Group #30685913 — subgroups

Subgroups — D₅×C₁₉×C₃⋊D₁₅ #30685913

Order 17100.

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

normal subgroups only

This group has 288 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	C₁₉×C₃⋊(C₃⋊(C₅⋊D₅))	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×C₁₉×C₃⋊D₁₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×C₅×D₅×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
4	D₅×C₁₉×D₁₅	—	3	4	no	maximal
5	C₃²×C₅²×C₁₉	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₂×C₁₉×C₃⋊D₁₅	—	5	1	no	maximal
7	D₅×C₃⋊D₃×C₁₉	—	5	1	no	maximal
8	C₁₉×C₃⋊(C₅⋊D₅)	—	3	4	no	—
9	C₅×C₁₉×D₁₅	—	3	4	no	—
10	C₃×C₅×D₅×C₁₉	D₃	1	4	yes	complemented
11	D₅²×C₁₉	—	9	1	no	Hall
12	C₂×C₃²×C₅×C₁₉	—	5	1	no	—
13	C₁₉×C₃⋊D₁₅	—	5	1	no	—
14	C₁₉×C₃⋊D₁₅	—	5	1	no	—
15	C₁₉×C₃⋊D₁₅	—	10	2	no	—
16	C₁₉×C₃⋊D₁₅	D₅	1	1	yes	characteristic, direct product factor
17	C₅×C₃⋊D₃×C₁₉	—	5	1	no	—
18	C₃²×D₅×C₁₉	D₅	1	1	yes	characteristic, complemented
19	C₃×C₅²×C₁₉	C₂×D₃	1	4	yes	complemented
20	C₂×C₁₉×D₁₅	—	15	4	no	—
21	D₃×D₅×C₁₉	—	15	4	no	—
22	C₁₉×C₅⋊D₅	—	9	1	no	—
23	C₅×D₅×C₁₉	—	9	1	no	—
24	C₅×D₅×C₁₉	C₃⋊D₃	1	1	yes	characteristic, complemented
25	D₅×C₃⋊D₁₅	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
26	C₃²×C₅×C₁₉	—	2	2	no	—
27	C₃²×C₅×C₁₉	C₂×D₅	1	1	yes	characteristic, complemented
28	C₃²×C₅×C₁₉	C₂×D₅	1	1	yes	characteristic, complemented
29	C₂×C₃⋊D₃×C₁₉	—	25	1	no	Hall
30	C₂×C₃×C₅×C₁₉	—	5	4	no	—
31	C₁₉×D₁₅	—	3	4	no	—
32	C₁₉×D₁₅	—	15	4	no	—
33	C₁₉×D₁₅	—	15	4	no	—
34	C₁₉×D₁₅	—	30	8	no	—
35	C₅×D₃×C₁₉	—	15	4	no	—
36	C₃×D₅×C₁₉	D₁₅	1	4	yes	complemented
37	C₅²×C₁₉	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
38	C₃⋊(C₃⋊(C₅⋊D₅))	C₂×C₁₉	1	1	yes	characteristic, complemented
39	C₅×C₃⋊D₁₅	C₂×C₁₉	1	1	yes	characteristic, complemented
40	C₃²×C₅×D₅	C₂×C₁₉	1	1	yes	characteristic, complemented
41	C₂×D₅×C₁₉	—	45	1	no	—
42	C₂×D₅×C₁₉	—	45	1	no	—
43	C₂×C₃²×C₁₉	—	5	1	no	—
44	C₃⋊D₃×C₁₉	—	5	1	no	—
45	C₃⋊D₃×C₁₉	—	25	1	no	—
46	D₅×D₁₅	—	3	4	no	—
47	C₃×C₅×C₁₉	—	2	8	no	—
48	C₃×C₅×C₁₉	D₃×D₅	1	4	yes	complemented
49	C₃×C₅×C₁₉	C₂×D₁₅	1	4	yes	complemented
50	C₂×D₃×C₁₉	—	75	4	no	—
51	C₃²×C₅²	C₂²×C₁₉	1	1	yes	characteristic, complemented, Hall
52	C₂×C₅×C₁₉	—	5	1	no	—
53	C₂×C₅×C₁₉	—	45	1	no	—
54	D₅×C₁₉	—	9	1	no	—
55	D₅×C₁₉	—	45	1	no	—
56	D₅×C₁₉	—	45	1	no	—
57	D₅×C₁₉	—	90	2	no	—
