Group #37561585 — subgroups

Subgroups — C₁₉×(C₃⋊D₃)⋊D₃₅ #37561585

Order 23940.

176 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 50 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×C₁₉×D₁₀₅	C₂	1	2	yes	maximal, complemented
2	C₅×C₇×C₃⋊D₃×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
3	D₃×C₁₉×D₃₅	—	3	2	no	maximal
4	C₃²×C₅×C₇×C₁₉	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₁₉×(C₃⋊D₃)⋊D₇	—	5	1	no	maximal, Hall
6	C₁₉×D₁₀₅	D₃	1	2	yes	complemented
7	C₅×D₃×C₇×C₁₉	—	3	2	no	—
8	C₅×D₃×C₇×C₁₉	—	6	1	no	—
9	C₃×C₁₉×D₃₅	—	3	2	no	—
10	C₁₉×(C₃⋊D₃)⋊D₅	—	7	1	no	maximal, Hall
11	C₂×C₁₉×D₃₅	—	9	1	no	Hall
12	C₇×C₃⋊D₃×C₁₉	D₅	1	1	yes	characteristic, complemented
13	C₃×C₁₉×D₂₁	—	5	2	no	—
14	C₃×C₅×C₇×C₁₉	—	2	1	no	—
15	C₃×C₅×C₇×C₁₉	C₂×D₃	1	2	yes	complemented
16	C₅×C₃⋊D₃×C₁₉	D₇	1	1	yes	characteristic, complemented
17	C₃×C₁₉×D₁₅	—	7	2	no	—
18	D₃×D₇×C₁₉	—	15	2	no	—
19	C₂×C₅×C₇×C₁₉	—	9	1	no	—
20	C₁₉×D₃₅	—	3	2	no	—
21	(C₃⋊D₃)⋊D₃₅	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₃²×C₇×C₁₉	C₂×D₅	1	1	yes	characteristic, complemented, Hall
23	D₃×D₅×C₁₉	—	21	2	no	—
24	C₃²×C₅×C₁₉	C₂×D₇	1	1	yes	characteristic, complemented, Hall
25	C₁₉×D₂₁	—	5	2	no	—
26	D₃×C₇×C₁₉	—	3	2	no	—
27	D₃×C₇×C₁₉	—	6	1	no	—
28	C₃×D₇×C₁₉	—	15	2	no	—
29	D₃²×C₁₉	—	35	1	no	Hall
30	C₅×C₇×C₁₉	D₃²	1	1	yes	characteristic, complemented, Hall
31	C₅×C₇×C₃⋊D₃	C₂×C₁₉	1	1	yes	characteristic, complemented
32	C₃×D₁₀₅	C₂×C₁₉	1	2	yes	complemented
33	C₁₉×D₁₅	—	7	2	no	—
34	C₅×D₃×C₁₉	—	3	2	no	—
35	C₅×D₃×C₁₉	—	6	1	no	—
36	C₃×D₅×C₁₉	—	21	2	no	—
37	C₂×D₇×C₁₉	—	45	1	no	Hall
38	D₃×D₃₅	—	3	2	no	—
39	C₃×C₇×C₁₉	—	2	1	no	—
40	C₃×C₇×C₁₉	D₃×D₅	1	2	yes	complemented
41	C₂×D₅×C₁₉	—	63	1	no	Hall
42	C₃⋊D₃×C₁₉	D₃₅	1	1	yes	characteristic, complemented
43	C₃×D₃×C₁₉	—	35	2	no	—
44	C₃²×C₅×C₇	C₂²×C₁₉	1	1	yes	characteristic, complemented, Hall
45	C₃×C₅×C₁₉	—	2	1	no	—
46	C₃×C₅×C₁₉	D₃×D₇	1	2	yes	complemented
47	C₂×C₇×C₁₉	—	9	1	no	—
48	D₇×C₁₉	—	15	2	no	—
49	(C₃⋊D₃)⋊D₇	—	5	1	no	Hall
50	C₂×D₃×C₁₉	—	105	2	no	—
51	D₁₀₅	D₃×C₁₉	1	2	yes	complemented
52	C₅×D₃×C₇	—	3	2	no	—
53	C₅×D₃×C₇	—	6	1	no	—
