Group #49178384 — subgroups

Subgroups — C₅×D₉×D₂₀₉ #49178384

Order 37620.

120 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 42 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	C₅×D₁₈₈₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×C₁₁×D₉×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×C₉×D₂₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×D₃×D₂₀₉	—	3	1	no	maximal
5	C₅×C₉×C₁₁×C₁₉	C₂²	1	1	yes	characteristic, complemented, Hall
6	D₉×D₂₀₉	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₅×D₆₂₇	—	3	1	no	—
8	C₅×D₃×C₁₁×C₁₉	—	3	1	no	—
9	C₃×C₅×D₂₀₉	D₃	1	1	yes	characteristic
10	C₂×C₅×D₂₀₉	—	9	1	no	Hall
11	D₁₈₈₁	C₂×C₅	1	1	yes	characteristic, complemented
12	C₁₁×D₉×C₁₉	C₂×C₅	1	1	yes	characteristic, complemented
13	C₉×D₂₀₉	C₂×C₅	1	1	yes	characteristic, complemented
14	C₅×D₉×D₁₉	—	11	1	no	maximal, Hall
15	C₃×C₅×C₁₁×C₁₉	C₂×D₃	1	1	yes	characteristic
16	D₃×D₂₀₉	—	3	1	no	—
17	C₅×D₂₀₉	—	9	1	no	—
18	C₅×D₂₀₉	D₉	1	1	yes	characteristic, direct product factor
19	C₂×C₅×C₁₁×C₁₉	—	9	1	no	—
20	C₅×D₉×D₁₁	—	19	1	no	maximal, Hall
21	C₉×C₁₁×C₁₉	C₂²×C₅	1	1	yes	characteristic, complemented, Hall
22	C₅×D₁₇₁	—	11	1	no	—
23	C₅×D₉×C₁₉	D₁₁	1	1	yes	characteristic, complemented
24	C₅×C₉×D₁₉	—	11	1	no	—
25	D₆₂₇	—	3	1	no	—
26	D₃×C₁₁×C₁₉	—	3	1	no	—
27	C₃×D₂₀₉	C₅×D₃	1	1	yes	characteristic
28	C₅×D₃×D₁₉	—	33	1	no	—
29	C₅×C₁₁×C₁₉	C₂×D₉	1	1	yes	characteristic, complemented, Hall
30	C₅×D₉₉	—	19	1	no	—
31	C₅×C₁₁×D₉	D₁₉	1	1	yes	characteristic, complemented
32	C₅×C₉×D₁₁	—	19	1	no	—
33	C₅×C₉×C₁₉	C₂×D₁₁	1	1	yes	characteristic, complemented, Hall
34	C₂×D₂₀₉	—	9	1	no	Hall
35	D₉×D₁₉	—	11	1	no	Hall
36	C₅×D₃×D₁₁	—	57	1	no	—
37	C₃×C₁₁×C₁₉	C₂×C₅×D₃	1	1	yes	characteristic
38	C₅×D₅₇	—	33	1	no	—
39	C₅×D₃×C₁₉	—	3	1	no	—
40	C₃×C₅×D₁₉	—	11	1	no	—
41	C₅×C₉×C₁₁	C₂×D₁₉	1	1	yes	characteristic, complemented, Hall
42	C₂×C₁₁×C₁₉	—	9	1	no	—
43	D₂₀₉	—	9	1	no	—
44	D₂₀₉	C₅×D₉	1	1	yes	characteristic, direct product factor
45	D₉×D₁₁	—	19	1	no	Hall
46	C₂×C₅×D₁₉	—	99	1	no	Hall
47	D₁₇₁	—	11	1	no	—
48	C₉×D₁₉	—	11	1	no	—
49	D₉×C₁₉	C₅×D₁₁	1	1	yes	characteristic, complemented
50	C₅×D₃₃	—	57	1	no	—
51	C₅×D₃×C₁₁	—	3	1	no	—
52	C₃×C₅×D₁₁	—	19	1	no	—