58	D₅×C₁₉	C₃⋊D₁₅	1	1	yes	characteristic, direct product factor
59	C₂×C₃⋊D₁₅	—	5	1	no	—
60	D₅×C₃⋊D₃	—	5	1	no	—
61	C₃²×C₁₉	D₅²	1	1	yes	characteristic, complemented, Hall
62	C₃⋊(C₅⋊D₅)	—	3	4	no	—
63	C₅×D₁₅	—	3	4	no	—
64	C₃×C₅×D₅	D₃×C₁₉	1	4	yes	complemented
65	C₂×C₃×C₁₉	—	5	4	no	—
66	D₃×C₁₉	—	15	4	no	—
67	D₃×C₁₉	—	75	4	no	—
68	D₅²	—	9	1	no	Hall
69	C₅×C₁₉	—	2	2	no	—
70	C₅×C₁₉	D₅×C₃⋊D₃	1	1	yes	characteristic, complemented
71	C₅×C₁₉	C₂×C₃⋊D₁₅	1	1	yes	characteristic, complemented
72	C₂×C₃²×C₅	—	5	1	no	—
73	C₃⋊D₁₅	—	5	1	no	—
74	C₃⋊D₁₅	—	5	1	no	—
75	C₃⋊D₁₅	—	10	2	no	—
76	C₃⋊D₁₅	D₅×C₁₉	1	1	yes	characteristic, direct product factor
77	C₅×C₃⋊D₃	—	5	1	no	—
78	C₃²×D₅	D₅×C₁₉	1	1	yes	characteristic, complemented
79	C₂²×C₁₉	—	225	1	no	Hall
80	C₃×C₅²	C₂×D₃×C₁₉	1	4	yes	complemented
81	C₂×D₁₅	—	15	4	no	—
82	D₃×D₅	—	15	4	no	—
83	C₃×C₁₉	D₅×D₁₅	1	4	yes	complemented
84	C₅⋊D₅	—	9	1	no	—
85	C₅×D₅	—	9	1	no	—
86	C₅×D₅	C₃⋊D₃×C₁₉	1	1	yes	characteristic, complemented
87	C₃²×C₅	—	2	2	no	—
88	C₃²×C₅	C₂×D₅×C₁₉	1	1	yes	characteristic, complemented
89	C₃²×C₅	C₂×D₅×C₁₉	1	1	yes	characteristic, complemented
90	C₂×C₁₉	—	5	1	no	—
91	C₂×C₁₉	—	45	1	no	—
92	C₂×C₁₉	—	225	1	no	—
93	C₂×C₃⋊D₃	—	25	1	no	Hall
94	C₂×C₃×C₅	—	5	4	no	—
95	D₁₅	—	3	4	no	—
96	D₁₅	—	15	4	no	—
97	D₁₅	—	15	4	no	—
98	D₁₅	—	30	8	no	—
99	C₃×D₅	C₁₉×D₁₅	1	4	yes	complemented
100	C₅×D₃	—	15	4	no	—
101	C₅²	C₂×C₃⋊D₃×C₁₉	1	1	yes	characteristic, complemented, Sylow
102	C₂×D₅	—	45	1	no	—
103	C₂×D₅	—	45	1	no	—
104	C₁₉	D₅×C₃⋊D₁₅	1	1	yes	characteristic, direct product factor, Sylow
105	C₂×C₃²	—	5	1	no	—
106	C₃⋊D₃	—	5	1	no	—
107	C₃⋊D₃	—	25	1	no	—
108	C₃×C₅	—	2	8	no	—
109	C₃×C₅	D₃×D₅×C₁₉	1	4	yes	complemented
110	C₃×C₅	C₂×C₁₉×D₁₅	1	4	yes	complemented
111	C₂×D₃	—	75	4	no	—
112	C₂×C₅	—	5	1	no	—
113	C₂×C₅	—	45	1	no	—
114	D₅	—	9	1	no	—
115	D₅	—	45	1	no	—
116	D₅	—	45	1	no	—
117	D₅	—	90	2	no	—
118	D₅	C₁₉×C₃⋊D₁₅	1	1	yes	characteristic, direct product factor
119	C₃²	D₅²×C₁₉	1	1	yes	characteristic, complemented, Sylow
120	C₂×C₃	—	5	4	no	—
121	D₃	—	15	4	no	—
122	D₃	—	75	4	no	—
123	C₅	—	2	2	no	—
124	C₅	D₅×C₃⋊D₃×C₁₉	1	1	yes	characteristic, complemented
125	C₅	C₂×C₁₉×C₃⋊D₁₅	1	1	yes	characteristic, complemented
126	C₂²	—	225	1	no	Sylow
127	C₃	D₅×C₁₉×D₁₅	1	4	yes	complemented
128	C₂	—	5	1	no	—
129	C₂	—	45	1	no	—
130	C₂	—	225	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×C₃⋊(C₃⋊(C₅⋊D₅))

