Group #28352821 — subgroups

Subgroups — C₁₉×#10624625 #28352821

Order 13851.

200 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 74 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×#6895	C₃	1	1	yes	maximal, characteristic, complemented
2	C₁₉×C₉⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₉×C₁₉×3₋¹⁺²	C₃	1	2	yes	maximal, complemented
4	C₁₉×#6862	C₃	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#6861	C₃	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#6859	C₃	1	2	yes	maximal
7	C₁₉×#6858	C₃	1	2	yes	maximal, complemented
8	C₁₉×#6858	C₃	1	2	yes	maximal, complemented
9	C₁₉×#6857	C₃	1	1	yes	maximal, characteristic
10	C₃×C₁₉×3₋¹⁺²	—	3	1	no	—
11	C₃×C₁₉×3₋¹⁺²	C₃²	1	2	yes	—
12	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
13	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
14	C₁₉×C₉⋊C₉	—	3	2	no	—
15	C₁₉×C₉⋊C₉	—	3	2	no	—
16	C₁₉×C₉⋊C₉	—	3	2	no	—
17	C₁₉×C₉⋊C₉	—	3	1	no	—
18	C₁₉×C₃²⋊C₉	—	3	2	no	—
19	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
20	C₁₉×C₃²⋊C₉	C₃²	1	2	yes	—
21	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
22	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
23	C₁₉×C₃²⋊C₉	C₃²	1	2	yes	—
24	C₁₉×C₃²⋊C₉	C₃²	1	2	yes	—
25	C₉²×C₁₉	—	3	2	no	—
26	#10624625	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₃³×C₁₉	—	3	1	no	—
28	C₃³×C₁₉	C₃³	1	1	yes	characteristic
29	C₁₉×3₋¹⁺²	—	3	6	no	—
30	C₁₉×3₋¹⁺²	—	9	3	no	—
31	C₁₉×H(3,𝔽₃)	—	9	1	no	—
32	C₃×C₉×C₁₉	—	3	2	no	—
33	C₃×C₉×C₁₉	—	3	2	no	—
34	C₃×C₉×C₁₉	—	3	2	no	—
35	C₃×C₉×C₁₉	—	3	1	no	—
36	C₃×C₉×C₁₉	—	3	1	no	—
37	C₃×C₉×C₁₉	—	3	1	no	—
38	C₃×C₉×C₁₉	—	3	2	no	—
39	C₃×C₉×C₁₉	—	9	1	no	—
40	C₃×C₉×C₁₉	H(3,𝔽₃)	1	2	yes	—
41	C₃×C₉×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
42	#6895	C₃×C₁₉	1	1	yes	characteristic, complemented
43	C₉⋊H(3,𝔽₃)	C₃×C₁₉	1	1	yes	characteristic
44	C₉×3₋¹⁺²	C₃×C₁₉	1	2	yes	complemented
45	#6862	C₃×C₁₉	1	1	yes	characteristic, complemented
46	#6861	C₃×C₁₉	1	1	yes	characteristic, complemented
47	#6859	C₃×C₁₉	1	2	yes	—
48	#6858	C₃×C₁₉	1	2	yes	complemented
49	#6858	C₃×C₁₉	1	2	yes	complemented
50	#6857	C₃×C₁₉	1	1	yes	characteristic
51	C₃²×C₁₉	—	3	1	no	—
52	C₃²×C₁₉	—	3	2	no	—
53	C₃²×C₁₉	—	3	1	no	—
54	C₃²×C₁₉	—	9	1	no	—
55	C₃²×C₁₉	—	9	2	no	—
56	C₃²×C₁₉	—	9	1	no	—
57	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
58	C₉×C₁₉	—	3	6	no	—
59	C₉×C₁₉	—	3	2	no	—
60	C₉×C₁₉	—	3	1	no	—
61	C₉×C₁₉	—	9	2	no	—
62	C₉×C₁₉	—	9	1	no	—
63	C₉×C₁₉	—	9	2	no	—
64	C₉×C₁₉	—	9	2	no	—
65	C₉×C₁₉	—	9	1	no	—
66	C₉×C₁₉	—	9	1	no	—
67	C₃×3₋¹⁺²	—	3	1	no	—
68	C₃×3₋¹⁺²	C₃²×C₁₉	1	2	yes	—
69	C₃×H(3,𝔽₃)	C₃²×C₁₉	1	1	yes	characteristic
70	C₃²×C₉	C₃²×C₁₉	1	1	yes	characteristic
71	C₉⋊C₉	—	3	2	no	—
72	C₉⋊C₉	—	3	2	no	—
73	C₉⋊C₉	—	3	2	no	—
74	C₉⋊C₉	—	3	1	no	—
75	C₃²⋊C₉	—	3	2	no	—
76	C₃²⋊C₉	C₃²×C₁₉	1	1	yes	characteristic
77	C₃²⋊C₉	C₃²×C₁₉	1	2	yes	—
78	C₃²⋊C₉	C₃²×C₁₉	1	1	yes	characteristic
79	C₃²⋊C₉	C₃²×C₁₉	1	1	yes	characteristic
80	C₃²⋊C₉	C₃²×C₁₉	1	2	yes	—
81	C₃²⋊C₉	C₃²×C₁₉	1	2	yes	—
82	C₉²	—	3	2	no	—
83	C₃×C₁₉	—	9	1	no	—
84	C₃×C₁₉	—	27	1	no	—
85	C₃×C₁₉	#6908	1	2	yes	—
86	C₃×C₁₉	#6910	1	1	yes	characteristic
87	C₃×C₁₉	#6913	1	1	yes	characteristic
88	C₃³	—	3	1	no	—
89	C₃³	C₃³×C₁₉	1	1	yes	characteristic
90	3₋¹⁺²	—	3	6	no	—
91	3₋¹⁺²	—	9	3	no	—
92	H(3,𝔽₃)	—	9	1	no	—
93	C₃×C₉	—	3	2	no	—
94	C₃×C₉	—	3	2	no	—
95	C₃×C₉	—	3	2	no	—
96	C₃×C₉	—	3	1	no	—
97	C₃×C₉	—	3	1	no	—
98	C₃×C₉	—	3	2	no	—
99	C₃×C₉	—	3	1	no	—
100	C₃×C₉	—	9	1	no	—
101	C₃×C₉	C₁₉×H(3,𝔽₃)	1	2	yes	—
102	C₃×C₉	C₁₉×H(3,𝔽₃)	1	1	yes	characteristic
103	C₁₉	#10624625	1	1	yes	characteristic, direct product factor, Sylow
104	C₃²	—	3	2	no	—
105	C₃²	—	3	1	no	—
106	C₃²	—	3	1	no	—
107	C₃²	—	9	2	no	—
108	C₃²	—	9	1	no	—
109	C₃²	—	9	1	no	—
110	C₃²	C₃×C₁₉×H(3,𝔽₃)	1	1	yes	characteristic
111	C₉	—	3	6	no	—
112	C₉	—	3	1	no	—
113	C₉	—	3	2	no	—
114	C₉	—	9	2	no	—
115	C₉	—	9	2	no	—
116	C₉	—	9	1	no	—
117	C₉	—	9	1	no	—
118	C₉	—	9	2	no	—
119	C₉	—	9	1	no	—
120	C₃	—	9	1	no	—
121	C₃	—	27	1	no	—
122	C₃	C₁₉×#6908	1	2	yes	—
123	C₃	C₁₉×#6910	1	1	yes	characteristic
124	C₃	C₁₉×#6913	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#6895

