Group #41004448 — subgroups

Subgroups — C₃₇×#10624659 #41004448

Order 26973.

236 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 74 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	C₃₇×#6891	C₃	1	1	yes	maximal, characteristic
2	C₃₇×#6891	C₃	1	1	yes	maximal, characteristic
3	C₃₇×#6891	C₃	1	2	yes	maximal
4	C₃₇×#6858	C₃	1	9	yes	maximal, complemented
5	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
6	C₃×3₋¹⁺²×C₃₇	—	3	1	no	—
7	C₃×3₋¹⁺²×C₃₇	—	3	2	no	—
8	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
9	C₃₇×C₃²⋊C₉	—	3	1	no	—
10	C₃₇×C₃²⋊C₉	—	3	2	no	—
11	C₃₇×C₃²⋊C₉	—	3	2	no	—
12	C₃₇×C₃²⋊C₉	—	3	2	no	—
13	C₃₇×C₃²⋊C₉	—	3	2	no	—
14	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
15	C₃₇×C₃²⋊C₉	C₃²	1	6	yes	—
16	C₃³×C₃₇	—	9	1	no	—
17	C₃³×C₃₇	—	9	1	no	—
18	C₃³×C₃₇	—	9	2	no	—
19	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
20	C₃³×C₃₇	H(3,𝔽₃)	1	2	yes	—
21	C₃³×C₃₇	C₃³	1	1	yes	characteristic
22	3₋¹⁺²×C₃₇	—	9	3	no	—
23	3₋¹⁺²×C₃₇	—	9	6	no	—
24	3₋¹⁺²×C₃₇	—	9	3	no	—
25	C₃×C₉×C₃₇	—	3	3	no	—
26	C₃×C₉×C₃₇	—	3	3	no	—
27	C₃×C₉×C₃₇	—	3	6	no	—
28	#10624659	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₃²×C₃₇	—	3	1	no	—
30	C₃²×C₃₇	—	3	2	no	—
31	C₃²×C₃₇	—	3	1	no	—
32	C₃²×C₃₇	—	3	1	no	—
33	C₃²×C₃₇	—	3	1	no	—
34	C₃²×C₃₇	—	3	2	no	—
35	C₃²×C₃₇	—	3	2	no	—
36	C₃²×C₃₇	—	3	2	no	—
37	C₃²×C₃₇	—	3	2	no	—
38	C₃²×C₃₇	—	3	2	no	—
39	C₃²×C₃₇	—	9	9	no	—
40	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
41	C₉×C₃₇	—	9	3	no	—
42	C₉×C₃₇	—	9	3	no	—
43	C₉×C₃₇	—	9	6	no	—
44	#6891	C₃×C₃₇	1	1	yes	characteristic
45	#6891	C₃×C₃₇	1	1	yes	characteristic
46	#6891	C₃×C₃₇	1	2	yes	—
47	#6858	C₃×C₃₇	1	9	yes	complemented
48	C₃×C₃₇	—	9	1	no	—
49	C₃×C₃₇	—	9	2	no	—
50	C₃×C₃₇	—	9	1	no	—
51	C₃×C₃₇	#6909	1	1	yes	characteristic
52	C₃×C₃₇	#6910	1	2	yes	—
53	C₃×C₃₇	#6912	1	1	yes	characteristic
54	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
55	C₃×3₋¹⁺²	—	3	1	no	—
56	C₃×3₋¹⁺²	—	3	2	no	—
57	C₃×3₋¹⁺²	C₃²×C₃₇	1	3	yes	—
58	C₃²⋊C₉	—	3	1	no	—
59	C₃²⋊C₉	—	3	2	no	—
60	C₃²⋊C₉	—	3	2	no	—
61	C₃²⋊C₉	—	3	2	no	—
62	C₃²⋊C₉	—	3	2	no	—
63	C₃²⋊C₉	C₃²×C₃₇	1	3	yes	—
64	C₃²⋊C₉	C₃²×C₃₇	1	6	yes	—
65	C₃₇	#10624659	1	1	yes	characteristic, direct product factor, Sylow
66	C₃³	—	9	1	no	—
67	C₃³	—	9	2	no	—
68	C₃³	—	9	1	no	—
69	C₃³	H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
70	C₃³	H(3,𝔽₃)×C₃₇	1	2	yes	—
71	C₃³	C₃³×C₃₇	1	1	yes	characteristic
72	3₋¹⁺²	—	9	3	no	—
73	3₋¹⁺²	—	9	6	no	—
74	3₋¹⁺²	—	9	3	no	—
75	C₃×C₉	—	3	3	no	—
76	C₃×C₉	—	3	3	no	—
77	C₃×C₉	—	3	6	no	—
78	C₃²	—	3	1	no	—
79	C₃²	—	3	1	no	—
80	C₃²	—	3	2	no	—
81	C₃²	—	3	2	no	—
82	C₃²	—	3	2	no	—
83	C₃²	—	3	2	no	—
84	C₃²	—	3	2	no	—
85	C₃²	—	3	1	no	—
86	C₃²	—	3	2	no	—
87	C₃²	—	3	1	no	—
88	C₃²	—	9	9	no	—
89	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
90	C₉	—	9	3	no	—
91	C₉	—	9	3	no	—
92	C₉	—	9	6	no	—
93	C₃	—	9	1	no	—
94	C₃	—	9	1	no	—
95	C₃	—	9	2	no	—
96	C₃	C₃₇×#6909	1	1	yes	characteristic
97	C₃	C₃₇×#6910	1	2	yes	—
98	C₃	C₃₇×#6912	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₇×#6891

