Group #28356930 — subgroups

Subgroups — #28356930

Order 13851.

312 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 84 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×#13095343	C₃	1	1	yes	maximal, characteristic, complemented
2	(C₁₉⋊C₃)⋊(C₂₇⋊C₃)	C₃	1	2	yes	maximal, complemented
3	#17987569	C₃	1	6	yes	maximal, complemented
4	#17987568	C₃	1	3	yes	maximal, complemented
5	C₁₉⋊C₃×#737	C₃	1	1	yes	maximal, characteristic
6	#13095344	—	3	2	no	—
7	#13095343	—	3	2	no	—
8	#13095343	—	3	1	no	—
9	#13095343	—	3	2	no	—
10	#13095342	C₃²	1	6	yes	complemented
11	(C₁₉⋊C₃)⋊C₂₇	—	3	2	no	—
12	C₃×#10587020	C₃²	1	3	yes	complemented
13	#13095333	—	3	2	no	—
14	#13095333	—	3	2	no	—
15	#13095333	C₉	1	2	yes	—
16	#13095333	C₃²	1	2	yes	—
17	#13095332	—	3	2	no	—
18	#13095332	C₉	1	2	yes	—
19	#13095331	C₉	1	1	yes	characteristic
20	#13095331	C₃²	1	1	yes	characteristic
21	3₋¹⁺²×C₁₉⋊C₃	—	3	2	no	—
22	3₋¹⁺²×C₁₉⋊C₃	C₉	1	2	yes	—
23	H(3,𝔽₃)×C₁₉⋊C₃	C₉	1	1	yes	characteristic
24	C₃×C₉×C₁₉⋊C₃	—	3	1	no	—
25	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
26	C₁₉×#737	C₉	1	1	yes	characteristic
27	#10624704	—	19	1	no	maximal, Sylow
28	C₃²×C₁₉⋊C₃	—	3	1	no	—
29	C₃²×C₁₉⋊C₃	C₃×C₉	1	1	yes	characteristic
30	C₁₉×3₋¹⁺²	—	3	2	no	—
31	C₁₉×3₋¹⁺²	C₃×C₉	1	2	yes	—
32	C₁₉×H(3,𝔽₃)	C₃×C₉	1	1	yes	characteristic
33	C₃×C₉×C₁₉	—	3	1	no	—
34	C₃×C₉×C₁₉	C₃×C₉	1	1	yes	characteristic
35	C₃²⋊(C₁₉⋊C₃)	—	3	2	no	—
36	C₃²⋊(C₁₉⋊C₃)	C₃×C₉	1	2	yes	—
37	#10587025	—	3	2	no	—
38	#10587025	—	3	2	no	—
39	#10587025	C₃×C₉	1	2	yes	—
40	#10587025	C₃×C₉	1	2	yes	—
41	C₃×#4294	—	3	1	no	—
42	C₃×#4294	—	3	1	no	—
43	C₃×#4294	C₃×C₉	1	1	yes	characteristic
44	C₃×#4294	C₃³	1	1	yes	characteristic
45	#10587023	—	3	2	no	—
46	#10587023	—	3	2	no	—
47	#10587023	—	3	2	no	—
48	#10587023	—	3	2	no	—
49	#10587022	—	3	2	no	—
50	#10587022	—	3	2	no	—
51	#10587022	—	3	2	no	—
52	#10587022	—	3	2	no	—
53	C₉×C₁₉⋊C₃	—	3	2	no	—
54	C₉×C₁₉⋊C₃	—	3	2	no	—
55	C₉×C₁₉⋊C₃	—	3	2	no	—
56	C₉×C₁₉⋊C₃	—	3	2	no	—
57	C₉×C₁₉⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
58	C₉×C₁₉⋊C₃	H(3,𝔽₃)	1	2	yes	—
59	C₉×C₁₉⋊C₃	3₋¹⁺²	1	2	yes	—
60	C₉×C₁₉⋊C₃	3₋¹⁺²	1	2	yes	—
61	C₉×C₁₉⋊C₃	3₋¹⁺²	1	2	yes	—
62	#10587020	—	3	3	no	—
63	#10587020	—	3	6	no	—
64	C₃×#737	—	19	1	no	—
65	C₃×C₂₇⋊C₃	—	19	1	no	—
66	C₃×C₂₇⋊C₃	—	19	2	no	—
67	#6873	—	19	3	no	—
68	#6873	—	19	6	no	—
69	C₃²×C₁₉	—	3	1	no	—
70	C₃²×C₁₉	C₃²×C₉	1	1	yes	characteristic
71	C₃×C₁₉⋊C₃	—	3	2	no	—
72	C₃×C₁₉⋊C₃	—	9	1	no	—
73	C₃×C₁₉⋊C₃	—	9	3	no	—