C₃

yes

maximal, characteristic, complemented

C₁₉×C₉⋊H(3,𝔽₃)

C₃

yes

maximal, characteristic

C₉×C₁₉×3₋¹⁺²

C₃

yes

maximal, complemented

C₁₉×#6862

C₃

yes

maximal, characteristic, complemented

C₁₉×#6861

C₃

yes

maximal, characteristic, complemented

C₁₉×#6859

C₃

yes

maximal

C₁₉×#6858

C₃

yes

maximal, complemented

C₁₉×#6858

C₃

yes

maximal, complemented

C₁₉×#6857

C₃

yes

maximal, characteristic

C₃×C₁₉×3₋¹⁺²

—

C₃×C₁₉×3₋¹⁺²

C₃²

yes

—

C₃×C₁₉×H(3,𝔽₃)

C₃²

yes

characteristic

C₃²×C₉×C₁₉

C₃²

yes

characteristic

C₁₉×C₉⋊C₉

—

C₁₉×C₉⋊C₉

—

C₁₉×C₉⋊C₉

—

C₁₉×C₉⋊C₉

—

C₁₉×C₃²⋊C₉

—

C₁₉×C₃²⋊C₉

C₃²

yes

characteristic

C₁₉×C₃²⋊C₉

C₃²

yes

—

C₁₉×C₃²⋊C₉

C₃²

yes

characteristic

C₁₉×C₃²⋊C₉

C₃²

yes

characteristic

C₁₉×C₃²⋊C₉

C₃²

yes

—

C₁₉×C₃²⋊C₉

C₃²

yes

—

C₉²×C₁₉

—

#10624625

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

C₃³

yes

characteristic

C₁₉×3₋¹⁺²

—

C₁₉×3₋¹⁺²

—

C₁₉×H(3,𝔽₃)

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

H(3,𝔽₃)

yes

—

C₃×C₉×C₁₉

H(3,𝔽₃)

yes

characteristic

#6895

C₃×C₁₉

yes

characteristic, complemented

C₉⋊H(3,𝔽₃)