54	C₃×D₃₅	—	3	2	no	—
55	C₂×C₅×C₁₉	—	9	1	no	—
56	D₅×C₁₉	—	21	2	no	—
57	(C₃⋊D₃)⋊D₅	—	7	1	no	Hall
58	C₃²×C₁₉	C₂×D₃₅	1	1	yes	characteristic, complemented, Hall
59	C₂×D₃₅	—	9	1	no	Hall
60	C₇×C₁₉	(C₃⋊D₃)⋊D₅	1	1	yes	characteristic, complemented, Hall
61	C₇×C₃⋊D₃	D₅×C₁₉	1	1	yes	characteristic, complemented
62	C₃×D₂₁	—	5	2	no	—
63	C₂×C₃×C₁₉	—	105	2	no	—
64	D₃×C₁₉	—	3	2	no	—
65	D₃×C₁₉	—	6	1	no	—
66	D₃×C₁₉	—	35	2	no	—
67	C₃×C₅×C₇	—	2	1	no	—
68	C₃×C₅×C₇	C₂×D₃×C₁₉	1	2	yes	complemented
69	C₅×C₁₉	(C₃⋊D₃)⋊D₇	1	1	yes	characteristic, complemented, Hall
70	C₅×C₃⋊D₃	D₇×C₁₉	1	1	yes	characteristic, complemented
71	C₃×D₁₅	—	7	2	no	—
72	D₃×D₇	—	15	2	no	—
73	C₂²×C₁₉	—	315	1	no	Hall
74	C₂×C₅×C₇	—	9	1	no	—
75	D₃₅	—	3	2	no	—
76	C₃²×C₇	C₂×D₅×C₁₉	1	1	yes	characteristic, complemented, Hall
77	D₃×D₅	—	21	2	no	—
78	C₃×C₁₉	—	2	1	no	—
79	C₃×C₁₉	D₃×D₃₅	1	2	yes	complemented
80	C₃²×C₅	C₂×D₇×C₁₉	1	1	yes	characteristic, complemented, Hall
81	D₂₁	—	5	2	no	—
82	C₃×D₇	—	15	2	no	—
83	D₃×C₇	—	3	2	no	—
84	D₃×C₇	—	6	1	no	—
85	C₂×C₁₉	—	9	1	no	—
86	C₂×C₁₉	—	105	2	no	—
87	D₃²	—	35	1	no	Hall
88	C₅×C₇	D₃²×C₁₉	1	1	yes	characteristic, complemented, Hall
89	D₁₅	—	7	2	no	—
90	C₃×D₅	—	21	2	no	—
91	C₅×D₃	—	3	2	no	—
92	C₅×D₃	—	6	1	no	—
93	C₂×D₇	—	45	1	no	Hall
94	C₃×C₇	—	2	1	no	—
95	C₃×C₇	D₃×D₅×C₁₉	1	2	yes	complemented
96	C₂×D₅	—	63	1	no	Hall
97	C₁₉	(C₃⋊D₃)⋊D₃₅	1	1	yes	characteristic, direct product factor, Sylow
98	C₃⋊D₃	C₁₉×D₃₅	1	1	yes	characteristic, complemented
99	C₃×D₃	—	35	2	no	—
100	C₃×C₅	—	2	1	no	—
101	C₃×C₅	D₃×D₇×C₁₉	1	2	yes	complemented
102	C₂×C₇	—	9	1	no	—
103	D₇	—	15	2	no	—
104	C₂×D₃	—	105	2	no	—
105	C₂×C₅	—	9	1	no	—
106	D₅	—	21	2	no	—
107	C₃²	C₂×C₁₉×D₃₅	1	1	yes	characteristic, complemented, Sylow
108	C₇	C₁₉×(C₃⋊D₃)⋊D₅	1	1	yes	characteristic, complemented, Sylow
109	C₂×C₃	—	105	2	no	—
110	D₃	—	3	2	no	—
111	D₃	—	6	1	no	—
112	D₃	—	35	2	no	—
113	C₅	C₁₉×(C₃⋊D₃)⋊D₇	1	1	yes	characteristic, complemented, Sylow
114	C₂²	—	315	1	no	Sylow
115	C₃	—	2	1	no	—
116	C₃	D₃×C₁₉×D₃₅	1	2	yes	complemented
117	C₂	—	9	1	no	—
118	C₂	—	105	2	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₁₉×D₁₀₅

