Group #28356595 — subgroups

Subgroups — #28356595

Order 13851.

960 conjugacy classes of subgroups; only the 268 normal subgroups are shown.

normal subgroups only

This group has 960 conjugacy classes of subgroups — too many to draw. Only its 268 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×#13095333	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#13095331	C₃	1	3	yes	maximal, complemented
3	#17987525	C₃	1	9	yes	maximal, complemented
4	#17987524	C₃	1	9	yes	maximal, complemented
5	#17987519	C₃	1	2	yes	maximal, complemented
6	#17987509	C₃	1	3	yes	maximal, complemented
7	#17987504	C₃	1	3	yes	maximal, complemented
8	#17987498	C₃	1	9	yes	maximal, complemented
9	C₁₉×#6911	C₃	1	1	yes	maximal, characteristic, complemented
10	#13095333	—	3	3	no	—
11	#13095333	—	3	6	no	—
12	#13095331	—	3	6	no	—
13	#13095331	—	3	6	no	—
14	#13095331	—	3	6	no	—
15	#13095331	C₃²	1	9	yes	complemented
16	#13095331	C₃²	1	9	yes	complemented
17	#13095331	C₃²	1	9	yes	complemented
18	C₃×#10587025	—	3	6	no	—
19	C₃×#10587025	C₃²	1	6	yes	—
20	C₃×#10587025	C₃²	1	1	yes	characteristic
21	C₃×#10587022	C₃²	1	3	yes	complemented
22	C₃×#10587023	C₃²	1	3	yes	complemented
23	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	1	yes	characteristic
24	C₃²×#4294	—	3	3	no	—
25	C₃²×#4294	C₃²	1	1	yes	characteristic
26	C₃×C₉×C₁₉⋊C₃	C₃²	1	3	yes	complemented
27	#13095322	C₃²	1	18	yes	complemented
28	#13095316	C₃²	1	27	yes	complemented
29	#13095313	C₃²	1	27	yes	complemented
30	C₃×C₁₉×3₋¹⁺²	C₃²	1	1	yes	characteristic
31	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	3	yes	—
32	C₁₉×#732	C₃²	1	9	yes	—
33	#10624929	—	19	1	no	maximal, Sylow
34	C₃³×C₁₉	—	3	3	no	—
35	C₃³×C₁₉	C₃³	1	1	yes	characteristic
36	C₃²×C₁₉⋊C₃	C₃³	1	3	yes	—
37	C₁₉×3₋¹⁺²	—	3	3	no	—
38	C₁₉×H(3,𝔽₃)	—	3	6	no	—
39	C₁₉×H(3,𝔽₃)	C₃³	1	9	yes	—
40	C₃×C₉×C₁₉	C₃³	1	3	yes	—
41	C₃²⋊(C₁₉⋊C₃)	—	3	3	no	—
42	#10587025	—	3	12	no	—
43	#10587025	—	3	3	no	—
44	#10587025	—	3	18	no	—
45	#10587025	—	3	18	no	—
46	#10587025	—	3	18	no	—
47	#10587025	C₃³	1	18	yes	—
48	C₃×#4294	—	3	6	no	—
49	C₃×#4294	—	3	9	no	—
50	C₃×#4294	—	3	9	no	—
51	C₃×#4294	—	3	9	no	—
52	C₃×#4294	H(3,𝔽₃)	1	3	yes	—
53	C₃×#4294	H(3,𝔽₃)	1	6	yes	—
54	C₃×#4294	C₃³	1	3	yes	—
55	C₃×#4294	C₃³	1	1	yes	characteristic
56	C₃×#4294	C₃³	1	1	yes	characteristic
57	C₃×#4294	C₃³	1	1	yes	characteristic
58	#10587023	—	3	9	no	—
59	#10587022	—	3	9	no	—
60	C₉×C₁₉⋊C₃	—	3	3	no	—
61	C₉×C₁₉⋊C₃	—	3	6	no	—
62	C₃×#737	—	19	1	no	—
63	C₃×#737	—	19	3	no	—
64	#6913	—	19	9	no	—
65	#6913	—	19	9	no	—
66	#6912	—	19	2	no	—
67	#6911	C₁₉⋊C₃	1	1	yes	characteristic, complemented
68	#6910	—	19	3	no	—
69	#6909	—	19	3	no	—
70	#6908	—	19	9	no	—
71	C₃²×C₁₉	—	3	9	no	—
72	C₃²×C₁₉	—	9	1	no	—
73	C₃²×C₁₉	—	9	9	no	—
74	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	3	yes	—
75	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
76	C₃×C₁₉⋊C₃	—	3	3	no	—
77	C₃×C₁₉⋊C₃	—	9	3	no	—
78	C₉×C₁₉	—	3	3	no	—
79	#4294	—	3	3	no	—
80	#4294	—	3	18	no	—
81	#4294	C₃×H(3,𝔽₃)	1	1	yes	characteristic
82	#4294	C₃×H(3,𝔽₃)	1	2	yes	—
83	#4294	C₃×H(3,𝔽₃)	1	3	yes	—
84	#4294	C₃×H(3,𝔽₃)	1	3	yes	—
85	#737	—	19	9	no	—
86	#737	—	19	9	no	—
87	#737	—	19	9	no	—
88	#737	—	57	6	no	—
89	#737	—	57	6	no	—
90	#737	—	57	6	no	—
91	#737	—	57	3	no	—
92	#737	—	57	6	no	—
93	C₃×3₋¹⁺²	—	19	1	no	—
94	C₃×3₋¹⁺²	—	19	3	no	—
95	C₃×3₋¹⁺²	—	19	3	no	—
96	C₃×3₋¹⁺²	—	19	6	no	—
97	C₃×3₋¹⁺²	—	57	6	no	—
98	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
99	C₃×H(3,𝔽₃)	—	19	1	no	—
100	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	3	yes	—
101	C₃²×C₉	—	19	1	no	—
102	C₃²×C₉	—	19	3	no	—
103	C₃²×C₉	—	57	3	no	—
104	#733	—	19	18	no	—
105	#732	C₃×C₁₉⋊C₃	1	9	yes	—
106	#731	—	19	27	no	—
107	AΣL(1,𝔽₂₇)	—	19	27	no	—
108	C₃×C₁₉	—	3	1	no	—
109	C₃×C₁₉	—	3	3	no	—
110	C₃×C₁₉	—	9	9	no	—
111	C₃×C₁₉	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
112	C₁₉⋊C₃	—	9	3	no	—
113	C₃³	—	3	3	no	—
114	C₃³	—	19	3	no	—
115	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
116	3₋¹⁺²	—	3	3	no	—
117	3₋¹⁺²	—	19	18	no	—
118	3₋¹⁺²	—	57	9	no	—
119	3₋¹⁺²	—	57	9	no	—
120	3₋¹⁺²	—	57	18	no	—
121	3₋¹⁺²	—	57	18	no	—
122	3₋¹⁺²	—	57	18	no	—
123	3₋¹⁺²	—	57	3	no	—
124	3₋¹⁺²	—	57	12	no	—
125	H(3,𝔽₃)	—	3	6	no	—
126	H(3,𝔽₃)	—	57	3	no	—
127	H(3,𝔽₃)	C₃²×C₁₉⋊C₃	1	9	yes	—
128	C₃×C₉	—	19	3	no	—
129	C₃×C₉	—	19	6	no	—
130	C₃×C₉	—	19	1	no	—
131	C₃×C₉	—	19	1	no	—
132	C₃×C₉	—	19	1	no	—
133	C₃×C₉	—	19	3	no	—
134	C₃×C₉	—	57	3	no	—
135	C₃×C₉	—	57	6	no	—
136	C₃×C₉	—	57	9	no	—
137	C₃×C₉	—	57	9	no	—
138	C₃×C₉	—	57	9	no	—
139	C₃×C₉	—	57	6	no	—
140	C₃×C₉	C₃²×C₁₉⋊C₃	1	3	yes	—
141	C₁₉	#10624929	1	1	yes	characteristic, complemented, Sylow
142	C₃²	—	3	9	no	—
143	C₃²	—	9	1	no	—
144	C₃²	—	9	9	no	—
145	C₃²	—	57	3	no	—
146	C₃²	—	171	3	no	—
147	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	3	yes	—
148	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
149	C₉	—	3	3	no	—
150	C₉	—	19	1	no	—
151	C₉	—	19	3	no	—
152	C₉	—	19	3	no	—
153	C₉	—	19	2	no	—
154	C₉	—	57	3	no	—
155	C₉	—	57	18	no	—
156	C₃	—	3	3	no	—
157	C₃	—	3	1	no	—
158	C₃	—	9	9	no	—
159	C₃	—	171	3	no	—
160	C₃	C₃×H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#13095333

