Group #28354470 — subgroups

Subgroups — #28354470

Order 13851.

278 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 49 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	#17987419	C₃	1	1	yes	maximal, characteristic, complemented
2	#17987419	C₃	1	2	yes	maximal, complemented
3	(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
4	#17987320	C₃	1	4	yes	maximal, complemented
5	#17987319	C₃	1	2	yes	maximal, complemented
6	#17987318	C₃	1	2	yes	maximal, complemented
7	C₁₉×#6859	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃³×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
9	C₃×#10587023	—	3	1	no	—
10	C₃×#10587023	—	3	2	no	—
11	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	2	no	—
12	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	1	no	—
13	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	2	yes	—
14	C₃²⋊#4294	C₃²	1	4	yes	—
15	C₃²⋊#4294	C₃²	1	2	yes	—
16	#13095303	—	3	2	no	—
17	#13095303	—	3	2	no	—
18	#13095303	—	3	2	no	—
19	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
20	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
21	C₁₉×C₃²⋊C₉	C₃²	1	2	yes	—
22	#10624667	—	19	1	no	maximal, Sylow
23	C₃³×C₁₉	—	3	1	no	—
24	C₃³×C₁₉	C₃³	1	1	yes	characteristic
25	C₃²×C₁₉⋊C₃	—	3	2	no	—
26	C₃²×C₁₉⋊C₃	—	9	1	no	—
27	C₃²×C₁₉⋊C₃	—	9	2	no	—
28	C₃²×C₁₉⋊C₃	—	9	1	no	—
29	C₃²×C₁₉⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
30	C₃²×C₁₉⋊C₃	H(3,𝔽₃)	1	2	yes	—
31	C₁₉×H(3,𝔽₃)	—	9	1	no	—
32	C₃×C₉×C₁₉	—	3	1	no	—
33	C₃×C₉×C₁₉	—	3	2	no	—
34	C₃²⋊(C₁₉⋊C₃)	—	9	6	no	—
35	C₃²⋊(C₁₉⋊C₃)	—	9	3	no	—
36	C₃²⋊(C₁₉⋊C₃)	—	9	2	no	—
37	C₃×#4294	—	3	2	no	—
38	C₃×#4294	—	3	4	no	—
39	#10587023	—	9	3	no	—
40	#10587023	—	9	6	no	—
41	#6891	—	19	2	no	—
42	#6891	—	19	1	no	—
43	C₃²⋊H(3,𝔽₃)	—	19	1	no	—
44	#6859	—	19	2	no	—
45	#6859	—	19	2	no	—
46	#6859	—	19	4	no	—
47	#6859	C₁₉⋊C₃	1	1	yes	characteristic, complemented
48	C₃²×C₁₉	—	3	1	no	—
49	C₃²×C₁₉	—	3	2	no	—
50	C₃²×C₁₉	—	3	1	no	—
51	C₃²×C₁₉	—	9	1	no	—
52	C₃²×C₁₉	—	9	2	no	—
53	C₃²×C₁₉	—	9	1	no	—
54	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
55	C₃×C₁₉⋊C₃	—	3	1	no	—
56	C₃×C₁₉⋊C₃	—	3	2	no	—
57	C₃×C₁₉⋊C₃	—	3	2	no	—
58	C₃×C₁₉⋊C₃	—	3	2	no	—
