Group #30685746 — subgroups

Subgroups — C₃×D₅²×C₁₉⋊C₃ #30685746

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	C₃×C₅⋊D₅×C₁₉⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×D₅×C₁₉⋊C₃	C₂	1	2	yes	maximal, complemented
3	C₃×D₅²×C₁₉	C₃	1	1	yes	maximal, characteristic, complemented
4	D₅²×C₁₉⋊C₃	C₃	1	3	yes	maximal, direct product factor
5	C₃×C₅²×C₁₉⋊C₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₂×C₃×D₅×C₁₉⋊C₃	—	5	2	no	maximal
7	C₃×C₁₉×C₅⋊D₅	C₂×C₃	1	1	yes	characteristic, complemented
8	C₃×C₅×D₅×C₁₉	C₂×C₃	1	2	yes	complemented
9	C₅⋊D₅×C₁₉⋊C₃	C₂×C₃	1	3	yes	complemented
10	C₅×D₅×C₁₉⋊C₃	C₂×C₃	1	6	yes	complemented
11	D₅²×C₁₉	C₃²	1	1	yes	characteristic, complemented, Hall
12	C₂×C₃×C₅×C₁₉⋊C₃	—	5	2	no	—
13	C₃×D₅×C₁₉⋊C₃	—	5	2	no	—
14	C₃×D₅×C₁₉⋊C₃	—	10	2	no	—
15	C₃×D₅×C₁₉⋊C₃	D₅	1	2	yes	direct product factor
16	C₃×C₅²×C₁₉	C₂²×C₃	1	1	yes	characteristic, complemented
17	C₅²×C₁₉⋊C₃	C₂²×C₃	1	3	yes	complemented
18	C₂×C₃×D₅×C₁₉	—	5	2	no	—
19	C₂×D₅×C₁₉⋊C₃	—	5	6	no	—
20	C₁₉×C₅⋊D₅	C₂×C₃²	1	1	yes	characteristic, complemented
21	C₅×D₅×C₁₉	C₂×C₃²	1	2	yes	complemented
22	C₃²×D₅²	—	19	1	no	maximal, Hall
23	C₃×C₅×C₁₉⋊C₃	—	2	2	no	—
24	C₃×C₅×C₁₉⋊C₃	C₂×D₅	1	2	yes	complemented
25	C₂²×C₃×C₁₉⋊C₃	—	25	1	no	Hall
26	C₂×C₃×C₅×C₁₉	—	5	2	no	—
27	C₃×D₅×C₁₉	—	5	2	no	—
28	C₃×D₅×C₁₉	—	10	2	no	—
29	C₃×D₅×C₁₉	C₃×D₅	1	2	yes	complemented
30	C₂×C₅×C₁₉⋊C₃	—	5	6	no	—
31	D₅×C₁₉⋊C₃	—	5	6	no	—
32	D₅×C₁₉⋊C₃	—	10	6	no	—
33	D₅×C₁₉⋊C₃	C₃×D₅	1	6	yes	direct product factor
34	C₅²×C₁₉	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
35	C₃²×C₅⋊D₅	—	19	1	no	—
36	C₃²×C₅×D₅	—	19	2	no	—
37	C₂×D₅×C₁₉	—	5	2	no	—
38	C₂×C₃×C₁₉⋊C₃	—	5	2	no	—
39	C₂×C₃×C₁₉⋊C₃	—	25	1	no	—
40	C₃×D₅²	—	19	3	no	—
41	C₃×D₅²	C₁₉⋊C₃	1	1	yes	characteristic, direct product factor
42	C₃×C₅×C₁₉	—	2	2	no	—
43	C₃×C₅×C₁₉	C₂×C₃×D₅	1	2	yes	complemented
44	C₅×C₁₉⋊C₃	—	2	6	no	—
45	C₅×C₁₉⋊C₃	C₂×C₃×D₅	1	6	yes	complemented
46	C₂²×C₃×C₁₉	—	25	1	no	—
47	C₂²×C₁₉⋊C₃	—	25	3	no	—
48	C₃²×C₅²	—	19	1	no	Hall
49	C₂×C₅×C₁₉	—	5	2	no	—
50	D₅×C₁₉	—	5	2	no	—
51	D₅×C₁₉	—	10	2	no	—
52	D₅×C₁₉	C₃²×D₅	1	2	yes	complemented
53	C₂×C₃²×D₅	—	95	2	no	—
