Group #28353550 — subgroups

Subgroups — #28353550

Order 13851.

608 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 142 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×#10587023	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₉×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic
3	C₃×#13095303	C₃	1	2	yes	maximal
4	#17987308	C₃	1	4	yes	maximal, complemented
5	#17987308	C₃	1	2	yes	maximal, complemented
6	#17987308	C₃	1	2	yes	maximal, complemented
7	C₁₉×#6855	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃³×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
9	C₃×#10587023	—	3	3	no	—
10	C₃×#10587023	—	3	6	no	—
11	C₃×#10587023	C₉	1	9	yes	complemented
12	C₃²×#4294	C₃²	1	4	yes	—
13	C₃²×#4294	C₃²	1	2	yes	—
14	C₃²×#4294	C₃²	1	2	yes	—
15	C₃×C₉×C₁₉⋊C₃	—	3	3	no	—
16	C₃×C₉×C₁₉⋊C₃	—	3	3	no	—
17	C₃×C₉×C₁₉⋊C₃	—	3	6	no	—
18	#13095303	—	3	6	no	—
19	C₃²×C₉×C₁₉	C₃²	1	2	yes	—
20	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
21	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
22	#10624558	—	19	1	no	maximal, Sylow
23	C₃³×C₁₉	C₃³	1	1	yes	characteristic
24	C₃²×C₁₉⋊C₃	—	3	2	no	—
25	C₃²×C₁₉⋊C₃	—	3	1	no	—
26	C₃²×C₁₉⋊C₃	—	3	2	no	—
27	C₃²×C₁₉⋊C₃	—	3	4	no	—
28	C₃²×C₁₉⋊C₃	C₃×C₉	1	3	yes	—
29	C₃²×C₁₉⋊C₃	H(3,𝔽₃)	1	3	yes	—
30	C₃²×C₁₉⋊C₃	3₋¹⁺²	1	6	yes	—
31	C₃×C₉×C₁₉	—	3	1	no	—
32	C₃×C₉×C₁₉	—	3	1	no	—
33	C₃×C₉×C₁₉	—	3	2	no	—
34	C₃×C₉×C₁₉	—	3	2	no	—
35	C₃×C₉×C₁₉	—	3	2	no	—
36	C₃×C₉×C₁₉	—	3	2	no	—
37	C₃×C₉×C₁₉	—	3	4	no	—
38	C₃×C₉×C₁₉	C₃×C₉	1	3	yes	complemented
39	C₃×C₉×C₁₉	3₋¹⁺²	1	3	yes	complemented
40	C₃×#4294	—	3	2	no	—
41	C₃×#4294	—	3	2	no	—
42	C₃×#4294	—	3	2	no	—
43	C₃×#4294	—	3	2	no	—
44	C₃×#4294	—	3	4	no	—
45	C₃×#4294	—	3	4	no	—
46	C₃×#4294	—	3	4	no	—
47	C₃×#4294	—	3	8	no	—
48	C₃×#4294	C₃×C₉	1	6	yes	complemented
49	C₃×#4294	3₋¹⁺²	1	6	yes	complemented
50	#10587023	—	3	27	no	—
51	C₉×C₁₉⋊C₃	—	3	27	no	—
52	C₃²×3₋¹⁺²	—	19	1	no	—
53	C₃³×C₉	—	19	1	no	—
54	C₃×C₃²⋊C₉	—	19	2	no	—
55	#6855	—	19	4	no	—
56	#6855	—	19	2	no	—
57	#6855	—	19	2	no	—
58	#6855	C₁₉⋊C₃	1	1	yes	characteristic, complemented
59	C₃²×C₁₉	C₃²×C₉	1	1	yes	characteristic
60	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
61	C₃²×C₁₉	C₃×3₋¹⁺²	1	2	yes	—
62	C₃²×C₁₉	C₃×3₋¹⁺²	1	2	yes	—
63	C₃²×C₁₉	C₃×3₋¹⁺²	1	1	yes	characteristic
64	C₃²×C₁₉	#737	1	2	yes	—
65	C₃²×C₁₉	#737	1	4	yes	—
66	C₃×C₁₉⋊C₃	—	3	6	no	—
67	C₃×C₁₉⋊C₃	—	3	3	no	—
68	C₃×C₁₉⋊C₃	—	3	3	no	—
69	C₃×C₁₉⋊C₃	—	3	6	no	—
70	C₃×C₁₉⋊C₃	—	3	6	no	—