53	C₃×C₅×C₁₉	D₃×D₁₁	1	1	yes	characteristic
54	D₃×D₁₉	—	33	1	no	—
55	C₂×C₅×D₁₁	—	171	1	no	Hall
56	C₁₁×C₁₉	C₂×C₅×D₉	1	1	yes	characteristic, complemented, Hall
57	D₉₉	—	19	1	no	—
58	C₉×D₁₁	—	19	1	no	—
59	C₁₁×D₉	C₅×D₁₉	1	1	yes	characteristic, complemented
60	C₂×C₅×C₁₉	—	9	1	no	—
61	C₅×D₁₉	—	11	1	no	—
62	C₅×D₁₉	—	99	1	no	—
63	C₂×C₅×D₉	—	209	1	no	Hall
64	C₉×C₁₉	C₂×C₅×D₁₁	1	1	yes	characteristic, complemented, Hall
65	C₃×C₅×C₁₁	D₃×D₁₉	1	1	yes	characteristic
66	D₃×D₁₁	—	57	1	no	—
67	D₅₇	—	33	1	no	—
68	C₃×D₁₉	—	11	1	no	—
69	D₃×C₁₉	—	3	1	no	—
70	C₂×C₅×C₁₁	—	9	1	no	—
71	C₅×D₁₁	—	19	1	no	—
72	C₅×D₁₁	—	171	1	no	—
73	C₉×C₁₁	C₂×C₅×D₁₉	1	1	yes	characteristic, complemented, Hall
74	C₅×C₁₉	D₉×D₁₁	1	1	yes	characteristic, complemented, Hall
75	C₂×C₅×C₉	—	209	1	no	—
76	C₅×D₉	—	209	1	no	—
77	C₅×D₉	D₂₀₉	1	1	yes	characteristic, direct product factor
78	C₂×D₁₉	—	99	1	no	Hall
79	D₃₃	—	57	1	no	—
80	C₃×D₁₁	—	19	1	no	—
81	D₃×C₁₁	—	3	1	no	—
82	C₂×C₅×D₃	—	627	1	no	—
83	C₃×C₁₉	C₅×D₃×D₁₁	1	1	yes	characteristic
84	C₅×C₁₁	D₉×D₁₉	1	1	yes	characteristic, complemented, Hall
85	C₅×C₉	C₂×D₂₀₉	1	1	yes	characteristic, complemented, Hall
86	C₂×D₁₁	—	171	1	no	Hall
87	C₂×C₁₉	—	9	1	no	—
88	D₁₉	—	11	1	no	—
89	D₁₉	—	99	1	no	—
90	C₂×D₉	—	209	1	no	Hall
91	C₃×C₁₁	C₅×D₃×D₁₉	1	1	yes	characteristic
92	C₂×C₃×C₅	—	209	1	no	—
93	C₅×D₃	—	3	1	no	—
94	C₅×D₃	—	627	1	no	—
95	C₂×C₁₁	—	9	1	no	—
96	D₁₁	—	19	1	no	—
97	D₁₁	—	171	1	no	—
98	C₂²×C₅	—	1881	1	no	Hall
99	C₁₉	C₅×D₉×D₁₁	1	1	yes	characteristic, complemented, Sylow
100	C₂×C₉	—	209	1	no	—
101	D₉	—	209	1	no	—
102	D₉	C₅×D₂₀₉	1	1	yes	characteristic, direct product factor
103	C₃×C₅	D₃×D₂₀₉	1	1	yes	characteristic
104	C₂×D₃	—	627	1	no	—
105	C₁₁	C₅×D₉×D₁₉	1	1	yes	characteristic, complemented, Sylow
106	C₂×C₅	—	9	1	no	—
107	C₂×C₅	—	1881	1	no	—
108	C₂×C₅	—	209	1	no	—
109	C₉	C₂×C₅×D₂₀₉	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₃	—	209	1	no	—
111	D₃	—	3	1	no	—
112	D₃	—	627	1	no	—
113	C₅	D₉×D₂₀₉	1	1	yes	characteristic, direct product factor, Sylow
114	C₂²	—	1881	1	no	Sylow
115	C₃	C₅×D₃×D₂₀₉	1	1	yes	characteristic
116	C₂	—	9	1	no	—
117	C₂	—	1881	1	no	—
118	C₂	—	209	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×D₁₈₈₁

