Group #41004334 — subgroups

Subgroups — C₉×C₃₇×C₃²⋊C₉ #41004334

Order 26973.

662 conjugacy classes of subgroups; only the 332 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 332 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	C₃³×C₉×C₃₇	C₃	1	1	yes	maximal, characteristic
2	C₃×C₃₇×C₃²⋊C₉	C₃	1	3	yes	maximal
3	C₃×C₉²×C₃₇	C₃	1	3	yes	maximal, complemented
4	C₃₇×#6855	C₃	1	6	yes	maximal, complemented
5	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
6	C₃²×C₉×C₃₇	—	3	3	no	—
7	C₃²×C₉×C₃₇	—	3	6	no	—
8	C₃²×C₉×C₃₇	C₉	1	9	yes	complemented
9	C₃²×C₉×C₃₇	C₃²	1	2	yes	—
10	C₃²×C₉×C₃₇	C₃²	1	9	yes	—
11	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
12	C₃₇×C₃²⋊C₉	C₉	1	27	yes	direct product factor
13	C₉²×C₃₇	—	3	9	no	—
14	C₃³×C₃₇	—	3	1	no	—
15	C₃³×C₃₇	—	3	2	no	—
16	C₃³×C₃₇	—	3	6	no	—
17	C₃³×C₃₇	C₃×C₉	1	9	yes	—
18	C₃³×C₃₇	C₃×C₉	1	3	yes	—
19	C₃³×C₃₇	C₃³	1	1	yes	characteristic
20	C₃×C₉×C₃₇	—	3	9	no	—
21	C₃×C₉×C₃₇	—	3	2	no	—
22	C₃×C₉×C₃₇	—	3	4	no	—
23	C₃×C₉×C₃₇	—	3	18	no	—
24	C₃×C₉×C₃₇	—	3	27	no	—
25	C₃×C₉×C₃₇	C₃×C₉	1	6	yes	—
26	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
27	C₃×C₉×C₃₇	C₃×C₉	1	27	yes	complemented
28	C₃×C₉×C₃₇	H(3,𝔽₃)	1	3	yes	—
29	C₃×C₉×C₃₇	3₋¹⁺²	1	6	yes	—
30	C₉×C₃²⋊C₉	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₃²×C₃₇	—	3	3	no	—
32	C₃²×C₃₇	—	3	6	no	—
33	C₃²×C₃₇	—	3	9	no	—
34	C₃²×C₃₇	—	3	18	no	—
35	C₃²×C₃₇	C₉²	1	9	yes	complemented
36	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
37	C₃²×C₃₇	C₃²×C₉	1	3	yes	—
38	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
39	C₃²×C₃₇	C₃×3₋¹⁺²	1	2	yes	—
40	C₃²×C₃₇	#737	1	6	yes	—
41	C₉×C₃₇	—	3	6	no	—
42	C₉×C₃₇	—	3	27	no	—
43	C₉×C₃₇	C₃²⋊C₉	1	9	yes	direct product factor
44	C₃³×C₉	C₃×C₃₇	1	1	yes	characteristic
45	C₃×C₃²⋊C₉	C₃×C₃₇	1	3	yes	—
46	C₃×C₉²	C₃×C₃₇	1	3	yes	complemented
47	#6855	C₃×C₃₇	1	6	yes	complemented
48	C₃×C₃₇	—	3	9	no	—
49	C₃×C₃₇	C₃×C₉²	1	1	yes	characteristic
50	C₃×C₃₇	C₃×C₃²⋊C₉	1	1	yes	characteristic
51	C₃×C₃₇	#6887	1	2	yes	—
52	C₃×C₃₇	C₉×H(3,𝔽₃)	1	3	yes	—
53	C₃×C₃₇	C₉×3₋¹⁺²	1	6	yes	—
54	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
55	C₃²×C₉	—	3	3	no	—
56	C₃²×C₉	—	3	6	no	—
57	C₃²×C₉	C₉×C₃₇	1	9	yes	complemented
58	C₃²×C₉	C₃²×C₃₇	1	2	yes	—
59	C₃²×C₉	C₃²×C₃₇	1	9	yes	—
60	C₃²×C₉	C₃²×C₃₇	1	1	yes	characteristic
61	C₃²⋊C₉	C₉×C₃₇	1	27	yes	direct product factor
62	C₉²	—	3	9	no	—
63	C₃₇	C₉×C₃²⋊C₉	1	1	yes	characteristic, direct product factor, Sylow
64	C₃³	—	3	1	no	—
65	C₃³	—	3	2	no	—
66	C₃³	—	3	6	no	—
67	C₃³	C₃×C₉×C₃₇	1	9	yes	—
68	C₃³	C₃×C₉×C₃₇	1	3	yes	—
69	C₃³	C₃³×C₃₇	1	1	yes	characteristic
70	C₃×C₉	—	3	9	no	—
71	C₃×C₉	—	3	2	no	—
72	C₃×C₉	—	3	4	no	—
73	C₃×C₉	—	3	18	no	—
74	C₃×C₉	—	3	27	no	—
75	C₃×C₉	C₃×C₉×C₃₇	1	6	yes	—
76	C₃×C₉	C₃×C₉×C₃₇	1	3	yes	—
77	C₃×C₉	C₃×C₉×C₃₇	1	27	yes	complemented
78	C₃×C₉	H(3,𝔽₃)×C₃₇	1	3	yes	—
79	C₃×C₉	3₋¹⁺²×C₃₇	1	6	yes	—
80	C₃²	—	3	3	no	—
81	C₃²	—	3	6	no	—
82	C₃²	—	3	9	no	—
83	C₃²	—	3	18	no	—
84	C₃²	C₉²×C₃₇	1	9	yes	complemented
85	C₃²	C₃²×C₉×C₃₇	1	1	yes	characteristic
86	C₃²	C₃²×C₉×C₃₇	1	3	yes	—
87	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
88	C₃²	C₃×3₋¹⁺²×C₃₇	1	2	yes	—
89	C₃²	C₃₇×#737	1	6	yes	—
90	C₉	—	3	6	no	—
91	C₉	—	3	27	no	—
92	C₉	C₃₇×C₃²⋊C₉	1	9	yes	direct product factor
93	C₃	—	3	9	no	—
94	C₃	C₃×C₉²×C₃₇	1	1	yes	characteristic
95	C₃	C₃×C₃₇×C₃²⋊C₉	1	1	yes	characteristic
96	C₃	C₃₇×#6887	1	2	yes	—
97	C₃	C₉×H(3,𝔽₃)×C₃₇	1	3	yes	—
98	C₃	C₉×3₋¹⁺²×C₃₇	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃³×C₉×C₃₇

