Group #28353818 — subgroups

Subgroups — #28353818

Order 13851.

260 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 55 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	#17987420	C₃	1	1	yes	maximal, characteristic
2	#17987419	C₃	1	1	yes	maximal, characteristic
3	C₁₉⋊C₃×C₃²⋊C₉	C₃	1	2	yes	maximal
4	C₁₉⋊#6862	C₃	1	4	yes	maximal, complemented
5	C₁₉⋊#6862	C₃	1	2	yes	maximal, complemented
6	C₁₉⋊#6862	C₃	1	2	yes	maximal, complemented
7	C₁₉×#6862	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃³×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
9	C₃×#10587022	—	3	1	no	—
10	C₃×#10587023	—	3	1	no	—
11	C₃×C₉×C₁₉⋊C₃	—	3	2	no	—
12	C₃²⋊#4294	C₃²	1	4	yes	—
13	C₃²⋊#4294	C₃²	1	2	yes	—
14	C₃²⋊#4294	C₃²	1	2	yes	—
15	#13095303	—	3	2	no	—
16	#13095303	—	3	2	no	—
17	#13095303	—	3	2	no	—
18	#13095303	—	3	2	no	—
19	C₁₉×C₃²⋊C₉	C₃²	1	2	yes	—
20	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
21	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
22	#10624603	—	19	1	no	maximal, Sylow
23	C₃³×C₁₉	C₃³	1	1	yes	characteristic
24	C₃²×C₁₉⋊C₃	—	3	6	no	—
25	C₃²×C₁₉⋊C₃	—	9	1	no	—
26	C₃²×C₁₉⋊C₃	—	9	1	no	—
27	C₃²×C₁₉⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
28	C₃²×C₁₉⋊C₃	H(3,𝔽₃)	1	2	yes	—
29	C₃×C₉×C₁₉	—	3	2	no	—
30	C₃×C₉×C₁₉	—	3	1	no	—
31	C₃×C₉×C₁₉	—	3	1	no	—
32	C₃×#4294	—	3	4	no	—
33	C₃×#4294	—	3	2	no	—
34	C₃×#4294	—	3	2	no	—
35	#10587023	—	9	3	no	—
36	#10587022	—	9	3	no	—
37	C₉×C₁₉⋊C₃	—	9	6	no	—
38	#6891	—	19	1	no	—
39	#6891	—	19	1	no	—
40	C₃×C₃²⋊C₉	—	19	2	no	—
41	#6862	—	19	2	no	—
42	#6862	—	19	2	no	—
43	#6862	—	19	4	no	—
44	#6862	C₁₉⋊C₃	1	1	yes	characteristic, complemented
45	C₃²×C₁₉	—	3	2	no	—
46	C₃²×C₁₉	—	3	1	no	—
47	C₃²×C₁₉	—	3	1	no	—
48	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
49	C₃×C₁₉⋊C₃	—	3	2	no	—
50	C₃×C₁₉⋊C₃	—	3	2	no	—
51	C₃×C₁₉⋊C₃	—	3	2	no	—
52	C₃×C₁₉⋊C₃	—	3	2	no	—
53	C₃×C₁₉⋊C₃	—	3	1	no	—
54	C₃×C₁₉⋊C₃	—	3	1	no	—
55	C₃×C₁₉⋊C₃	—	9	9	no	—
56	C₃×C₁₉⋊C₃	#731	1	6	yes	—
57	C₉×C₁₉	—	9	2	no	—
58	C₉×C₁₉	—	9	1	no	—
59	C₉×C₁₉	—	9	1	no	—
60	#4294	—	9	4	no	—
61	#4294	—	9	2	no	—
62	#4294	—	9	2	no	—
63	C₃⁴	—	19	1	no	—
64	C₃×3₋¹⁺²	—	57	1	no	—
65	C₃×3₋¹⁺²	—	57	1	no	—
66	C₃²×C₉	—	57	2	no	—
67	C₃²⋊C₉	—	19	4	no	—
68	C₃²⋊C₉	—	19	2	no	—
69	C₃²⋊C₉	—	19	2	no	—
70	C₃²⋊C₉	—	57	2	no	—
71	C₃²⋊C₉	—	57	2	no	—
72	C₃²⋊C₉	—	57	2	no	—
73	C₃²⋊C₉	—	57	2	no	—
74	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	2	yes	—
75	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
76	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
77	C₃×C₁₉	—	9	1	no	—
78	C₃×C₁₉	C₃×#731	1	2	yes	—
79	C₃×C₁₉	#6910	1	1	yes	characteristic
80	C₃×C₁₉	#6910	1	1	yes	characteristic
81	C₁₉⋊C₃	—	3	6	no	—
82	C₁₉⋊C₃	—	9	1	no	—
83	C₃³	—	19	2	no	—
84	C₃³	—	19	1	no	—
85	C₃³	—	57	6	no	—
86	C₃³	—	171	1	no	—
87	C₃³	—	171	1	no	—
88	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
89	3₋¹⁺²	—	171	3	no	—
90	3₋¹⁺²	—	171	3	no	—
91	C₃×C₉	—	3	2	no	—
92	C₃×C₉	—	3	1	no	—
93	C₃×C₉	—	3	1	no	—
94	C₃×C₉	—	57	4	no	—
95	C₃×C₉	—	57	2	no	—
96	C₃×C₉	—	57	2	no	—
97	C₃×C₉	—	171	6	no	—
98	C₁₉	#10624603	1	1	yes	characteristic, complemented, Sylow
99	C₃²	—	3	2	no	—
100	C₃²	—	3	1	no	—
101	C₃²	—	3	1	no	—
102	C₃²	—	19	6	no	—
103	C₃²	—	57	2	no	—
104	C₃²	—	57	2	no	—
105	C₃²	—	57	2	no	—
106	C₃²	—	57	2	no	—
107	C₃²	—	57	1	no	—
108	C₃²	—	57	1	no	—
109	C₃²	—	171	9	no	—
110	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
111	C₉	—	9	2	no	—
112	C₉	—	9	1	no	—
113	C₉	—	9	1	no	—
114	C₉	—	171	4	no	—
115	C₉	—	171	2	no	—
116	C₉	—	171	2	no	—
117	C₃	—	9	1	no	—
118	C₃	—	57	6	no	—
119	C₃	—	171	1	no	—
120	C₃	C₁₉⋊C₃×#731	1	2	yes	—
121	C₃	#17987507	1	1	yes	characteristic
122	C₃	#17987508	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17987420