C₂

yes

maximal, characteristic, complemented

C₅×C₁₁×D₉×C₁₉

C₂

yes

maximal, characteristic, complemented

C₅×C₉×D₂₀₉

C₂

yes

maximal, characteristic, complemented

C₅×D₃×D₂₀₉

—

maximal

C₅×C₉×C₁₁×C₁₉

C₂²

yes

characteristic, complemented, Hall

D₉×D₂₀₉

C₅

yes

maximal, characteristic, direct product factor, Hall

C₅×D₆₂₇

—

C₅×D₃×C₁₁×C₁₉

—

C₃×C₅×D₂₀₉

D₃

yes

characteristic

C₂×C₅×D₂₀₉

—

Hall

D₁₈₈₁

C₂×C₅

yes

characteristic, complemented

C₁₁×D₉×C₁₉

C₂×C₅

yes

characteristic, complemented

C₉×D₂₀₉

C₂×C₅

yes

characteristic, complemented

C₅×D₉×D₁₉

—

maximal, Hall

C₃×C₅×C₁₁×C₁₉

C₂×D₃

yes

characteristic

D₃×D₂₀₉

—

C₅×D₂₀₉

—

C₅×D₂₀₉

D₉

yes

characteristic, direct product factor

C₂×C₅×C₁₁×C₁₉

—

C₅×D₉×D₁₁

—

maximal, Hall

C₉×C₁₁×C₁₉

C₂²×C₅

yes

characteristic, complemented, Hall

C₅×D₁₇₁

—

C₅×D₉×C₁₉

D₁₁

yes

characteristic, complemented

C₅×C₉×D₁₉

—

D₆₂₇

—

D₃×C₁₁×C₁₉

—

C₃×D₂₀₉

C₅×D₃

yes

characteristic

C₅×D₃×D₁₉

—

C₅×C₁₁×C₁₉

C₂×D₉

yes

characteristic, complemented, Hall

C₅×D₉₉

—

C₅×C₁₁×D₉

D₁₉

yes

characteristic, complemented

C₅×C₉×D₁₁

—

C₅×C₉×C₁₉

C₂×D₁₁

yes

characteristic, complemented, Hall

C₂×D₂₀₉

—

Hall

D₉×D₁₉

—

Hall

C₅×D₃×D₁₁

—

C₃×C₁₁×C₁₉

C₂×C₅×D₃

yes

characteristic

C₅×D₅₇

—

C₅×D₃×C₁₉

—

C₃×C₅×D₁₉

—

C₅×C₉×C₁₁

C₂×D₁₉

yes

characteristic, complemented, Hall

C₂×C₁₁×C₁₉

—

D₂₀₉

—

D₂₀₉

C₅×D₉

yes

characteristic, direct product factor

D₉×D₁₁

—

Hall

C₂×C₅×D₁₉

—

Hall

D₁₇₁

—

C₉×D₁₉

—

D₉×C₁₉

C₅×D₁₁

yes

characteristic, complemented

C₅×D₃₃

—

C₅×D₃×C₁₁

—

C₃×C₅×D₁₁

—

C₃×C₅×C₁₉

D₃×D₁₁

yes

characteristic

D₃×D₁₉

—

C₂×C₅×D₁₁

—

171

Hall

C₁₁×C₁₉

C₂×C₅×D₉

yes

characteristic, complemented, Hall

D₉₉

—

C₉×D₁₁

—

C₁₁×D₉

C₅×D₁₉

yes