59	C₃×C₁₉⋊C₃	—	3	2	no	—
60	C₃×C₁₉⋊C₃	—	3	2	no	—
61	C₃×C₁₉⋊C₃	—	3	1	no	—
62	C₃×C₁₉⋊C₃	—	9	9	no	—
63	C₃×C₁₉⋊C₃	—	9	2	no	—
64	C₃×C₁₉⋊C₃	—	9	4	no	—
65	C₃×C₁₉⋊C₃	—	9	2	no	—
66	C₉×C₁₉	—	9	1	no	—
67	C₉×C₁₉	—	9	2	no	—
68	#4294	—	9	2	no	—
69	#4294	—	9	4	no	—
70	C₃⁴	—	19	1	no	—
71	C₃×3₋¹⁺²	—	57	1	no	—
72	C₃×3₋¹⁺²	—	57	2	no	—
73	C₃×H(3,𝔽₃)	—	19	2	no	—
74	C₃×H(3,𝔽₃)	—	57	2	no	—
75	C₃×H(3,𝔽₃)	—	57	1	no	—
76	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	1	yes	characteristic
77	C₃²⋊C₉	—	19	4	no	—
78	C₃²⋊C₉	—	19	2	no	—
79	C₃²⋊C₉	—	57	2	no	—
80	C₃²⋊C₉	—	57	2	no	—
81	C₃²⋊C₉	—	57	2	no	—
82	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
83	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	2	yes	—
84	C₃×C₁₉	—	9	1	no	—
85	C₃×C₁₉	—	27	1	no	—
86	C₃×C₁₉	#6909	1	1	yes	characteristic
87	C₃×C₁₉	#6910	1	2	yes	—
88	C₃×C₁₉	#6911	1	1	yes	characteristic
89	C₁₉⋊C₃	—	9	1	no	—
90	C₁₉⋊C₃	—	9	2	no	—
91	C₁₉⋊C₃	—	27	2	no	—
92	C₃³	—	3	1	no	—
93	C₃³	—	19	1	no	—
94	C₃³	—	19	2	no	—
95	C₃³	—	57	2	no	—
96	C₃³	—	171	1	no	—
97	C₃³	—	171	2	no	—
98	C₃³	—	171	1	no	—
99	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
100	3₋¹⁺²	—	171	3	no	—
101	3₋¹⁺²	—	171	6	no	—
102	H(3,𝔽₃)	—	9	1	no	—
103	H(3,𝔽₃)	—	171	2	no	—
104	H(3,𝔽₃)	—	171	6	no	—
105	H(3,𝔽₃)	—	171	3	no	—
106	C₃×C₉	—	3	1	no	—
107	C₃×C₉	—	3	2	no	—
108	C₃×C₉	—	57	2	no	—
109	C₃×C₉	—	57	4	no	—
110	C₁₉	#10624667	1	1	yes	characteristic, complemented, Sylow
111	C₃²	—	3	1	no	—
112	C₃²	—	3	2	no	—
113	C₃²	—	3	1	no	—
114	C₃²	—	9	1	no	—
115	C₃²	—	9	2	no	—
116	C₃²	—	9	1	no	—
117	C₃²	—	57	1	no	—
118	C₃²	—	57	2	no	—
119	C₃²	—	57	2	no	—
120	C₃²	—	57	2	no	—
121	C₃²	—	57	2	no	—
122	C₃²	—	57	1	no	—
123	C₃²	—	57	2	no	—
124	C₃²	—	171	2	no	—
125	C₃²	—	171	4	no	—
126	C₃²	—	171	2	no	—
127	C₃²	—	171	9	no	—
128	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
129	C₉	—	9	1	no	—
130	C₉	—	9	2	no	—
131	C₉	—	171	2	no	—
132	C₉	—	171	4	no	—
133	C₃	—	513	2	no	—
134	C₃	—	9	1	no	—
135	C₃	—	27	1	no	—
136	C₃	—	171	1	no	—
137	C₃	—	171	2	no	—
138	C₃	#17987503	1	1	yes	characteristic
139	C₃	#17987507	1	2	yes	—
140	C₃	#17987513	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17987419

