Group #28353791 — subgroups

Subgroups — #28353791

Order 13851.

344 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 64 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	#17987417	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×C₃²⋊#4294	C₃	1	2	yes	maximal, complemented
3	#17987309	C₃	1	9	yes	maximal, complemented
4	C₁₉×C₃²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	1	no	—
6	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	2	no	—
7	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	3	yes	—
8	C₃²×#4294	—	3	2	no	—
9	C₃²⋊#4294	—	3	2	no	—
10	C₃²⋊#4294	—	3	2	no	—
11	C₃²⋊#4294	C₃²	1	6	yes	—
12	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
13	C₃×C₁₉×H(3,𝔽₃)	—	3	1	no	—
14	C₃×C₁₉×H(3,𝔽₃)	—	3	2	no	—
15	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	3	yes	—
16	#10624599	—	19	1	no	maximal, Sylow
17	C₃³×C₁₉	—	3	6	no	—
18	C₃³×C₁₉	—	3	3	no	—
19	C₃³×C₁₉	—	9	1	no	—
20	C₃³×C₁₉	—	9	1	no	—
21	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
22	C₃³×C₁₉	H(3,𝔽₃)	1	2	yes	—
23	C₃³×C₁₉	C₃³	1	1	yes	characteristic
24	C₃²×C₁₉⋊C₃	—	3	3	no	—
25	C₁₉×H(3,𝔽₃)	—	9	3	no	—
26	C₁₉×H(3,𝔽₃)	—	9	3	no	—
27	C₁₉×H(3,𝔽₃)	—	9	6	no	—
28	C₃²⋊(C₁₉⋊C₃)	—	9	3	no	—
29	C₃²⋊(C₁₉⋊C₃)	—	9	6	no	—
30	C₃²⋊(C₁₉⋊C₃)	—	9	3	no	—
31	C₃×#4294	—	3	6	no	—
32	C₃×#4294	—	9	6	no	—
33	C₃²⋊H(3,𝔽₃)	—	19	1	no	—
34	C₃²⋊H(3,𝔽₃)	C₁₉⋊C₃	1	1	yes	characteristic, complemented
35	C₃×C₃²⋊C₉	—	19	2	no	—
36	#6856	—	19	9	no	—
37	C₃²×C₁₉	—	3	2	no	—
38	C₃²×C₁₉	—	3	2	no	—
39	C₃²×C₁₉	—	3	2	no	—
40	C₃²×C₁₉	—	3	2	no	—
41	C₃²×C₁₉	—	3	2	no	—
42	C₃²×C₁₉	—	3	1	no	—
43	C₃²×C₁₉	—	3	1	no	—
44	C₃²×C₁₉	—	3	1	no	—
45	C₃²×C₁₉	—	3	1	no	—
46	C₃²×C₁₉	—	9	9	no	—
47	C₃²×C₁₉	—	9	6	no	—
48	C₃²×C₁₉	—	9	3	no	—
49	C₃²×C₁₉	—	9	3	no	—
50	C₃²×C₁₉	#732	1	6	yes	—
51	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
52	C₃×C₁₉⋊C₃	—	9	6	no	—
53	C₃×C₁₉⋊C₃	—	9	3	no	—
54	C₃×C₁₉⋊C₃	—	9	3	no	—
55	#4294	—	9	6	no	—
56	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
57	C₃×H(3,𝔽₃)	—	3	1	no	—
58	C₃×H(3,𝔽₃)	—	3	2	no	—
59	C₃×H(3,𝔽₃)	—	19	3	no	—
60	C₃×H(3,𝔽₃)	—	57	1	no	—
61	C₃×H(3,𝔽₃)	—	57	2	no	—
62	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	3	yes	—
63	C₃²×C₉	—	57	2	no	—
64	C₃²⋊C₉	—	19	6	no	—
65	C₃²⋊C₉	—	57	2	no	—
66	C₃²⋊C₉	—	57	2	no	—
67	C₃×C₁₉	—	3	6	no	—
68	C₃×C₁₉	—	9	1	no	—
69	C₃×C₁₉	—	9	1	no	—
70	C₃×C₁₉	—	27	3	no	—
71	C₃×C₁₉	C₃×#732	1	2	yes	—
72	C₃×C₁₉	#6909	1	1	yes	characteristic
73	C₃×C₁₉	#6909	1	1	yes	characteristic
74	C₁₉⋊C₃	—	27	3	no	—
75	C₃³	—	3	6	no	—
76	C₃³	—	3	3	no	—
77	C₃³	—	9	1	no	—
78	C₃³	—	9	1	no	—
79	C₃³	—	57	3	no	—
80	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
81	C₃³	C₃²⋊(C₁₉⋊C₃)	1	2	yes	—
82	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
83	H(3,𝔽₃)	—	9	3	no	—
84	H(3,𝔽₃)	—	9	3	no	—
85	H(3,𝔽₃)	—	9	6	no	—
86	H(3,𝔽₃)	—	171	3	no	—
87	H(3,𝔽₃)	—	171	3	no	—
88	H(3,𝔽₃)	—	171	6	no	—
89	C₃×C₉	—	57	6	no	—
90	C₃×C₉	—	171	6	no	—
91	C₁₉	#10624599	1	1	yes	characteristic, complemented, Sylow
92	C₃²	—	3	2	no	—
93	C₃²	—	3	2	no	—
94	C₃²	—	3	2	no	—
95	C₃²	—	3	2	no	—
96	C₃²	—	3	2	no	—
97	C₃²	—	3	1	no	—
98	C₃²	—	3	1	no	—
99	C₃²	—	3	1	no	—
100	C₃²	—	3	1	no	—
101	C₃²	—	9	9	no	—
102	C₃²	—	9	6	no	—
103	C₃²	—	9	3	no	—
104	C₃²	—	9	3	no	—
105	C₃²	—	171	6	no	—
106	C₃²	—	171	3	no	—
107	C₃²	—	171	3	no	—
108	C₃²	#13095319	1	6	yes	—
109	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
110	C₉	—	171	6	no	—
111	C₃	—	513	3	no	—
112	C₃	—	3	6	no	—
113	C₃	—	9	1	no	—
114	C₃	—	9	1	no	—
115	C₃	—	27	3	no	—
116	C₃	C₃×#13095319	1	2	yes	—
117	C₃	#17987504	1	1	yes	characteristic
118	C₃	#17987505	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17987417

