Group #45686246 — subgroups

Subgroups — D₃×D₃₇×C₅²⋊C₃ #45686246

Order 33300.

128 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 30 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	C₅²⋊C₃×D₁₁₁	C₂	1	1	yes	maximal, characteristic, complemented
2	D₃×C₃₇×C₅²⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₃₇×C₅²⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅²×D₃×D₃₇	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂×D₃₇×C₅²⋊C₃	—	3	1	no	maximal
6	C₃×C₃₇×C₅²⋊C₃	C₂²	1	1	yes	characteristic, complemented, Hall
7	C₅²×D₁₁₁	C₂×C₃	1	1	yes	characteristic, complemented
8	C₅²×D₃×C₃₇	C₂×C₃	1	1	yes	characteristic, complemented
9	C₃×C₅²×D₃₇	C₂×C₃	1	1	yes	characteristic, complemented
10	C₂×C₃₇×C₅²⋊C₃	—	3	1	no	—
11	D₃₇×C₅²⋊C₃	—	2	1	no	—
12	D₃₇×C₅²⋊C₃	—	3	1	no	—
13	D₃₇×C₅²⋊C₃	D₃	1	1	yes	characteristic, direct product factor
14	C₂×C₅²×D₃₇	—	3	1	no	Hall
15	C₃₇×C₅²⋊C₃	—	2	1	no	—
16	C₃₇×C₅²⋊C₃	C₂×D₃	1	1	yes	characteristic, complemented
17	C₃×C₅²×C₃₇	C₂²×C₃	1	1	yes	characteristic, complemented
18	C₅×D₃×D₃₇	—	3	2	no	—
19	C₂×C₅²×C₃₇	—	3	1	no	—
20	C₅²×D₃₇	—	3	1	no	—
21	C₅²×D₃₇	C₃×D₃	1	1	yes	characteristic, complemented
22	C₃×D₃×D₃₇	—	25	1	no	maximal, Hall
23	C₅×D₁₁₁	—	3	2	no	—
24	C₅×D₃×C₃₇	—	3	2	no	—
25	C₃×C₅×D₃₇	—	3	2	no	—
26	C₅²×C₃₇	C₂×C₃×D₃	1	1	yes	characteristic, complemented, Hall
27	C₂×D₃×C₅²⋊C₃	—	37	1	no	maximal, Hall
28	C₂×C₅×D₃₇	—	9	2	no	—
29	C₃×D₁₁₁	—	25	1	no	—
30	C₃×D₃×C₃₇	—	25	1	no	—
31	C₃²×D₃₇	—	25	1	no	—
32	C₃×C₅×C₃₇	—	3	2	no	—
33	C₂×C₃×C₅²⋊C₃	—	37	1	no	—
34	D₃×C₅²⋊C₃	—	37	1	no	—
35	D₃×C₅²⋊C₃	D₃₇	1	1	yes	characteristic, direct product factor
36	C₂×C₃×D₃₇	—	75	1	no	—
37	D₃×D₃₇	C₅²⋊C₃	1	1	yes	characteristic, direct product factor
38	C₂×C₅×C₃₇	—	9	2	no	—
39	C₅×D₃₇	—	3	2	no	—
40	C₅×D₃₇	—	9	2	no	—
41	C₃²×C₃₇	—	25	1	no	Hall
42	C₂×C₅²×D₃	—	37	1	no	—
43	C₂²×C₅²⋊C₃	—	111	1	no	—
44	C₃×C₅²⋊C₃	C₂×D₃₇	1	1	yes	characteristic, complemented, Hall
