Group #49178404 — subgroups

Subgroups — D₉×C₁₉×D₅₅ #49178404

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	C₁₉×D₄₉₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×C₁₁×D₉×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×C₁₉×D₅₅	C₂	1	1	yes	maximal, characteristic, complemented
4	D₃×C₁₉×D₅₅	—	3	1	no	maximal
5	C₅×C₉×C₁₁×C₁₉	C₂²	1	1	yes	characteristic, complemented, Hall
6	D₉×C₁₉×D₁₁	—	5	1	no	maximal, Hall
7	C₁₉×D₁₆₅	—	3	1	no	—
8	C₅×D₃×C₁₁×C₁₉	—	3	1	no	—
9	C₃×C₁₉×D₅₅	D₃	1	1	yes	characteristic
10	C₂×C₁₉×D₅₅	—	9	1	no	Hall
11	C₁₉×D₉₉	—	5	1	no	—
12	C₁₁×D₉×C₁₉	D₅	1	1	yes	characteristic, complemented
13	C₉×C₁₉×D₁₁	—	5	1	no	—
14	D₅×D₉×C₁₉	—	11	1	no	maximal, Hall
15	C₃×C₅×C₁₁×C₁₉	C₂×D₃	1	1	yes	characteristic
16	D₃×C₁₉×D₁₁	—	15	1	no	—
17	C₁₉×D₅₅	—	9	1	no	—
18	C₁₉×D₅₅	D₉	1	1	yes	characteristic, direct product factor
19	C₂×C₅×C₁₁×C₁₉	—	9	1	no	—
20	D₉×D₅₅	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₉×C₁₁×C₁₉	C₂×D₅	1	1	yes	characteristic, complemented, Hall
22	C₁₉×D₄₅	—	11	1	no	—
23	C₅×D₉×C₁₉	D₁₁	1	1	yes	characteristic, complemented
24	C₉×D₅×C₁₉	—	11	1	no	—
25	C₁₉×D₃₃	—	15	1	no	—
26	D₃×C₁₁×C₁₉	—	3	1	no	—
27	C₃×C₁₉×D₁₁	—	5	1	no	—
28	D₃×D₅×C₁₉	—	33	1	no	—
29	C₅×C₁₁×C₁₉	C₂×D₉	1	1	yes	characteristic, complemented, Hall
30	D₄₉₅	C₂×C₁₉	1	1	yes	characteristic, complemented
31	C₅×C₁₁×D₉	C₂×C₁₉	1	1	yes	characteristic, complemented
32	C₉×D₅₅	C₂×C₁₉	1	1	yes	characteristic, complemented
33	C₅×C₉×C₁₉	C₂×D₁₁	1	1	yes	characteristic, complemented, Hall
34	C₂×C₁₉×D₁₁	—	45	1	no	Hall
35	C₂×D₉×C₁₉	—	55	1	no	Hall
36	D₃×D₅₅	—	3	1	no	—
37	C₃×C₁₁×C₁₉	D₃×D₅	1	1	yes	characteristic
38	C₁₉×D₁₅	—	33	1	no	—
39	C₅×D₃×C₁₉	—	3	1	no	—
40	C₃×D₅×C₁₉	—	11	1	no	—
41	C₅×C₉×C₁₁	C₂²×C₁₉	1	1	yes	characteristic, complemented, Hall
42	C₂×C₁₁×C₁₉	—	9	1	no	—
43	C₁₉×D₁₁	—	5	1	no	—
44	C₁₉×D₁₁	—	45	1	no	—
45	D₉×D₁₁	—	5	1	no	Hall
46	C₂×D₅×C₁₉	—	99	1	no	Hall
47	C₂×C₉×C₁₉	—	55	1	no	—
48	D₉×C₁₉	—	55	1	no	—
49	D₉×C₁₉	D₅₅	1	1	yes	characteristic, direct product factor
50	D₁₆₅	—	3	1	no	—
51	C₅×D₃×C₁₁	—	3	1	no	—
52	C₃×D₅₅	D₃×C₁₉	1	1	yes	characteristic
53	C₃×C₅×C₁₉	D₃×D₁₁	1	1	yes	characteristic