C₃

yes

maximal, characteristic

C₃×C₃₇×C₃²⋊C₉

C₃

yes

maximal

C₃×C₉²×C₃₇

C₃

yes

maximal, complemented

C₃₇×#6855

C₃

yes

maximal, complemented

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃²×C₉×C₃₇

—

C₃²×C₉×C₃₇

—

C₃²×C₉×C₃₇

C₉

yes

complemented

C₃²×C₉×C₃₇

C₃²

yes

—

C₃²×C₉×C₃₇

C₃²

yes

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×C₃²⋊C₉

C₉

yes

direct product factor

C₉²×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

C₃×C₉

yes

—

C₃³×C₃₇

C₃×C₉

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃×C₉

yes

—

C₃×C₉×C₃₇

C₃×C₉

yes

—

C₃×C₉×C₃₇

C₃×C₉

yes

complemented

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

—

C₃×C₉×C₃₇

3₋¹⁺²

yes

—

C₉×C₃²⋊C₉

C₃₇

yes

maximal, characteristic, direct product factor, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₉²

yes

complemented

C₃²×C₃₇

C₃²×C₉

yes

characteristic

C₃²×C₃₇

C₃²×C₉

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₃²×C₃₇

#737

yes

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

C₃²⋊C₉

yes

direct product factor

C₃³×C₉

C₃×C₃₇

yes

characteristic

C₃×C₃²⋊C₉

C₃×C₃₇

yes

—

C₃×C₉²

C₃×C₃₇

yes

complemented

#6855

C₃×C₃₇

yes

complemented

C₃×C₃₇

—

C₃×C₃₇

C₃×C₉²

yes

characteristic

C₃×C₃₇

C₃×C₃²⋊C₉

yes

characteristic

C₃×C₃₇

#6887

yes

—

C₃×C₃₇

C₉×H(3,𝔽₃)

yes

—

C₃×C₃₇

C₉×3₋¹⁺²

yes

—

C₃⁴

C₃²×C₃₇

yes

characteristic

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₉×C₃₇

yes

complemented

C₃²×C₉

C₃²×C₃₇

yes

—

C₃²×C₉

C₃²×C₃₇

yes

—

C₃²×C₉

C₃²×C₃₇

yes

characteristic

C₃²⋊C₉

C₉×C₃₇

yes

direct product factor

C₉²

—

C₃₇

C₉×C₃²⋊C₉

yes

characteristic, direct product factor, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×C₉×C₃₇