45	C₂×C₃×C₃₇	—	75	1	no	—
46	D₁₁₁	C₂×C₅²⋊C₃	1	1	yes	characteristic, complemented
47	C₃×D₃₇	—	25	1	no	—
48	C₃×D₃₇	—	50	1	no	—
49	C₃×D₃₇	—	75	1	no	—
50	C₃×D₃₇	C₂×C₅²⋊C₃	1	1	yes	characteristic, complemented
51	D₃×C₃₇	C₂×C₅²⋊C₃	1	1	yes	characteristic, complemented
52	C₅×C₃₇	—	3	2	no	—
53	C₂×C₃×C₅²	—	37	1	no	—
54	C₅²×D₃	—	37	1	no	—
55	C₅²×D₃	C₃×D₃₇	1	1	yes	characteristic, complemented
56	C₂×C₅²⋊C₃	—	3	1	no	—
57	C₂×C₅²⋊C₃	—	37	1	no	—
58	C₂×C₅²⋊C₃	—	74	1	no	—
59	C₂×C₅²⋊C₃	—	111	1	no	—
60	C₂×D₃₇	—	3	1	no	Hall
61	C₃×C₃₇	—	25	1	no	—
62	C₃×C₃₇	—	50	1	no	—
63	C₃×C₃₇	C₂²×C₅²⋊C₃	1	1	yes	characteristic, complemented
64	C₂²×C₅²	—	111	1	no	Hall
65	C₃×C₅²	C₂×C₃×D₃₇	1	1	yes	characteristic, complemented
66	C₅²⋊C₃	—	2	1	no	—
67	C₅²⋊C₃	D₃×D₃₇	1	1	yes	characteristic, direct product factor
68	C₂×C₃₇	—	3	1	no	—
69	D₃₇	—	3	1	no	—
70	D₃₇	D₃×C₅²⋊C₃	1	1	yes	characteristic, direct product factor
71	C₂×C₅×D₃	—	111	2	no	—
72	C₂×C₅²	—	3	1	no	—
73	C₂×C₅²	—	37	1	no	—
74	C₂×C₅²	—	111	1	no	—
75	C₃₇	C₂×D₃×C₅²⋊C₃	1	1	yes	characteristic, complemented, Sylow
76	C₂×C₃×D₃	—	925	1	no	Hall
77	C₂×C₃×C₅	—	111	2	no	—
78	C₅×D₃	—	3	2	no	—
79	C₅×D₃	—	111	2	no	—
80	C₅²	C₃×D₃×D₃₇	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₅	—	333	2	no	—
82	C₂×C₃²	—	925	1	no	—
83	C₃×D₃	—	25	1	no	—
84	C₃×D₃	—	925	1	no	—
85	C₃×C₅	—	3	2	no	—
86	C₂²×C₃	—	2775	1	no	—
87	C₂×D₃	—	37	1	no	—
88	C₂×C₅	—	9	2	no	—
89	C₂×C₅	—	333	2	no	—
90	C₂×C₅	—	111	2	no	—
91	C₃²	—	25	1	no	Sylow
92	C₂×C₃	—	37	1	no	—
93	C₂×C₃	—	1850	1	no	—
94	C₂×C₃	—	75	1	no	—
95	C₂×C₃	—	925	1	no	—
96	C₂×C₃	—	2775	1	no	—
97	D₃	—	37	1	no	—
98	D₃	D₃₇×C₅²⋊C₃	1	1	yes	characteristic, direct product factor
99	C₅	—	3	2	no	—
100	C₂²	—	111	1	no	Sylow
101	C₃	—	25	1	no	—
102	C₃	—	50	1	no	—
103	C₃	C₂×D₃₇×C₅²⋊C₃	1	1	yes	characteristic, complemented
104	C₂	—	3	1	no	—
105	C₂	—	37	1	no	—
106	C₂	—	111	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²⋊C₃×D₁₁₁