C₂

yes

maximal, complemented

C₅×C₇×C₃⋊D₃×C₁₉

C₂

yes

maximal, characteristic, complemented

D₃×C₁₉×D₃₅

—

maximal

C₃²×C₅×C₇×C₁₉

C₂²

yes

characteristic, complemented, Hall

C₁₉×(C₃⋊D₃)⋊D₇

—

maximal, Hall

C₁₉×D₁₀₅

D₃

yes

complemented

C₅×D₃×C₇×C₁₉

—

C₅×D₃×C₇×C₁₉

—

C₃×C₁₉×D₃₅

—

C₁₉×(C₃⋊D₃)⋊D₅

—

maximal, Hall

C₂×C₁₉×D₃₅

—

Hall

C₇×C₃⋊D₃×C₁₉

D₅

yes

characteristic, complemented

C₃×C₁₉×D₂₁

—

C₃×C₅×C₇×C₁₉

—

C₃×C₅×C₇×C₁₉

C₂×D₃

yes

complemented

C₅×C₃⋊D₃×C₁₉

D₇

yes

characteristic, complemented

C₃×C₁₉×D₁₅

—

D₃×D₇×C₁₉

—

C₂×C₅×C₇×C₁₉

—

C₁₉×D₃₅

—

(C₃⋊D₃)⋊D₃₅

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

C₁₉×D₂₁

—

D₃×C₇×C₁₉

—

D₃×C₇×C₁₉

—

C₃×D₇×C₁₉

—

D₃²×C₁₉

—

Hall

C₅×C₇×C₁₉

D₃²

yes

characteristic, complemented, Hall

C₅×C₇×C₃⋊D₃

C₂×C₁₉

yes

characteristic, complemented

C₃×D₁₀₅

C₂×C₁₉

yes

complemented

C₁₉×D₁₅

—

C₅×D₃×C₁₉

—

C₅×D₃×C₁₉

—

C₃×D₅×C₁₉

—

C₂×D₇×C₁₉

—

Hall

D₃×D₃₅

—

C₃×C₇×C₁₉

—

C₃×C₇×C₁₉

D₃×D₅

yes

complemented

C₂×D₅×C₁₉

—

Hall

C₃⋊D₃×C₁₉

D₃₅

yes

characteristic, complemented

C₃×D₃×C₁₉

—

C₃²×C₅×C₇

C₂²×C₁₉

yes

characteristic, complemented, Hall

C₃×C₅×C₁₉

—

C₃×C₅×C₁₉

D₃×D₇

yes

complemented

C₂×C₇×C₁₉

—

D₇×C₁₉

—

(C₃⋊D₃)⋊D₇

—

Hall

C₂×D₃×C₁₉

—

105

—

D₁₀₅

D₃×C₁₉

yes

complemented

C₅×D₃×C₇

—

C₅×D₃×C₇

—

C₃×D₃₅

—

C₂×C₅×C₁₉

—

D₅×C₁₉

—

(C₃⋊D₃)⋊D₅

—

Hall

C₃²×C₁₉

C₂×D₃₅

yes

characteristic, complemented, Hall

C₂×D₃₅

—

Hall

C₇×C₁₉

(C₃⋊D₃)⋊D₅

yes

characteristic, complemented, Hall

C₇×C₃⋊D₃

D₅×C₁₉

yes

characteristic, complemented

C₃×D₂₁

—

C₂×C₃×C₁₉

—

105

—

D₃×C₁₉

—

D₃×C₁₉

—

D₃×C₁₉

—

C₃×C₅×C₇

—

C₃×C₅×C₇

C₂×D₃×C₁₉

yes

complemented

C₅×C₁₉

(C₃⋊D₃)⋊D₇

yes

characteristic, complemented, Hall

C₅×C₃⋊D₃

D₇×C₁₉

yes

characteristic, complemented

C₃×D₁₅

—

D₃×D₇

—

C₂²×C₁₉

—

315

Hall

C₂×C₅×C₇

—

D₃₅

—

C₃²×C₇

C₂×D₅×C₁₉

yes

characteristic, complemented, Hall

D₃×D₅

—

C₃×C₁₉

—

C₃×C₁₉

D₃×D₃₅

yes

complemented

C₃²×C₅

C₂×D₇×C₁₉

yes

characteristic, complemented, Hall

D₂₁

—

C₃×D₇

—

D₃×C₇

—

D₃×C₇

—

C₂×C₁₉

—

C₂×C₁₉

—

105

