Group #37561608 — subgroups

Subgroups — D₅×C₁₉×C₃⋊D₂₁ #37561608

Order 23940.

240 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 80 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	C₃²×C₇×D₅×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×C₁₉×C₃⋊D₂₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×C₃⋊D₁₀₅	C₂	1	1	yes	maximal, characteristic, complemented
4	D₅×C₁₉×D₂₁	—	3	4	no	maximal
5	C₃²×C₅×C₇×C₁₉	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₂×C₁₉×C₃⋊D₂₁	—	5	1	no	maximal, Hall
7	C₁₉×D₁₀₅	—	3	4	no	—
8	C₅×C₁₉×D₂₁	—	3	4	no	—
9	C₃×C₇×D₅×C₁₉	D₃	1	4	yes	complemented
10	D₅×C₃⋊D₃×C₁₉	—	7	1	no	maximal, Hall
11	D₅×D₇×C₁₉	—	9	1	no	Hall
12	C₁₉×C₃⋊D₂₁	—	5	1	no	—
13	C₁₉×C₃⋊D₂₁	D₅	1	1	yes	characteristic, direct product factor
14	C₂×C₃²×C₇×C₁₉	—	5	1	no	—
15	C₃×C₅×C₇×C₁₉	C₂×D₃	1	4	yes	complemented
16	C₁₉×C₃⋊D₁₅	—	7	1	no	—
17	C₅×C₃⋊D₃×C₁₉	—	7	1	no	—
18	C₃²×D₅×C₁₉	D₇	1	1	yes	characteristic, complemented
19	C₂×C₁₉×D₂₁	—	15	4	no	—
20	C₁₉×D₃₅	—	9	1	no	—
21	C₇×D₅×C₁₉	C₃⋊D₃	1	1	yes	characteristic, complemented
22	C₅×D₇×C₁₉	—	9	1	no	—
23	D₅×C₃⋊D₂₁	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
24	C₃²×C₇×C₁₉	C₂×D₅	1	1	yes	characteristic, complemented, Hall
25	D₃×D₅×C₁₉	—	21	4	no	—
26	C₃²×C₅×C₁₉	C₂×D₇	1	1	yes	characteristic, complemented, Hall
27	C₂×C₃×C₇×C₁₉	—	5	4	no	—
28	C₁₉×D₂₁	—	3	4	no	—
29	C₁₉×D₂₁	—	15	4	no	—
30	C₂×C₃⋊D₃×C₁₉	—	35	1	no	Hall
31	C₅×C₇×C₁₉	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
32	C₃⋊D₁₀₅	C₂×C₁₉	1	1	yes	characteristic, complemented
33	C₅×C₃⋊D₂₁	C₂×C₁₉	1	1	yes	characteristic, complemented
34	C₃²×C₇×D₅	C₂×C₁₉	1	1	yes	characteristic, complemented
35	C₁₉×D₁₅	—	21	4	no	—
36	C₅×D₃×C₁₉	—	21	4	no	—
37	C₃×D₅×C₁₉	D₂₁	1	4	yes	complemented
38	C₂×D₇×C₁₉	—	45	1	no	Hall
39	D₅×D₂₁	—	3	4	no	—
40	C₃×C₇×C₁₉	D₃×D₅	1	4	yes	complemented
41	C₂×D₅×C₁₉	—	63	1	no	Hall
42	C₂×C₃²×C₁₉	—	5	1	no	—
43	C₃⋊D₃×C₁₉	—	7	1	no	—
44	C₃⋊D₃×C₁₉	—	35	1	no	—
45	C₃²×C₅×C₇	C₂²×C₁₉	1	1	yes	characteristic, complemented, Hall
46	C₃×C₅×C₁₉	C₂×D₂₁	1	4	yes	complemented
47	C₂×C₇×C₁₉	—	5	1	no	—
48	D₇×C₁₉	—	9	1	no	—
49	D₇×C₁₉	—	45	1	no	—
50	C₂×C₃⋊D₂₁	—	5	1	no	Hall
51	C₂×D₃×C₁₉	—	105	4	no	—
52	D₁₀₅	—	3	4	no	—
53	C₅×D₂₁	—	3	4	no	—
54	C₃×C₇×D₅	D₃×C₁₉	1	4	yes	complemented