C₂

yes

maximal, characteristic, complemented

D₃×C₃₇×C₅²⋊C₃

C₂

yes

maximal, characteristic, complemented

C₃×D₃₇×C₅²⋊C₃

C₂

yes

maximal, characteristic, complemented

C₅²×D₃×D₃₇

C₃

yes

maximal, characteristic, complemented

C₂×D₃₇×C₅²⋊C₃

—

maximal

C₃×C₃₇×C₅²⋊C₃

C₂²

yes

characteristic, complemented, Hall

C₅²×D₁₁₁

C₂×C₃

yes

characteristic, complemented

C₅²×D₃×C₃₇

C₂×C₃

yes

characteristic, complemented

C₃×C₅²×D₃₇

C₂×C₃

yes

characteristic, complemented

C₂×C₃₇×C₅²⋊C₃

—

D₃₇×C₅²⋊C₃

—

D₃₇×C₅²⋊C₃

—

D₃₇×C₅²⋊C₃

D₃

yes

characteristic, direct product factor

C₂×C₅²×D₃₇

—

Hall

C₃₇×C₅²⋊C₃

—

C₃₇×C₅²⋊C₃

C₂×D₃

yes

characteristic, complemented

C₃×C₅²×C₃₇

C₂²×C₃

yes

characteristic, complemented

C₅×D₃×D₃₇

—

C₂×C₅²×C₃₇

—

C₅²×D₃₇

—

C₅²×D₃₇

C₃×D₃

yes

characteristic, complemented

C₃×D₃×D₃₇

—

maximal, Hall

C₅×D₁₁₁

—

C₅×D₃×C₃₇

—

C₃×C₅×D₃₇

—

C₅²×C₃₇

C₂×C₃×D₃

yes

characteristic, complemented, Hall

C₂×D₃×C₅²⋊C₃

—

maximal, Hall

C₂×C₅×D₃₇

—

C₃×D₁₁₁

—

C₃×D₃×C₃₇

—

C₃²×D₃₇

—

C₃×C₅×C₃₇

—

C₂×C₃×C₅²⋊C₃

—

D₃×C₅²⋊C₃

—

D₃×C₅²⋊C₃

D₃₇

yes

characteristic, direct product factor

C₂×C₃×D₃₇

—

D₃×D₃₇

C₅²⋊C₃

yes

characteristic, direct product factor

C₂×C₅×C₃₇

—

C₅×D₃₇

—

C₅×D₃₇

—

C₃²×C₃₇

—

Hall

C₂×C₅²×D₃

—

C₂²×C₅²⋊C₃

—

111

—

C₃×C₅²⋊C₃

C₂×D₃₇

yes

characteristic, complemented, Hall

C₂×C₃×C₃₇

—

D₁₁₁

C₂×C₅²⋊C₃

yes

characteristic, complemented

C₃×D₃₇

—

C₃×D₃₇

—

C₃×D₃₇

—

C₃×D₃₇

C₂×C₅²⋊C₃

yes

characteristic, complemented

D₃×C₃₇

C₂×C₅²⋊C₃

yes

characteristic, complemented

C₅×C₃₇

—

C₂×C₃×C₅²

—

C₅²×D₃

—

C₅²×D₃

C₃×D₃₇

yes

characteristic, complemented

C₂×C₅²⋊C₃

—

C₂×C₅²⋊C₃

—

C₂×C₅²⋊C₃

—

C₂×C₅²⋊C₃

—

111

—

C₂×D₃₇

—

Hall

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₂²×C₅²⋊C₃

yes

characteristic, complemented

C₂²×C₅²

—

111

Hall

C₃×C₅²

C₂×C₃×D₃₇

yes

characteristic, complemented

C₅²⋊C₃

—

C₅²⋊C₃

D₃×D₃₇

yes

characteristic, direct product factor

C₂×C₃₇

—

D₃₇

—

D₃₇

D₃×C₅²⋊C₃

yes

characteristic, direct product factor

C₂×C₅×D₃

—

111

—

C₂×C₅²

—

C₂×C₅²

—

C₂×C₅²

—

111

—

C₃₇

C₂×D₃×C₅²⋊C₃

yes

characteristic, complemented, Sylow

C₂×C₃×D₃

—

925

Hall

C₂×C₃×C₅

—

111

—

C₅×D₃

—

C₅×D₃

—

111

—

C₅²

C₃×D₃×D₃₇

yes

characteristic, complemented, Sylow

C₂²×C₅

—

333

—

C₂×C₃²

—

925

—

C₃×D₃

—

C₃×D₃

—

925

—

C₃×C₅

—

C₂²×C₃

—

2775

—

C₂×D₃

—

C₂×C₅

—

C₂×C₅

—

333

—

C₂×C₅

—

111

—

C₃²

—

Sylow

C₂×C₃

—

C₂×C₃

—

1850

—

C₂×C₃

—

C₂×C₃

—

925

—

C₂×C₃

—

2775

—

D₃

—

D₃

D₃₇×C₅²⋊C₃

yes

characteristic, direct product factor

C₅

—

100

C₂²

—

111

Sylow

101

C₃

—

102

C₃

—

103

C₃

C₂×D₃₇×C₅²⋊C₃

yes

characteristic, complemented

104

C₂

—

105

C₂

—

106

C₂

—

111

—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅²×C₃₇	C₂²×C₃
LCS	G ▷ C₃×C₅²×C₃₇
UCS	C₁

