Group #30685891 — subgroups

Subgroups — C₃×C₁₉×D₁₅⋊D₅ #30685891

Order 17100.

224 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 60 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	C₃×C₅×C₁₉×D₁₅	C₂	1	2	yes	maximal, complemented
2	C₃²×C₁₉×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×D₁₅⋊D₅	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₃×D₅²×C₁₉	—	3	1	no	maximal
5	C₃²×C₅²×C₁₉	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₃×D₃×D₅×C₁₉	—	5	2	no	maximal
7	C₅×C₁₉×D₁₅	C₂×C₃	1	2	yes	complemented
8	C₃×C₁₉×C₅⋊D₅	—	2	1	no	—
9	C₃×C₁₉×C₅⋊D₅	D₃	1	1	yes	characteristic, complemented
10	C₃×C₁₉×C₅⋊D₅	C₂×C₃	1	1	yes	characteristic, complemented
11	C₃×C₅×D₅×C₁₉	—	3	2	no	—
12	D₅²×C₁₉	—	3	1	no	Hall
13	C₃×C₁₉×D₁₅	D₅	1	2	yes	complemented
14	C₃×C₅×D₃×C₁₉	—	5	2	no	—
15	C₃²×D₅×C₁₉	—	5	2	no	—
16	C₃²×D₅×C₁₉	—	10	2	no	—
17	C₃×C₅²×C₁₉	—	2	1	no	—
18	C₃×C₅²×C₁₉	C₂×D₃	1	1	yes	characteristic, complemented
19	C₃×C₅²×C₁₉	C₂²×C₃	1	1	yes	characteristic, complemented
20	C₂×C₃×D₅×C₁₉	—	15	2	no	—
21	D₃×D₅×C₁₉	—	5	2	no	—
22	C₁₉×C₅⋊D₅	C₃×D₃	1	1	yes	characteristic, complemented
23	C₅×D₅×C₁₉	—	3	2	no	—
24	C₃×D₁₅⋊D₅	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₃²×C₅×C₁₉	—	2	2	no	—
26	C₃²×C₅×C₁₉	C₂×D₅	1	2	yes	complemented
27	C₂×C₃×D₃×C₁₉	—	25	1	no	Hall
28	C₂×C₃×C₅×C₁₉	—	15	2	no	—
29	C₁₉×D₁₅	C₃×D₅	1	2	yes	complemented
30	C₅×D₃×C₁₉	—	5	2	no	—
31	C₃×D₅×C₁₉	—	3	2	no	—
32	C₃×D₅×C₁₉	—	5	2	no	—
33	C₃×D₅×C₁₉	—	5	2	no	—
34	C₃×D₅×C₁₉	—	10	2	no	—
35	C₃×D₅×C₁₉	—	10	2	no	—
36	C₃×D₅×C₁₉	—	10	2	no	—
37	C₃×D₅×C₁₉	—	10	4	no	—
38	C₅²×C₁₉	C₂×C₃×D₃	1	1	yes	characteristic, complemented, Hall
39	C₃×C₅×D₁₅	C₂×C₁₉	1	2	yes	complemented
40	C₃²×C₅⋊D₅	C₂×C₁₉	1	1	yes	characteristic, complemented
41	C₂×D₅×C₁₉	—	15	2	no	—
42	C₂×C₃²×C₁₉	—	25	1	no	—
43	C₃×D₃×C₁₉	—	5	2	no	—
44	D₁₅⋊D₅	C₃×C₁₉	1	1	yes	characteristic, direct product factor
45	C₃×D₅²	—	3	1	no	—
46	C₃×C₅×C₁₉	—	2	4	no	—
47	C₃×C₅×C₁₉	—	2	2	no	—
48	C₃×C₅×C₁₉	—	2	2	no	—
49	C₃×C₅×C₁₉	—	2	2	no	—
50	C₃×C₅×C₁₉	D₃×D₅	1	2	yes	complemented
51	C₃×C₅×C₁₉	C₂×C₃×D₅	1	2	yes	complemented
52	C₂²×C₃×C₁₉	—	75	1	no	—
53	C₂×D₃×C₁₉	—	25	1	no	—
54	C₃²×C₅²	C₂²×C₁₉	1	1	yes	characteristic, complemented, Hall
55	C₂×C₅×C₁₉	—	15	2	no	—
56	D₅×C₁₉	—	3	2	no	—
57	D₅×C₁₉	—	5	2	no	—
58	D₅×C₁₉	—	10	2	no	—
59	C₃×D₃×D₅	—	5	2	no	—
60	C₃²×C₁₉	D₅²	1	1	yes	characteristic, complemented, Hall
