Group #28353605 — subgroups

Subgroups — #28353605

Order 13851.

470 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 112 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×#10587022	C₃	1	1	yes	maximal, characteristic
2	C₃×#13095305	C₃	1	3	yes	maximal, complemented
3	C₃×#13095303	C₃	1	3	yes	maximal
4	#17987308	C₃	1	3	yes	maximal, complemented
5	#17987308	C₃	1	1	yes	maximal, characteristic, complemented
6	#17987308	C₃	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#6855	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃³×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
9	C₃×#10587022	—	3	3	no	—
10	C₃×#10587022	—	3	3	no	—
11	C₃×#10587022	—	3	3	no	—
12	C₃×#10587022	—	3	3	no	—
13	C₃²×#4294	C₃²	1	3	yes	—
14	C₃²×#4294	C₃²	1	1	yes	characteristic
15	C₃²×#4294	C₃²	1	1	yes	characteristic
16	C₃²×#4294	C₃²	1	3	yes	—
17	#13095305	—	3	9	no	—
18	#13095303	—	3	9	no	—
19	C₃²×C₉×C₁₉	C₃²	1	3	yes	—
20	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
21	#10624565	—	19	1	no	maximal, Sylow
22	C₃³×C₁₉	C₃³	1	1	yes	characteristic
23	C₃²×C₁₉⋊C₃	—	3	1	no	—
24	C₃²×C₁₉⋊C₃	—	3	3	no	—
25	C₃²×C₁₉⋊C₃	—	3	1	no	—
26	C₃²×C₁₉⋊C₃	—	3	1	no	—
27	C₃²×C₁₉⋊C₃	—	3	3	no	—
28	C₃²×C₁₉⋊C₃	—	3	3	no	—
29	C₃²×C₁₉⋊C₃	3₋¹⁺²	1	3	yes	—
30	C₃×C₉×C₁₉	—	3	1	no	—
31	C₃×C₉×C₁₉	—	3	1	no	—
32	C₃×C₉×C₁₉	—	3	1	no	—
33	C₃×C₉×C₁₉	—	3	3	no	—
34	C₃×C₉×C₁₉	—	3	3	no	—
35	C₃×C₉×C₁₉	—	3	3	no	—
36	C₃×C₉×C₁₉	—	3	3	no	—
37	C₃×C₉×C₁₉	3₋¹⁺²	1	3	yes	—
38	C₃×#4294	—	3	3	no	—
39	C₃×#4294	—	3	1	no	—
40	C₃×#4294	—	3	1	no	—
41	C₃×#4294	—	3	1	no	—
42	C₃×#4294	—	3	1	no	—
43	C₃×#4294	—	3	1	no	—
44	C₃×#4294	—	3	1	no	—
45	C₃×#4294	—	3	3	no	—
46	C₃×#4294	—	3	3	no	—
47	C₃×#4294	—	3	3	no	—
48	C₃×#4294	—	3	3	no	—
49	C₃×#4294	—	3	3	no	—
50	C₃×#4294	—	3	3	no	—
51	C₃×#4294	H(3,𝔽₃)	1	3	yes	—
52	C₃×#4294	3₋¹⁺²	1	3	yes	—
53	C₃×#4294	3₋¹⁺²	1	9	yes	complemented
54	#10587022	—	3	27	no	—
55	C₃²×3₋¹⁺²	—	19	1	no	—
56	C₃×C₉⋊C₉	—	19	3	no	—
57	C₃×C₃²⋊C₉	—	19	3	no	—
58	#6855	—	19	3	no	—
59	#6855	—	19	1	no	—
60	#6855	—	19	1	no	—
61	#6855	C₁₉⋊C₃	1	1	yes	characteristic, complemented
62	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
63	C₃²×C₁₉	C₃×3₋¹⁺²	1	1	yes	characteristic
64	C₃²×C₁₉	C₃×3₋¹⁺²	1	1	yes	characteristic
65	C₃²×C₁₉	C₃×3₋¹⁺²	1	3	yes	—
66	C₃²×C₁₉	C₃×3₋¹⁺²	1	1	yes	characteristic
67	C₃²×C₁₉	#737	1	3	yes	—
68	C₃²×C₁₉	#737	1	3	yes	—
69	C₃×C₁₉⋊C₃	—	3	3	no	—
70	C₃×C₁₉⋊C₃	—	3	9	no	—
71	C₃×C₁₉⋊C₃	—	9	3	no	—
72	C₃×C₁₉⋊C₃	—	9	3	no	—
73	C₃×C₁₉⋊C₃	—	9	1	no	—