54	C₃×C₁₉⋊C₃	D₅²	1	1	yes	characteristic, direct product factor, Hall
55	C₃×C₅⋊D₅	—	19	3	no	—
56	C₃×C₅⋊D₅	C₂×C₁₉⋊C₃	1	1	yes	characteristic, complemented
57	C₃×C₅×D₅	—	19	6	no	—
58	C₃×C₅×D₅	C₂×C₁₉⋊C₃	1	2	yes	complemented
59	C₂×C₃×C₁₉	—	5	2	no	—
60	C₂×C₃×C₁₉	—	25	1	no	—
61	C₂×C₁₉⋊C₃	—	5	6	no	—
62	C₂×C₁₉⋊C₃	—	25	3	no	—
63	D₅²	C₃×C₁₉⋊C₃	1	1	yes	characteristic, direct product factor, Hall
64	C₅×C₁₉	—	2	2	no	—
65	C₅×C₁₉	C₂×C₃²×D₅	1	2	yes	complemented
66	C₂×C₃²×C₅	—	95	2	no	—
67	C₃²×D₅	—	19	2	no	—
68	C₃²×D₅	—	95	2	no	—
69	C₃²×D₅	—	190	2	no	—
70	C₂²×C₁₉	—	25	1	no	Hall
71	C₃×C₅²	—	19	3	no	—
72	C₃×C₅²	C₂²×C₁₉⋊C₃	1	1	yes	characteristic, complemented
73	C₂×C₃×D₅	—	5	2	no	—
74	C₂×C₃×D₅	—	95	6	no	—
75	C₃×C₁₉	C₃×D₅²	1	1	yes	characteristic, complemented
76	C₁₉⋊C₃	C₃×D₅²	1	3	yes	direct product factor
77	C₅⋊D₅	C₂×C₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented
78	C₅×D₅	C₂×C₃×C₁₉⋊C₃	1	2	yes	complemented
79	C₃²×C₅	—	19	2	no	—
80	C₃²×C₅	—	38	2	no	—
81	C₂×C₁₉	—	5	2	no	—
82	C₂×C₁₉	—	25	1	no	—
83	C₂²×C₃²	—	475	1	no	Hall
84	C₂×C₃×C₅	—	5	2	no	—
85	C₂×C₃×C₅	—	95	6	no	—
86	C₃×D₅	—	5	2	no	—
87	C₃×D₅	—	10	2	no	—
88	C₃×D₅	—	19	6	no	—
89	C₃×D₅	—	95	6	no	—
90	C₃×D₅	—	190	6	no	—
91	C₃×D₅	D₅×C₁₉⋊C₃	1	2	yes	direct product factor
92	C₅²	C₂²×C₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented, Sylow
93	C₂×D₅	—	5	2	no	—
94	C₁₉	C₃²×D₅²	1	1	yes	characteristic, complemented, Sylow
95	C₂×C₃²	—	95	2	no	—
96	C₂×C₃²	—	475	1	no	—
97	C₃×C₅	—	2	2	no	—
98	C₃×C₅	—	19	6	no	—
99	C₃×C₅	—	38	6	no	—
100	C₃×C₅	C₂×D₅×C₁₉⋊C₃	1	2	yes	complemented
101	C₂²×C₃	—	25	1	no	—
102	C₂²×C₃	—	475	3	no	—
103	C₂×C₅	—	5	2	no	—
104	D₅	—	5	2	no	—
105	D₅	—	10	2	no	—
106	D₅	C₃×D₅×C₁₉⋊C₃	1	2	yes	direct product factor
107	C₃²	—	19	1	no	Sylow
108	C₂×C₃	—	5	2	no	—
109	C₂×C₃	—	25	1	no	—
110	C₂×C₃	—	95	6	no	—
111	C₂×C₃	—	475	3	no	—
112	C₅	—	2	2	no	—
113	C₅	C₂×C₃×D₅×C₁₉⋊C₃	1	2	yes	complemented
114	C₂²	—	25	1	no	Sylow
115	C₃	—	19	3	no	—
116	C₃	D₅²×C₁₉⋊C₃	1	1	yes	characteristic, direct product factor
117	C₂	—	5	2	no	—
118	C₂	—	25	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₅⋊D₅×C₁₉⋊C₃

