Group #28353860 — subgroups

Subgroups — #28353860

Order 13851.

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

normal subgroups only

This group has 398 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, complemented
2	#17987421	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₃²⋊#4294	C₃	1	1	yes	maximal, characteristic, complemented
4	#17987312	C₃	1	9	yes	maximal, complemented
5	C₁₉×C₃²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
6	C₃×#10587025	—	3	1	no	—
7	C₃×#10587025	—	3	1	no	—
8	C₃×#10587025	—	3	9	no	—
9	C₃×#10587025	C₃²	1	3	yes	—
10	C₃×#10587025	C₃²	1	3	yes	—
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	—	3	1	no	—
16	C₃²⋊#4294	—	3	1	no	—
17	C₃²⋊#4294	—	3	1	no	—
18	C₃²⋊#4294	C₃²	1	3	yes	—
19	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
20	C₃×C₁₉×H(3,𝔽₃)	—	3	1	no	—
21	C₃×C₁₉×H(3,𝔽₃)	—	3	1	no	—
22	C₃×C₁₉×H(3,𝔽₃)	—	3	1	no	—
23	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	3	yes	—
24	#10624609	—	19	1	no	maximal, Sylow
25	C₃³×C₁₉	—	3	3	no	—
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×H(3,𝔽₃)	—	9	3	no	—
35	C₁₉×H(3,𝔽₃)	—	9	3	no	—
36	C₁₉×H(3,𝔽₃)	—	9	3	no	—
37	C₁₉×H(3,𝔽₃)	—	9	3	no	—
38	#10587025	—	3	9	no	—
39	#10587025	—	3	9	no	—
40	#10587025	—	3	9	no	—
41	#10587025	—	9	3	no	—
42	C₃×#4294	—	3	9	no	—
43	C₃×#4294	—	3	3	no	—
44	C₃×#4294	—	3	3	no	—
45	C₃×#4294	—	3	3	no	—
46	C₃×#4294	—	9	3	no	—
47	C₃²×3₋¹⁺²	—	19	1	no	—
48	#6891	—	19	1	no	—
49	C₃²⋊H(3,𝔽₃)	C₁₉⋊C₃	1	1	yes	characteristic
50	C₃×C₃²⋊C₉	—	19	1	no	—
51	#6857	—	19	9	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	—	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	—	9	9	no	—
67	C₃²×C₁₉	—	9	3	no	—
68	C₃²×C₁₉	—	9	3	no	—
69	C₃²×C₁₉	—	9	3	no	—
70	C₃²×C₁₉	—	9	3	no	—
71	C₃²×C₁₉	AΣL(1,𝔽₂₇)	1	3	yes	—
72	C₃²×C₁₉	#731	1	3	yes	—
73	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
74	#4294	—	3	9	no	—
75	#4294	—	9	3	no	—
76	#4294	—	9	3	no	—
77	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
78	C₃×3₋¹⁺²	—	19	3	no	—
79	C₃×3₋¹⁺²	—	19	3	no	—
80	C₃×3₋¹⁺²	—	57	1	no	—
81	C₃×3₋¹⁺²	—	57	1	no	—
82	C₃×3₋¹⁺²	—	57	9	no	—
83	C₃×H(3,𝔽₃)	—	3	1	no	—
84	C₃×H(3,𝔽₃)	—	3	1	no	—
85	C₃×H(3,𝔽₃)	—	3	1	no	—
86	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	3	yes	—
87	C₃²×C₉	—	57	1	no	—
88	C₃²×C₉	—	57	1	no	—
89	C₃²⋊C₉	—	19	3	no	—
90	C₃²⋊C₉	—	57	1	no	—
91	C₃²⋊C₉	—	57	1	no	—
92	C₃²⋊C₉	—	57	1	no	—
93	C₃²⋊C₉	—	57	1	no	—
94	C₃²⋊C₉	—	57	1	no	—
95	C₃×C₁₉	—	3	3	no	—
96	C₃×C₁₉	—	3	3	no	—
97	C₃×C₁₉	—	9	1	no	—
98	C₃×C₁₉	—	9	1	no	—
99	C₃×C₁₉	—	27	3	no	—
100	C₃×C₁₉	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
101	C₃×C₁₉	C₃×#731	1	1	yes	characteristic
102	C₃×C₁₉	#6909	1	1	yes	characteristic
103	C₃×C₁₉	#6909	1	1	yes	characteristic
104	C₃³	—	3	3	no	—
105	C₃³	—	3	3	no	—
106	C₃³	—	3	3	no	—
107	C₃³	—	9	1	no	—
108	C₃³	—	9	1	no	—
109	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
110	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
111	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
112	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
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	—	171	3	no	—
117	H(3,𝔽₃)	—	9	3	no	—
118	H(3,𝔽₃)	—	9	3	no	—
119	H(3,𝔽₃)	—	9	3	no	—
120	H(3,𝔽₃)	—	9	3	no	—
121	C₃×C₉	—	57	9	no	—
122	C₃×C₉	—	57	3	no	—
123	C₃×C₉	—	57	3	no	—
124	C₃×C₉	—	57	3	no	—
125	C₃×C₉	—	171	3	no	—
126	C₁₉	#10624609	1	1	yes	characteristic, complemented, Sylow
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	—	3	1	no	—
140	C₃²	—	3	1	no	—
141	C₃²	—	9	9	no	—
142	C₃²	—	9	3	no	—
143	C₃²	—	9	3	no	—
144	C₃²	—	9	3	no	—
145	C₃²	—	9	3	no	—
146	C₃²	#13095313	1	3	yes	—
147	C₃²	#13095316	1	3	yes	—
148	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
149	C₉	—	57	9	no	—
150	C₉	—	171	3	no	—
151	C₉	—	171	3	no	—
152	C₃	—	3	3	no	—
153	C₃	—	3	3	no	—
154	C₃	—	9	1	no	—
155	C₃	—	9	1	no	—
156	C₃	—	27	3	no	—
157	C₃	C₃×#13095313	1	1	yes	characteristic
158	C₃	C₃×#13095316	1	1	yes	characteristic
159	C₃	#17987504	1	1	yes	characteristic
160	C₃	#17987505	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#10587025