characteristic, complemented

C₂×C₅×C₁₉

—

C₅×D₁₉

—

C₅×D₁₉

—

C₂×C₅×D₉

—

209

Hall

C₉×C₁₉

C₂×C₅×D₁₁

yes

characteristic, complemented, Hall

C₃×C₅×C₁₁

D₃×D₁₉

yes

characteristic

D₃×D₁₁

—

D₅₇

—

C₃×D₁₉

—

D₃×C₁₉

—

C₂×C₅×C₁₁

—

C₅×D₁₁

—

C₅×D₁₁

—

171

—

C₉×C₁₁

C₂×C₅×D₁₉

yes

characteristic, complemented, Hall

C₅×C₁₉

D₉×D₁₁

yes

characteristic, complemented, Hall

C₂×C₅×C₉

—

209

—

C₅×D₉

—

209

—

C₅×D₉

D₂₀₉

yes

characteristic, direct product factor

C₂×D₁₉

—

Hall

D₃₃

—

C₃×D₁₁

—

D₃×C₁₁

—

C₂×C₅×D₃

—

627

—

C₃×C₁₉

C₅×D₃×D₁₁

yes

characteristic

C₅×C₁₁

D₉×D₁₉

yes

characteristic, complemented, Hall

C₅×C₉

C₂×D₂₀₉

yes

characteristic, complemented, Hall

C₂×D₁₁

—

171

Hall

C₂×C₁₉

—

D₁₉

—

D₁₉

—

C₂×D₉

—

209

Hall

C₃×C₁₁

C₅×D₃×D₁₉

yes

characteristic

C₂×C₃×C₅

—

209

—

C₅×D₃

—

C₅×D₃

—

627

—

C₂×C₁₁

—

D₁₁

—

D₁₁

—

171

—

C₂²×C₅

—

1881

Hall

C₁₉

C₅×D₉×D₁₁

yes

characteristic, complemented, Sylow

100

C₂×C₉

—

209

—

101

D₉

—

209

—

102

D₉

C₅×D₂₀₉

yes

characteristic, direct product factor

103

C₃×C₅

D₃×D₂₀₉

yes

characteristic

104

C₂×D₃

—

627

—

105

C₁₁

C₅×D₉×D₁₉

yes

characteristic, complemented, Sylow

106

C₂×C₅

—

107

C₂×C₅

—

1881

—

108

C₂×C₅

—

209

—

109

C₉

C₂×C₅×D₂₀₉

yes

characteristic, complemented, Sylow

110

C₂×C₃

—

209

—

111

D₃

—

112

D₃

—

627

—

113

C₅

D₉×D₂₀₉

yes

characteristic, direct product factor, Sylow

114

C₂²

—

1881

Sylow

115

C₃

C₅×D₃×D₂₀₉

yes

characteristic

116

C₂

—

117

C₂

—

1881

—

118

C₂

—

209

—

	subgroup	G/H
center, hypercenter	C₅	D₉×D₂₀₉
commutator, cosocle, hypocenter	C₉×C₁₁×C₁₉	C₂²×C₅
Frattini	C₃	C₅×D₃×D₂₀₉
Fitting	C₅×C₉×C₁₁×C₁₉	C₂²
socle	C₃×C₅×C₁₁×C₁₉	C₂×D₃
LCS	G ▷ C₉×C₁₁×C₁₉
UCS	C₁ ◁ C₅