C₃×C₁₉

yes

characteristic

C₉×3₋¹⁺²

C₃×C₁₉

yes

complemented

#6862

C₃×C₁₉

yes

characteristic, complemented

#6861

C₃×C₁₉

yes

characteristic, complemented

#6859

C₃×C₁₉

yes

—

#6858

C₃×C₁₉

yes

complemented

#6858

C₃×C₁₉

yes

complemented

#6857

C₃×C₁₉

yes

characteristic

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₁₉

—

C₉×C₁₉

—

C₉×C₁₉

—

C₉×C₁₉

—

C₉×C₁₉

—

C₉×C₁₉

—

C₉×C₁₉

—

C₉×C₁₉

—

C₉×C₁₉

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃²×C₁₉

yes

—

C₃×H(3,𝔽₃)

C₃²×C₁₉

yes

characteristic

C₃²×C₉

C₃²×C₁₉

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃²×C₁₉

yes

characteristic

C₃²⋊C₉

C₃²×C₁₉

yes

—

C₃²⋊C₉

C₃²×C₁₉

yes

characteristic

C₃²⋊C₉

C₃²×C₁₉

yes

characteristic

C₃²⋊C₉

C₃²×C₁₉

yes

—

C₃²⋊C₉

C₃²×C₁₉

yes

—

C₉²

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

#6908

yes

—

C₃×C₁₉

yes

C₃×C₁₉

yes

C₃³

—

C₃³

C₃³×C₁₉

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

100

C₃×C₉

—

101

C₃×C₉

C₁₉×H(3,𝔽₃)

yes

—

102

C₃×C₉

C₁₉×H(3,𝔽₃)

yes

characteristic

103

C₁₉

#10624625

yes

characteristic, direct product factor, Sylow

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

C₃×C₁₉×H(3,𝔽₃)

yes

characteristic

111

C₉

—

112

C₉

—

113

C₉

—

114

C₉

—

115

C₉

—

116

C₉

—

117

C₉

—

118

C₉

—

119

C₉

—

120

C₃

—

121

C₃

—

122

C₃

C₁₉×#6908

yes

—

123

C₃

C₁₉×#6910

yes

characteristic

124

C₃

C₁₉×#6913

yes

characteristic

	subgroup	G/H
center, socle	C₃²×C₁₉	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₁₉
prime radical	C₃×C₁₉×H(3,𝔽₃)	C₃²
LCS	G ▷ C₃³ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₁₉ ◁ C₃²×C₉×C₁₉ ◁ G

subgroup

G/H

center, socle

C₃²×C₁₉

C₃×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₃³×C₁₉

prime radical

C₃×C₁₉×H(3,𝔽₃)

C₃²

LCS

G ▷ C₃³ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃²×C₁₉ ◁ C₃²×C₉×C₁₉ ◁ G

N	H ≅ G/N	type
1. C₁₉×#6895	C₃	semidirect
3. C₉×C₁₉×3₋¹⁺²	C₃	semidirect
4. C₁₉×#6862	C₃	semidirect
5. C₁₉×#6861	C₃	semidirect
7. C₁₉×#6858	C₃	semidirect
8. C₁₉×#6858	C₃	semidirect
26. #10624625	C₁₉	direct (N × H)
42. #6895	C₃×C₁₉	semidirect
44. C₉×3₋¹⁺²	C₃×C₁₉	semidirect
45. #6862	C₃×C₁₉	semidirect
46. #6861	C₃×C₁₉	semidirect
48. #6858	C₃×C₁₉	semidirect
49. #6858	C₃×C₁₉	semidirect
103. C₁₉	#10624625	direct (N × H)

H ≅ G/N

type

1. C₁₉×#6895

C₃

semidirect

3. C₉×C₁₉×3₋¹⁺²

C₃

semidirect

4. C₁₉×#6862

C₃

semidirect

5. C₁₉×#6861

C₃

semidirect

7. C₁₉×#6858

C₃

semidirect

8. C₁₉×#6858

C₃

semidirect

26. #10624625

C₁₉

direct (N × H)

42. #6895

C₃×C₁₉

semidirect

44. C₉×3₋¹⁺²

C₃×C₁₉

semidirect

45. #6862

C₃×C₁₉

semidirect

46. #6861

C₃×C₁₉

semidirect

48. #6858

C₃×C₁₉

semidirect

49. #6858

C₃×C₁₉

semidirect

103. C₁₉

#10624625

direct (N × H)

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

subgroup	index
C₁₉×#6895	3
C₁₉×C₉⋊H(3,𝔽₃)	3
C₉×C₁₉×3₋¹⁺²	3
C₁₉×#6862	3
C₁₉×#6861	3
C₁₉×#6859	3
C₁₉×#6858	3
C₁₉×#6857	3
#10624625	19

subgroup

index

C₁₉×#6895

C₁₉×C₉⋊H(3,𝔽₃)

C₉×C₁₉×3₋¹⁺²

C₁₉×#6862

C₁₉×#6861

C₁₉×#6859

C₁₉×#6858

C₁₉×#6857

#10624625

Subgroups — C₁₉×#10624625 #28352821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#10624625 #28352821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#10624625 #28352821