C₃

yes

maximal, characteristic, complemented

#17987421

C₃

yes

maximal, characteristic, complemented

C₃×C₃²⋊#4294

C₃

yes

maximal, characteristic, complemented

#17987312

C₃

yes

maximal, complemented

C₁₉×C₃²⋊H(3,𝔽₃)

C₃

yes

maximal, characteristic

—

—

—

C₃²

yes

—

C₃×#10587025

C₃²

yes

—

C₃²×#4294

—

C₃²×#4294

—

C₃²⋊#4294

—

C₃²⋊#4294

—

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)

—

C₃×C₁₉×H(3,𝔽₃)

C₃²

yes

—

#10624609

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

yes

C₃³×C₁₉

yes

C₃³×C₁₉

yes

C₃³×C₁₉

C₃³

yes

characteristic

C₁₉×H(3,𝔽₃)

—

C₁₉×H(3,𝔽₃)

—

C₁₉×H(3,𝔽₃)

—

C₁₉×H(3,𝔽₃)

—

—

—

—

—

—

—

—

—

—

C₃²×3₋¹⁺²

—

#6891

—

C₃²⋊H(3,𝔽₃)

C₁₉⋊C₃

yes

characteristic

C₃×C₃²⋊C₉

—

#6857

—

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

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₁₉

#731

yes

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

—

—

—

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×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

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

100

C₃×C₁₉

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

101

C₃×C₁₉

C₃×#731

yes

characteristic

102

C₃×C₁₉

#6909

yes

characteristic

103

C₃×C₁₉

#6909

yes

characteristic

104

C₃³

—

105

C₃³

—

106

C₃³

—

107

C₃³

—

108

C₃³

—

109

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

110

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

111

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

112

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

113

3₋¹⁺²

—

114

3₋¹⁺²

—

115

3₋¹⁺²

—

116

3₋¹⁺²

—

171

—

117

H(3,𝔽₃)

—

118

H(3,𝔽₃)

—

119

H(3,𝔽₃)

—

120

H(3,𝔽₃)

—

121

C₃×C₉

—

122

C₃×C₉

—

123

C₃×C₉

—

124

C₃×C₉

—

125

C₃×C₉

—

171

—

126

C₁₉

#10624609

yes

characteristic, complemented, Sylow

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

—

141

C₃²

—

142

C₃²

—

143

C₃²

—

144

C₃²

—

145

C₃²

—

146

C₃²

#13095313

yes

—

147

C₃²

#13095316

yes

—

148

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

149

C₉

—

150

C₉

—

171

—

151

C₉

—

171

—

152

C₃

—

153

C₃

—

154

C₃

—

155

C₃

—

156

C₃

—

157

C₃

C₃×#13095313

yes

characteristic

158

C₃

C₃×#13095316

yes

characteristic

159

C₃

#17987504

yes

characteristic

160

C₃

#17987505

yes

characteristic

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)
commutator, cosocle	C₃³×C₁₉	C₃³
Frattini	C₃³	C₃²×C₁₉⋊C₃
Fitting, prime radical	C₁₉×C₃²⋊H(3,𝔽₃)	C₃
socle	C₃²×C₁₉	C₃×H(3,𝔽₃)
hypercenter	C₃²⋊H(3,𝔽₃)	C₁₉⋊C₃
hypocenter	C₁₉	#10624609
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, prime radical

C₁₉×C₃²⋊H(3,𝔽₃)

C₃

socle

C₃²×C₁₉

C₃×H(3,𝔽₃)

hypercenter

C₃²⋊H(3,𝔽₃)

C₁₉⋊C₃

hypocenter

C₁₉

#10624609

LCS

G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²⋊H(3,𝔽₃)

N	H ≅ G/N	type
1. C₃²×#10587025	C₃	semidirect
2. #17987421	C₃	semidirect
3. C₃×C₃²⋊#4294	C₃	semidirect
4. #17987312	C₃	semidirect
126. C₁₉	#10624609	semidirect

H ≅ G/N

type

1. C₃²×#10587025

C₃

semidirect

2. #17987421

C₃

semidirect

3. C₃×C₃²⋊#4294

C₃

semidirect

4. #17987312

C₃

semidirect

126. C₁₉

#10624609

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
C₃×C₃²⋊#4294	3
#17987312	3
C₁₉×C₃²⋊H(3,𝔽₃)	3
#10624609	19

subgroup

index

C₃×C₃²⋊#4294

C₁₉×C₃²⋊H(3,𝔽₃)

Subgroups — #28353860

Conjugacy classes of subgroups

Complements

Maximal subgroups