Group #28354467 — subgroups

Subgroups — #28354467

Order 13851.

236 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 52 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	#17987421	C₃	1	1	yes	maximal, characteristic
2	#17987421	C₃	1	1	yes	maximal, characteristic
3	#17987421	C₃	1	1	yes	maximal, characteristic
4	#17987324	C₃	1	9	yes	maximal, complemented
5	C₁₉×#6891	C₃	1	1	yes	maximal, characteristic
6	C₃×#10587025	—	3	1	no	—
7	C₃×#10587025	—	3	1	no	—
8	C₃×#10587025	—	3	1	no	—
9	C₃²⋊#4294	—	3	1	no	—
10	C₃²⋊#4294	—	3	1	no	—
11	C₃²⋊#4294	—	3	1	no	—
12	C₃²⋊#4294	—	3	1	no	—
13	C₃²⋊#4294	—	3	1	no	—
14	C₃²⋊#4294	—	3	1	no	—
15	C₃²⋊#4294	C₃²	1	3	yes	—
16	C₃²⋊#4294	C₃²	1	3	yes	—
17	C₃²⋊#4294	C₃²	1	3	yes	—
18	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
19	C₃×C₁₉×3₋¹⁺²	—	3	1	no	—
20	C₁₉×C₃²⋊C₉	—	3	1	no	—
21	C₁₉×C₃²⋊C₉	—	3	1	no	—
22	C₁₉×C₃²⋊C₉	C₃²	1	3	yes	—
23	#10624666	—	19	1	no	maximal, Sylow
24	C₃³×C₁₉	—	9	2	no	—
25	C₃³×C₁₉	—	9	1	no	—
26	C₃³×C₁₉	—	9	1	no	—
27	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
28	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
29	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
30	C₃³×C₁₉	C₃³	1	1	yes	characteristic
31	C₁₉×3₋¹⁺²	—	9	3	no	—
32	C₃×C₉×C₁₉	—	3	3	no	—
33	#10587025	—	9	3	no	—
34	#10587025	—	9	3	no	—
35	#10587025	—	9	3	no	—
36	C₃×#4294	—	3	3	no	—
37	C₃×#4294	—	3	3	no	—
38	C₃×#4294	—	3	3	no	—
39	#6891	—	19	1	no	—
40	#6891	—	19	1	no	—
41	#6891	—	19	1	no	—
42	#6891	C₁₉⋊C₃	1	1	yes	characteristic
43	#6861	—	19	9	no	—
44	C₃²×C₁₉	—	3	2	no	—
45	C₃²×C₁₉	—	3	2	no	—
46	C₃²×C₁₉	—	3	2	no	—
47	C₃²×C₁₉	—	3	2	no	—
48	C₃²×C₁₉	—	3	1	no	—
49	C₃²×C₁₉	—	3	1	no	—
50	C₃²×C₁₉	—	3	1	no	—
51	C₃²×C₁₉	—	3	1	no	—
52	C₃²×C₁₉	—	3	1	no	—
53	C₃²×C₁₉	—	3	1	no	—
54	C₃²×C₁₉	—	3	1	no	—
55	C₃²×C₁₉	—	3	1	no	—
56	C₃²×C₁₉	—	9	9	no	—
57	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
58	C₉×C₁₉	—	9	3	no	—
59	#4294	—	9	3	no	—
60	#4294	—	9	3	no	—
61	#4294	—	9	3	no	—
62	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
63	C₃×3₋¹⁺²	—	3	1	no	—
64	C₃×3₋¹⁺²	—	57	1	no	—
65	C₃×3₋¹⁺²	—	57	1	no	—
66	C₃×3₋¹⁺²	—	57	1	no	—
67	C₃²⋊C₉	—	3	1	no	—
68	C₃²⋊C₉	—	3	1	no	—
69	C₃²⋊C₉	—	19	3	no	—
70	C₃²⋊C₉	—	19	3	no	—
71	C₃²⋊C₉	—	19	3	no	—
72	C₃²⋊C₉	—	57	1	no	—
73	C₃²⋊C₉	—	57	1	no	—
74	C₃²⋊C₉	—	57	1	no	—
75	C₃²⋊C₉	—	57	1	no	—
76	C₃²⋊C₉	—	57	1	no	—
77	C₃²⋊C₉	—	57	1	no	—
78	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	—
79	C₃×C₁₉	—	9	1	no	—
80	C₃×C₁₉	—	9	1	no	—
81	C₃×C₁₉	—	9	1	no	—
82	C₃×C₁₉	—	9	1	no	—
83	C₃×C₁₉	#6910	1	1	yes	characteristic
84	C₃×C₁₉	#6911	1	1	yes	characteristic
85	C₃×C₁₉	#6912	1	2	yes	—
86	C₃³	—	9	1	no	—
87	C₃³	—	9	1	no	—
88	C₃³	—	9	2	no	—
89	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
90	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
91	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
92	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
93	3₋¹⁺²	—	9	3	no	—
94	3₋¹⁺²	—	171	3	no	—
95	3₋¹⁺²	—	171	3	no	—
96	3₋¹⁺²	—	171	3	no	—
97	C₃×C₉	—	3	3	no	—
98	C₃×C₉	—	57	3	no	—
99	C₃×C₉	—	57	3	no	—
100	C₃×C₉	—	57	3	no	—
101	C₁₉	#10624666	1	1	yes	characteristic, complemented, Sylow
102	C₃²	—	3	1	no	—
103	C₃²	—	3	1	no	—
104	C₃²	—	3	1	no	—
105	C₃²	—	3	1	no	—
106	C₃²	—	3	1	no	—
107	C₃²	—	3	1	no	—
108	C₃²	—	3	1	no	—
109	C₃²	—	3	1	no	—
110	C₃²	—	3	2	no	—
111	C₃²	—	3	2	no	—
112	C₃²	—	3	2	no	—
113	C₃²	—	3	2	no	—
114	C₃²	—	9	9	no	—
115	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
116	C₉	—	9	3	no	—
117	C₉	—	171	3	no	—
118	C₉	—	171	3	no	—
119	C₉	—	171	3	no	—
120	C₃	—	9	1	no	—
121	C₃	—	9	1	no	—
122	C₃	—	9	1	no	—
123	C₃	—	9	1	no	—
124	C₃	#17987510	1	1	yes	characteristic
125	C₃	#17987514	1	1	yes	characteristic
126	C₃	#17987518	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17987421