61	C₅×D₁₅	C₂×C₃×C₁₉	1	2	yes	complemented
62	C₃×C₅⋊D₅	—	2	1	no	—
63	C₃×C₅⋊D₅	D₃×C₁₉	1	1	yes	characteristic, complemented
64	C₃×C₅⋊D₅	C₂×C₃×C₁₉	1	1	yes	characteristic, complemented
65	C₃×C₅×D₅	—	3	2	no	—
66	C₂×C₃×C₁₉	—	15	2	no	—
67	C₂×C₃×C₁₉	—	25	1	no	—
68	C₂×C₃×C₁₉	—	25	1	no	—
69	C₂×C₃×C₁₉	—	50	1	no	—
70	D₃×C₁₉	—	5	2	no	—
71	D₅²	—	3	1	no	Hall
72	C₅×C₁₉	—	2	2	no	—
73	C₅×C₁₉	C₃×D₃×D₅	1	2	yes	complemented
74	C₃×D₁₅	D₅×C₁₉	1	2	yes	complemented
75	C₃×C₅×D₃	—	5	2	no	—
76	C₃²×D₅	—	5	2	no	—
77	C₃²×D₅	—	10	2	no	—
78	C₂²×C₁₉	—	75	1	no	Hall
79	C₃×C₅²	—	2	1	no	—
80	C₃×C₅²	C₂×D₃×C₁₉	1	1	yes	characteristic, complemented
81	C₃×C₅²	C₂²×C₃×C₁₉	1	1	yes	characteristic, complemented
82	C₂×C₃×D₅	—	15	2	no	—
83	D₃×D₅	—	5	2	no	—
84	C₃×C₁₉	—	2	1	no	—
85	C₃×C₁₉	C₃×D₅²	1	1	yes	characteristic, complemented
86	C₃×C₁₉	D₁₅⋊D₅	1	1	yes	characteristic, direct product factor
87	C₅⋊D₅	C₃×D₃×C₁₉	1	1	yes	characteristic, complemented
88	C₅×D₅	—	3	2	no	—
89	C₃²×C₅	—	2	2	no	—
90	C₃²×C₅	C₂×D₅×C₁₉	1	2	yes	complemented
91	C₂×C₁₉	—	15	2	no	—
92	C₂×C₁₉	—	25	1	no	—
93	C₂×C₃×D₃	—	25	1	no	Hall
94	C₂×C₃×C₅	—	15	2	no	—
95	D₁₅	C₃×D₅×C₁₉	1	2	yes	complemented
96	C₃×D₅	—	3	2	no	—
97	C₃×D₅	—	5	2	no	—
98	C₃×D₅	—	5	2	no	—
99	C₃×D₅	—	10	2	no	—
100	C₃×D₅	—	10	2	no	—
101	C₃×D₅	—	10	2	no	—
102	C₃×D₅	—	10	4	no	—
103	C₅×D₃	—	5	2	no	—
104	C₅²	C₂×C₃×D₃×C₁₉	1	1	yes	characteristic, complemented, Sylow
105	C₂×D₅	—	15	2	no	—
106	C₁₉	C₃×D₁₅⋊D₅	1	1	yes	characteristic, direct product factor, Sylow
107	C₂×C₃²	—	25	1	no	—
108	C₃×D₃	—	5	2	no	—
109	C₃×C₅	—	2	4	no	—
110	C₃×C₅	—	2	2	no	—
111	C₃×C₅	—	2	2	no	—
112	C₃×C₅	—	2	2	no	—
113	C₃×C₅	D₃×D₅×C₁₉	1	2	yes	complemented
114	C₃×C₅	C₂×C₃×D₅×C₁₉	1	2	yes	complemented
115	C₂²×C₃	—	75	1	no	—
116	C₂×D₃	—	25	1	no	—
117	C₂×C₅	—	15	2	no	—
118	D₅	—	3	2	no	—
119	D₅	—	5	2	no	—
120	D₅	—	10	2	no	—
121	C₃²	D₅²×C₁₉	1	1	yes	characteristic, complemented, Sylow
122	C₂×C₃	—	15	2	no	—
123	C₂×C₃	—	25	1	no	—
124	C₂×C₃	—	25	1	no	—
125	C₂×C₃	—	50	1	no	—
126	D₃	—	5	2	no	—
127	C₅	—	2	2	no	—
128	C₅	C₃×D₃×D₅×C₁₉	1	2	yes	complemented
129	C₂²	—	75	1	no	Sylow
130	C₃	—	2	1	no	—
131	C₃	C₃×D₅²×C₁₉	1	1	yes	characteristic, complemented
132	C₃	C₁₉×D₁₅⋊D₅	1	1	yes	characteristic, direct product factor
133	C₂	—	15	2	no	—
134	C₂	—	25	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₅×C₁₉×D₁₅