55	C₂×C₅×C₁₉	—	63	1	no	—
56	D₅×C₁₉	—	63	1	no	—
57	D₅×C₁₉	C₃⋊D₂₁	1	1	yes	characteristic, direct product factor
58	D₅×C₃⋊D₃	—	7	1	no	Hall
59	C₃²×C₁₉	D₅×D₇	1	1	yes	characteristic, complemented, Hall
60	D₅×D₇	—	9	1	no	Hall
61	C₇×C₁₉	D₅×C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
62	C₂×C₃²×C₇	—	5	1	no	—
63	C₃⋊D₂₁	—	5	1	no	—
64	C₃⋊D₂₁	D₅×C₁₉	1	1	yes	characteristic, direct product factor
65	C₂×C₃×C₁₉	—	5	4	no	—
66	D₃×C₁₉	—	21	4	no	—
67	D₃×C₁₉	—	105	4	no	—
68	C₃×C₅×C₇	C₂×D₃×C₁₉	1	4	yes	complemented
69	C₅×C₁₉	C₂×C₃⋊D₂₁	1	1	yes	characteristic, complemented, Hall
70	C₃⋊D₁₅	—	7	1	no	—
71	C₅×C₃⋊D₃	—	7	1	no	—
72	C₃²×D₅	D₇×C₁₉	1	1	yes	characteristic, complemented
73	C₂×D₂₁	—	15	4	no	—
74	C₂²×C₁₉	—	315	1	no	Hall
75	D₃₅	—	9	1	no	—
76	C₅×D₇	—	9	1	no	—
77	C₇×D₅	C₃⋊D₃×C₁₉	1	1	yes	characteristic, complemented
78	C₃²×C₇	C₂×D₅×C₁₉	1	1	yes	characteristic, complemented, Hall
79	D₃×D₅	—	21	4	no	—
80	C₃×C₁₉	D₅×D₂₁	1	4	yes	complemented
81	C₃²×C₅	C₂×D₇×C₁₉	1	1	yes	characteristic, complemented, Hall
82	C₂×C₃×C₇	—	5	4	no	—
83	D₂₁	—	3	4	no	—
84	D₂₁	—	15	4	no	—
85	C₂×C₁₉	—	5	1	no	—
86	C₂×C₁₉	—	315	1	no	—
87	C₂×C₁₉	—	63	1	no	—
88	C₂×C₃⋊D₃	—	35	1	no	Hall
89	C₅×C₇	C₂×C₃⋊D₃×C₁₉	1	1	yes	characteristic, complemented, Hall
90	D₁₅	—	21	4	no	—
91	C₃×D₅	C₁₉×D₂₁	1	4	yes	complemented
92	C₅×D₃	—	21	4	no	—
93	C₂×D₇	—	45	1	no	Hall
94	C₃×C₇	D₃×D₅×C₁₉	1	4	yes	complemented
95	C₂×D₅	—	63	1	no	Hall
96	C₁₉	D₅×C₃⋊D₂₁	1	1	yes	characteristic, direct product factor, Sylow
97	C₂×C₃²	—	5	1	no	—
98	C₃⋊D₃	—	7	1	no	—
99	C₃⋊D₃	—	35	1	no	—
100	C₃×C₅	C₂×C₁₉×D₂₁	1	4	yes	complemented
101	C₂×C₇	—	5	1	no	—
102	D₇	—	9	1	no	—
103	D₇	—	45	1	no	—
104	C₂×D₃	—	105	4	no	—
105	C₂×C₅	—	63	1	no	—
106	D₅	—	63	1	no	—
107	D₅	C₁₉×C₃⋊D₂₁	1	1	yes	characteristic, direct product factor
108	C₃²	D₅×D₇×C₁₉	1	1	yes	characteristic, complemented, Sylow
109	C₇	D₅×C₃⋊D₃×C₁₉	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₃	—	5	4	no	—
111	D₃	—	21	4	no	—
112	D₃	—	105	4	no	—
113	C₅	C₂×C₁₉×C₃⋊D₂₁	1	1	yes	characteristic, complemented, Sylow
114	C₂²	—	315	1	no	Sylow
115	C₃	D₅×C₁₉×D₂₁	1	4	yes	complemented
116	C₂	—	5	1	no	—
117	C₂	—	315	1	no	—
118	C₂	—	63	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₇×D₅×C₁₉