subgroup

G/H

center, Frattini, hypercenter

C₁

commutator, Fitting, socle, cosocle, hypocenter

C₃×C₅²×C₃₇

C₂²×C₃

LCS

G ▷ C₃×C₅²×C₃₇

UCS

C₁

N	H ≅ G/N	type
1. C₅²⋊C₃×D₁₁₁	C₂	semidirect
2. D₃×C₃₇×C₅²⋊C₃	C₂	semidirect
3. C₃×D₃₇×C₅²⋊C₃	C₂	semidirect
4. C₅²×D₃×D₃₇	C₃	semidirect
6. C₃×C₃₇×C₅²⋊C₃	C₂²	semidirect
7. C₅²×D₁₁₁	C₂×C₃	semidirect
8. C₅²×D₃×C₃₇	C₂×C₃	semidirect
9. C₃×C₅²×D₃₇	C₂×C₃	semidirect
13. D₃₇×C₅²⋊C₃	D₃	direct (N × H)
16. C₃₇×C₅²⋊C₃	C₂×D₃	semidirect
17. C₃×C₅²×C₃₇	C₂²×C₃	semidirect
21. C₅²×D₃₇	C₃×D₃	semidirect
26. C₅²×C₃₇	C₂×C₃×D₃	semidirect
35. D₃×C₅²⋊C₃	D₃₇	direct (N × H)
37. D₃×D₃₇	C₅²⋊C₃	direct (N × H)
44. C₃×C₅²⋊C₃	C₂×D₃₇	semidirect
46. D₁₁₁	C₂×C₅²⋊C₃	semidirect
50. C₃×D₃₇	C₂×C₅²⋊C₃	semidirect
51. D₃×C₃₇	C₂×C₅²⋊C₃	semidirect
55. C₅²×D₃	C₃×D₃₇	semidirect
63. C₃×C₃₇	C₂²×C₅²⋊C₃	semidirect
65. C₃×C₅²	C₂×C₃×D₃₇	semidirect
67. C₅²⋊C₃	D₃×D₃₇	direct (N × H)
70. D₃₇	D₃×C₅²⋊C₃	direct (N × H)
75. C₃₇	C₂×D₃×C₅²⋊C₃	semidirect
80. C₅²	C₃×D₃×D₃₇	semidirect
98. D₃	D₃₇×C₅²⋊C₃	direct (N × H)
103. C₃	C₂×D₃₇×C₅²⋊C₃	semidirect

H ≅ G/N

type

1. C₅²⋊C₃×D₁₁₁

C₂

semidirect

2. D₃×C₃₇×C₅²⋊C₃

C₂

semidirect

3. C₃×D₃₇×C₅²⋊C₃

C₂

semidirect

4. C₅²×D₃×D₃₇

C₃

semidirect

6. C₃×C₃₇×C₅²⋊C₃

C₂²

semidirect

7. C₅²×D₁₁₁

C₂×C₃

semidirect

8. C₅²×D₃×C₃₇

C₂×C₃

semidirect

9. C₃×C₅²×D₃₇

C₂×C₃

semidirect

13. D₃₇×C₅²⋊C₃

D₃

direct (N × H)

16. C₃₇×C₅²⋊C₃

C₂×D₃

semidirect

17. C₃×C₅²×C₃₇

C₂²×C₃

semidirect

21. C₅²×D₃₇

C₃×D₃

semidirect

26. C₅²×C₃₇

C₂×C₃×D₃

semidirect

35. D₃×C₅²⋊C₃

D₃₇

direct (N × H)

37. D₃×D₃₇

C₅²⋊C₃

direct (N × H)

44. C₃×C₅²⋊C₃

C₂×D₃₇

semidirect

46. D₁₁₁

C₂×C₅²⋊C₃

semidirect

50. C₃×D₃₇

C₂×C₅²⋊C₃

semidirect

51. D₃×C₃₇

C₂×C₅²⋊C₃

semidirect

55. C₅²×D₃

C₃×D₃₇

semidirect

63. C₃×C₃₇

C₂²×C₅²⋊C₃

semidirect

65. C₃×C₅²

C₂×C₃×D₃₇

semidirect

67. C₅²⋊C₃

D₃×D₃₇

direct (N × H)

70. D₃₇

D₃×C₅²⋊C₃

direct (N × H)

75. C₃₇

C₂×D₃×C₅²⋊C₃

semidirect

80. C₅²

C₃×D₃×D₃₇

semidirect

98. D₃

D₃₇×C₅²⋊C₃

direct (N × H)

103. C₃

C₂×D₃₇×C₅²⋊C₃

semidirect

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

subgroup	index
C₅²⋊C₃×D₁₁₁	2
D₃×C₃₇×C₅²⋊C₃	2
C₃×D₃₇×C₅²⋊C₃	2
C₅²×D₃×D₃₇	3
C₂×D₃₇×C₅²⋊C₃	3
C₃×D₃×D₃₇	25
C₂×D₃×C₅²⋊C₃	37

subgroup

index

C₅²⋊C₃×D₁₁₁

D₃×C₃₇×C₅²⋊C₃

C₃×D₃₇×C₅²⋊C₃

C₅²×D₃×D₃₇

C₂×D₃₇×C₅²⋊C₃

C₃×D₃×D₃₇

C₂×D₃×C₅²⋊C₃

Subgroups — D₃×D₃₇×C₅²⋊C₃ #45686246

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D3×D37×C52⋊C3 #45686246

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×D₃₇×C₅²⋊C₃ #45686246