71	C₃×C₁₉⋊C₃	—	3	12	no	—
72	C₃×C₁₉⋊C₃	C₃²⋊C₉	1	9	yes	—
73	C₉×C₁₉	—	3	3	no	—
74	C₉×C₁₉	—	3	3	no	—
75	C₉×C₁₉	—	9	2	no	—
76	#4294	—	3	6	no	—
77	#4294	—	3	6	no	—
78	#4294	—	9	4	no	—
79	C₃⁴	—	19	1	no	—
80	C₃×3₋¹⁺²	—	19	9	no	—
81	C₃×3₋¹⁺²	—	57	3	no	—
82	C₃×3₋¹⁺²	—	57	6	no	—
83	C₃²×C₉	—	19	2	no	—
84	C₃²×C₉	—	19	2	no	—
85	C₃²×C₉	—	19	4	no	—
86	C₃²×C₉	—	57	6	no	—
87	C₃²×C₉	—	57	3	no	—
88	C₃²×C₉	—	57	3	no	—
89	C₃²×C₉	C₃×C₁₉⋊C₃	1	2	yes	—
90	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
91	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
92	C₃²⋊C₉	—	57	6	no	—
93	C₃×C₁₉	C₃×C₃²⋊C₉	1	1	yes	characteristic
94	C₃×C₁₉	C₉×3₋¹⁺²	1	2	yes	—
95	C₃×C₁₉	C₉×3₋¹⁺²	1	1	yes	characteristic
96	C₃×C₁₉	#6891	1	1	yes	characteristic
97	C₃×C₁₉	#6893	1	2	yes	—
98	C₃×C₁₉	#6894	1	2	yes	—
99	C₃×C₁₉	#6894	1	4	yes	—
100	C₁₉⋊C₃	—	3	9	no	—
101	C₃³	—	19	3	no	—
102	C₃³	—	19	3	no	—
103	C₃³	—	19	6	no	—
104	C₃³	—	57	2	no	—
105	C₃³	—	57	1	no	—
106	C₃³	—	57	2	no	—
107	C₃³	—	57	4	no	—
108	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
109	3₋¹⁺²	—	57	27	no	—
110	C₃×C₉	—	3	1	no	—
111	C₃×C₉	—	3	1	no	—
112	C₃×C₉	—	3	2	no	—
113	C₃×C₉	—	3	2	no	—
114	C₃×C₉	—	3	2	no	—
115	C₃×C₉	—	3	2	no	—
116	C₃×C₉	—	3	4	no	—
117	C₃×C₉	—	19	6	no	—
118	C₃×C₉	—	19	6	no	—
119	C₃×C₉	—	57	27	no	—
120	C₃×C₉	—	57	2	no	—
121	C₃×C₉	—	57	2	no	—
122	C₃×C₉	—	57	2	no	—
123	C₃×C₉	—	57	4	no	—
124	C₃×C₉	—	57	2	no	—
125	C₃×C₉	—	57	4	no	—
126	C₃×C₉	—	57	4	no	—
127	C₃×C₉	—	57	8	no	—
128	C₃×C₉	C₉×C₁₉⋊C₃	1	3	yes	complemented
129	C₃×C₉	#10587023	1	3	yes	complemented
130	C₁₉	#10624558	1	1	yes	characteristic, complemented, Sylow
131	C₃²	—	19	9	no	—
132	C₃²	—	57	6	no	—
133	C₃²	—	57	3	no	—
134	C₃²	—	57	3	no	—
135	C₃²	—	57	6	no	—
136	C₃²	—	57	6	no	—
137	C₃²	—	57	12	no	—
138	C₃²	C₃×C₉×C₁₉⋊C₃	1	1	yes	characteristic
139	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
140	C₃²	3₋¹⁺²×C₁₉⋊C₃	1	2	yes	—
141	C₃²	3₋¹⁺²×C₁₉⋊C₃	1	2	yes	—
142	C₃²	C₃×#10587023	1	1	yes	characteristic
143	C₃²	#13095332	1	2	yes	—
144	C₃²	#13095333	1	4	yes	—
145	C₉	—	3	3	no	—
146	C₉	—	3	3	no	—
147	C₉	—	9	2	no	—
148	C₉	—	57	6	no	—
149	C₉	—	57	6	no	—
150	C₉	—	171	4	no	—
151	C₃	—	57	9	no	—
152	C₃	C₁₉⋊C₃×C₃²⋊C₉	1	1	yes	characteristic
153	C₃	#17987409	1	2	yes	—
154	C₃	C₉×#10587023	1	1	yes	characteristic
155	C₃	#17987419	1	1	yes	characteristic
156	C₃	#17987430	1	2	yes	—
157	C₃	#17987433	1	2	yes	—
158	C₃	#17987436	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#10587023