74	C₃×C₁₉⋊C₃	—	9	1	no	—
75	C₃×C₁₉⋊C₃	—	9	1	no	—
76	C₉×C₁₉	—	3	3	no	—
77	C₉×C₁₉	—	9	3	no	—
78	#4294	—	3	3	no	—
79	#4294	—	3	3	no	—
80	#4294	—	3	9	no	—
81	#4294	—	9	3	no	—
82	C₃⁴	—	19	1	no	—
83	C₃×3₋¹⁺²	—	57	3	no	—
84	C₃×3₋¹⁺²	—	57	3	no	—
85	C₃×3₋¹⁺²	—	57	3	no	—
86	C₃×3₋¹⁺²	—	57	3	no	—
87	C₃²×C₉	—	19	1	no	—
88	C₃²×C₉	—	19	1	no	—
89	C₃²×C₉	—	19	3	no	—
90	C₃²×C₉	—	19	3	no	—
91	C₃²×C₉	C₃×C₁₉⋊C₃	1	3	yes	—
92	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
93	C₉⋊C₉	—	57	9	no	—
94	C₃²⋊C₉	—	57	9	no	—
95	C₃×C₁₉	#6891	1	1	yes	characteristic
96	C₃×C₁₉	C₉⋊H(3,𝔽₃)	1	3	yes	—
97	C₃×C₁₉	#6894	1	3	yes	—
98	C₃×C₁₉	#6896	1	3	yes	—
99	C₃×C₁₉	#6899	1	1	yes	characteristic
100	C₃×C₁₉	#6899	1	1	yes	characteristic
101	C₃×C₁₉	#6900	1	1	yes	characteristic
102	C₁₉⋊C₃	—	9	3	no	—
103	C₃³	—	19	3	no	—
104	C₃³	—	57	1	no	—
105	C₃³	—	57	3	no	—
106	C₃³	—	57	1	no	—
107	C₃³	—	57	1	no	—
108	C₃³	—	57	3	no	—
109	C₃³	—	57	3	no	—
110	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
111	3₋¹⁺²	—	57	27	no	—
112	C₃×C₉	—	3	1	no	—
113	C₃×C₉	—	3	1	no	—
114	C₃×C₉	—	3	1	no	—
115	C₃×C₉	—	3	3	no	—
116	C₃×C₉	—	3	3	no	—
117	C₃×C₉	—	3	3	no	—
118	C₃×C₉	—	3	3	no	—
119	C₃×C₉	—	19	3	no	—
120	C₃×C₉	—	19	3	no	—
121	C₃×C₉	—	19	9	no	—
122	C₃×C₉	—	57	3	no	—
123	C₃×C₉	—	57	1	no	—
124	C₃×C₉	—	57	1	no	—
125	C₃×C₉	—	57	1	no	—
126	C₃×C₉	—	57	1	no	—
127	C₃×C₉	—	57	1	no	—
128	C₃×C₉	—	57	1	no	—
129	C₃×C₉	—	57	3	no	—
130	C₃×C₉	—	57	3	no	—
131	C₃×C₉	—	57	3	no	—
132	C₃×C₉	—	57	3	no	—
133	C₃×C₉	—	57	3	no	—
134	C₃×C₉	—	57	3	no	—
135	C₃×C₉	#10587022	1	3	yes	—
136	C₁₉	#10624565	1	1	yes	characteristic, complemented, Sylow
137	C₃²	—	57	3	no	—
138	C₃²	—	57	9	no	—
139	C₃²	—	171	3	no	—
140	C₃²	—	171	3	no	—
141	C₃²	—	171	1	no	—
142	C₃²	—	171	1	no	—
143	C₃²	—	171	1	no	—
144	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
145	C₃²	3₋¹⁺²×C₁₉⋊C₃	1	1	yes	characteristic
146	C₃²	3₋¹⁺²×C₁₉⋊C₃	1	1	yes	characteristic
147	C₃²	3₋¹⁺²×C₁₉⋊C₃	1	3	yes	—
148	C₃²	C₃×#10587022	1	1	yes	characteristic
149	C₃²	#13095332	1	3	yes	—
150	C₃²	#13095333	1	3	yes	—
151	C₉	—	3	3	no	—
152	C₉	—	9	3	no	—
153	C₉	—	57	3	no	—
154	C₉	—	57	3	no	—
155	C₉	—	57	9	no	—
156	C₉	—	171	3	no	—
157	C₃	—	171	3	no	—
158	C₃	#17987420	1	1	yes	characteristic
159	C₃	#17987424	1	3	yes	—
160	C₃	#17987437	1	3	yes	—
161	C₃	#17987449	1	3	yes	—
162	C₃	#17987463	1	1	yes	characteristic
163	C₃	#17987466	1	1	yes	characteristic
164	C₃	#17987472	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#10587022