yes

—

C₃³

C₃×C₉×C₃₇

yes

—

C₃³

C₃³×C₃₇

yes

characteristic

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃×C₉×C₃₇

yes

—

C₃×C₉

C₃×C₉×C₃₇

yes

—

C₃×C₉

C₃×C₉×C₃₇

yes

complemented

C₃×C₉

H(3,𝔽₃)×C₃₇

yes

—

C₃×C₉

3₋¹⁺²×C₃₇

yes

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₉²×C₃₇

yes

complemented

C₃²

C₃²×C₉×C₃₇

yes

characteristic

C₃²

C₃²×C₉×C₃₇

yes

—

C₃²

C₃×H(3,𝔽₃)×C₃₇

yes

characteristic

C₃²

C₃×3₋¹⁺²×C₃₇

yes

—

C₃²

C₃₇×#737

yes

—

C₉

—

C₉

—

C₉

C₃₇×C₃²⋊C₉

yes

direct product factor

C₃

—

C₃

C₃×C₉²×C₃₇

yes

characteristic

C₃

C₃×C₃₇×C₃²⋊C₉

yes

characteristic

C₃

C₃₇×#6887

yes

—

C₃

C₉×H(3,𝔽₃)×C₃₇

yes

—

C₃

C₉×3₋¹⁺²×C₃₇

yes

—

	subgroup	G/H
center	C₃²×C₉×C₃₇	C₃²
commutator	C₃	C₃×C₉²×C₃₇
Frattini, cosocle	C₃³	C₃³×C₃₇
socle	C₃³×C₃₇	C₃³
prime radical	C₃⁴×C₃₇	C₃²
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₉×C₃₇ ◁ G

subgroup

G/H

center

C₃²×C₉×C₃₇

C₃²

commutator

C₃

C₃×C₉²×C₃₇

Frattini, cosocle

C₃³

C₃³×C₃₇

socle

C₃³×C₃₇

C₃³

prime radical

C₃⁴×C₃₇

C₃²

LCS

G ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃²×C₉×C₃₇ ◁ G

N	H ≅ G/N	type
3. C₃×C₉²×C₃₇	C₃	semidirect
4. C₃₇×#6855	C₃	semidirect
8. C₃²×C₉×C₃₇	C₉	semidirect
12. C₃₇×C₃²⋊C₉	C₉	direct (N × H)
27. C₃×C₉×C₃₇	C₃×C₉	semidirect
30. C₉×C₃²⋊C₉	C₃₇	direct (N × H)
35. C₃²×C₃₇	C₉²	semidirect
43. C₉×C₃₇	C₃²⋊C₉	direct (N × H)
46. C₃×C₉²	C₃×C₃₇	semidirect
47. #6855	C₃×C₃₇	semidirect
57. C₃²×C₉	C₉×C₃₇	semidirect
61. C₃²⋊C₉	C₉×C₃₇	direct (N × H)
63. C₃₇	C₉×C₃²⋊C₉	direct (N × H)
77. C₃×C₉	C₃×C₉×C₃₇	semidirect
84. C₃²	C₉²×C₃₇	semidirect
92. C₉	C₃₇×C₃²⋊C₉	direct (N × H)

H ≅ G/N

type

3. C₃×C₉²×C₃₇

C₃

semidirect

4. C₃₇×#6855

C₃

semidirect

8. C₃²×C₉×C₃₇

C₉

semidirect

12. C₃₇×C₃²⋊C₉

C₉

direct (N × H)

27. C₃×C₉×C₃₇

C₃×C₉

semidirect

30. C₉×C₃²⋊C₉

C₃₇

direct (N × H)

35. C₃²×C₃₇

C₉²

semidirect

43. C₉×C₃₇

C₃²⋊C₉

direct (N × H)

46. C₃×C₉²

C₃×C₃₇

semidirect

47. #6855

C₃×C₃₇

semidirect

57. C₃²×C₉

C₉×C₃₇

semidirect

61. C₃²⋊C₉

C₉×C₃₇

direct (N × H)

63. C₃₇

C₉×C₃²⋊C₉

direct (N × H)

77. C₃×C₉

C₃×C₉×C₃₇

semidirect

84. C₃²

C₉²×C₃₇

semidirect

92. C₉

C₃₇×C₃²⋊C₉

direct (N × H)

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

subgroup	index
C₃³×C₉×C₃₇	3
C₃×C₃₇×C₃²⋊C₉	3
C₃×C₉²×C₃₇	3
C₃₇×#6855	3
C₉×C₃²⋊C₉	37

subgroup

index

C₃³×C₉×C₃₇

C₃×C₃₇×C₃²⋊C₉

C₃×C₉²×C₃₇

C₃₇×#6855

C₉×C₃²⋊C₉

Subgroups — C₉×C₃₇×C₃²⋊C₉ #41004334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C9×C37×C32⋊C9 #41004334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×C₃₇×C₃²⋊C₉ #41004334