Group #28353849 — subgroups

Subgroups — #28353849

Order 13851.

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

normal subgroups only

This group has 452 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	C₃²×#10587025	C₃	1	1	yes	maximal, characteristic
2	C₃²×#10587025	C₃	1	1	yes	maximal, characteristic
3	#17987421	C₃	1	1	yes	maximal, characteristic
4	#17987322	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	C₃×#10587025	—	3	1	no	—
10	C₃×#10587025	—	3	9	no	—
11	C₃×#10587025	—	3	9	no	—
12	C₃×#10587025	—	3	1	no	—
13	C₃×#10587025	C₃²	1	3	yes	—
14	C₃×#10587025	C₃²	1	3	yes	—
15	C₃²×#4294	—	3	1	no	—
16	C₃²×#4294	—	3	1	no	—
17	C₃²⋊#4294	—	3	1	no	—
18	C₃²⋊#4294	—	3	1	no	—
19	C₃²⋊#4294	C₃²	1	3	yes	—
20	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
21	C₃×C₁₉×3₋¹⁺²	—	3	1	no	—
22	C₁₉×C₃²⋊C₉	—	3	1	no	—
23	C₁₉×C₃²⋊C₉	—	3	1	no	—
24	C₁₉×C₃²⋊C₉	C₃²	1	3	yes	—
25	#10624607	—	19	1	no	maximal, Sylow
26	C₃³×C₁₉	—	3	3	no	—
27	C₃³×C₁₉	—	3	3	no	—
28	C₃³×C₁₉	—	9	1	no	—
29	C₃³×C₁₉	—	9	1	no	—
30	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
31	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
32	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
33	C₃³×C₁₉	C₃³	1	1	yes	characteristic
34	C₁₉×3₋¹⁺²	—	9	3	no	—
35	C₃×C₉×C₁₉	—	3	3	no	—
36	#10587025	—	3	9	no	—
37	#10587025	—	3	9	no	—
38	#10587025	—	3	9	no	—
39	#10587025	—	3	9	no	—
40	#10587025	—	3	9	no	—
41	#10587025	—	3	9	no	—
42	#10587025	—	9	3	no	—
43	C₃×#4294	—	3	9	no	—
44	C₃×#4294	—	3	9	no	—
45	C₃×#4294	—	3	3	no	—
46	C₃×#4294	—	3	3	no	—
47	C₃×#4294	—	3	3	no	—
48	C₃²×3₋¹⁺²	—	19	1	no	—
49	C₃²×3₋¹⁺²	—	19	1	no	—
50	#6891	—	19	1	no	—
51	#6891	C₁₉⋊C₃	1	1	yes	characteristic
52	#6860	—	19	9	no	—
53	C₃²×C₁₉	—	3	1	no	—
54	C₃²×C₁₉	—	3	1	no	—
55	C₃²×C₁₉	—	3	1	no	—
56	C₃²×C₁₉	—	3	1	no	—
57	C₃²×C₁₉	—	3	1	no	—
58	C₃²×C₁₉	—	3	1	no	—
59	C₃²×C₁₉	—	3	1	no	—
60	C₃²×C₁₉	—	3	1	no	—
61	C₃²×C₁₉	—	3	1	no	—
62	C₃²×C₁₉	—	3	1	no	—
63	C₃²×C₁₉	—	3	1	no	—
64	C₃²×C₁₉	—	3	1	no	—
65	C₃²×C₁₉	—	3	1	no	—
66	C₃²×C₁₉	—	3	1	no	—
67	C₃²×C₁₉	—	9	9	no	—
68	C₃²×C₁₉	AΣL(1,𝔽₂₇)	1	3	yes	—
69	C₃²×C₁₉	AΣL(1,𝔽₂₇)	1	3	yes	—
70	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
71	C₉×C₁₉	—	9	3	no	—
72	#4294	—	3	9	no	—
73	#4294	—	3	9	no	—
74	#4294	—	9	3	no	—
75	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
76	C₃×3₋¹⁺²	—	3	1	no	—
77	C₃×3₋¹⁺²	—	19	3	no	—
78	C₃×3₋¹⁺²	—	19	3	no	—
79	C₃×3₋¹⁺²	—	57	1	no	—
80	C₃×3₋¹⁺²	—	57	1	no	—
81	C₃×3₋¹⁺²	—	57	1	no	—
82	C₃×3₋¹⁺²	—	57	1	no	—
83	C₃×3₋¹⁺²	—	57	9	no	—
84	C₃×3₋¹⁺²	—	57	9	no	—
85	C₃×3₋¹⁺²	—	57	1	no	—
86	C₃²×C₉	—	57	1	no	—
87	C₃²×C₉	—	57	1	no	—
88	C₃²⋊C₉	—	3	1	no	—
89	C₃²⋊C₉	—	3	1	no	—
90	C₃²⋊C₉	—	19	3	no	—
91	C₃²⋊C₉	—	57	1	no	—
92	C₃²⋊C₉	—	57	1	no	—
93	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	—
94	C₃×C₁₉	—	3	3	no	—
95	C₃×C₁₉	—	3	3	no	—
96	C₃×C₁₉	—	9	1	no	—
97	C₃×C₁₉	—	9	1	no	—
98	C₃×C₁₉	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
99	C₃×C₁₉	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
100	C₃×C₁₉	#6912	1	1	yes	characteristic
101	C₃×C₁₉	#6912	1	1	yes	characteristic
102	C₃³	—	3	3	no	—
103	C₃³	—	3	3	no	—
104	C₃³	—	9	1	no	—
105	C₃³	—	9	1	no	—
106	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
107	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
108	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
109	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
110	3₋¹⁺²	—	9	3	no	—
111	3₋¹⁺²	—	57	9	no	—
112	3₋¹⁺²	—	57	9	no	—
113	3₋¹⁺²	—	57	9	no	—
114	3₋¹⁺²	—	57	9	no	—
115	3₋¹⁺²	—	57	9	no	—
116	3₋¹⁺²	—	57	9	no	—
117	3₋¹⁺²	—	171	3	no	—
118	C₃×C₉	—	3	3	no	—
119	C₃×C₉	—	57	9	no	—
120	C₃×C₉	—	57	9	no	—
121	C₃×C₉	—	57	3	no	—
122	C₃×C₉	—	57	3	no	—
123	C₃×C₉	—	57	3	no	—
124	C₁₉	#10624607	1	1	yes	characteristic, complemented, Sylow
125	C₃²	—	3	1	no	—
126	C₃²	—	3	1	no	—
127	C₃²	—	3	1	no	—
128	C₃²	—	3	1	no	—
129	C₃²	—	3	1	no	—
130	C₃²	—	3	1	no	—
131	C₃²	—	3	1	no	—
132	C₃²	—	3	1	no	—
133	C₃²	—	3	1	no	—
134	C₃²	—	3	1	no	—
135	C₃²	—	3	1	no	—
136	C₃²	—	3	1	no	—
137	C₃²	—	3	1	no	—
138	C₃²	—	3	1	no	—
139	C₃²	—	9	9	no	—
140	C₃²	#13095313	1	3	yes	—
141	C₃²	#13095315	1	3	yes	—
142	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
143	C₉	—	9	3	no	—
144	C₉	—	57	9	no	—
145	C₉	—	57	9	no	—
146	C₉	—	171	3	no	—
147	C₃	—	3	3	no	—
148	C₃	—	3	3	no	—
149	C₃	—	9	1	no	—
150	C₃	—	9	1	no	—
151	C₃	C₃×#13095313	1	1	yes	characteristic
152	C₃	C₃×#13095315	1	1	yes	characteristic
153	C₃	#17987518	1	1	yes	characteristic
154	C₃	#17987519	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#10587025