C₃

yes

maximal, characteristic

C₃²×C₉×C₁₉⋊C₃

C₃

yes

maximal, characteristic

C₃×#13095303

C₃

yes

maximal

#17987308

C₃

yes

maximal, complemented

#17987308

C₃

yes

maximal, complemented

#17987308

C₃

yes

maximal, complemented

C₁₉×#6855

C₃

yes

maximal, characteristic, complemented

C₃³×C₁₉⋊C₃

C₃²

yes

—

—

C₉

yes

complemented

C₃²×#4294

C₃²

yes

—

C₃²×#4294

C₃²

yes

—

C₃²×#4294

C₃²

yes

—

C₃×C₉×C₁₉⋊C₃

—

C₃×C₉×C₁₉⋊C₃

—

C₃×C₉×C₁₉⋊C₃

—

#13095303

—

C₃²×C₉×C₁₉

C₃²

yes

—

C₃²×C₉×C₁₉

C₃²

yes

characteristic

C₃²×C₉×C₁₉

C₃²

yes

characteristic

#10624558

—

maximal, Sylow

C₃³×C₁₉

C₃³

yes

characteristic

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

C₃×C₉

yes

—

C₃²×C₁₉⋊C₃

H(3,𝔽₃)

yes

—

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

C₃×C₉

yes

complemented

C₃×C₉×C₁₉

3₋¹⁺²

yes

—

—

—

—

—

—

—

—

C₃×C₉

yes

complemented

C₃×#4294

3₋¹⁺²

yes

complemented

#10587023

—

C₉×C₁₉⋊C₃

—

C₃²×3₋¹⁺²

—

C₃³×C₉

—

C₃×C₃²⋊C₉

—

—

—

—

C₁₉⋊C₃

yes

characteristic, complemented

C₃²×C₁₉

C₃²×C₉

yes

characteristic

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₉

C₃×3₋¹⁺²

yes

—

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

C₃²⋊C₉

yes

—

C₉×C₁₉

—

C₉×C₁₉

—

C₉×C₁₉

—

#4294

—

#4294

—

#4294

—

C₃⁴

—

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

C₃×C₁₉⋊C₃

yes

—

C₃²×C₉

C₃×C₁₉⋊C₃

yes

characteristic

C₃²×C₉

C₃×C₁₉⋊C₃

yes

characteristic

C₃²⋊C₉

—

C₃×C₁₉

C₃×C₃²⋊C₉

yes

characteristic

C₃×C₁₉

C₉×3₋¹⁺²

yes

—

C₃×C₁₉

C₉×3₋¹⁺²

yes

C₃×C₁₉

yes

C₃×C₁₉

yes

—

C₃×C₁₉

#6894

yes

—

C₃×C₁₉

#6894

yes

—

100

C₁₉⋊C₃

—

101

C₃³

—

102

C₃³

—

103

C₃³

—

104

C₃³

—

105

C₃³

—

106

C₃³

—

107

C₃³

—

108

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

109

3₋¹⁺²

—

110

C₃×C₉

—

111

C₃×C₉

—

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

C₉×C₁₉⋊C₃

yes

complemented

129

C₃×C₉

#10587023

yes

complemented

130

C₁₉

#10624558

yes

characteristic, complemented, Sylow

131

C₃²

—

132

C₃²

—

133

C₃²

—

134

C₃²

—

135

C₃²

—

136

C₃²

—

137

C₃²

—

138

C₃²

C₃×C₉×C₁₉⋊C₃

yes