C₃

yes

maximal, characteristic

C₃₇×#6891

C₃

yes

maximal, characteristic

C₃₇×#6891

C₃

yes

maximal

C₃₇×#6858

C₃

yes

maximal, complemented

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₇

—

C₃×3₋¹⁺²×C₃₇

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃₇×C₃²⋊C₉

—

C₃₇×C₃²⋊C₉

—

C₃₇×C₃²⋊C₉

—

C₃₇×C₃²⋊C₉

—

C₃₇×C₃²⋊C₉

—

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

H(3,𝔽₃)

yes

characteristic

C₃³×C₃₇

H(3,𝔽₃)

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

#10624659

C₃₇

yes

maximal, characteristic, direct product factor, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

#6891

C₃×C₃₇

yes

characteristic

#6891

C₃×C₃₇

yes

characteristic

#6891

C₃×C₃₇

yes

—

#6858

C₃×C₃₇

yes

complemented

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

#6909

yes

characteristic

C₃×C₃₇

#6910

yes

—

C₃×C₃₇

#6912

yes

characteristic

C₃⁴

C₃²×C₃₇

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃²×C₃₇

yes

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃²×C₃₇

yes

—

C₃²⋊C₉

C₃²×C₃₇

yes

—

C₃₇

#10624659

yes

characteristic, direct product factor, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

H(3,𝔽₃)×C₃₇

yes

characteristic

C₃³

H(3,𝔽₃)×C₃₇

yes

—

C₃³

C₃³×C₃₇

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃×H(3,𝔽₃)×C₃₇

yes

characteristic

C₉

—

C₉

—

C₉

—

C₃

—

C₃

—

C₃

—

C₃

C₃₇×#6909

yes

characteristic

C₃

C₃₇×#6910

yes

—

C₃

C₃₇×#6912

yes

characteristic

	subgroup	G/H
center, socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₃₇
prime radical	C₃⁴×C₃₇	C₃²
LCS	G ▷ C₃³ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₃₇ ◁ C₃⁴×C₃₇ ◁ G

subgroup

G/H

center, socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₃³×C₃₇

prime radical

C₃⁴×C₃₇

C₃²

LCS

G ▷ C₃³ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃²×C₃₇ ◁ C₃⁴×C₃₇ ◁ G

N	H ≅ G/N	type
4. C₃₇×#6858	C₃	semidirect
28. #10624659	C₃₇	direct (N × H)
47. #6858	C₃×C₃₇	semidirect
65. C₃₇	#10624659	direct (N × H)

H ≅ G/N

type

4. C₃₇×#6858

C₃

semidirect

28. #10624659

C₃₇

direct (N × H)

47. #6858

C₃×C₃₇

semidirect

65. C₃₇

#10624659

direct (N × H)

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

subgroup	index
C₃₇×#6891	3
C₃₇×#6858	3
#10624659	37

subgroup

index

C₃₇×#6891

C₃₇×#6858

#10624659

Subgroups — C₃₇×#10624659 #41004448

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C37×#10624659 #41004448

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624659 #41004448