54	C₂×D₃×C₁₉	—	165	1	no	—
55	C₂×D₅₅	—	9	1	no	Hall
56	C₁₁×C₁₉	D₅×D₉	1	1	yes	characteristic, complemented, Hall
57	D₉₉	—	5	1	no	—
58	C₉×D₁₁	—	5	1	no	—
59	C₁₁×D₉	D₅×C₁₉	1	1	yes	characteristic, complemented
60	C₂×C₅×C₁₉	—	9	1	no	—
61	D₅×C₁₉	—	11	1	no	—
62	D₅×C₁₉	—	99	1	no	—
63	D₅×D₉	—	11	1	no	Hall
64	C₉×C₁₉	C₂×D₅₅	1	1	yes	characteristic, complemented, Hall
65	C₃×C₅×C₁₁	C₂×D₃×C₁₉	1	1	yes	characteristic
66	D₃×D₁₁	—	15	1	no	—
67	C₂×C₃×C₁₉	—	55	1	no	—
68	D₃×C₁₉	—	3	1	no	—
69	D₃×C₁₉	—	165	1	no	—
70	C₂×C₅×C₁₁	—	9	1	no	—
71	D₅₅	—	9	1	no	—
72	D₅₅	D₉×C₁₉	1	1	yes	characteristic, direct product factor
73	C₉×C₁₁	C₂×D₅×C₁₉	1	1	yes	characteristic, complemented, Hall
74	C₅×C₁₉	D₉×D₁₁	1	1	yes	characteristic, complemented, Hall
75	D₄₅	—	11	1	no	—
76	C₉×D₅	—	11	1	no	—
77	C₅×D₉	C₁₉×D₁₁	1	1	yes	characteristic, complemented
78	C₂²×C₁₉	—	495	1	no	Hall
79	D₃₃	—	15	1	no	—
80	C₃×D₁₁	—	5	1	no	—
81	D₃×C₁₁	—	3	1	no	—
82	D₃×D₅	—	33	1	no	—
83	C₃×C₁₉	D₃×D₅₅	1	1	yes	characteristic
84	C₅×C₁₁	C₂×D₉×C₁₉	1	1	yes	characteristic, complemented, Hall
85	C₅×C₉	C₂×C₁₉×D₁₁	1	1	yes	characteristic, complemented, Hall
86	C₂×D₁₁	—	45	1	no	Hall
87	C₂×C₁₉	—	9	1	no	—
88	C₂×C₁₉	—	55	1	no	—
89	C₂×C₁₉	—	495	1	no	—
90	C₂×D₉	—	55	1	no	Hall
91	C₃×C₁₁	D₃×D₅×C₁₉	1	1	yes	characteristic
92	D₁₅	—	33	1	no	—
93	C₃×D₅	—	11	1	no	—
94	C₅×D₃	—	3	1	no	—
95	C₂×C₁₁	—	9	1	no	—
96	D₁₁	—	5	1	no	—
97	D₁₁	—	45	1	no	—
98	C₂×D₅	—	99	1	no	Hall
99	C₁₉	D₉×D₅₅	1	1	yes	characteristic, direct product factor, Sylow
100	C₂×C₉	—	55	1	no	—
101	D₉	—	55	1	no	—
102	D₉	C₁₉×D₅₅	1	1	yes	characteristic, direct product factor
103	C₃×C₅	D₃×C₁₉×D₁₁	1	1	yes	characteristic
104	C₂×D₃	—	165	1	no	—
105	C₁₁	D₅×D₉×C₁₉	1	1	yes	characteristic, complemented, Sylow
106	C₂×C₅	—	9	1	no	—
107	D₅	—	11	1	no	—
108	D₅	—	99	1	no	—
109	C₉	C₂×C₁₉×D₅₅	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₃	—	55	1	no	—
111	D₃	—	3	1	no	—
112	D₃	—	165	1	no	—
113	C₅	D₉×C₁₉×D₁₁	1	1	yes	characteristic, complemented, Sylow
114	C₂²	—	495	1	no	Sylow
115	C₃	D₃×C₁₉×D₅₅	1	1	yes	characteristic
116	C₂	—	9	1	no	—
117	C₂	—	55	1	no	—
118	C₂	—	495	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×D₄₉₅