characteristic

139

C₃²

H(3,𝔽₃)×C₁₉⋊C₃

yes

characteristic

140

C₃²

3₋¹⁺²×C₁₉⋊C₃

yes

—

141

C₃²

3₋¹⁺²×C₁₉⋊C₃

yes

—

142

C₃²

C₃×#10587023

yes

characteristic

143

C₃²

#13095332

yes

—

144

C₃²

#13095333

yes

—

145

C₉

—

146

C₉

—

147

C₉

—

148

C₉

—

149

C₉

—

150

C₉

—

171

—

151

C₃

—

152

C₃

C₁₉⋊C₃×C₃²⋊C₉

yes

characteristic

153

C₃

#17987409

yes

—

154

C₃

C₉×#10587023

yes

characteristic

155

C₃

#17987419

yes

characteristic

156

C₃

#17987430

yes

—

157

C₃

#17987433

yes

—

158

C₃

#17987436

yes

—

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₁₉⋊C₃
commutator	C₃²×C₁₉	C₃²×C₉
Fitting	C₁₉×#6855	C₃
socle, cosocle	C₃³×C₁₉	C₃³
prime radical	C₃³×C₁₉⋊C₃	C₃²
hypercenter	#6855	C₁₉⋊C₃
hypocenter	C₁₉	#10624558
LCS	G ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃³ ◁ #6855

subgroup

G/H

center, Frattini

C₃³

C₃²×C₁₉⋊C₃

commutator

C₃²×C₁₉

C₃²×C₉

Fitting

C₁₉×#6855

C₃

socle, cosocle

C₃³×C₁₉

C₃³

prime radical

C₃³×C₁₉⋊C₃

C₃²

C₁₉⋊C₃

C₁₉

G ▷ C₃²×C₁₉ ▷ C₁₉

C₁ ◁ C₃³ ◁ #6855

N	H ≅ G/N	type
4. #17987308	C₃	semidirect
5. #17987308	C₃	semidirect
6. #17987308	C₃	semidirect
7. C₁₉×#6855	C₃	semidirect
11. C₃×#10587023	C₉	semidirect
38. C₃×C₉×C₁₉	C₃×C₉	semidirect
39. C₃×C₉×C₁₉	3₋¹⁺²	semidirect
48. C₃×#4294	C₃×C₉	semidirect
49. C₃×#4294	3₋¹⁺²	semidirect
58. #6855	C₁₉⋊C₃	semidirect
128. C₃×C₉	C₉×C₁₉⋊C₃	semidirect
129. C₃×C₉	#10587023	semidirect
130. C₁₉	#10624558	semidirect

H ≅ G/N

type

4. #17987308

C₃

semidirect

5. #17987308

C₃

semidirect

6. #17987308

C₃

semidirect

7. C₁₉×#6855

C₃

semidirect

11. C₃×#10587023

C₉

semidirect

38. C₃×C₉×C₁₉

C₃×C₉

semidirect

39. C₃×C₉×C₁₉

3₋¹⁺²

semidirect

48. C₃×#4294

C₃×C₉

semidirect

49. C₃×#4294

3₋¹⁺²

semidirect

58. #6855

C₁₉⋊C₃

semidirect

128. C₃×C₉

C₉×C₁₉⋊C₃

semidirect

129. C₃×C₉

#10587023

semidirect

130. C₁₉

#10624558

semidirect

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

subgroup	index
C₃²×#10587023	3
C₃²×C₉×C₁₉⋊C₃	3
C₃×#13095303	3
#17987308	3
C₁₉×#6855	3
#10624558	19

subgroup

index

C₃²×C₉×C₁₉⋊C₃

C₁₉×#6855

Subgroups — #28353550

Conjugacy classes of subgroups

Complements

Maximal subgroups