C₃

yes

maximal, characteristic

#17987419

C₃

yes

maximal, characteristic

C₁₉⋊C₃×C₃²⋊C₉

C₃

yes

maximal

C₁₉⋊#6862

C₃

yes

maximal, complemented

C₁₉⋊#6862

C₃

yes

maximal, complemented

C₁₉⋊#6862

C₃

yes

maximal, complemented

C₁₉×#6862

C₃

yes

maximal, characteristic, complemented

C₃³×C₁₉⋊C₃

C₃²

yes

characteristic

C₃×#10587022

—

C₃×#10587023

—

C₃×C₉×C₁₉⋊C₃

—

C₃²⋊#4294

C₃²

yes

—

C₃²⋊#4294

C₃²

yes

—

C₃²⋊#4294

C₃²

yes

—

—

—

—

—

C₁₉×C₃²⋊C₉

C₃²

yes

—

C₁₉×C₃²⋊C₉

C₃²

yes

characteristic

C₁₉×C₃²⋊C₉

C₃²

yes

characteristic

#10624603

—

maximal, Sylow

C₃³×C₁₉

C₃³

yes

characteristic

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

H(3,𝔽₃)

yes

characteristic

C₃²×C₁₉⋊C₃

H(3,𝔽₃)

yes

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

—

—

—

—

—

C₉×C₁₉⋊C₃

—

#6891

—

#6891

—

C₃×C₃²⋊C₉

—

—

—

—

C₁₉⋊C₃

yes

characteristic, complemented

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

#731

yes

—

C₉×C₁₉

—

C₉×C₁₉

—

C₉×C₁₉

—

#4294

—

#4294

—

#4294

—

C₃⁴

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₁₉⋊C₃

yes

—

C₃²⋊C₉

C₃×C₁₉⋊C₃

yes

characteristic

C₃²⋊C₉

C₃×C₁₉⋊C₃

yes

characteristic

C₃×C₁₉

—

C₃×C₁₉

C₃×#731

yes

—

C₃×C₁₉

yes

C₃×C₁₉

yes

C₁₉⋊C₃

—

C₁₉⋊C₃

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

171

—

C₃³

—

171

—

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

3₋¹⁺²

—

171

—

3₋¹⁺²

—

171

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

171

—

C₁₉

#10624603

yes

characteristic, complemented, Sylow

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

171

—

110

C₃²

H(3,𝔽₃)×C₁₉⋊C₃

yes

characteristic

111

C₉

—

112

C₉

—

113

C₉

—

114

C₉

—

171

—

115

C₉

—

171

—

116

C₉

—

171

—

117

C₃

—

118

C₃

—

119

C₃

—

171

—

120

C₃

C₁₉⋊C₃×#731

yes

—

121

C₃

#17987507

yes

characteristic

122

C₃

#17987508

yes

characteristic

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₁₉⋊C₃
commutator, cosocle	C₃³×C₁₉	C₃³
Frattini	C₃³	C₃²×C₁₉⋊C₃
Fitting	C₁₉×#6862	C₃
socle	C₃²×C₁₉	C₃×H(3,𝔽₃)
prime radical	C₃³×C₁₉⋊C₃	C₃²
hypercenter	#6862	C₁₉⋊C₃
hypocenter	C₁₉	#10624603
LCS	G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ #6862

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₁₉⋊C₃

commutator, cosocle

C₃³×C₁₉

C₃³

Frattini

C₃³

C₃²×C₁₉⋊C₃

Fitting

C₁₉×#6862

C₃

socle

C₃²×C₁₉

C₃×H(3,𝔽₃)

prime radical

C₃³×C₁₉⋊C₃

C₃²

C₁₉⋊C₃

C₁₉

G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ #6862

N	H ≅ G/N	type
4. C₁₉⋊#6862	C₃	semidirect
5. C₁₉⋊#6862	C₃	semidirect
6. C₁₉⋊#6862	C₃	semidirect
7. C₁₉×#6862	C₃	semidirect
44. #6862	C₁₉⋊C₃	semidirect
98. C₁₉	#10624603	semidirect

H ≅ G/N

type

4. C₁₉⋊#6862

C₃

semidirect

5. C₁₉⋊#6862

C₃

semidirect

6. C₁₉⋊#6862

C₃

semidirect

7. C₁₉×#6862

C₃

semidirect

44. #6862

C₁₉⋊C₃

semidirect

98. C₁₉

#10624603

semidirect

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

subgroup	index
#17987420	3
#17987419	3
C₁₉⋊C₃×C₃²⋊C₉	3
C₁₉⋊#6862	3
C₁₉×#6862	3
#10624603	19

subgroup

index

#17987420

#17987419

C₁₉⋊C₃×C₃²⋊C₉

C₁₉⋊#6862

C₁₉×#6862

#10624603

Subgroups — #28353818

Conjugacy classes of subgroups

Complements

Maximal subgroups