C₂

yes

maximal, complemented

C₃²×C₁₉×C₅⋊D₅

C₂

yes

maximal, characteristic, complemented

C₁₉×D₁₅⋊D₅

C₃

yes

maximal, characteristic, direct product factor

C₃×D₅²×C₁₉

—

maximal

C₃²×C₅²×C₁₉

C₂²

yes

characteristic, complemented, Hall

C₃×D₃×D₅×C₁₉

—

maximal

C₅×C₁₉×D₁₅

C₂×C₃

yes

complemented

C₃×C₁₉×C₅⋊D₅

—

C₃×C₁₉×C₅⋊D₅

D₃

yes

characteristic, complemented

C₃×C₁₉×C₅⋊D₅

C₂×C₃

yes

characteristic, complemented

C₃×C₅×D₅×C₁₉

—

D₅²×C₁₉

—

Hall

C₃×C₁₉×D₁₅

D₅

yes

complemented

C₃×C₅×D₃×C₁₉

—

C₃²×D₅×C₁₉

—

C₃²×D₅×C₁₉

—

C₃×C₅²×C₁₉

—

C₃×C₅²×C₁₉

C₂×D₃

yes

characteristic, complemented

C₃×C₅²×C₁₉

C₂²×C₃

yes

characteristic, complemented

C₂×C₃×D₅×C₁₉

—

D₃×D₅×C₁₉

—

C₁₉×C₅⋊D₅

C₃×D₃

yes

characteristic, complemented

C₅×D₅×C₁₉

—

C₃×D₁₅⋊D₅

C₁₉

yes

maximal, characteristic, direct product factor, Hall

C₃²×C₅×C₁₉

—

C₃²×C₅×C₁₉

C₂×D₅

yes

complemented

C₂×C₃×D₃×C₁₉

—

Hall

C₂×C₃×C₅×C₁₉

—

C₁₉×D₁₅

C₃×D₅

yes

complemented

C₅×D₃×C₁₉

—

C₃×D₅×C₁₉

—

C₃×D₅×C₁₉

—

C₃×D₅×C₁₉

—

C₃×D₅×C₁₉

—

C₃×D₅×C₁₉

—

C₃×D₅×C₁₉

—

C₃×D₅×C₁₉

—

C₅²×C₁₉

C₂×C₃×D₃

yes

characteristic, complemented, Hall

C₃×C₅×D₁₅

C₂×C₁₉

yes

complemented

C₃²×C₅⋊D₅

C₂×C₁₉

yes

characteristic, complemented

C₂×D₅×C₁₉

—

C₂×C₃²×C₁₉

—

C₃×D₃×C₁₉

—

D₁₅⋊D₅

C₃×C₁₉

yes

characteristic, direct product factor

C₃×D₅²

—

C₃×C₅×C₁₉

—

C₃×C₅×C₁₉

—

C₃×C₅×C₁₉

—

C₃×C₅×C₁₉

—

C₃×C₅×C₁₉

D₃×D₅

yes

complemented

C₃×C₅×C₁₉

C₂×C₃×D₅

yes

complemented

C₂²×C₃×C₁₉

—

C₂×D₃×C₁₉

—

C₃²×C₅²

C₂²×C₁₉

yes

characteristic, complemented, Hall

C₂×C₅×C₁₉

—

D₅×C₁₉

—

D₅×C₁₉

—

D₅×C₁₉

—

C₃×D₃×D₅

—

C₃²×C₁₉

D₅²

yes

characteristic, complemented, Hall