74	C₃×C₁₉⋊C₃	C₃²⋊C₉	1	1	yes	characteristic
75	C₃×C₁₉⋊C₃	C₃²⋊C₉	1	2	yes	—
76	C₉×C₁₉	—	3	2	no	—
77	C₉×C₁₉	C₃²⋊C₉	1	1	yes	characteristic
78	C₉×C₁₉	C₃²⋊C₉	1	2	yes	—
79	#4294	—	3	2	no	—
80	#4294	—	3	2	no	—
81	#4294	C₃²⋊C₉	1	1	yes	characteristic
82	#4294	C₃²⋊C₉	1	2	yes	—
83	#4294	C₃×H(3,𝔽₃)	1	1	yes	characteristic
84	#4294	C₃×3₋¹⁺²	1	2	yes	—
85	#737	—	19	2	no	—
86	#737	—	19	2	no	—
87	#737	—	19	1	no	—
88	#737	—	19	1	no	—
89	#737	—	19	2	no	—
90	#737	—	57	2	no	—
91	#737	—	57	2	no	—
92	#737	—	57	2	no	—
93	#737	#4296	1	1	yes	characteristic
94	C₃×3₋¹⁺²	—	19	2	no	—
95	C₃×3₋¹⁺²	—	57	2	no	—
96	C₃×H(3,𝔽₃)	—	19	1	no	—
97	C₃²×C₉	—	19	1	no	—
98	C₃²×C₉	—	57	1	no	—
99	C₂₇⋊C₃	—	19	6	no	—
100	C₂₇⋊C₃	—	57	2	no	—
101	C₂₇⋊C₃	—	57	1	no	—
102	C₂₇⋊C₃	—	57	2	no	—
103	C₂₇⋊C₃	—	57	2	no	—
104	C₃×C₂₇	—	19	3	no	—
105	C₃×C₂₇	—	57	2	no	—
106	C₃×C₁₉	—	3	1	no	—
107	C₃×C₁₉	—	9	1	no	—
108	C₃×C₁₉	C₃×C₃²⋊C₉	1	1	yes	characteristic
109	C₁₉⋊C₃	—	3	1	no	—
110	C₁₉⋊C₃	—	3	2	no	—
111	C₁₉⋊C₃	—	9	2	no	—
112	C₃³	—	19	1	no	—
113	C₃³	—	57	1	no	—
114	3₋¹⁺²	—	3	2	no	—
115	3₋¹⁺²	—	19	2	no	—
116	3₋¹⁺²	—	19	2	no	—
117	3₋¹⁺²	—	57	2	no	—
118	3₋¹⁺²	—	57	2	no	—
119	3₋¹⁺²	—	57	2	no	—
120	3₋¹⁺²	—	57	2	no	—
121	3₋¹⁺²	—	57	2	no	—
122	3₋¹⁺²	—	57	2	no	—
123	3₋¹⁺²	—	57	2	no	—
124	3₋¹⁺²	—	57	2	no	—
125	3₋¹⁺²	—	57	2	no	—
126	3₋¹⁺²	—	57	2	no	—
127	3₋¹⁺²	C₃×#4296	1	2	yes	—
128	H(3,𝔽₃)	—	19	2	no	—
129	H(3,𝔽₃)	—	57	2	no	—
130	H(3,𝔽₃)	C₃×#4296	1	1	yes	characteristic
131	C₃×C₉	—	3	1	no	—
132	C₃×C₉	—	19	1	no	—
133	C₃×C₉	—	19	1	no	—
134	C₃×C₉	—	19	2	no	—
135	C₃×C₉	—	19	1	no	—
136	C₃×C₉	—	19	2	no	—
137	C₃×C₉	—	19	2	no	—
138	C₃×C₉	—	19	2	no	—
139	C₃×C₉	—	57	2	no	—
140	C₃×C₉	—	57	2	no	—
141	C₃×C₉	—	57	2	no	—
142	C₃×C₉	—	57	1	no	—
143	C₃×C₉	—	57	1	no	—
144	C₃×C₉	—	57	2	no	—
145	C₃×C₉	C₃×#4296	1	1	yes	characteristic
146	C₂₇	—	57	3	no	—
147	C₂₇	—	57	6	no	—
148	C₁₉	#10624704	1	1	yes	characteristic, complemented, Sylow
149	C₃²	—	3	1	no	—
150	C₃²	—	19	1	no	—
151	C₃²	—	19	2	no	—
152	C₃²	—	57	2	no	—
153	C₃²	—	171	1	no	—
154	C₃²	—	171	3	no	—
155	C₃²	C₃²×#4296	1	1	yes	characteristic
156	C₉	—	3	2	no	—
157	C₉	—	19	1	no	—
158	C₉	—	19	1	no	—
159	C₉	—	19	2	no	—
160	C₉	—	19	2	no	—
161	C₉	—	57	2	no	—
162	C₉	—	57	2	no	—
163	C₉	C₃²⋊#4296	1	1	yes	characteristic
164	C₉	C₃²⋊#4296	1	2	yes	—
165	C₃	—	3	1	no	—
166	C₃	—	9	1	no	—
167	C₃	—	57	1	no	—
168	C₃	—	57	2	no	—
169	C₃	—	171	2	no	—
170	C₃	C₃×C₃²⋊#4296	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#13095343