C₂

yes

maximal, characteristic, complemented

C₅×C₁₉×C₃⋊D₂₁

C₂

yes

maximal, characteristic, complemented

C₁₉×C₃⋊D₁₀₅

C₂

yes

maximal, characteristic, complemented

D₅×C₁₉×D₂₁

—

maximal

C₃²×C₅×C₇×C₁₉

C₂²

yes

characteristic, complemented, Hall

C₂×C₁₉×C₃⋊D₂₁

—

maximal, Hall

C₁₉×D₁₀₅

—

C₅×C₁₉×D₂₁

—

C₃×C₇×D₅×C₁₉

D₃

yes

complemented

D₅×C₃⋊D₃×C₁₉

—

maximal, Hall

D₅×D₇×C₁₉

—

Hall

C₁₉×C₃⋊D₂₁

—

C₁₉×C₃⋊D₂₁

D₅

yes

characteristic, direct product factor

C₂×C₃²×C₇×C₁₉

—

C₃×C₅×C₇×C₁₉

C₂×D₃

yes

complemented

C₁₉×C₃⋊D₁₅

—

C₅×C₃⋊D₃×C₁₉

—

C₃²×D₅×C₁₉

D₇

yes

characteristic, complemented

C₂×C₁₉×D₂₁

—

C₁₉×D₃₅

—

C₇×D₅×C₁₉

C₃⋊D₃

yes

characteristic, complemented

C₅×D₇×C₁₉

—

D₅×C₃⋊D₂₁

C₁₉

yes

maximal, characteristic, direct product factor, Hall

C₃²×C₇×C₁₉

C₂×D₅

yes

characteristic, complemented, Hall

D₃×D₅×C₁₉

—

C₃²×C₅×C₁₉

C₂×D₇

yes

characteristic, complemented, Hall

C₂×C₃×C₇×C₁₉

—

C₁₉×D₂₁

—

C₁₉×D₂₁

—

C₂×C₃⋊D₃×C₁₉

—

Hall

C₅×C₇×C₁₉

C₂×C₃⋊D₃

yes

characteristic, complemented, Hall

C₃⋊D₁₀₅

C₂×C₁₉

yes

characteristic, complemented

C₅×C₃⋊D₂₁

C₂×C₁₉

yes

characteristic, complemented

C₃²×C₇×D₅

C₂×C₁₉

yes

characteristic, complemented

C₁₉×D₁₅

—

C₅×D₃×C₁₉

—

C₃×D₅×C₁₉

D₂₁

yes

complemented

C₂×D₇×C₁₉

—

Hall

D₅×D₂₁

—

C₃×C₇×C₁₉

D₃×D₅

yes

complemented

C₂×D₅×C₁₉

—

Hall

C₂×C₃²×C₁₉

—

C₃⋊D₃×C₁₉

—

C₃⋊D₃×C₁₉

—

C₃²×C₅×C₇

C₂²×C₁₉

yes

characteristic, complemented, Hall

C₃×C₅×C₁₉

C₂×D₂₁

yes

complemented

C₂×C₇×C₁₉

—

D₇×C₁₉

—

D₇×C₁₉

—

C₂×C₃⋊D₂₁

—

Hall

C₂×D₃×C₁₉

—

105

—

D₁₀₅

—

C₅×D₂₁

—

C₃×C₇×D₅

D₃×C₁₉

yes

complemented

C₂×C₅×C₁₉

—

D₅×C₁₉

—

D₅×C₁₉

C₃⋊D₂₁

yes

characteristic, direct product factor

D₅×C₃⋊D₃

—

Hall

C₃²×C₁₉

D₅×D₇

yes