C₂

yes

maximal, characteristic, complemented

C₃×C₅×D₅×C₁₉⋊C₃

C₂

yes

maximal, complemented

C₃×D₅²×C₁₉

C₃

yes

maximal, characteristic, complemented

D₅²×C₁₉⋊C₃

C₃

yes

maximal, direct product factor

C₃×C₅²×C₁₉⋊C₃

C₂²

yes

characteristic, complemented, Hall

C₂×C₃×D₅×C₁₉⋊C₃

—

maximal

C₃×C₁₉×C₅⋊D₅

C₂×C₃

yes

characteristic, complemented

C₃×C₅×D₅×C₁₉

C₂×C₃

yes

complemented

C₅⋊D₅×C₁₉⋊C₃

C₂×C₃

yes

complemented

C₅×D₅×C₁₉⋊C₃

C₂×C₃

yes

complemented

D₅²×C₁₉

C₃²

yes

characteristic, complemented, Hall

C₂×C₃×C₅×C₁₉⋊C₃

—

C₃×D₅×C₁₉⋊C₃

—

C₃×D₅×C₁₉⋊C₃

—

C₃×D₅×C₁₉⋊C₃

D₅

yes

direct product factor

C₃×C₅²×C₁₉

C₂²×C₃

yes

characteristic, complemented

C₅²×C₁₉⋊C₃

C₂²×C₃

yes

complemented

C₂×C₃×D₅×C₁₉

—

C₂×D₅×C₁₉⋊C₃

—

C₁₉×C₅⋊D₅

C₂×C₃²

yes

characteristic, complemented

C₅×D₅×C₁₉

C₂×C₃²

yes

complemented

C₃²×D₅²

—

maximal, Hall

C₃×C₅×C₁₉⋊C₃

—

C₃×C₅×C₁₉⋊C₃

C₂×D₅

yes

complemented

C₂²×C₃×C₁₉⋊C₃

—

Hall

C₂×C₃×C₅×C₁₉

—

C₃×D₅×C₁₉

—

C₃×D₅×C₁₉

—

C₃×D₅×C₁₉

C₃×D₅

yes

complemented

C₂×C₅×C₁₉⋊C₃

—

D₅×C₁₉⋊C₃

—

D₅×C₁₉⋊C₃

—

D₅×C₁₉⋊C₃

C₃×D₅

yes

direct product factor

C₅²×C₁₉

C₂²×C₃²

yes

characteristic, complemented, Hall

C₃²×C₅⋊D₅

—

C₃²×C₅×D₅

—

C₂×D₅×C₁₉

—

C₂×C₃×C₁₉⋊C₃

—

C₂×C₃×C₁₉⋊C₃

—

C₃×D₅²

—

C₃×D₅²

C₁₉⋊C₃

yes

characteristic, direct product factor

C₃×C₅×C₁₉

—

C₃×C₅×C₁₉

C₂×C₃×D₅

yes

complemented

C₅×C₁₉⋊C₃

—

C₅×C₁₉⋊C₃

C₂×C₃×D₅

yes

complemented

C₂²×C₃×C₁₉

—

C₂²×C₁₉⋊C₃

—

C₃²×C₅²

—

Hall

C₂×C₅×C₁₉

—

D₅×C₁₉

—

D₅×C₁₉

—

D₅×C₁₉

C₃²×D₅

yes

complemented

C₂×C₃²×D₅

—

C₃×C₁₉⋊C₃

D₅²

yes

characteristic, direct product factor, Hall

C₃×C₅⋊D₅

—

C₃×C₅⋊D₅

C₂×C₁₉⋊C₃

yes

characteristic, complemented

C₃×C₅×D₅

—

C₃×C₅×D₅

C₂×C₁₉⋊C₃

yes

complemented

C₂×C₃×C₁₉

—

C₂×C₃×C₁₉

—

C₂×C₁₉⋊C₃

—

C₂×C₁₉⋊C₃

—

D₅²

C₃×C₁₉⋊C₃

yes

characteristic, direct product factor, Hall

C₅×C₁₉

—

C₅×C₁₉

C₂×C₃²×D₅

yes

complemented

C₂×C₃²×C₅

—

C₃²×D₅

—

C₃²×D₅

—

C₃²×D₅

—

190

—

C₂²×C₁₉

—

Hall

C₃×C₅²

—

C₃×C₅²

C₂²×C₁₉⋊C₃

yes

characteristic, complemented

C₂×C₃×D₅

—

C₂×C₃×D₅

—

C₃×C₁₉

C₃×D₅²

yes

characteristic, complemented

C₁₉⋊C₃

C₃×D₅²

yes

direct product factor

C₅⋊D₅

C₂×C₃×C₁₉⋊C₃

yes

characteristic, complemented

C₅×D₅

C₂×C₃×C₁₉⋊C₃

yes

complemented

C₃²×C₅

—

C₃²×C₅

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂²×C₃²

—

475

Hall

C₂×C₃×C₅

—

C₂×C₃×C₅

—

C₃×D₅