C₂

yes

maximal, characteristic, complemented

C₅×C₁₁×D₉×C₁₉

C₂

yes

maximal, characteristic, complemented

C₉×C₁₉×D₅₅

C₂

yes

maximal, characteristic, complemented

D₃×C₁₉×D₅₅

—

maximal

C₅×C₉×C₁₁×C₁₉

C₂²

yes

characteristic, complemented, Hall

D₉×C₁₉×D₁₁

—

maximal, Hall

C₁₉×D₁₆₅

—

C₅×D₃×C₁₁×C₁₉

—

C₃×C₁₉×D₅₅

D₃

yes

characteristic

C₂×C₁₉×D₅₅

—

Hall

C₁₉×D₉₉

—

C₁₁×D₉×C₁₉

D₅

yes

characteristic, complemented

C₉×C₁₉×D₁₁

—

D₅×D₉×C₁₉

—

maximal, Hall

C₃×C₅×C₁₁×C₁₉

C₂×D₃

yes

characteristic

D₃×C₁₉×D₁₁

—

C₁₉×D₅₅

—

C₁₉×D₅₅

D₉

yes

characteristic, direct product factor

C₂×C₅×C₁₁×C₁₉

—

D₉×D₅₅

C₁₉

yes

maximal, characteristic, direct product factor, Hall

C₉×C₁₁×C₁₉

C₂×D₅

yes

characteristic, complemented, Hall

C₁₉×D₄₅

—

C₅×D₉×C₁₉

D₁₁

yes

characteristic, complemented

C₉×D₅×C₁₉

—

C₁₉×D₃₃

—

D₃×C₁₁×C₁₉

—

C₃×C₁₉×D₁₁

—

D₃×D₅×C₁₉

—

C₅×C₁₁×C₁₉

C₂×D₉

yes

characteristic, complemented, Hall

D₄₉₅

C₂×C₁₉

yes

characteristic, complemented

C₅×C₁₁×D₉

C₂×C₁₉

yes

characteristic, complemented

C₉×D₅₅

C₂×C₁₉

yes

characteristic, complemented

C₅×C₉×C₁₉

C₂×D₁₁

yes

characteristic, complemented, Hall

C₂×C₁₉×D₁₁

—

Hall

C₂×D₉×C₁₉

—

Hall

D₃×D₅₅

—

C₃×C₁₁×C₁₉

D₃×D₅

yes

characteristic

C₁₉×D₁₅

—

C₅×D₃×C₁₉

—

C₃×D₅×C₁₉

—

C₅×C₉×C₁₁

C₂²×C₁₉

yes

characteristic, complemented, Hall

C₂×C₁₁×C₁₉

—

C₁₉×D₁₁

—

C₁₉×D₁₁

—

D₉×D₁₁

—

Hall

C₂×D₅×C₁₉

—

Hall

C₂×C₉×C₁₉

—

D₉×C₁₉

—

D₉×C₁₉

D₅₅

yes

characteristic, direct product factor

D₁₆₅

—

C₅×D₃×C₁₁

—

C₃×D₅₅

D₃×C₁₉

yes

characteristic

C₃×C₅×C₁₉

D₃×D₁₁

yes

characteristic

C₂×D₃×C₁₉

—

165

—

C₂×D₅₅

—

Hall

C₁₁×C₁₉

D₅×D₉

yes

characteristic, complemented, Hall

D₉₉

—

C₉×D₁₁

—

C₁₁×D₉

D₅×C₁₉