characteristic, complemented, Hall

D₅×D₇

—

Hall

C₇×C₁₉

D₅×C₃⋊D₃

yes

characteristic, complemented, Hall

C₂×C₃²×C₇

—

C₃⋊D₂₁

—

C₃⋊D₂₁

D₅×C₁₉

yes

characteristic, direct product factor

C₂×C₃×C₁₉

—

D₃×C₁₉

—

D₃×C₁₉

—

105

—

C₃×C₅×C₇

C₂×D₃×C₁₉

yes

complemented

C₅×C₁₉

C₂×C₃⋊D₂₁

yes

characteristic, complemented, Hall

C₃⋊D₁₅

—

C₅×C₃⋊D₃

—

C₃²×D₅

D₇×C₁₉

yes

characteristic, complemented

C₂×D₂₁

—

C₂²×C₁₉

—

315

Hall

D₃₅

—

C₅×D₇

—

C₇×D₅

C₃⋊D₃×C₁₉

yes

characteristic, complemented

C₃²×C₇

C₂×D₅×C₁₉

yes

characteristic, complemented, Hall

D₃×D₅

—

C₃×C₁₉

D₅×D₂₁

yes

complemented

C₃²×C₅

C₂×D₇×C₁₉

yes

characteristic, complemented, Hall

C₂×C₃×C₇

—

D₂₁

—

D₂₁

—

C₂×C₁₉

—

C₂×C₁₉

—

315

—

C₂×C₁₉

—

C₂×C₃⋊D₃

—

Hall

C₅×C₇

C₂×C₃⋊D₃×C₁₉

yes

characteristic, complemented, Hall

D₁₅

—

C₃×D₅

C₁₉×D₂₁

yes

complemented

C₅×D₃

—

C₂×D₇

—

Hall

C₃×C₇

D₃×D₅×C₁₉

yes

complemented

C₂×D₅

—

Hall

C₁₉

D₅×C₃⋊D₂₁

yes

characteristic, direct product factor, Sylow

C₂×C₃²

—

C₃⋊D₃

—

C₃⋊D₃

—

100

C₃×C₅

C₂×C₁₉×D₂₁

yes

complemented

101

C₂×C₇

—

102

D₇

—

103

D₇

—

104

C₂×D₃

—

105

—

105

C₂×C₅

—

106

D₅

—

107

D₅

C₁₉×C₃⋊D₂₁

yes

characteristic, direct product factor

108

C₃²

D₅×D₇×C₁₉

yes

characteristic, complemented, Sylow

109

C₇

D₅×C₃⋊D₃×C₁₉

yes

characteristic, complemented, Sylow

110

C₂×C₃

—

111

D₃

—

112

D₃

—

105

—

113

C₅

C₂×C₁₉×C₃⋊D₂₁

yes

characteristic, complemented, Sylow

114

C₂²

—

315

Sylow

115

C₃

D₅×C₁₉×D₂₁

yes

complemented

116

C₂

—

117

C₂

—

315

—

118

C₂

—

	subgroup	G/H
center, hypercenter	C₁₉	D₅×C₃⋊D₂₁
commutator, cosocle, hypocenter	C₃²×C₅×C₇	C₂²×C₁₉
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₇×C₁₉	C₂²
LCS	G ▷ C₃²×C₅×C₇
UCS	C₁ ◁ C₁₉