C₅×D₁₅

C₂×C₃×C₁₉

yes

complemented

C₃×C₅⋊D₅

—

C₃×C₅⋊D₅

D₃×C₁₉

yes

characteristic, complemented

C₃×C₅⋊D₅

C₂×C₃×C₁₉

yes

characteristic, complemented

C₃×C₅×D₅

—

C₂×C₃×C₁₉

—

C₂×C₃×C₁₉

—

C₂×C₃×C₁₉

—

C₂×C₃×C₁₉

—

D₃×C₁₉

—

D₅²

—

Hall

C₅×C₁₉

—

C₅×C₁₉

C₃×D₃×D₅

yes

complemented

C₃×D₁₅

D₅×C₁₉

yes

complemented

C₃×C₅×D₃

—

C₃²×D₅

—

C₃²×D₅

—

C₂²×C₁₉

—

Hall

C₃×C₅²

—

C₃×C₅²

C₂×D₃×C₁₉

yes

characteristic, complemented

C₃×C₅²

C₂²×C₃×C₁₉

yes

characteristic, complemented

C₂×C₃×D₅

—

D₃×D₅

—

C₃×C₁₉

—

C₃×C₁₉

C₃×D₅²

yes

characteristic, complemented

C₃×C₁₉

D₁₅⋊D₅

yes

characteristic, direct product factor

C₅⋊D₅

C₃×D₃×C₁₉

yes

characteristic, complemented

C₅×D₅

—

C₃²×C₅

—

C₃²×C₅

C₂×D₅×C₁₉

yes

complemented

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₃×D₃

—

Hall

C₂×C₃×C₅

—

D₁₅

C₃×D₅×C₁₉

yes

complemented

C₃×D₅

—

C₃×D₅

—

C₃×D₅

—

C₃×D₅

—

100

C₃×D₅

—

101

C₃×D₅

—

102

C₃×D₅

—

103

C₅×D₃

—

104

C₅²

C₂×C₃×D₃×C₁₉

yes

characteristic, complemented, Sylow

105

C₂×D₅

—

106

C₁₉

C₃×D₁₅⋊D₅

yes

characteristic, direct product factor, Sylow

107

C₂×C₃²

—

108

C₃×D₃

—

109

C₃×C₅

—

110

C₃×C₅

—

111

C₃×C₅

—

112

C₃×C₅

—

113

C₃×C₅

D₃×D₅×C₁₉

yes

complemented

114

C₃×C₅

C₂×C₃×D₅×C₁₉

yes

complemented

115

C₂²×C₃

—

116

C₂×D₃

—

117

C₂×C₅

—

118

D₅

—

119

D₅

—

120

D₅

—

121

C₃²

D₅²×C₁₉

yes

characteristic, complemented, Sylow

122

C₂×C₃

—

123

C₂×C₃

—

124

C₂×C₃

—

125

C₂×C₃

—

126

D₃

—

127

C₅

—

128

C₅

C₃×D₃×D₅×C₁₉

yes

complemented

129

C₂²

—

Sylow

130

C₃

—

131

C₃

C₃×D₅²×C₁₉

yes

characteristic, complemented