subgroup

G/H

center, hypercenter

C₅

D₉×D₂₀₉

commutator, cosocle, hypocenter

C₉×C₁₁×C₁₉

C₂²×C₅

Frattini

C₃

C₅×D₃×D₂₀₉

Fitting

C₅×C₉×C₁₁×C₁₉

C₂²

socle

C₃×C₅×C₁₁×C₁₉

C₂×D₃

LCS

G ▷ C₉×C₁₁×C₁₉

UCS

C₁ ◁ C₅

N	H ≅ G/N	type
1. C₅×D₁₈₈₁	C₂	semidirect
2. C₅×C₁₁×D₉×C₁₉	C₂	semidirect
3. C₅×C₉×D₂₀₉	C₂	semidirect
5. C₅×C₉×C₁₁×C₁₉	C₂²	semidirect
6. D₉×D₂₀₉	C₅	direct (N × H)
11. D₁₈₈₁	C₂×C₅	semidirect
12. C₁₁×D₉×C₁₉	C₂×C₅	semidirect
13. C₉×D₂₀₉	C₂×C₅	semidirect
18. C₅×D₂₀₉	D₉	direct (N × H)
21. C₉×C₁₁×C₁₉	C₂²×C₅	semidirect
23. C₅×D₉×C₁₉	D₁₁	semidirect
29. C₅×C₁₁×C₁₉	C₂×D₉	semidirect
31. C₅×C₁₁×D₉	D₁₉	semidirect
33. C₅×C₉×C₁₉	C₂×D₁₁	semidirect
41. C₅×C₉×C₁₁	C₂×D₁₉	semidirect
44. D₂₀₉	C₅×D₉	direct (N × H)
49. D₉×C₁₉	C₅×D₁₁	semidirect
56. C₁₁×C₁₉	C₂×C₅×D₉	semidirect
59. C₁₁×D₉	C₅×D₁₉	semidirect
64. C₉×C₁₉	C₂×C₅×D₁₁	semidirect
73. C₉×C₁₁	C₂×C₅×D₁₉	semidirect
74. C₅×C₁₉	D₉×D₁₁	semidirect
77. C₅×D₉	D₂₀₉	direct (N × H)
84. C₅×C₁₁	D₉×D₁₉	semidirect
85. C₅×C₉	C₂×D₂₀₉	semidirect
99. C₁₉	C₅×D₉×D₁₁	semidirect
102. D₉	C₅×D₂₀₉	direct (N × H)
105. C₁₁	C₅×D₉×D₁₉	semidirect
109. C₉	C₂×C₅×D₂₀₉	semidirect
113. C₅	D₉×D₂₀₉	direct (N × H)

H ≅ G/N

type

1. C₅×D₁₈₈₁

C₂

semidirect

2. C₅×C₁₁×D₉×C₁₉

C₂

semidirect

3. C₅×C₉×D₂₀₉

C₂

semidirect

5. C₅×C₉×C₁₁×C₁₉

C₂²

semidirect

6. D₉×D₂₀₉

C₅

direct (N × H)

11. D₁₈₈₁

C₂×C₅

semidirect

12. C₁₁×D₉×C₁₉

C₂×C₅

semidirect

13. C₉×D₂₀₉

C₂×C₅

semidirect

18. C₅×D₂₀₉

D₉

direct (N × H)

21. C₉×C₁₁×C₁₉

C₂²×C₅

semidirect

23. C₅×D₉×C₁₉

D₁₁

semidirect

29. C₅×C₁₁×C₁₉

C₂×D₉

semidirect

31. C₅×C₁₁×D₉

D₁₉

semidirect

33. C₅×C₉×C₁₉

C₂×D₁₁

semidirect

41. C₅×C₉×C₁₁

C₂×D₁₉

semidirect

44. D₂₀₉

C₅×D₉

direct (N × H)

49. D₉×C₁₉

C₅×D₁₁

semidirect

56. C₁₁×C₁₉

C₂×C₅×D₉

semidirect

59. C₁₁×D₉

C₅×D₁₉

semidirect

64. C₉×C₁₉

C₂×C₅×D₁₁

semidirect

73. C₉×C₁₁

C₂×C₅×D₁₉

semidirect

74. C₅×C₁₉

D₉×D₁₁

semidirect

77. C₅×D₉

D₂₀₉

direct (N × H)

84. C₅×C₁₁

D₉×D₁₉

semidirect

85. C₅×C₉

C₂×D₂₀₉

semidirect

99. C₁₉

C₅×D₉×D₁₁

semidirect

102. D₉

C₅×D₂₀₉

direct (N × H)

105. C₁₁

C₅×D₉×D₁₉

semidirect

109. C₉

C₂×C₅×D₂₀₉

semidirect

113. C₅

D₉×D₂₀₉

direct (N × H)

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

subgroup	index
C₅×D₁₈₈₁	2
C₅×C₁₁×D₉×C₁₉	2
C₅×C₉×D₂₀₉	2
C₅×D₃×D₂₀₉	3
D₉×D₂₀₉	5
C₅×D₉×D₁₉	11
C₅×D₉×D₁₁	19

subgroup

index

C₅×D₁₈₈₁

C₅×C₁₁×D₉×C₁₉

C₅×C₉×D₂₀₉

C₅×D₃×D₂₀₉

D₉×D₂₀₉

C₅×D₉×D₁₉

C₅×D₉×D₁₁

Subgroups — C₅×D₉×D₂₀₉ #49178384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×D9×D209 #49178384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×D₉×D₂₀₉ #49178384