Group #28356892 — subgroups

Subgroups — #28356892

Order 13851.

332 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 90 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	C₃×#13095344	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#13095343	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×(C₁₉⋊C₃)⋊C₂₇	C₃	1	1	yes	maximal, characteristic, complemented
4	#17987557	C₃	1	3	yes	maximal, complemented
5	#17987557	C₃	1	3	yes	maximal, complemented
6	C₃²⋊#10587020	C₃	1	3	yes	maximal, complemented
7	C₃×#13095333	C₃	1	1	yes	maximal, characteristic
8	#13095344	—	3	3	no	—
9	#13095343	—	3	3	no	—
10	(C₁₉⋊C₃)⋊C₂₇	—	3	3	no	—
11	C₃×#10587020	C₃²	1	3	yes	—
12	C₃×#10587020	C₃²	1	3	yes	—
13	C₃×#10587020	C₃²	1	3	yes	—
14	#13095333	—	3	3	no	—
15	#13095333	—	3	6	no	—
16	C₃×#10587025	C₉	1	1	yes	characteristic
17	C₃×#10587022	C₉	1	1	yes	characteristic
18	C₃×#10587022	C₉	1	1	yes	characteristic
19	C₃×#10587022	C₉	1	1	yes	characteristic
20	C₃×#10587023	C₉	1	1	yes	characteristic
21	C₃×#10587023	C₉	1	1	yes	characteristic
22	C₃×#10587023	C₉	1	1	yes	characteristic
23	C₃×C₃²⋊(C₁₉⋊C₃)	C₉	1	1	yes	characteristic
24	C₃²×#4294	C₃²	1	1	yes	characteristic
25	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
26	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
27	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
28	C₃×C₁₉×3₋¹⁺²	C₉	1	1	yes	characteristic
29	#10624684	—	19	1	no	maximal, Sylow
30	C₃³×C₁₉	C₃×C₉	1	1	yes	characteristic
31	C₃²×C₁₉⋊C₃	C₃×C₉	1	1	yes	characteristic
32	C₃²×C₁₉⋊C₃	C₃×C₉	1	1	yes	characteristic
33	C₃²×C₁₉⋊C₃	C₃×C₉	1	1	yes	characteristic
34	C₁₉×3₋¹⁺²	—	3	3	no	—
35	C₃×C₉×C₁₉	C₃×C₉	1	1	yes	characteristic
36	C₃×C₉×C₁₉	C₃×C₉	1	1	yes	characteristic
37	C₃×C₉×C₁₉	C₃×C₉	1	1	yes	characteristic
38	C₃²⋊(C₁₉⋊C₃)	—	3	3	no	—
39	#10587025	—	3	3	no	—
40	C₃×#4294	—	3	1	no	—
41	C₃×#4294	—	3	2	no	—
42	C₃×#4294	C₃×C₉	1	1	yes	characteristic
43	C₃×#4294	C₃×C₉	1	1	yes	characteristic
44	C₃×#4294	C₃×C₉	1	1	yes	characteristic
45	C₃×#4294	C₃×C₉	1	1	yes	characteristic
46	C₃×#4294	C₃×C₉	1	1	yes	characteristic
47	C₃×#4294	C₃³	1	1	yes	characteristic
48	#10587023	—	3	3	no	—
49	#10587023	—	3	3	no	—
50	#10587023	—	3	3	no	—
51	#10587022	—	3	3	no	—
52	#10587022	—	3	3	no	—
53	#10587022	—	3	3	no	—
54	C₉×C₁₉⋊C₃	—	3	1	no	—
55	C₉×C₁₉⋊C₃	—	3	1	no	—
56	C₉×C₁₉⋊C₃	—	3	2	no	—
57	C₉×C₁₉⋊C₃	—	3	2	no	—
58	C₉×C₁₉⋊C₃	H(3,𝔽₃)	1	3	yes	—
59	C₉×C₁₉⋊C₃	3₋¹⁺²	1	6	yes	—
60	#10587020	—	3	3	no	—
61	#10587020	—	3	3	no	—
62	#10587020	—	3	3	no	—
63	C₃×#737	—	19	1	no	—
64	C₃×C₂₇⋊C₃	—	19	1	no	—
65	C₃×C₂₇⋊C₃	—	19	1	no	—
66	C₃²×C₂₇	—	19	1	no	—
67	C₃²⋊C₂₇	—	19	3	no	—
68	C₃²⋊C₂₇	—	19	3	no	—
69	C₃²⋊C₂₇	—	19	3	no	—