C₃

yes

maximal, characteristic, complemented

#17987419

C₃

yes

maximal, complemented

(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)

C₃

yes

maximal, characteristic

#17987320

C₃

yes

maximal, complemented

#17987319

C₃

yes

maximal, complemented

#17987318

C₃

yes

maximal, complemented

C₁₉×#6859

C₃

yes

maximal, characteristic, complemented

C₃³×C₁₉⋊C₃

C₃²

yes

characteristic

C₃×#10587023

—

C₃×#10587023

—

C₃×C₃²⋊(C₁₉⋊C₃)

—

C₃×C₃²⋊(C₁₉⋊C₃)

—

C₃×C₃²⋊(C₁₉⋊C₃)

C₃²

yes

—

C₃²⋊#4294

C₃²

yes

—

C₃²⋊#4294

C₃²

yes

—

#13095303

—

#13095303

—

#13095303

—

C₃×C₁₉×H(3,𝔽₃)

C₃²

yes

characteristic

C₁₉×C₃²⋊C₉

C₃²

yes

characteristic

C₁₉×C₃²⋊C₉

C₃²

yes

—

#10624667

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

C₃³

yes

characteristic

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

H(3,𝔽₃)

yes

characteristic

C₃²×C₁₉⋊C₃

H(3,𝔽₃)

yes

—

C₁₉×H(3,𝔽₃)

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

—

—

—

—

—

—

C₃²⋊H(3,𝔽₃)

—

—

—

—

C₁₉⋊C₃

yes

characteristic, complemented

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

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

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₉×C₁₉

—

C₉×C₁₉

—

#4294

—

#4294

—

C₃⁴

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×C₁₉⋊C₃

yes

characteristic

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₁₉⋊C₃

yes

characteristic

C₃²⋊C₉

C₃×C₁₉⋊C₃

yes

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

#6909

yes

characteristic

C₃×C₁₉

#6910

yes

—

C₃×C₁₉

#6911

yes

characteristic

C₁₉⋊C₃

—

C₁₉⋊C₃

—

C₁₉⋊C₃

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

171

—

C₃³

—

171

—

C₃³

—

171

—

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

100

3₋¹⁺²

—

171

—

101

3₋¹⁺²

—

171

—

102

H(3,𝔽₃)

—

103

H(3,𝔽₃)

—

171

—

104

H(3,𝔽₃)

—

171

—

105

H(3,𝔽₃)

—

171

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

—

110

C₁₉

#10624667

yes

characteristic, complemented, Sylow

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

—

171

—

125

C₃²

—

171

—

126

C₃²

—

171

—

127

C₃²

—

171

—

128

C₃²

H(3,𝔽₃)×C₁₉⋊C₃

yes

characteristic

129

C₉

—

130

C₉

—

131

C₉

—

171

—

132

C₉

—

171

—

133

C₃

—

513

—

134

C₃

—

135

C₃

—

136

C₃

—

171

—

137

C₃

—

171

—

138

C₃

#17987503

yes

characteristic

139

C₃

#17987507

yes

—

140

C₃

#17987513

yes

characteristic

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₁₉⋊C₃
commutator, cosocle	C₃³×C₁₉	C₃³
Frattini	C₃³	C₃²×C₁₉⋊C₃
Fitting	C₁₉×#6859	C₃
socle	C₃²×C₁₉	C₃×H(3,𝔽₃)
prime radical	(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)	C₃
hypercenter	#6859	C₁₉⋊C₃
hypocenter	C₁₉	#10624667
LCS	G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ #6859

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₁₉⋊C₃

commutator, cosocle

C₃³×C₁₉

C₃³

Frattini

C₃³

C₃²×C₁₉⋊C₃

Fitting

C₁₉×#6859

C₃

socle

C₃²×C₁₉

C₃×H(3,𝔽₃)

prime radical

(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)

C₃

C₁₉⋊C₃

C₁₉

G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ #6859

N	H ≅ G/N	type
1. #17987419	C₃	semidirect
2. #17987419	C₃	semidirect
4. #17987320	C₃	semidirect
5. #17987319	C₃	semidirect
6. #17987318	C₃	semidirect
7. C₁₉×#6859	C₃	semidirect
47. #6859	C₁₉⋊C₃	semidirect
110. C₁₉	#10624667	semidirect

H ≅ G/N

type

1. #17987419

C₃

semidirect

2. #17987419

C₃

semidirect

4. #17987320

C₃

semidirect

5. #17987319

C₃

semidirect

6. #17987318

C₃

semidirect

7. C₁₉×#6859

C₃

semidirect

47. #6859

C₁₉⋊C₃

semidirect

110. C₁₉

#10624667

semidirect

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

subgroup	index
#17987419	3
(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)	3
#17987320	3
#17987319	3
#17987318	3
C₁₉×#6859	3
#10624667	19

subgroup

index

(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)

C₁₉×#6859

Subgroups — #28354470

Conjugacy classes of subgroups

Complements

Maximal subgroups