132

C₃

C₁₉×D₁₅⋊D₅

yes

characteristic, direct product factor

133

C₂

—

134

C₂

—

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

subgroup

G/H

center, hypercenter

C₃×C₁₉

D₁₅⋊D₅

commutator, cosocle, hypocenter

C₃×C₅²

C₂²×C₃×C₁₉

Frattini

C₁

Fitting, socle

C₃²×C₅²×C₁₉

C₂²

LCS

G ▷ C₃×C₅²

UCS

C₁ ◁ C₃×C₁₉

N	H ≅ G/N	type
1. C₃×C₅×C₁₉×D₁₅	C₂	semidirect
2. C₃²×C₁₉×C₅⋊D₅	C₂	semidirect
3. C₁₉×D₁₅⋊D₅	C₃	direct (N × H)
5. C₃²×C₅²×C₁₉	C₂²	semidirect
7. C₅×C₁₉×D₁₅	C₂×C₃	semidirect
9. C₃×C₁₉×C₅⋊D₅	D₃	semidirect
10. C₃×C₁₉×C₅⋊D₅	C₂×C₃	semidirect
13. C₃×C₁₉×D₁₅	D₅	semidirect
18. C₃×C₅²×C₁₉	C₂×D₃	semidirect
19. C₃×C₅²×C₁₉	C₂²×C₃	semidirect
22. C₁₉×C₅⋊D₅	C₃×D₃	semidirect
24. C₃×D₁₅⋊D₅	C₁₉	direct (N × H)
26. C₃²×C₅×C₁₉	C₂×D₅	semidirect
29. C₁₉×D₁₅	C₃×D₅	semidirect
38. C₅²×C₁₉	C₂×C₃×D₃	semidirect
39. C₃×C₅×D₁₅	C₂×C₁₉	semidirect
40. C₃²×C₅⋊D₅	C₂×C₁₉	semidirect
44. D₁₅⋊D₅	C₃×C₁₉	direct (N × H)
50. C₃×C₅×C₁₉	D₃×D₅	semidirect
51. C₃×C₅×C₁₉	C₂×C₃×D₅	semidirect
54. C₃²×C₅²	C₂²×C₁₉	semidirect
60. C₃²×C₁₉	D₅²	semidirect
61. C₅×D₁₅	C₂×C₃×C₁₉	semidirect
63. C₃×C₅⋊D₅	D₃×C₁₉	semidirect
64. C₃×C₅⋊D₅	C₂×C₃×C₁₉	semidirect
73. C₅×C₁₉	C₃×D₃×D₅	semidirect
74. C₃×D₁₅	D₅×C₁₉	semidirect
80. C₃×C₅²	C₂×D₃×C₁₉	semidirect
81. C₃×C₅²	C₂²×C₃×C₁₉	semidirect
85. C₃×C₁₉	C₃×D₅²	semidirect
86. C₃×C₁₉	D₁₅⋊D₅	direct (N × H)
87. C₅⋊D₅	C₃×D₃×C₁₉	semidirect
90. C₃²×C₅	C₂×D₅×C₁₉	semidirect
95. D₁₅	C₃×D₅×C₁₉	semidirect
104. C₅²	C₂×C₃×D₃×C₁₉	semidirect
106. C₁₉	C₃×D₁₅⋊D₅	direct (N × H)
113. C₃×C₅	D₃×D₅×C₁₉	semidirect
114. C₃×C₅	C₂×C₃×D₅×C₁₉	semidirect
121. C₃²	D₅²×C₁₉	semidirect
128. C₅	C₃×D₃×D₅×C₁₉	semidirect
131. C₃	C₃×D₅²×C₁₉	semidirect
132. C₃	C₁₉×D₁₅⋊D₅	direct (N × H)