70	C₃²×C₁₉	—	3	1	no	—
71	C₃²×C₁₉	—	3	3	no	—
72	C₃²×C₁₉	C₃²×C₉	1	1	yes	characteristic
73	C₃×C₁₉⋊C₃	—	3	3	no	—
74	C₃×C₁₉⋊C₃	—	3	1	no	—
75	C₃×C₁₉⋊C₃	—	3	1	no	—
76	C₃×C₁₉⋊C₃	—	3	3	no	—
77	C₃×C₁₉⋊C₃	—	3	3	no	—
78	C₃×C₁₉⋊C₃	C₃²⋊C₉	1	3	yes	—
79	C₉×C₁₉	—	3	1	no	—
80	C₉×C₁₉	—	3	1	no	—
81	C₉×C₁₉	C₃²⋊C₉	1	3	yes	—
82	#4294	—	3	1	no	—
83	#4294	—	3	1	no	—
84	#4294	—	3	1	no	—
85	#4294	—	3	1	no	—
86	#4294	C₃²⋊C₉	1	3	yes	—
87	#4294	C₃×H(3,𝔽₃)	1	1	yes	characteristic
88	#4294	C₃×3₋¹⁺²	1	2	yes	—
89	#737	—	57	3	no	—
90	#737	—	57	6	no	—
91	C₃×3₋¹⁺²	—	19	1	no	—
92	C₃×3₋¹⁺²	—	19	1	no	—
93	C₃×3₋¹⁺²	—	19	1	no	—
94	C₃×3₋¹⁺²	—	19	1	no	—
95	C₃×3₋¹⁺²	—	19	1	no	—
96	C₃×3₋¹⁺²	—	19	1	no	—
97	C₃×3₋¹⁺²	—	19	1	no	—
98	C₃×3₋¹⁺²	#4296	1	1	yes	characteristic
99	C₃×H(3,𝔽₃)	—	19	1	no	—
100	C₃²×C₉	—	19	1	no	—
101	C₃²×C₉	—	19	1	no	—
102	C₃²×C₉	—	19	1	no	—
103	C₃²×C₉	—	19	1	no	—
104	C₂₇⋊C₃	—	57	3	no	—
105	C₂₇⋊C₃	—	57	3	no	—
106	C₃×C₂₇	—	19	3	no	—
107	C₃×C₂₇	—	19	3	no	—
108	C₃×C₂₇	—	19	3	no	—
109	C₃×C₂₇	—	57	3	no	—
110	C₃×C₁₉	—	9	1	no	—
111	C₃×C₁₉	C₃×C₃²⋊C₉	1	1	yes	characteristic
112	C₃×C₁₉	#6903	1	3	yes	—
113	C₁₉⋊C₃	—	3	3	no	—
114	C₁₉⋊C₃	—	9	1	no	—
115	C₁₉⋊C₃	—	9	1	no	—
116	C₃³	—	19	1	no	—
117	C₃³	—	19	1	no	—
118	C₃³	—	19	1	no	—
119	C₃³	C₃×#4296	1	1	yes	characteristic
120	3₋¹⁺²	—	3	3	no	—
121	3₋¹⁺²	—	57	3	no	—
122	3₋¹⁺²	—	57	3	no	—
123	3₋¹⁺²	—	57	3	no	—
124	3₋¹⁺²	—	57	3	no	—
125	3₋¹⁺²	—	57	3	no	—
126	3₋¹⁺²	—	57	3	no	—
127	3₋¹⁺²	—	57	3	no	—
128	H(3,𝔽₃)	—	57	3	no	—
129	C₃×C₉	—	19	1	no	—
130	C₃×C₉	—	19	1	no	—
131	C₃×C₉	—	19	3	no	—
132	C₃×C₉	—	19	6	no	—
133	C₃×C₉	—	19	1	no	—
134	C₃×C₉	—	19	1	no	—
135	C₃×C₉	—	19	1	no	—
136	C₃×C₉	—	19	1	no	—
137	C₃×C₉	—	57	1	no	—
138	C₃×C₉	—	57	1	no	—
139	C₃×C₉	—	57	2	no	—
140	C₃×C₉	—	57	2	no	—
141	C₃×C₉	—	57	1	no	—
142	C₃×C₉	—	57	2	no	—
143	C₃×C₉	C₃×#4296	1	1	yes	characteristic
144	C₃×C₉	C₃×#4296	1	1	yes	characteristic
145	C₃×C₉	C₃×#4296	1	1	yes	characteristic
146	C₂₇	—	57	3	no	—
147	C₂₇	—	57	3	no	—
148	C₂₇	—	57	3	no	—
149	C₁₉	#10624684	1	1	yes	characteristic, complemented, Sylow
150	C₃²	—	3	1	no	—
151	C₃²	—	3	3	no	—
152	C₃²	—	19	3	no	—
153	C₃²	—	57	3	no	—
154	C₃²	—	57	1	no	—
155	C₃²	—	57	1	no	—
156	C₃²	—	57	3	no	—
157	C₃²	—	57	3	no	—
158	C₃²	C₃²×#4296	1	1	yes	characteristic
159	C₉	—	3	1	no	—
160	C₉	—	3	1	no	—
161	C₉	—	19	1	no	—
162	C₉	—	19	2	no	—
163	C₉	—	19	3	no	—
164	C₉	—	57	1	no	—
165	C₉	—	57	1	no	—
166	C₉	—	57	1	no	—
167	C₉	—	57	1	no	—
168	C₉	C₃²⋊#4296	1	3	yes	—
169	C₃	—	9	1	no	—
170	C₃	—	57	3	no	—
171	C₃	—	171	1	no	—
172	C₃	—	171	1	no	—
173	C₃	C₃×C₃²⋊#4296	1	1	yes	characteristic
174	C₃	#17987599	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#13095344