C₂

yes

maximal, characteristic, complemented

C₅×C₁₉×C₃⋊D₁₅

C₂

yes

maximal, characteristic, complemented

C₃²×C₅×D₅×C₁₉

C₂

yes

maximal, characteristic, complemented

D₅×C₁₉×D₁₅

—

maximal

C₃²×C₅²×C₁₉

C₂²

yes

characteristic, complemented, Hall

C₂×C₁₉×C₃⋊D₁₅

—

maximal

D₅×C₃⋊D₃×C₁₉

—

maximal

C₁₉×C₃⋊(C₅⋊D₅)

—

C₅×C₁₉×D₁₅

—

C₃×C₅×D₅×C₁₉

D₃

yes

complemented

D₅²×C₁₉

—

Hall

C₂×C₃²×C₅×C₁₉

—

C₁₉×C₃⋊D₁₅

—

C₁₉×C₃⋊D₁₅

—

C₁₉×C₃⋊D₁₅

—

C₁₉×C₃⋊D₁₅

D₅

yes

characteristic, direct product factor

C₅×C₃⋊D₃×C₁₉

—

C₃²×D₅×C₁₉

D₅

yes

characteristic, complemented

C₃×C₅²×C₁₉

C₂×D₃

yes

complemented

C₂×C₁₉×D₁₅

—

D₃×D₅×C₁₉

—

C₁₉×C₅⋊D₅

—

C₅×D₅×C₁₉

—

C₅×D₅×C₁₉

C₃⋊D₃

yes

characteristic, complemented

D₅×C₃⋊D₁₅

C₁₉

yes

maximal, characteristic, direct product factor, Hall

C₃²×C₅×C₁₉

—

C₃²×C₅×C₁₉

C₂×D₅

yes

characteristic, complemented

C₃²×C₅×C₁₉

C₂×D₅

yes

characteristic, complemented

C₂×C₃⋊D₃×C₁₉

—

Hall

C₂×C₃×C₅×C₁₉

—

C₁₉×D₁₅

—

C₁₉×D₁₅

—

C₁₉×D₁₅

—

C₁₉×D₁₅

—

C₅×D₃×C₁₉

—

C₃×D₅×C₁₉

D₁₅

yes

complemented

C₅²×C₁₉

C₂×C₃⋊D₃

yes

characteristic, complemented, Hall

C₃⋊(C₃⋊(C₅⋊D₅))