C₃

yes

maximal, characteristic, complemented

(C₁₉⋊C₃)⋊(C₂₇⋊C₃)

C₃

yes

maximal, complemented

#17987569

C₃

yes

maximal, complemented

#17987568

C₃

yes

maximal, complemented

C₁₉⋊C₃×#737

C₃

yes

maximal, characteristic

—

—

—

—

C₃²

yes

complemented

(C₁₉⋊C₃)⋊C₂₇

—

C₃×#10587020

C₃²

yes

—

—

C₉

yes

—

#13095333

C₃²

yes

—

#13095332

—

#13095332

C₉

yes

—

#13095331

C₉

yes

characteristic

#13095331

C₃²

yes

characteristic

3₋¹⁺²×C₁₉⋊C₃

—

3₋¹⁺²×C₁₉⋊C₃

C₉

yes

—

H(3,𝔽₃)×C₁₉⋊C₃

C₉

yes

characteristic

C₃×C₉×C₁₉⋊C₃

—

C₃×C₉×C₁₉⋊C₃

C₃²

yes

characteristic

C₁₉×#737

C₉

yes

characteristic

#10624704

—

maximal, Sylow

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

C₃×C₉

yes

characteristic

C₁₉×3₋¹⁺²

—

C₁₉×3₋¹⁺²

C₃×C₉

yes

—

C₁₉×H(3,𝔽₃)

C₃×C₉

yes

characteristic

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

C₃×C₉

yes

characteristic

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

C₃×C₉

yes

—

#10587025

—

#10587025

—

#10587025

C₃×C₉

yes

—

#10587025

C₃×C₉

yes

—

C₃×#4294

—

C₃×#4294

—

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

characteristic

C₉×C₁₉⋊C₃

H(3,𝔽₃)

yes

—

C₉×C₁₉⋊C₃

3₋¹⁺²

yes

—

C₉×C₁₉⋊C₃

3₋¹⁺²

yes

—

C₉×C₁₉⋊C₃

3₋¹⁺²

yes

—

#10587020

—

#10587020

—

C₃×#737

—

C₃×C₂₇⋊C₃

—

C₃×C₂₇⋊C₃

—

#6873

—

#6873

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃²×C₉

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

C₃²⋊C₉

yes

characteristic

C₃×C₁₉⋊C₃

C₃²⋊C₉

yes

—

C₉×C₁₉

—

C₉×C₁₉

C₃²⋊C₉

yes

characteristic

C₉×C₁₉

C₃²⋊C₉

yes

—

#4294

—

#4294

—

#4294

C₃²⋊C₉

yes

characteristic

#4294

C₃²⋊C₉

yes

—

#4294

C₃×H(3,𝔽₃)

