C₃×#15688515	C₃	1	2	yes	maximal, complemented
5	C₃₇×H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃×#11758608	C₃²	1	4	yes	complemented
7	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	4	yes	complemented
8	C₃²×#66524	C₃²	1	1	yes	characteristic, complemented
9	#15688518	—	3	6	no	—
10	#15688518	—	3	12	no	—
11	#15688515	—	3	6	no	—
12	#15688512	—	3	12	no	—
13	C₃×H(3,𝔽₃)×C₃₇	—	3	6	no	—
14	C₃×H(3,𝔽₃)×C₃₇	—	3	6	no	—
15	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	2	yes	—
16	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	2	yes	—
17	C₃³×C₃₇	—	3	9	no	—
18	C₃³×C₃₇	—	3	2	no	—
19	C₃³×C₃₇	—	3	2	no	—
20	C₃³×C₃₇	C₃³	1	1	yes	characteristic
21	C₃²×C₃₇⋊C₃	—	3	4	no	—
22	H(3,𝔽₃)×C₃₇	—	3	18	no	—
23	H(3,𝔽₃)×C₃₇	—	3	4	no	—
24	H(3,𝔽₃)×C₃₇	—	3	4	no	—
25	H(3,𝔽₃)×C₃₇	H(3,𝔽₃)	1	6	yes	—
26	H(3,𝔽₃)×C₃₇	H(3,𝔽₃)	1	6	yes	—
27	C₃²⋊(C₃₇⋊C₃)	—	3	8	no	—
28	C₃²⋊(C₃₇⋊C₃)	—	3	4	no	—
29	#11758608	—	3	8	no	—
30	#11758608	—	3	4	no	—
31	C₃×#66524	—	3	2	no	—
32	C₃×#66524	—	3	2	no	—
33	C₃×#66524	—	3	4	no	—
34	#10624760	—	37	1	no	maximal, Sylow
35	C₃²×C₃₇	—	3	6	no	—
36	C₃²×C₃₇	—	3	6	no	—
37	C₃²×C₃₇	—	9	1	no	—
38	C₃²×C₃₇	—	27	2	no	—
39	C₃²×C₃₇	—	27	2	no	—
40	C₃²×C₃₇	—	27	9	no	—
41	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	2	yes	—
42	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	2	yes	—
43	C₃×C₃₇⋊C₃	—	9	4	no	—
44	C₃×C₃₇⋊C₃	—	27	4	no	—
45	#66524	—	9	1	no	—
46	#66524	—	9	4	no	—
47	H(5,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic, complemented
48	#6911	—	37	4	no	—
49	#6909	—	37	4	no	—
50	C₃×#732	—	37	2	no	—
51	C₃×#731	—	37	2	no	—
52	C₃×C₃₇	—	3	2	no	—
53	C₃×C₃₇	—	3	2	no	—
54	C₃×C₃₇	—	9	6	no	—
55	C₃×C₃₇	—	9	6	no	—
56	C₃×C₃₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
57	C₃₇⋊C₃	—	27	4	no	—
58	C₃×3₋¹⁺²	—	37	4	no	—
59	C₃×H(3,𝔽₃)	—	3	6	no	—
60	C₃×H(3,𝔽₃)	—	3	6	no	—
61	C₃×H(3,𝔽₃)	—	37	4	no	—
62	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	2	yes	—
63	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	2	yes	—
64	C₃²×C₉	—	37	1	no	—
65	#732	—	111	6	no	—
66	#732	—	111	12	no	—
67	#731	—	111	6	no	—
68	AΣL(1,𝔽₂₇)	—	111	12	no	—
69	C₃₇	#10624760	1	1	yes	characteristic, complemented, Sylow
70	C₃³	—	3	9	no	—
71	C₃³	—	3	2	no	—
72	C₃³	—	3	2	no	—
73	C₃³	—	111	4	no	—
74	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
75	3₋¹⁺²	—	111	8	no	—
76	3₋¹⁺²	—	111	4	no	—
77	H(3,𝔽₃)	—	3	18	no	—
78	H(3,𝔽₃)	—	3	4	no	—
79	H(3,𝔽₃)	—	3	4	no	—
80	H(3,𝔽₃)	—	111	8	no	—
81	H(3,𝔽₃)	—	111	4	no	—
82	H(3,𝔽₃)	C₃²⋊(C₃₇⋊C₃)	1	6	yes	—
83	H(3,𝔽₃)	C₃²⋊(C₃₇⋊C₃)	1	6	yes	—
84	C₃×C₉	—	111	2	no	—
85	C₃×C₉	—	111	2	no	—
86	C₃×C₉	—	111	4	no	—
87	C₃²	—	3	6	no	—
88	C₃²	—	3	6	no	—
89	C₃²	—	9	1	no	—
90	C₃²	—	27	2	no	—
91	C₃²	—	27	2	no	—
92	C₃²	—	27	9	no	—
93	C₃²	—	333	4	no	—
94	C₃²	—	999	4	no	—
95	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
96	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
97	C₉	—	333	1	no	—
98	C₉	—	333	4	no	—
99	C₃	—	3	2	no	—
100	C₃	—	3	2	no	—
101	C₃	—	9	6	no	—
102	C₃	—	9	6	no	—
103	C₃	—	999	4	no	—
104	C₃	#22582550	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22582647