subgroup

G/H

center, hypercenter

C₁₉

D₅×C₃⋊D₂₁

commutator, cosocle, hypocenter

C₃²×C₅×C₇

C₂²×C₁₉

Frattini

C₁

Fitting, socle

C₃²×C₅×C₇×C₁₉

C₂²

LCS

G ▷ C₃²×C₅×C₇

UCS

C₁ ◁ C₁₉

N	H ≅ G/N	type
1. C₃²×C₇×D₅×C₁₉	C₂	semidirect
2. C₅×C₁₉×C₃⋊D₂₁	C₂	semidirect
3. C₁₉×C₃⋊D₁₀₅	C₂	semidirect
5. C₃²×C₅×C₇×C₁₉	C₂²	semidirect
9. C₃×C₇×D₅×C₁₉	D₃	semidirect
13. C₁₉×C₃⋊D₂₁	D₅	direct (N × H)
15. C₃×C₅×C₇×C₁₉	C₂×D₃	semidirect
18. C₃²×D₅×C₁₉	D₇	semidirect
21. C₇×D₅×C₁₉	C₃⋊D₃	semidirect
23. D₅×C₃⋊D₂₁	C₁₉	direct (N × H)
24. C₃²×C₇×C₁₉	C₂×D₅	semidirect
26. C₃²×C₅×C₁₉	C₂×D₇	semidirect
31. C₅×C₇×C₁₉	C₂×C₃⋊D₃	semidirect
32. C₃⋊D₁₀₅	C₂×C₁₉	semidirect
33. C₅×C₃⋊D₂₁	C₂×C₁₉	semidirect
34. C₃²×C₇×D₅	C₂×C₁₉	semidirect
37. C₃×D₅×C₁₉	D₂₁	semidirect
40. C₃×C₇×C₁₉	D₃×D₅	semidirect
45. C₃²×C₅×C₇	C₂²×C₁₉	semidirect
46. C₃×C₅×C₁₉	C₂×D₂₁	semidirect
54. C₃×C₇×D₅	D₃×C₁₉	semidirect
57. D₅×C₁₉	C₃⋊D₂₁	direct (N × H)
59. C₃²×C₁₉	D₅×D₇	semidirect
61. C₇×C₁₉	D₅×C₃⋊D₃	semidirect
64. C₃⋊D₂₁	D₅×C₁₉	direct (N × H)
68. C₃×C₅×C₇	C₂×D₃×C₁₉	semidirect
69. C₅×C₁₉	C₂×C₃⋊D₂₁	semidirect
72. C₃²×D₅	D₇×C₁₉	semidirect
77. C₇×D₅	C₃⋊D₃×C₁₉	semidirect
78. C₃²×C₇	C₂×D₅×C₁₉	semidirect
80. C₃×C₁₉	D₅×D₂₁	semidirect
81. C₃²×C₅	C₂×D₇×C₁₉	semidirect
89. C₅×C₇	C₂×C₃⋊D₃×C₁₉	semidirect
91. C₃×D₅	C₁₉×D₂₁	semidirect
94. C₃×C₇	D₃×D₅×C₁₉	semidirect
96. C₁₉	D₅×C₃⋊D₂₁	direct (N × H)
100. C₃×C₅	C₂×C₁₉×D₂₁	semidirect
107. D₅	C₁₉×C₃⋊D₂₁	direct (N × H)
108. C₃²	D₅×D₇×C₁₉	semidirect
109. C₇	D₅×C₃⋊D₃×C₁₉	semidirect
113. C₅	C₂×C₁₉×C₃⋊D₂₁	semidirect
115. C₃	D₅×C₁₉×D₂₁	semidirect

H ≅ G/N

type

1. C₃²×C₇×D₅×C₁₉

C₂

semidirect

2. C₅×C₁₉×C₃⋊D₂₁

C₂

semidirect

3. C₁₉×C₃⋊D₁₀₅

C₂

semidirect

5. C₃²×C₅×C₇×C₁₉

C₂²

semidirect

9. C₃×C₇×D₅×C₁₉

D₃

semidirect

13. C₁₉×C₃⋊D₂₁

D₅

direct (N × H)