C₃

yes

maximal, characteristic

C₃×#13095305

C₃

yes

maximal, complemented

C₃×#13095303

C₃

yes

maximal

#17987308

C₃

yes

maximal, complemented

#17987308

C₃

yes

maximal, characteristic, complemented

#17987308

C₃

yes

maximal, characteristic, complemented

C₁₉×#6855

C₃

yes

maximal, characteristic, complemented

C₃³×C₁₉⋊C₃

C₃²

yes

—

—

—

—

C₃²×#4294

C₃²

yes

—

C₃²×#4294

C₃²

yes

characteristic

C₃²×#4294

C₃²

yes

characteristic

C₃²×#4294

C₃²

yes

—

#13095305

—

#13095303

—

C₃²×C₉×C₁₉

C₃²

yes

—

C₃²×C₉×C₁₉

C₃²

yes

characteristic

#10624565

—

maximal, Sylow

C₃³×C₁₉

C₃³

yes

characteristic

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

3₋¹⁺²

yes

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

3₋¹⁺²

yes

—

—

—

—

—

—

—

—

—

—

—

—

—

—

yes

—

C₃×#4294

3₋¹⁺²

yes

—

C₃×#4294

3₋¹⁺²

yes

complemented

#10587022

—

C₃²×3₋¹⁺²

—

C₃×C₉⋊C₉

—

C₃×C₃²⋊C₉

—

—

—

—

C₁₉⋊C₃

yes

characteristic, complemented

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₉

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₁₉

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₁₉

C₃×3₋¹⁺²

yes

—

C₃²×C₁₉

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₁₉

#737

yes

—

C₃²×C₁₉

#737

yes

—

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

—

—

—

—

—

C₃⁴

—

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

C₃×C₁₉⋊C₃

yes

—

C₃²×C₉

C₃×C₁₉⋊C₃

yes

characteristic

C₉⋊C₉

—

C₃²⋊C₉

—

C₃×C₁₉

#6891

yes

characteristic

C₃×C₁₉

C₉⋊H(3,𝔽₃)