H ≅ G/N

type

1. C₃×C₅×C₁₉×D₁₅

C₂

semidirect

2. C₃²×C₁₉×C₅⋊D₅

C₂

semidirect

3. C₁₉×D₁₅⋊D₅

C₃

direct (N × H)

5. C₃²×C₅²×C₁₉

C₂²

semidirect

7. C₅×C₁₉×D₁₅

C₂×C₃

semidirect

9. C₃×C₁₉×C₅⋊D₅

D₃

semidirect

10. C₃×C₁₉×C₅⋊D₅

C₂×C₃

semidirect

13. C₃×C₁₉×D₁₅

D₅

semidirect

18. C₃×C₅²×C₁₉

C₂×D₃

semidirect

19. C₃×C₅²×C₁₉

C₂²×C₃

semidirect

22. C₁₉×C₅⋊D₅

C₃×D₃

semidirect

24. C₃×D₁₅⋊D₅

C₁₉

direct (N × H)

26. C₃²×C₅×C₁₉

C₂×D₅

semidirect

29. C₁₉×D₁₅

C₃×D₅

semidirect

38. C₅²×C₁₉

C₂×C₃×D₃

semidirect

39. C₃×C₅×D₁₅

C₂×C₁₉

semidirect

40. C₃²×C₅⋊D₅

C₂×C₁₉

semidirect

44. D₁₅⋊D₅

C₃×C₁₉

direct (N × H)

50. C₃×C₅×C₁₉

D₃×D₅

semidirect

51. C₃×C₅×C₁₉

C₂×C₃×D₅

semidirect

54. C₃²×C₅²

C₂²×C₁₉

semidirect

60. C₃²×C₁₉

D₅²

semidirect

61. C₅×D₁₅

C₂×C₃×C₁₉

semidirect

63. C₃×C₅⋊D₅

D₃×C₁₉

semidirect

64. C₃×C₅⋊D₅

C₂×C₃×C₁₉

semidirect

73. C₅×C₁₉

C₃×D₃×D₅

semidirect

74. C₃×D₁₅

D₅×C₁₉

semidirect

80. C₃×C₅²

C₂×D₃×C₁₉

semidirect

81. C₃×C₅²

C₂²×C₃×C₁₉

semidirect

85. C₃×C₁₉

C₃×D₅²

semidirect

86. C₃×C₁₉

D₁₅⋊D₅

direct (N × H)

87. C₅⋊D₅

C₃×D₃×C₁₉

semidirect

90. C₃²×C₅

C₂×D₅×C₁₉

semidirect

95. D₁₅

C₃×D₅×C₁₉

semidirect

104. C₅²

C₂×C₃×D₃×C₁₉

semidirect

106. C₁₉

C₃×D₁₅⋊D₅

direct (N × H)

113. C₃×C₅

D₃×D₅×C₁₉

semidirect

114. C₃×C₅

C₂×C₃×D₅×C₁₉

semidirect

121. C₃²

D₅²×C₁₉

semidirect

128. C₅

C₃×D₃×D₅×C₁₉

semidirect

131. C₃

C₃×D₅²×C₁₉

semidirect

132. C₃

C₁₉×D₁₅⋊D₅

direct (N × H)

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

subgroup	index
C₃×C₅×C₁₉×D₁₅	2
C₃²×C₁₉×C₅⋊D₅	2
C₁₉×D₁₅⋊D₅	3
C₃×D₅²×C₁₉	3
C₃×D₃×D₅×C₁₉	5
C₃×D₁₅⋊D₅	19

subgroup

index

C₃×C₅×C₁₉×D₁₅

C₃²×C₁₉×C₅⋊D₅

C₁₉×D₁₅⋊D₅

C₃×D₅²×C₁₉

C₃×D₃×D₅×C₁₉

C₃×D₁₅⋊D₅

Subgroups — C₃×C₁₉×D₁₅⋊D₅ #30685891

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×C19×D15⋊D5 #30685891

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₁₉×D₁₅⋊D₅ #30685891