C₂×C₁₉

yes

characteristic, complemented

C₅×C₃⋊D₁₅

C₂×C₁₉

yes

characteristic, complemented

C₃²×C₅×D₅

C₂×C₁₉

yes

characteristic, complemented

C₂×D₅×C₁₉

—

C₂×D₅×C₁₉

—

C₂×C₃²×C₁₉

—

C₃⋊D₃×C₁₉

—

C₃⋊D₃×C₁₉

—

D₅×D₁₅

—

C₃×C₅×C₁₉

—

C₃×C₅×C₁₉

D₃×D₅

yes

complemented

C₃×C₅×C₁₉

C₂×D₁₅

yes

complemented

C₂×D₃×C₁₉

—

C₃²×C₅²

C₂²×C₁₉

yes

characteristic, complemented, Hall

C₂×C₅×C₁₉

—

C₂×C₅×C₁₉

—

D₅×C₁₉

—

D₅×C₁₉

—

D₅×C₁₉

—

D₅×C₁₉

—

D₅×C₁₉

C₃⋊D₁₅

yes

characteristic, direct product factor

C₂×C₃⋊D₁₅

—

D₅×C₃⋊D₃

—

C₃²×C₁₉

D₅²

yes

characteristic, complemented, Hall

C₃⋊(C₅⋊D₅)

—

C₅×D₁₅

—

C₃×C₅×D₅

D₃×C₁₉

yes

complemented

C₂×C₃×C₁₉

—

D₃×C₁₉

—

D₃×C₁₉

—

D₅²

—

Hall

C₅×C₁₉

—

C₅×C₁₉

D₅×C₃⋊D₃

yes

characteristic, complemented

C₅×C₁₉

C₂×C₃⋊D₁₅

yes

characteristic, complemented

C₂×C₃²×C₅

—

C₃⋊D₁₅

—

C₃⋊D₁₅

—

C₃⋊D₁₅

—

C₃⋊D₁₅

D₅×C₁₉

yes

characteristic, direct product factor

C₅×C₃⋊D₃

—

C₃²×D₅

D₅×C₁₉

yes

characteristic, complemented

C₂²×C₁₉

—

225

Hall

C₃×C₅²

C₂×D₃×C₁₉

yes

complemented

C₂×D₁₅

—

D₃×D₅

—

C₃×C₁₉

D₅×D₁₅

yes

complemented

C₅⋊D₅

—

C₅×D₅

—

C₅×D₅

C₃⋊D₃×C₁₉

yes

characteristic, complemented

C₃²×C₅

—

C₃²×C₅

C₂×D₅×C₁₉

yes

characteristic, complemented

C₃²×C₅

C₂×D₅×C₁₉

yes

characteristic, complemented

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

—

225

—

C₂×C₃⋊D₃

—

Hall

C₂×C₃×C₅

—

D₁₅

—

D₁₅

—

D₁₅

—

D₁₅

—

C₃×D₅

C₁₉×D₁₅

yes

complemented

100

C₅×D₃

—

101

C₅²

C₂×C₃⋊D₃×C₁₉

yes

characteristic, complemented, Sylow

102

C₂×D₅

—

103

C₂×D₅

—

104

C₁₉

D₅×C₃⋊D₁₅

yes

characteristic, direct product factor, Sylow

105

C₂×C₃²

—

106

C₃⋊D₃

—

107

C₃⋊D₃

—

108

C₃×C₅

—

109

C₃×C₅

D₃×D₅×C₁₉

yes

complemented

110

C₃×C₅

C₂×C₁₉×D₁₅

yes

complemented

111

C₂×D₃

—

112

C₂×C₅

—

113

C₂×C₅

—

114

D₅

—

115

D₅

—

116

D₅

—

117

D₅

—

118

D₅

C₁₉×C₃⋊D₁₅

yes

characteristic, direct product factor

119

C₃²

D₅²×C₁₉

yes

characteristic, complemented, Sylow

120

C₂×C₃

—

121

D₃

—

122

D₃

—

123

C₅

—

124

C₅

D₅×C₃⋊D₃×C₁₉

yes

characteristic, complemented

125

C₅

C₂×C₁₉×C₃⋊D₁₅

yes

characteristic, complemented

126

C₂²

—

225

Sylow

127

C₃

D₅×C₁₉×D₁₅

yes

complemented

128

C₂

—

129

C₂

—

130

C₂

—

225

—

	subgroup	G/H
center, hypercenter	C₁₉	D₅×C₃⋊D₁₅
commutator, cosocle, hypocenter	C₃²×C₅²	C₂²×C₁₉
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₁₉	C₂²
LCS	G ▷ C₃²×C₅²
UCS	C₁ ◁ C₁₉