C₃

yes

maximal, complemented

#22582637

C₃

yes

maximal, complemented

C₃×#15688518

C₃

yes

maximal, complemented

C₃×#15688515

C₃

yes

maximal, complemented

C₃₇×H(5,𝔽₃)

C₃

yes

maximal, characteristic, complemented

C₃×#11758608

C₃²

yes

complemented

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

complemented

C₃²×#66524

C₃²

yes

characteristic, complemented

—

—

—

—

C₃×H(3,𝔽₃)×C₃₇

—

C₃×H(3,𝔽₃)×C₃₇

—

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

—

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×C₃₇

H(3,𝔽₃)

yes

—

H(3,𝔽₃)×C₃₇

H(3,𝔽₃)

yes

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

—

—

—

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

#66524

—

#66524

—

H(5,𝔽₃)

C₃₇⋊C₃

yes

characteristic, complemented

—

—

—

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃²⋊H(3,𝔽₃)

yes

characteristic

C₃₇⋊C₃

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×C₃₇⋊C₃

yes

—

C₃×H(3,𝔽₃)

C₃×C₃₇⋊C₃

yes

—

C₃²×C₉

—

#732

—

111

—

#732

—

111

—

#731

—

111

—

AΣL(1,𝔽₂₇)

—

111

—

C₃₇

#10624760

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

—

111

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

—

—

—

—

111

—

H(3,𝔽₃)

—

111

—

H(3,𝔽₃)

C₃²⋊(C₃₇⋊C₃)

yes

—

H(3,𝔽₃)

C₃²⋊(C₃₇⋊C₃)

yes

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

333

—

C₃²

—

999

—

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

—

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

—

C₉

—

333

—

C₉

—

333

—

C₃

—

100

C₃

—

101

C₃

—

102

C₃

—

103

C₃

—

999

—

104

C₃

#22582550

yes

characteristic

	subgroup	G/H
center	C₃	#22582550
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×H(5,𝔽₃)	C₃
socle	C₃×C₃₇	C₃²⋊H(3,𝔽₃)
hypercenter	H(5,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	#10624760
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ H(5,𝔽₃)

subgroup

G/H

center

C₃

#22582550

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃₇×H(5,𝔽₃)

C₃

socle

C₃×C₃₇

C₃²⋊H(3,𝔽₃)

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃ ◁ C₃³ ◁ H(5,𝔽₃)

N	H ≅ G/N	type
1. #22582647	C₃	semidirect
2. #22582637	C₃	semidirect
3. C₃×#15688518	C₃	semidirect
4. C₃×#15688515	C₃	semidirect
5. C₃₇×H(5,𝔽₃)	C₃	semidirect
6. C₃×#11758608	C₃²	semidirect
7. C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	semidirect
8. C₃²×#66524	C₃²	semidirect
47. H(5,𝔽₃)	C₃₇⋊C₃	semidirect
69. C₃₇	#10624760	semidirect

H ≅ G/N

type

1. #22582647

C₃

semidirect

2. #22582637

C₃

semidirect

3. C₃×#15688518

C₃

semidirect

4. C₃×#15688515

C₃

semidirect

5. C₃₇×H(5,𝔽₃)

C₃

semidirect

6. C₃×#11758608

C₃²

semidirect

7. C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

semidirect

8. C₃²×#66524

C₃²

semidirect

47. H(5,𝔽₃)

C₃₇⋊C₃

semidirect

69. C₃₇

#10624760

semidirect

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

subgroup	index
#22582647	3
#22582637	3
C₃×#15688518	3
C₃×#15688515	3
C₃₇×H(5,𝔽₃)	3
#10624760	37

subgroup

index

C₃₇×H(5,𝔽₃)

Subgroups — #41006668

Conjugacy classes of subgroups

Complements

Maximal subgroups