yes

characteristic, complemented

C₂×C₅×C₁₉

—

D₅×C₁₉

—

D₅×C₁₉

—

D₅×D₉

—

Hall

C₉×C₁₉

C₂×D₅₅

yes

characteristic, complemented, Hall

C₃×C₅×C₁₁

C₂×D₃×C₁₉

yes

characteristic

D₃×D₁₁

—

C₂×C₃×C₁₉

—

D₃×C₁₉

—

D₃×C₁₉

—

165

—

C₂×C₅×C₁₁

—

D₅₅

—

D₅₅

D₉×C₁₉

yes

characteristic, direct product factor

C₉×C₁₁

C₂×D₅×C₁₉

yes

characteristic, complemented, Hall

C₅×C₁₉

D₉×D₁₁

yes

characteristic, complemented, Hall

D₄₅

—

C₉×D₅

—

C₅×D₉

C₁₉×D₁₁

yes

characteristic, complemented

C₂²×C₁₉

—

495

Hall

D₃₃

—

C₃×D₁₁

—

D₃×C₁₁

—

D₃×D₅

—

C₃×C₁₉

D₃×D₅₅

yes

characteristic

C₅×C₁₁

C₂×D₉×C₁₉

yes

characteristic, complemented, Hall

C₅×C₉

C₂×C₁₉×D₁₁

yes

characteristic, complemented, Hall

C₂×D₁₁

—

Hall

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

—

495

—

C₂×D₉

—

Hall

C₃×C₁₁

D₃×D₅×C₁₉

yes

characteristic

D₁₅

—

C₃×D₅

—

C₅×D₃

—

C₂×C₁₁

—

D₁₁

—

D₁₁

—

C₂×D₅

—

Hall

C₁₉

D₉×D₅₅

yes

characteristic, direct product factor, Sylow

100

C₂×C₉

—

101

D₉

—

102

D₉

C₁₉×D₅₅

yes

characteristic, direct product factor

103

C₃×C₅

D₃×C₁₉×D₁₁

yes

characteristic

104

C₂×D₃

—

165

—

105

C₁₁

D₅×D₉×C₁₉

yes

characteristic, complemented, Sylow

106

C₂×C₅

—

107

D₅

—

108

D₅

—

109

C₉

C₂×C₁₉×D₅₅

yes

characteristic, complemented, Sylow

110

C₂×C₃

—

111

D₃

—

112

D₃

—

165

—

113

C₅

D₉×C₁₉×D₁₁

yes

characteristic, complemented, Sylow

114

C₂²

—

495

Sylow

115

C₃

D₃×C₁₉×D₅₅

yes

characteristic

116

C₂

—

117

C₂

—

118

C₂

—

495

—

	subgroup	G/H
center, hypercenter	C₁₉	D₉×D₅₅
commutator, cosocle, hypocenter	C₅×C₉×C₁₁	C₂²×C₁₉
Frattini	C₃	D₃×C₁₉×D₅₅
Fitting	C₅×C₉×C₁₁×C₁₉	C₂²
socle	C₃×C₅×C₁₁×C₁₉	C₂×D₃
LCS	G ▷ C₅×C₉×C₁₁
UCS	C₁ ◁ C₁₉