subgroup

G/H

center, hypercenter

C₁₉

D₅×C₃⋊D₁₅

commutator, cosocle, hypocenter

C₃²×C₅²

C₂²×C₁₉

Frattini

C₁

Fitting, socle

C₃²×C₅²×C₁₉

C₂²

LCS

G ▷ C₃²×C₅²

UCS

C₁ ◁ C₁₉

N	H ≅ G/N	type
1. C₁₉×C₃⋊(C₃⋊(C₅⋊D₅))	C₂	semidirect
2. C₅×C₁₉×C₃⋊D₁₅	C₂	semidirect
3. C₃²×C₅×D₅×C₁₉	C₂	semidirect
5. C₃²×C₅²×C₁₉	C₂²	semidirect
10. C₃×C₅×D₅×C₁₉	D₃	semidirect
16. C₁₉×C₃⋊D₁₅	D₅	direct (N × H)
18. C₃²×D₅×C₁₉	D₅	semidirect
19. C₃×C₅²×C₁₉	C₂×D₃	semidirect
24. C₅×D₅×C₁₉	C₃⋊D₃	semidirect
25. D₅×C₃⋊D₁₅	C₁₉	direct (N × H)
27. C₃²×C₅×C₁₉	C₂×D₅	semidirect
28. C₃²×C₅×C₁₉	C₂×D₅	semidirect
36. C₃×D₅×C₁₉	D₁₅	semidirect
37. C₅²×C₁₉	C₂×C₃⋊D₃	semidirect
38. C₃⋊(C₃⋊(C₅⋊D₅))	C₂×C₁₉	semidirect
39. C₅×C₃⋊D₁₅	C₂×C₁₉	semidirect
40. C₃²×C₅×D₅	C₂×C₁₉	semidirect
48. C₃×C₅×C₁₉	D₃×D₅	semidirect
49. C₃×C₅×C₁₉	C₂×D₁₅	semidirect
51. C₃²×C₅²	C₂²×C₁₉	semidirect
58. D₅×C₁₉	C₃⋊D₁₅	direct (N × H)
61. C₃²×C₁₉	D₅²	semidirect
64. C₃×C₅×D₅	D₃×C₁₉	semidirect
70. C₅×C₁₉	D₅×C₃⋊D₃	semidirect
71. C₅×C₁₉	C₂×C₃⋊D₁₅	semidirect
76. C₃⋊D₁₅	D₅×C₁₉	direct (N × H)
78. C₃²×D₅	D₅×C₁₉	semidirect
80. C₃×C₅²	C₂×D₃×C₁₉	semidirect
83. C₃×C₁₉	D₅×D₁₅	semidirect
86. C₅×D₅	C₃⋊D₃×C₁₉	semidirect
88. C₃²×C₅	C₂×D₅×C₁₉	semidirect
89. C₃²×C₅	C₂×D₅×C₁₉	semidirect
99. C₃×D₅	C₁₉×D₁₅	semidirect
101. C₅²	C₂×C₃⋊D₃×C₁₉	semidirect
104. C₁₉	D₅×C₃⋊D₁₅	direct (N × H)
109. C₃×C₅	D₃×D₅×C₁₉	semidirect
110. C₃×C₅	C₂×C₁₉×D₁₅	semidirect
118. D₅	C₁₉×C₃⋊D₁₅	direct (N × H)
119. C₃²	D₅²×C₁₉	semidirect
124. C₅	D₅×C₃⋊D₃×C₁₉	semidirect
125. C₅	C₂×C₁₉×C₃⋊D₁₅	semidirect
127. C₃	D₅×C₁₉×D₁₅	semidirect