C₃

yes

maximal, characteristic, complemented

C₃×#13095343

C₃

yes

maximal, characteristic, complemented

C₃×(C₁₉⋊C₃)⋊C₂₇

C₃

yes

maximal, characteristic, complemented

#17987557

C₃

yes

maximal, complemented

#17987557

C₃

yes

maximal, complemented

C₃²⋊#10587020

C₃

yes

maximal, complemented

C₃×#13095333

C₃

yes

maximal, characteristic

#13095344

—

#13095343

—

(C₁₉⋊C₃)⋊C₂₇

—

C₃×#10587020

C₃²

yes

—

C₃×#10587020

C₃²

yes

—

C₃×#10587020

C₃²

yes

—

#13095333

—

#13095333

—

C₃×#10587025

C₉

yes

characteristic

C₃×#10587022

C₉

yes

characteristic

C₃×#10587022

C₉

yes

characteristic

C₃×#10587022

C₉

yes

characteristic

C₃×#10587023

C₉

yes

characteristic

C₃×#10587023

C₉

yes

characteristic

C₃×#10587023

C₉

yes

characteristic

C₃×C₃²⋊(C₁₉⋊C₃)

C₉

yes

characteristic

C₃²×#4294

C₃²

yes

characteristic

C₃×C₉×C₁₉⋊C₃

C₃²

yes

characteristic

C₃×C₉×C₁₉⋊C₃

C₃²

yes

characteristic

C₃×C₉×C₁₉⋊C₃

C₃²

yes

characteristic

C₃×C₁₉×3₋¹⁺²

C₉

yes

characteristic

#10624684

—

maximal, Sylow

C₃³×C₁₉

C₃×C₉

yes

characteristic

C₃²×C₁₉⋊C₃

C₃×C₉

yes

characteristic

C₃²×C₁₉⋊C₃

C₃×C₉

yes

characteristic

C₃²×C₁₉⋊C₃

C₃×C₉

yes

characteristic

C₁₉×3₋¹⁺²

—

C₃×C₉×C₁₉

C₃×C₉

yes

characteristic

C₃×C₉×C₁₉

C₃×C₉

yes

characteristic

C₃×C₉×C₁₉

C₃×C₉

yes

characteristic

C₃²⋊(C₁₉⋊C₃)

—

—

—

—

C₃×C₉

yes

characteristic

C₃×#4294

C₃×C₉

yes

characteristic

C₃×#4294

C₃×C₉

yes

characteristic

C₃×#4294

C₃×C₉

yes

characteristic

C₃×#4294

C₃×C₉

yes

characteristic

C₃×#4294

C₃³

yes

—

—

—

—

—

—

C₉×C₁₉⋊C₃

—

C₉×C₁₉⋊C₃

—

C₉×C₁₉⋊C₃

—

C₉×C₁₉⋊C₃

—

C₉×C₁₉⋊C₃

H(3,𝔽₃)

yes

—

C₉×C₁₉⋊C₃

3₋¹⁺²

yes

—

—

—

—

—

C₃×C₂₇⋊C₃

—

C₃×C₂₇⋊C₃

—

C₃²×C₂₇

—

C₃²⋊C₂₇

—

C₃²⋊C₂₇

—

C₃²⋊C₂₇

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃²×C₉

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

C₃²⋊C₉

yes

—

C₉×C₁₉

—

C₉×C₁₉

—

C₉×C₁₉

C₃²⋊C₉

yes

—

—

—

—

—

C₃²⋊C₉

yes

—

#4294

C₃×H(3,𝔽₃)

yes

characteristic

#4294

C₃×3₋¹⁺²

yes

—

#737

—

#737

—

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

#4296

yes

characteristic

C₃×H(3,𝔽₃)