C₃

yes

maximal, characteristic, complemented

C₃×C₃²⋊#4294

C₃

yes

maximal, complemented

#17987309

C₃

yes

maximal, complemented

C₁₉×C₃²⋊H(3,𝔽₃)

C₃

yes

maximal, characteristic, complemented

C₃×C₃²⋊(C₁₉⋊C₃)

—

C₃×C₃²⋊(C₁₉⋊C₃)

—

C₃×C₃²⋊(C₁₉⋊C₃)

C₃²

yes

—

C₃²×#4294

—

C₃²⋊#4294

—

C₃²⋊#4294

—

C₃²⋊#4294

C₃²

yes

—

C₃⁴×C₁₉

C₃²

yes

characteristic

C₃×C₁₉×H(3,𝔽₃)

—

C₃×C₁₉×H(3,𝔽₃)

—

C₃×C₁₉×H(3,𝔽₃)

C₃²

yes

—

#10624599

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

H(3,𝔽₃)

yes

characteristic

C₃³×C₁₉

H(3,𝔽₃)

yes

—

C₃³×C₁₉

C₃³

yes

characteristic

C₃²×C₁₉⋊C₃

—

C₁₉×H(3,𝔽₃)

—

C₁₉×H(3,𝔽₃)

—

C₁₉×H(3,𝔽₃)

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

C₃×#4294

—

C₃×#4294

—

C₃²⋊H(3,𝔽₃)

—

C₃²⋊H(3,𝔽₃)

C₁₉⋊C₃

yes

characteristic, complemented

C₃×C₃²⋊C₉

—

#6856

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

#732

yes

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

#4294

—

C₃⁴

C₃×C₁₉⋊C₃

yes

characteristic

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×C₁₉⋊C₃

yes

—

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

C₃×#732

yes

—

C₃×C₁₉

yes

C₃×C₁₉

yes

C₁₉⋊C₃

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

—

C₃³

C₃²×C₁₉⋊C₃

yes

—

—

—

—

171

—

H(3,𝔽₃)

—

171

—

H(3,𝔽₃)

—

171

—

C₃×C₉

—

C₃×C₉

—

171

—

C₁₉

#10624599

yes

characteristic, complemented, Sylow

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

171

—

106

C₃²

—

171

—

107

C₃²

—

171

—

108

C₃²

#13095319

yes

—

109

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

110

C₉

—

171

—

111

C₃

—

513

—

112

C₃

—

113

C₃

—

114

C₃

—

115

C₃

—

116

C₃

C₃×#13095319

yes

—

117

C₃

#17987504

yes

characteristic

118

C₃

#17987505

yes

characteristic

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)
commutator, cosocle	C₃³×C₁₉	C₃³
Frattini	C₃³	C₃²×C₁₉⋊C₃
Fitting	C₁₉×C₃²⋊H(3,𝔽₃)	C₃
socle	C₃²×C₁₉	C₃×H(3,𝔽₃)
hypercenter	C₃²⋊H(3,𝔽₃)	C₁₉⋊C₃
hypocenter	C₁₉	#10624599
LCS	G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²⋊H(3,𝔽₃)

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

commutator, cosocle

C₃³×C₁₉

C₃³

Frattini

C₃³

C₃²×C₁₉⋊C₃

Fitting

C₁₉×C₃²⋊H(3,𝔽₃)

C₃

socle

C₃²×C₁₉

C₃×H(3,𝔽₃)

hypercenter

C₃²⋊H(3,𝔽₃)

C₁₉⋊C₃

hypocenter

C₁₉

#10624599

LCS

G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²⋊H(3,𝔽₃)

N	H ≅ G/N	type
1. #17987417	C₃	semidirect
2. C₃×C₃²⋊#4294	C₃	semidirect
3. #17987309	C₃	semidirect
4. C₁₉×C₃²⋊H(3,𝔽₃)	C₃	semidirect
34. C₃²⋊H(3,𝔽₃)	C₁₉⋊C₃	semidirect
91. C₁₉	#10624599	semidirect

H ≅ G/N

type

1. #17987417

C₃

semidirect

2. C₃×C₃²⋊#4294

C₃

semidirect

3. #17987309

C₃

semidirect

4. C₁₉×C₃²⋊H(3,𝔽₃)

C₃

semidirect

34. C₃²⋊H(3,𝔽₃)

C₁₉⋊C₃

semidirect

91. C₁₉

#10624599

semidirect

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

subgroup	index
#17987417	3
C₃×C₃²⋊#4294	3
#17987309	3
C₁₉×C₃²⋊H(3,𝔽₃)	3
#10624599	19

subgroup

index

#17987417

C₃×C₃²⋊#4294

#17987309

C₁₉×C₃²⋊H(3,𝔽₃)

#10624599

Subgroups — #28353791

Conjugacy classes of subgroups

Complements

Maximal subgroups