yes

—

C₃×C₁₉

#6894

yes

—

C₃×C₁₉

#6896

yes

—

C₃×C₁₉

#6899

yes

characteristic

100

C₃×C₁₉

#6899

yes

characteristic

101

C₃×C₁₉

#6900

yes

characteristic

102

C₁₉⋊C₃

—

103

C₃³

—

104

C₃³

—

105

C₃³

—

106

C₃³

—

107

C₃³

—

108

C₃³

—

109

C₃³

—

110

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

111

3₋¹⁺²

—

112

C₃×C₉

—

113

C₃×C₉

—

114

C₃×C₉

—

115

C₃×C₉

—

116

C₃×C₉

—

117

C₃×C₉

—

118

C₃×C₉

—

119

C₃×C₉

—

120

C₃×C₉

—

121

C₃×C₉

—

122

C₃×C₉

—

123

C₃×C₉

—

124

C₃×C₉

—

125

C₃×C₉

—

126

C₃×C₉

—

127

C₃×C₉

—

128

C₃×C₉

—

129

C₃×C₉

—

130

C₃×C₉

—

131

C₃×C₉

—

132

C₃×C₉

—

133

C₃×C₉

—

134

C₃×C₉

—

135

C₃×C₉

#10587022

yes

—

136

C₁₉

#10624565

yes

characteristic, complemented, Sylow

137

C₃²

—

138

C₃²

—

139

C₃²

—

171

—

140

C₃²

—

171

—

141

C₃²

—

171

—

142

C₃²

—

171

—

143

C₃²

—

171

—

144

C₃²

H(3,𝔽₃)×C₁₉⋊C₃

yes

characteristic

145

C₃²

3₋¹⁺²×C₁₉⋊C₃

yes

characteristic

146

C₃²

3₋¹⁺²×C₁₉⋊C₃

yes

characteristic

147

C₃²

3₋¹⁺²×C₁₉⋊C₃

yes

—

148

C₃²

C₃×#10587022

yes

characteristic

149

C₃²

#13095332

yes

—

150

C₃²

#13095333

yes

—

151

C₉

—

152

C₉

—

153

C₉

—

154

C₉

—

155

C₉

—

156

C₉

—

171

—

157

C₃

—

171

—

158

C₃

#17987420

yes

characteristic

159

C₃

#17987424

yes

—

160

C₃

#17987437

yes

—

161

C₃

#17987449

yes

—

162

C₃

#17987463

yes

characteristic

163

C₃

#17987466

yes

characteristic

164

C₃

#17987472

yes

characteristic

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₁₉⋊C₃
commutator, socle, cosocle	C₃³×C₁₉	C₃³
Fitting	C₁₉×#6855	C₃
prime radical	C₃³×C₁₉⋊C₃	C₃²
hypercenter	#6855	C₁₉⋊C₃
hypocenter	C₁₉	#10624565
LCS	G ▷ C₃³×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃³ ◁ #6855

subgroup

G/H

center, Frattini

C₃³

C₃²×C₁₉⋊C₃

commutator, socle, cosocle

C₃³×C₁₉

C₃³

Fitting

C₁₉×#6855

C₃

prime radical

C₃³×C₁₉⋊C₃

C₃²

C₁₉⋊C₃

C₁₉

G ▷ C₃³×C₁₉ ▷ C₁₉

C₁ ◁ C₃³ ◁ #6855

N	H ≅ G/N	type
2. C₃×#13095305	C₃	semidirect
4. #17987308	C₃	semidirect
5. #17987308	C₃	semidirect
6. #17987308	C₃	semidirect
7. C₁₉×#6855	C₃	semidirect
53. C₃×#4294	3₋¹⁺²	semidirect
61. #6855	C₁₉⋊C₃	semidirect
136. C₁₉	#10624565	semidirect

H ≅ G/N

type

2. C₃×#13095305

C₃

semidirect

4. #17987308

C₃

semidirect

5. #17987308

C₃

semidirect

6. #17987308

C₃

semidirect

7. C₁₉×#6855

C₃

semidirect

53. C₃×#4294

3₋¹⁺²

semidirect

61. #6855

C₁₉⋊C₃

semidirect

136. C₁₉

#10624565

semidirect

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

subgroup	index
C₃²×#10587022	3
C₃×#13095305	3
C₃×#13095303	3
#17987308	3
C₁₉×#6855	3
#10624565	19

subgroup

index

C₁₉×#6855

Subgroups — #28353605

Conjugacy classes of subgroups

Complements

Maximal subgroups