C₃

yes

maximal, characteristic

#17987421

C₃

yes

maximal, characteristic

#17987421

C₃

yes

maximal, characteristic

#17987324

C₃

yes

maximal, complemented

C₁₉×#6891

C₃

yes

maximal, characteristic

C₃×#10587025

—

C₃×#10587025

—

C₃×#10587025

—

C₃²⋊#4294

—

C₃²⋊#4294

—

C₃²⋊#4294

—

C₃²⋊#4294

—

C₃²⋊#4294

—

C₃²⋊#4294

—

C₃²⋊#4294

C₃²

yes

—

C₃²⋊#4294

C₃²

yes

—

C₃²⋊#4294

C₃²

yes

—

C₃⁴×C₁₉

C₃²

yes

characteristic

C₃×C₁₉×3₋¹⁺²

—

C₁₉×C₃²⋊C₉

—

C₁₉×C₃²⋊C₉

—

C₁₉×C₃²⋊C₉

C₃²

yes

—

#10624666

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

yes

C₃³×C₁₉

yes

C₃³×C₁₉

yes

C₃³×C₁₉

C₃³

yes

characteristic

C₁₉×3₋¹⁺²

—

C₃×C₉×C₁₉

—

—

—

—

—

—

—

—

—

—

C₁₉⋊C₃

yes

characteristic

#6861

—

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

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₁₉

—

#4294

—

#4294

—

#4294

—

C₃⁴

C₃×C₁₉⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

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

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₁₉⋊C₃

yes

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

yes

C₃×C₁₉

yes

C₃×C₁₉

yes

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

171

—

3₋¹⁺²

—

171

—

3₋¹⁺²

—

171

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

100

C₃×C₉

—

101

C₁₉

#10624666

yes

characteristic, complemented, Sylow

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

115

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

116

C₉

—

117

C₉

—

171

—

118

C₉

—

171

—

119

C₉

—

171

—

120

C₃

—

121

C₃

—

122

C₃

—

123

C₃

—

124

C₃

#17987510

yes

characteristic

125

C₃

#17987514

yes

characteristic

126

C₃

#17987518

yes

—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)
commutator, cosocle	C₃³×C₁₉	C₃³
Frattini	C₃³	C₃²×C₁₉⋊C₃
Fitting	C₁₉×#6891	C₃
socle	C₃²×C₁₉	C₃×H(3,𝔽₃)
prime radical	C₃⁴×C₁₉	C₃²
hypercenter	#6891	C₁₉⋊C₃
hypocenter	C₁₉	#10624666
LCS	G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ #6891

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

commutator, cosocle

C₃³×C₁₉

C₃³

Frattini

C₃³

C₃²×C₁₉⋊C₃

Fitting

C₁₉×#6891

C₃

socle

C₃²×C₁₉

C₃×H(3,𝔽₃)

prime radical

C₃⁴×C₁₉

C₃²

C₁₉⋊C₃

C₁₉

G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ #6891

N	H ≅ G/N	type
4. #17987324	C₃	semidirect
101. C₁₉	#10624666	semidirect

H ≅ G/N

type

4. #17987324

C₃

semidirect

101. C₁₉

#10624666

semidirect

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

subgroup	index
#17987421	3
#17987324	3
C₁₉×#6891	3
#10624666	19

subgroup

index

#17987421

#17987324

C₁₉×#6891

#10624666

Subgroups — #28354467

Conjugacy classes of subgroups

Complements

Maximal subgroups