—

C₃×D₅

—

C₃×D₅

—

C₃×D₅

—

C₃×D₅

—

190

—

C₃×D₅

D₅×C₁₉⋊C₃

yes

direct product factor

C₅²

C₂²×C₃×C₁₉⋊C₃

yes

characteristic, complemented, Sylow

C₂×D₅

—

C₁₉

C₃²×D₅²

yes

characteristic, complemented, Sylow

C₂×C₃²

—

C₂×C₃²

—

475

—

C₃×C₅

—

C₃×C₅

—

C₃×C₅

—

100

C₃×C₅

C₂×D₅×C₁₉⋊C₃

yes

complemented

101

C₂²×C₃

—

102

C₂²×C₃

—

475

—

103

C₂×C₅

—

104

D₅

—

105

D₅

—

106

D₅

C₃×D₅×C₁₉⋊C₃

yes

direct product factor

107

C₃²

—

Sylow

108

C₂×C₃

—

109

C₂×C₃

—

110

C₂×C₃

—

111

C₂×C₃

—

475

—

112

C₅

—

113

C₅

C₂×C₃×D₅×C₁₉⋊C₃

yes

complemented

114

C₂²

—

Sylow

115

C₃

—

116

C₃

D₅²×C₁₉⋊C₃

yes

characteristic, direct product factor

117

C₂

—

118

C₂

—

	subgroup	G/H
center, hypercenter	C₃	D₅²×C₁₉⋊C₃
commutator, cosocle, hypocenter	C₅²×C₁₉	C₂²×C₃²
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₁₉	C₂²×C₃
LCS	G ▷ C₅²×C₁₉
UCS	C₁ ◁ C₃

subgroup

G/H

center, hypercenter

C₃

D₅²×C₁₉⋊C₃

commutator, cosocle, hypocenter

C₅²×C₁₉

C₂²×C₃²

Frattini

C₁

Fitting, socle

C₃×C₅²×C₁₉

C₂²×C₃

LCS

G ▷ C₅²×C₁₉

UCS

C₁ ◁ C₃

N	H ≅ G/N	type
1. C₃×C₅⋊D₅×C₁₉⋊C₃	C₂	semidirect
2. C₃×C₅×D₅×C₁₉⋊C₃	C₂	semidirect
3. C₃×D₅²×C₁₉	C₃	semidirect
4. D₅²×C₁₉⋊C₃	C₃	direct (N × H)
5. C₃×C₅²×C₁₉⋊C₃	C₂²	semidirect
7. C₃×C₁₉×C₅⋊D₅	C₂×C₃	semidirect
8. C₃×C₅×D₅×C₁₉	C₂×C₃	semidirect
9. C₅⋊D₅×C₁₉⋊C₃	C₂×C₃	semidirect
10. C₅×D₅×C₁₉⋊C₃	C₂×C₃	semidirect
11. D₅²×C₁₉	C₃²	semidirect
15. C₃×D₅×C₁₉⋊C₃	D₅	direct (N × H)
16. C₃×C₅²×C₁₉	C₂²×C₃	semidirect
17. C₅²×C₁₉⋊C₃	C₂²×C₃	semidirect
20. C₁₉×C₅⋊D₅	C₂×C₃²	semidirect
21. C₅×D₅×C₁₉	C₂×C₃²	semidirect
24. C₃×C₅×C₁₉⋊C₃	C₂×D₅	semidirect
29. C₃×D₅×C₁₉	C₃×D₅	semidirect
33. D₅×C₁₉⋊C₃	C₃×D₅	direct (N × H)
34. C₅²×C₁₉	C₂²×C₃²	semidirect
41. C₃×D₅²	C₁₉⋊C₃	direct (N × H)
43. C₃×C₅×C₁₉	C₂×C₃×D₅	semidirect
45. C₅×C₁₉⋊C₃	C₂×C₃×D₅	semidirect
52. D₅×C₁₉	C₃²×D₅	semidirect
54. C₃×C₁₉⋊C₃	D₅²	direct (N × H)
56. C₃×C₅⋊D₅	C₂×C₁₉⋊C₃	semidirect
58. C₃×C₅×D₅	C₂×C₁₉⋊C₃	semidirect
63. D₅²	C₃×C₁₉⋊C₃	direct (N × H)
65. C₅×C₁₉	C₂×C₃²×D₅	semidirect
72. C₃×C₅²	C₂²×C₁₉⋊C₃	semidirect
75. C₃×C₁₉	C₃×D₅²	semidirect
76. C₁₉⋊C₃	C₃×D₅²	direct (N × H)
77. C₅⋊D₅	C₂×C₃×C₁₉⋊C₃	semidirect
78. C₅×D₅	C₂×C₃×C₁₉⋊C₃	semidirect
91. C₃×D₅	D₅×C₁₉⋊C₃	direct (N × H)
92. C₅²	C₂²×C₃×C₁₉⋊C₃	semidirect
94. C₁₉	C₃²×D₅²	semidirect
100. C₃×C₅	C₂×D₅×C₁₉⋊C₃	semidirect
106. D₅	C₃×D₅×C₁₉⋊C₃	direct (N × H)
113. C₅	C₂×C₃×D₅×C₁₉⋊C₃	semidirect
116. C₃	D₅²×C₁₉⋊C₃	direct (N × H)