15. C₃×C₅×C₇×C₁₉

C₂×D₃

semidirect

18. C₃²×D₅×C₁₉

D₇

semidirect

21. C₇×D₅×C₁₉

C₃⋊D₃

semidirect

23. D₅×C₃⋊D₂₁

C₁₉

direct (N × H)

24. C₃²×C₇×C₁₉

C₂×D₅

semidirect

26. C₃²×C₅×C₁₉

C₂×D₇

semidirect

31. C₅×C₇×C₁₉

C₂×C₃⋊D₃

semidirect

32. C₃⋊D₁₀₅

C₂×C₁₉

semidirect

33. C₅×C₃⋊D₂₁

C₂×C₁₉

semidirect

34. C₃²×C₇×D₅

C₂×C₁₉

semidirect

37. C₃×D₅×C₁₉

D₂₁

semidirect

40. C₃×C₇×C₁₉

D₃×D₅

semidirect

45. C₃²×C₅×C₇

C₂²×C₁₉

semidirect

46. C₃×C₅×C₁₉

C₂×D₂₁

semidirect

54. C₃×C₇×D₅

D₃×C₁₉

semidirect

57. D₅×C₁₉

C₃⋊D₂₁

direct (N × H)

59. C₃²×C₁₉

D₅×D₇

semidirect

61. C₇×C₁₉

D₅×C₃⋊D₃

semidirect

64. C₃⋊D₂₁

D₅×C₁₉

direct (N × H)

68. C₃×C₅×C₇

C₂×D₃×C₁₉

semidirect

69. C₅×C₁₉

C₂×C₃⋊D₂₁

semidirect

72. C₃²×D₅

D₇×C₁₉

semidirect

77. C₇×D₅

C₃⋊D₃×C₁₉

semidirect

78. C₃²×C₇

C₂×D₅×C₁₉

semidirect

80. C₃×C₁₉

D₅×D₂₁

semidirect

81. C₃²×C₅

C₂×D₇×C₁₉

semidirect

89. C₅×C₇

C₂×C₃⋊D₃×C₁₉

semidirect

91. C₃×D₅

C₁₉×D₂₁

semidirect

94. C₃×C₇

D₃×D₅×C₁₉

semidirect

96. C₁₉

D₅×C₃⋊D₂₁

direct (N × H)

100. C₃×C₅

C₂×C₁₉×D₂₁

semidirect

107. D₅

C₁₉×C₃⋊D₂₁

direct (N × H)

108. C₃²

D₅×D₇×C₁₉

semidirect

109. C₇

D₅×C₃⋊D₃×C₁₉

semidirect

113. C₅

C₂×C₁₉×C₃⋊D₂₁

semidirect

115. C₃

D₅×C₁₉×D₂₁

semidirect

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

subgroup	index
C₃²×C₇×D₅×C₁₉	2
C₅×C₁₉×C₃⋊D₂₁	2
C₁₉×C₃⋊D₁₀₅	2
D₅×C₁₉×D₂₁	3
C₂×C₁₉×C₃⋊D₂₁	5
D₅×C₃⋊D₃×C₁₉	7
D₅×C₃⋊D₂₁	19

subgroup

index

C₃²×C₇×D₅×C₁₉

C₅×C₁₉×C₃⋊D₂₁

C₁₉×C₃⋊D₁₀₅

D₅×C₁₉×D₂₁

C₂×C₁₉×C₃⋊D₂₁

D₅×C₃⋊D₃×C₁₉

D₅×C₃⋊D₂₁

Subgroups — D₅×C₁₉×C₃⋊D₂₁ #37561608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D5×C19×C3⋊D21 #37561608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×C₁₉×C₃⋊D₂₁ #37561608