—

D₃²

—

Hall

C₅×C₇

D₃²×C₁₉

yes

characteristic, complemented, Hall

D₁₅

—

C₃×D₅

—

C₅×D₃

—

C₅×D₃

—

C₂×D₇

—

Hall

C₃×C₇

—

C₃×C₇

D₃×D₅×C₁₉

yes

complemented

C₂×D₅

—

Hall

C₁₉

(C₃⋊D₃)⋊D₃₅

yes

characteristic, direct product factor, Sylow

C₃⋊D₃

C₁₉×D₃₅

yes

characteristic, complemented

C₃×D₃

—

100

C₃×C₅

—

101

C₃×C₅

D₃×D₇×C₁₉

yes

complemented

102

C₂×C₇

—

103

D₇

—

104

C₂×D₃

—

105

—

105

C₂×C₅

—

106

D₅

—

107

C₃²

C₂×C₁₉×D₃₅

yes

characteristic, complemented, Sylow

108

C₇

C₁₉×(C₃⋊D₃)⋊D₅

yes

characteristic, complemented, Sylow

109

C₂×C₃

—

105

—

110

D₃

—

111

D₃

—

112

D₃

—

113

C₅

C₁₉×(C₃⋊D₃)⋊D₇

yes

characteristic, complemented, Sylow

114

C₂²

—

315

Sylow

115

C₃

—

116

C₃

D₃×C₁₉×D₃₅

yes

complemented

117

C₂

—

118

C₂

—

105

—

	subgroup	G/H
center, hypercenter	C₁₉	(C₃⋊D₃)⋊D₃₅
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

(C₃⋊D₃)⋊D₃₅

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. C₃×C₁₉×D₁₀₅	C₂	semidirect
2. C₅×C₇×C₃⋊D₃×C₁₉	C₂	semidirect
4. C₃²×C₅×C₇×C₁₉	C₂²	semidirect
6. C₁₉×D₁₀₅	D₃	semidirect
12. C₇×C₃⋊D₃×C₁₉	D₅	semidirect
15. C₃×C₅×C₇×C₁₉	C₂×D₃	semidirect
16. C₅×C₃⋊D₃×C₁₉	D₇	semidirect
21. (C₃⋊D₃)⋊D₃₅	C₁₉	direct (N × H)
22. C₃²×C₇×C₁₉	C₂×D₅	semidirect
24. C₃²×C₅×C₁₉	C₂×D₇	semidirect
30. C₅×C₇×C₁₉	D₃²	semidirect
31. C₅×C₇×C₃⋊D₃	C₂×C₁₉	semidirect
32. C₃×D₁₀₅	C₂×C₁₉	semidirect
40. C₃×C₇×C₁₉	D₃×D₅	semidirect
42. C₃⋊D₃×C₁₉	D₃₅	semidirect
44. C₃²×C₅×C₇	C₂²×C₁₉	semidirect
46. C₃×C₅×C₁₉	D₃×D₇	semidirect
51. D₁₀₅	D₃×C₁₉	semidirect
58. C₃²×C₁₉	C₂×D₃₅	semidirect
60. C₇×C₁₉	(C₃⋊D₃)⋊D₅	semidirect
61. C₇×C₃⋊D₃	D₅×C₁₉	semidirect
68. C₃×C₅×C₇	C₂×D₃×C₁₉	semidirect
69. C₅×C₁₉	(C₃⋊D₃)⋊D₇	semidirect
70. C₅×C₃⋊D₃	D₇×C₁₉	semidirect
76. C₃²×C₇	C₂×D₅×C₁₉	semidirect
79. C₃×C₁₉	D₃×D₃₅	semidirect
80. C₃²×C₅	C₂×D₇×C₁₉	semidirect
88. C₅×C₇	D₃²×C₁₉	semidirect
95. C₃×C₇	D₃×D₅×C₁₉	semidirect
97. C₁₉	(C₃⋊D₃)⋊D₃₅	direct (N × H)
98. C₃⋊D₃	C₁₉×D₃₅	semidirect
101. C₃×C₅	D₃×D₇×C₁₉	semidirect
107. C₃²	C₂×C₁₉×D₃₅	semidirect
108. C₇	C₁₉×(C₃⋊D₃)⋊D₅	semidirect
113. C₅	C₁₉×(C₃⋊D₃)⋊D₇	semidirect
116. C₃	D₃×C₁₉×D₃₅	semidirect