H ≅ G/N

type

1. C₁₉×C₃⋊(C₃⋊(C₅⋊D₅))

C₂

semidirect

2. C₅×C₁₉×C₃⋊D₁₅

C₂

semidirect

3. C₃²×C₅×D₅×C₁₉

C₂

semidirect

5. C₃²×C₅²×C₁₉

C₂²

semidirect

10. C₃×C₅×D₅×C₁₉

D₃

semidirect

16. C₁₉×C₃⋊D₁₅

D₅

direct (N × H)

18. C₃²×D₅×C₁₉

D₅

semidirect

19. C₃×C₅²×C₁₉

C₂×D₃

semidirect

24. C₅×D₅×C₁₉

C₃⋊D₃

semidirect

25. D₅×C₃⋊D₁₅

C₁₉

direct (N × H)

27. C₃²×C₅×C₁₉

C₂×D₅

semidirect

28. C₃²×C₅×C₁₉

C₂×D₅

semidirect

36. C₃×D₅×C₁₉

D₁₅

semidirect

37. C₅²×C₁₉

C₂×C₃⋊D₃

semidirect

38. C₃⋊(C₃⋊(C₅⋊D₅))

C₂×C₁₉

semidirect

39. C₅×C₃⋊D₁₅

C₂×C₁₉

semidirect

40. C₃²×C₅×D₅

C₂×C₁₉

semidirect

48. C₃×C₅×C₁₉

D₃×D₅

semidirect

49. C₃×C₅×C₁₉

C₂×D₁₅

semidirect

51. C₃²×C₅²

C₂²×C₁₉

semidirect

58. D₅×C₁₉

C₃⋊D₁₅

direct (N × H)

61. C₃²×C₁₉

D₅²

semidirect

64. C₃×C₅×D₅

D₃×C₁₉

semidirect

70. C₅×C₁₉

D₅×C₃⋊D₃

semidirect

71. C₅×C₁₉

C₂×C₃⋊D₁₅

semidirect

76. C₃⋊D₁₅

D₅×C₁₉

direct (N × H)

78. C₃²×D₅

D₅×C₁₉

semidirect

80. C₃×C₅²

C₂×D₃×C₁₉

semidirect

83. C₃×C₁₉

D₅×D₁₅

semidirect

86. C₅×D₅

C₃⋊D₃×C₁₉

semidirect

88. C₃²×C₅

C₂×D₅×C₁₉

semidirect

89. C₃²×C₅

C₂×D₅×C₁₉

semidirect

99. C₃×D₅

C₁₉×D₁₅

semidirect

101. C₅²

C₂×C₃⋊D₃×C₁₉

semidirect

104. C₁₉

D₅×C₃⋊D₁₅

direct (N × H)

109. C₃×C₅

D₃×D₅×C₁₉

semidirect

110. C₃×C₅

C₂×C₁₉×D₁₅

semidirect

118. D₅

C₁₉×C₃⋊D₁₅

direct (N × H)

119. C₃²

D₅²×C₁₉

semidirect

124. C₅

D₅×C₃⋊D₃×C₁₉

semidirect

125. C₅

C₂×C₁₉×C₃⋊D₁₅

semidirect

127. C₃

D₅×C₁₉×D₁₅

semidirect

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

subgroup	index
C₁₉×C₃⋊(C₃⋊(C₅⋊D₅))	2
C₅×C₁₉×C₃⋊D₁₅	2
C₃²×C₅×D₅×C₁₉	2
D₅×C₁₉×D₁₅	3
C₂×C₁₉×C₃⋊D₁₅	5
D₅×C₃⋊D₃×C₁₉	5
D₅×C₃⋊D₁₅	19

subgroup

index

C₁₉×C₃⋊(C₃⋊(C₅⋊D₅))

C₅×C₁₉×C₃⋊D₁₅

C₃²×C₅×D₅×C₁₉

D₅×C₁₉×D₁₅

C₂×C₁₉×C₃⋊D₁₅

D₅×C₃⋊D₃×C₁₉

D₅×C₃⋊D₁₅

Subgroups — D₅×C₁₉×C₃⋊D₁₅ #30685913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D5×C19×C3⋊D15 #30685913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×C₁₉×C₃⋊D₁₅ #30685913