—

100

C₃²×C₉

—

101

C₃²×C₉

—

102

C₃²×C₉

—

103

C₃²×C₉

—

104

C₂₇⋊C₃

—

105

C₂₇⋊C₃

—

106

C₃×C₂₇

—

107

C₃×C₂₇

—

108

C₃×C₂₇

—

109

C₃×C₂₇

—

110

C₃×C₁₉

—

111

C₃×C₁₉

C₃×C₃²⋊C₉

yes

characteristic

112

C₃×C₁₉

#6903

yes

—

113

C₁₉⋊C₃

—

114

C₁₉⋊C₃

—

115

C₁₉⋊C₃

—

116

C₃³

—

117

C₃³

—

118

C₃³

—

119

C₃³

C₃×#4296

yes

characteristic

120

3₋¹⁺²

—

121

3₋¹⁺²

—

122

3₋¹⁺²

—

123

3₋¹⁺²

—

124

3₋¹⁺²

—

125

3₋¹⁺²

—

126

3₋¹⁺²

—

127

3₋¹⁺²

—

128

H(3,𝔽₃)

—

129

C₃×C₉

—

130

C₃×C₉

—

131

C₃×C₉

—

132

C₃×C₉

—

133

C₃×C₉

—

134

C₃×C₉

—

135

C₃×C₉

—

136

C₃×C₉

—

137

C₃×C₉

—

138

C₃×C₉

—

139

C₃×C₉

—

140

C₃×C₉

—

141

C₃×C₉

—

142

C₃×C₉

—

143

C₃×C₉

C₃×#4296

yes

characteristic

144

C₃×C₉

C₃×#4296

yes

characteristic

145

C₃×C₉

C₃×#4296

yes

characteristic

146

C₂₇

—

147

C₂₇

—

148

C₂₇

—

149

C₁₉

#10624684

yes

characteristic, complemented, Sylow

150

C₃²

—

151

C₃²

—

152

C₃²

—

153

C₃²

—

154

C₃²

—

155

C₃²

—

156

C₃²

—

157

C₃²

—

158

C₃²

C₃²×#4296

yes

characteristic

159

C₉

—

160

C₉

—

161

C₉

—

162

C₉

—

163

C₉

—

164

C₉

—

165

C₉

—

166

C₉

—

167

C₉

—

168

C₉

C₃²⋊#4296

yes

—

169

C₃

—

170

C₃

—

171

C₃

—

171

—

172

C₃

—

171

—

173

C₃

C₃×C₃²⋊#4296

yes

characteristic

174

C₃

#17987599

yes

—

	subgroup	G/H
center, Frattini	C₃²	C₃²×#4296
commutator, socle	C₃²×C₁₉	C₃²×C₉
Fitting	C₃×C₁₉×3₋¹⁺²	C₉
prime radical	C₃×C₃²⋊(C₁₉⋊C₃)	C₉
cosocle	C₃×#4294	C₃³
hypercenter	C₃×3₋¹⁺²	#4296
hypocenter	C₁₉	#10624684
LCS	G ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃×3₋¹⁺²

subgroup

G/H

center, Frattini

C₃²

C₃²×#4296

commutator, socle

C₃²×C₁₉

C₃²×C₉

Fitting

C₃×C₁₉×3₋¹⁺²

C₉

prime radical

C₃×C₃²⋊(C₁₉⋊C₃)

C₉

C₃³

C₃×3₋¹⁺²

C₁₉

G ▷ C₃²×C₁₉ ▷ C₁₉

C₁ ◁ C₃² ◁ C₃×3₋¹⁺²

N	H ≅ G/N	type
1. C₃×#13095344	C₃	semidirect
2. C₃×#13095343	C₃	semidirect
3. C₃×(C₁₉⋊C₃)⋊C₂₇	C₃	semidirect
4. #17987557	C₃	semidirect
5. #17987557	C₃	semidirect
6. C₃²⋊#10587020	C₃	semidirect
149. C₁₉	#10624684	semidirect

H ≅ G/N

type

1. C₃×#13095344

C₃

semidirect

2. C₃×#13095343

C₃

semidirect

3. C₃×(C₁₉⋊C₃)⋊C₂₇

C₃

semidirect

4. #17987557

C₃

semidirect

5. #17987557

C₃

semidirect

6. C₃²⋊#10587020

C₃

semidirect

149. C₁₉

#10624684

semidirect

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

subgroup	index
C₃×#13095344	3
C₃×#13095343	3
C₃×(C₁₉⋊C₃)⋊C₂₇	3
#17987557	3
C₃²⋊#10587020	3
C₃×#13095333	3
#10624684	19

subgroup

index

C₃×(C₁₉⋊C₃)⋊C₂₇

Subgroups — #28356892

Conjugacy classes of subgroups

Complements

Maximal subgroups