H ≅ G/N

type

1. C₃×C₁₉×D₁₀₅

C₂

semidirect

2. C₅×C₇×C₃⋊D₃×C₁₉

C₂

semidirect

4. C₃²×C₅×C₇×C₁₉

C₂²

semidirect

6. C₁₉×D₁₀₅

D₃

semidirect

12. C₇×C₃⋊D₃×C₁₉

D₅

semidirect

15. C₃×C₅×C₇×C₁₉

C₂×D₃

semidirect

16. C₅×C₃⋊D₃×C₁₉

D₇

semidirect

21. (C₃⋊D₃)⋊D₃₅

C₁₉

direct (N × H)

22. C₃²×C₇×C₁₉

C₂×D₅

semidirect

24. C₃²×C₅×C₁₉

C₂×D₇

semidirect

30. C₅×C₇×C₁₉

D₃²

semidirect

31. C₅×C₇×C₃⋊D₃

C₂×C₁₉

semidirect

32. C₃×D₁₀₅

C₂×C₁₉

semidirect

40. C₃×C₇×C₁₉

D₃×D₅

semidirect

42. C₃⋊D₃×C₁₉

D₃₅

semidirect

44. C₃²×C₅×C₇

C₂²×C₁₉

semidirect

46. C₃×C₅×C₁₉

D₃×D₇

semidirect

51. D₁₀₅

D₃×C₁₉

semidirect

58. C₃²×C₁₉

C₂×D₃₅

semidirect

60. C₇×C₁₉

(C₃⋊D₃)⋊D₅

semidirect

61. C₇×C₃⋊D₃

D₅×C₁₉

semidirect

68. C₃×C₅×C₇

C₂×D₃×C₁₉

semidirect

69. C₅×C₁₉

(C₃⋊D₃)⋊D₇

semidirect

70. C₅×C₃⋊D₃

D₇×C₁₉

semidirect

76. C₃²×C₇

C₂×D₅×C₁₉

semidirect

79. C₃×C₁₉

D₃×D₃₅

semidirect

80. C₃²×C₅

C₂×D₇×C₁₉

semidirect

88. C₅×C₇

D₃²×C₁₉

semidirect

95. C₃×C₇

D₃×D₅×C₁₉

semidirect

97. C₁₉

(C₃⋊D₃)⋊D₃₅

direct (N × H)

98. C₃⋊D₃

C₁₉×D₃₅

semidirect

101. C₃×C₅

D₃×D₇×C₁₉

semidirect

107. C₃²

C₂×C₁₉×D₃₅

semidirect

108. C₇

C₁₉×(C₃⋊D₃)⋊D₅

semidirect

113. C₅

C₁₉×(C₃⋊D₃)⋊D₇

semidirect

116. C₃

D₃×C₁₉×D₃₅

semidirect

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

subgroup	index
C₃×C₁₉×D₁₀₅	2
C₅×C₇×C₃⋊D₃×C₁₉	2
D₃×C₁₉×D₃₅	3
C₁₉×(C₃⋊D₃)⋊D₇	5
C₁₉×(C₃⋊D₃)⋊D₅	7
(C₃⋊D₃)⋊D₃₅	19

subgroup

index

C₃×C₁₉×D₁₀₅

C₅×C₇×C₃⋊D₃×C₁₉

D₃×C₁₉×D₃₅

C₁₉×(C₃⋊D₃)⋊D₇

C₁₉×(C₃⋊D₃)⋊D₅

(C₃⋊D₃)⋊D₃₅

Subgroups — C₁₉×(C₃⋊D₃)⋊D₃₅ #37561585

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×(C3⋊D3)⋊D35 #37561585

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×(C₃⋊D₃)⋊D₃₅ #37561585