H ≅ G/N

type

1. C₃×C₅⋊D₅×C₁₉⋊C₃

C₂

semidirect

2. C₃×C₅×D₅×C₁₉⋊C₃

C₂

semidirect

3. C₃×D₅²×C₁₉

C₃

semidirect

4. D₅²×C₁₉⋊C₃

C₃

direct (N × H)

5. C₃×C₅²×C₁₉⋊C₃

C₂²

semidirect

7. C₃×C₁₉×C₅⋊D₅

C₂×C₃

semidirect

8. C₃×C₅×D₅×C₁₉

C₂×C₃

semidirect

9. C₅⋊D₅×C₁₉⋊C₃

C₂×C₃

semidirect

10. C₅×D₅×C₁₉⋊C₃

C₂×C₃

semidirect

11. D₅²×C₁₉

C₃²

semidirect

15. C₃×D₅×C₁₉⋊C₃

D₅

direct (N × H)

16. C₃×C₅²×C₁₉

C₂²×C₃

semidirect

17. C₅²×C₁₉⋊C₃

C₂²×C₃

semidirect

20. C₁₉×C₅⋊D₅

C₂×C₃²

semidirect

21. C₅×D₅×C₁₉

C₂×C₃²

semidirect

24. C₃×C₅×C₁₉⋊C₃

C₂×D₅

semidirect

29. C₃×D₅×C₁₉

C₃×D₅

semidirect

33. D₅×C₁₉⋊C₃

C₃×D₅

direct (N × H)

34. C₅²×C₁₉

C₂²×C₃²

semidirect

41. C₃×D₅²

C₁₉⋊C₃

direct (N × H)

43. C₃×C₅×C₁₉

C₂×C₃×D₅

semidirect

45. C₅×C₁₉⋊C₃

C₂×C₃×D₅

semidirect

52. D₅×C₁₉

C₃²×D₅

semidirect

54. C₃×C₁₉⋊C₃

D₅²

direct (N × H)

56. C₃×C₅⋊D₅

C₂×C₁₉⋊C₃

semidirect

58. C₃×C₅×D₅

C₂×C₁₉⋊C₃

semidirect

63. D₅²

C₃×C₁₉⋊C₃

direct (N × H)

65. C₅×C₁₉

C₂×C₃²×D₅

semidirect

72. C₃×C₅²

C₂²×C₁₉⋊C₃

semidirect

75. C₃×C₁₉

C₃×D₅²

semidirect

76. C₁₉⋊C₃

C₃×D₅²

direct (N × H)

77. C₅⋊D₅

C₂×C₃×C₁₉⋊C₃

semidirect

78. C₅×D₅

C₂×C₃×C₁₉⋊C₃

semidirect

91. C₃×D₅

D₅×C₁₉⋊C₃

direct (N × H)

92. C₅²

C₂²×C₃×C₁₉⋊C₃

semidirect

94. C₁₉

C₃²×D₅²

semidirect

100. C₃×C₅

C₂×D₅×C₁₉⋊C₃

semidirect

106. D₅

C₃×D₅×C₁₉⋊C₃

direct (N × H)

113. C₅

C₂×C₃×D₅×C₁₉⋊C₃

semidirect

116. C₃

D₅²×C₁₉⋊C₃

direct (N × H)

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

subgroup	index
C₃×C₅⋊D₅×C₁₉⋊C₃	2
C₃×C₅×D₅×C₁₉⋊C₃	2
C₃×D₅²×C₁₉	3
D₅²×C₁₉⋊C₃	3
C₂×C₃×D₅×C₁₉⋊C₃	5
C₃²×D₅²	19

subgroup

index

C₃×C₅⋊D₅×C₁₉⋊C₃

C₃×C₅×D₅×C₁₉⋊C₃

C₃×D₅²×C₁₉

D₅²×C₁₉⋊C₃

C₂×C₃×D₅×C₁₉⋊C₃

C₃²×D₅²

Subgroups — C₃×D₅²×C₁₉⋊C₃ #30685746

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×D52×C19⋊C3 #30685746

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₅²×C₁₉⋊C₃ #30685746