C₃

yes

maximal, characteristic

C₃²×#10587025

C₃

yes

maximal, characteristic

#17987421

C₃

yes

maximal, characteristic

#17987322

C₃

yes

maximal, complemented

C₁₉×#6891

C₃

yes

maximal, characteristic

—

—

—

—

—

—

—

C₃²

yes

—

C₃×#10587025

C₃²

yes

—

C₃²×#4294

—

C₃²×#4294

—

C₃²⋊#4294

—

C₃²⋊#4294

—

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

—

#10624607

—

maximal, Sylow

C₃³×C₁₉

—

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

—

—

—

—

—

—

—

—

—

—

—

—

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

—

#6891

—

#6891

C₁₉⋊C₃

yes

characteristic

#6860

—

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

—

C₃²×C₁₉

—

C₃²×C₁₉

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₁₉

AΣL(1,𝔽₂₇)

yes

—

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

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

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

C₃×C₁₉⋊C₃

yes

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

C₃×C₁₉

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

100

C₃×C₁₉

#6912

yes

characteristic

101

C₃×C₁₉

#6912

yes

characteristic

102

C₃³

—

103

C₃³

—

104

C₃³

—

105

C₃³

—

106

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

107

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

108

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

109

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

110

3₋¹⁺²

—

111

3₋¹⁺²

—

112

3₋¹⁺²

—

113

3₋¹⁺²

—

114

3₋¹⁺²

—

115

3₋¹⁺²

—

116

3₋¹⁺²

—

117

3₋¹⁺²

—

171

—

118

C₃×C₉

—

119

C₃×C₉

—

120

C₃×C₉

—

121

C₃×C₉

—

122

C₃×C₉

—

123

C₃×C₉

—

124

C₁₉

#10624607

yes

characteristic, complemented, Sylow

125

C₃²

—

126

C₃²

—

127

C₃²

—

128

C₃²

—

129

C₃²

—

130

C₃²

—

131

C₃²

—

132

C₃²

—

133

C₃²

—

134

C₃²

—

135

C₃²

—

136

C₃²

—

137

C₃²

—

138

C₃²

—

139

C₃²

—

140

C₃²

#13095313

yes

—

141

C₃²

#13095315

yes

—

142

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

143

C₉

—

144

C₉

—

145

C₉

—

146

C₉

—

171

—

147

C₃

—

148

C₃

—

149

C₃

—

150

C₃

—

151

C₃

C₃×#13095313

yes

characteristic

152

C₃

C₃×#13095315

yes

characteristic

153

C₃

#17987518

yes

characteristic

154

C₃

#17987519

yes

characteristic

	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	#10624607
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. #17987322	C₃	semidirect
124. C₁₉	#10624607	semidirect

H ≅ G/N

type

4. #17987322

C₃

semidirect

124. C₁₉

#10624607

semidirect

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

subgroup	index
C₃²×#10587025	3
#17987421	3
#17987322	3
C₁₉×#6891	3
#10624607	19

subgroup

index

C₁₉×#6891

Subgroups — #28353849

Conjugacy classes of subgroups

Complements

Maximal subgroups