subgroup

G/H

center, hypercenter

C₁₉

D₉×D₅₅

commutator, cosocle, hypocenter

C₅×C₉×C₁₁

C₂²×C₁₉

Frattini

C₃

D₃×C₁₉×D₅₅

Fitting

C₅×C₉×C₁₁×C₁₉

C₂²

socle

C₃×C₅×C₁₁×C₁₉

C₂×D₃

LCS

G ▷ C₅×C₉×C₁₁

UCS

C₁ ◁ C₁₉

N	H ≅ G/N	type
1. C₁₉×D₄₉₅	C₂	semidirect
2. C₅×C₁₁×D₉×C₁₉	C₂	semidirect
3. C₉×C₁₉×D₅₅	C₂	semidirect
5. C₅×C₉×C₁₁×C₁₉	C₂²	semidirect
12. C₁₁×D₉×C₁₉	D₅	semidirect
18. C₁₉×D₅₅	D₉	direct (N × H)
20. D₉×D₅₅	C₁₉	direct (N × H)
21. C₉×C₁₁×C₁₉	C₂×D₅	semidirect
23. C₅×D₉×C₁₉	D₁₁	semidirect
29. C₅×C₁₁×C₁₉	C₂×D₉	semidirect
30. D₄₉₅	C₂×C₁₉	semidirect
31. C₅×C₁₁×D₉	C₂×C₁₉	semidirect
32. C₉×D₅₅	C₂×C₁₉	semidirect
33. C₅×C₉×C₁₉	C₂×D₁₁	semidirect
41. C₅×C₉×C₁₁	C₂²×C₁₉	semidirect
49. D₉×C₁₉	D₅₅	direct (N × H)
56. C₁₁×C₁₉	D₅×D₉	semidirect
59. C₁₁×D₉	D₅×C₁₉	semidirect
64. C₉×C₁₉	C₂×D₅₅	semidirect
72. D₅₅	D₉×C₁₉	direct (N × H)
73. C₉×C₁₁	C₂×D₅×C₁₉	semidirect
74. C₅×C₁₉	D₉×D₁₁	semidirect
77. C₅×D₉	C₁₉×D₁₁	semidirect
84. C₅×C₁₁	C₂×D₉×C₁₉	semidirect
85. C₅×C₉	C₂×C₁₉×D₁₁	semidirect
99. C₁₉	D₉×D₅₅	direct (N × H)
102. D₉	C₁₉×D₅₅	direct (N × H)
105. C₁₁	D₅×D₉×C₁₉	semidirect
109. C₉	C₂×C₁₉×D₅₅	semidirect
113. C₅	D₉×C₁₉×D₁₁	semidirect

H ≅ G/N

type

1. C₁₉×D₄₉₅

C₂

semidirect

2. C₅×C₁₁×D₉×C₁₉

C₂

semidirect

3. C₉×C₁₉×D₅₅

C₂

semidirect

5. C₅×C₉×C₁₁×C₁₉

C₂²

semidirect

12. C₁₁×D₉×C₁₉

D₅

semidirect

18. C₁₉×D₅₅

D₉

direct (N × H)

20. D₉×D₅₅

C₁₉

direct (N × H)

21. C₉×C₁₁×C₁₉

C₂×D₅

semidirect

23. C₅×D₉×C₁₉

D₁₁

semidirect

29. C₅×C₁₁×C₁₉

C₂×D₉

semidirect

30. D₄₉₅

C₂×C₁₉

semidirect

31. C₅×C₁₁×D₉

C₂×C₁₉

semidirect

32. C₉×D₅₅

C₂×C₁₉

semidirect

33. C₅×C₉×C₁₉

C₂×D₁₁

semidirect

41. C₅×C₉×C₁₁

C₂²×C₁₉

semidirect

49. D₉×C₁₉

D₅₅

direct (N × H)

56. C₁₁×C₁₉

D₅×D₉

semidirect

59. C₁₁×D₉

D₅×C₁₉

semidirect

64. C₉×C₁₉

C₂×D₅₅

semidirect

72. D₅₅

D₉×C₁₉

direct (N × H)

73. C₉×C₁₁

C₂×D₅×C₁₉

semidirect

74. C₅×C₁₉

D₉×D₁₁

semidirect

77. C₅×D₉

C₁₉×D₁₁

semidirect

84. C₅×C₁₁

C₂×D₉×C₁₉

semidirect

85. C₅×C₉

C₂×C₁₉×D₁₁

semidirect

99. C₁₉

D₉×D₅₅

direct (N × H)

102. D₉

C₁₉×D₅₅

direct (N × H)

105. C₁₁

D₅×D₉×C₁₉

semidirect

109. C₉

C₂×C₁₉×D₅₅

semidirect

113. C₅

D₉×C₁₉×D₁₁

semidirect

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

subgroup	index
C₁₉×D₄₉₅	2
C₅×C₁₁×D₉×C₁₉	2
C₉×C₁₉×D₅₅	2
D₃×C₁₉×D₅₅	3
D₉×C₁₉×D₁₁	5
D₅×D₉×C₁₉	11
D₉×D₅₅	19

subgroup

index

C₁₉×D₄₉₅

C₅×C₁₁×D₉×C₁₉

C₉×C₁₉×D₅₅

D₃×C₁₉×D₅₅

D₉×C₁₉×D₁₁

D₅×D₉×C₁₉

D₉×D₅₅

Subgroups — D₉×C₁₉×D₅₅ #49178404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D9×C19×D55 #49178404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₉×C₁₉×D₅₅ #49178404