yes

characteristic

#4294

C₃×3₋¹⁺²

yes

—

—

—

—

—

—

—

—

—

yes

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

C₂₇⋊C₃

—

100

C₂₇⋊C₃

—

101

C₂₇⋊C₃

—

102

C₂₇⋊C₃

—

103

C₂₇⋊C₃

—

104

C₃×C₂₇

—

105

C₃×C₂₇

—

106

C₃×C₁₉

—

107

C₃×C₁₉

—

108

C₃×C₁₉

C₃×C₃²⋊C₉

yes

characteristic

109

C₁₉⋊C₃

—

110

C₁₉⋊C₃

—

111

C₁₉⋊C₃

—

112

C₃³

—

113

C₃³

—

114

3₋¹⁺²

—

115

3₋¹⁺²

—

116

3₋¹⁺²

—

117

3₋¹⁺²

—

118

3₋¹⁺²

—

119

3₋¹⁺²

—

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

C₃×#4296

yes

—

128

H(3,𝔽₃)

—

129

H(3,𝔽₃)

—

130

H(3,𝔽₃)

C₃×#4296

yes

characteristic

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

—

144

C₃×C₉

—

145

C₃×C₉

C₃×#4296

yes

characteristic

146

C₂₇

—

147

C₂₇

—

148

C₁₉

#10624704

yes

characteristic, complemented, Sylow

149

C₃²

—

150

C₃²

—

151

C₃²

—

152

C₃²

—

153

C₃²

—

171

—

154

C₃²

—

171

—

155

C₃²

C₃²×#4296

yes

characteristic

156

C₉

—

157

C₉

—

158

C₉

—

159

C₉

—

160

C₉

—

161

C₉

—

162

C₉

—

163

C₉

C₃²⋊#4296

yes

characteristic

164

C₉

C₃²⋊#4296

yes

—

165

C₃

—

166

C₃

—

167

C₃

—

168

C₃

—

169

C₃

—

171

—

170

C₃

C₃×C₃²⋊#4296

yes

characteristic

	subgroup	G/H
center	C₃	C₃×C₃²⋊#4296
commutator	C₃²×C₁₉	C₃²×C₉
Frattini	C₃²	C₃²×#4296
Fitting	C₁₉×#737	C₉
socle	C₃×C₁₉	C₃×C₃²⋊C₉
prime radical	H(3,𝔽₃)×C₁₉⋊C₃	C₉
cosocle	C₃×#4294	C₃³
hypercenter	#737	#4296
hypocenter	C₁₉	#10624704
LCS	G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃×C₉ ◁ #737

subgroup

G/H

center

C₃

C₃×C₃²⋊#4296

commutator

C₃²×C₁₉

C₃²×C₉

Frattini

C₃²

C₃²×#4296

Fitting

C₁₉×#737

C₉

socle

C₃×C₁₉

C₃×C₃²⋊C₉

prime radical

H(3,𝔽₃)×C₁₉⋊C₃

C₉

C₃³

C₁₉

G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃ ◁ C₃×C₉ ◁ #737

N	H ≅ G/N	type
1. C₃×#13095343	C₃	semidirect
2. (C₁₉⋊C₃)⋊(C₂₇⋊C₃)	C₃	semidirect
3. #17987569	C₃	semidirect
4. #17987568	C₃	semidirect
10. #13095342	C₃²	semidirect
12. C₃×#10587020	C₃²	semidirect
148. C₁₉	#10624704	semidirect

H ≅ G/N

type

1. C₃×#13095343

C₃

semidirect

2. (C₁₉⋊C₃)⋊(C₂₇⋊C₃)

C₃

semidirect

3. #17987569

C₃

semidirect

4. #17987568

C₃

semidirect

10. #13095342

C₃²

semidirect

12. C₃×#10587020

C₃²

semidirect

148. C₁₉

#10624704

semidirect

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

subgroup	index
C₃×#13095343	3
(C₁₉⋊C₃)⋊(C₂₇⋊C₃)	3
#17987569	3
#17987568	3
C₁₉⋊C₃×#737	3
#10624704	19

subgroup

index

(C₁₉⋊C₃)⋊(C₂₇⋊C₃)

C₁₉⋊C₃×#737

Subgroups — #28356930

Conjugacy classes of subgroups

Complements

Maximal subgroups