C₃

yes

maximal, characteristic, complemented

C₃×#13095331

C₃

yes

maximal, complemented

#17987525

C₃

yes

maximal, complemented

#17987524

C₃

yes

maximal, complemented

#17987519

C₃

yes

maximal, complemented

#17987509

C₃

yes

maximal, complemented

#17987504

C₃

yes

maximal, complemented

#17987498

C₃

yes

maximal, complemented

C₁₉×#6911

C₃

yes

maximal, characteristic, complemented

—

—

—

—

—

C₃²

yes

complemented

#13095331

C₃²

yes

complemented

#13095331

C₃²

yes

complemented

C₃×#10587025

—

C₃×#10587025

C₃²

yes

—

C₃×#10587025

C₃²

yes

characteristic

C₃×#10587022

C₃²

yes

complemented

C₃×#10587023

C₃²

yes

complemented

C₃×C₃²⋊(C₁₉⋊C₃)

C₃²

yes

characteristic

C₃²×#4294

—

C₃²×#4294

C₃²

yes

characteristic

C₃×C₉×C₁₉⋊C₃

C₃²

yes

complemented

#13095322

C₃²

yes

complemented

#13095316

C₃²

yes

complemented

#13095313

C₃²

yes

complemented

C₃×C₁₉×3₋¹⁺²

C₃²

yes

characteristic

C₃×C₁₉×H(3,𝔽₃)

C₃²

yes

—

C₁₉×#732

C₃²

yes

—

#10624929

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

C₃³

yes

characteristic

C₃²×C₁₉⋊C₃

C₃³

yes

—

C₁₉×3₋¹⁺²

—

C₁₉×H(3,𝔽₃)

—

C₁₉×H(3,𝔽₃)

C₃³

yes

—

C₃×C₉×C₁₉

C₃³

yes

—

C₃²⋊(C₁₉⋊C₃)

—

—

—

—

—

—

C₃³

yes

—

—

—

—

—

yes

—

C₃×#4294

H(3,𝔽₃)

yes

—

C₃×#4294

C₃³

yes

—

C₃×#4294

C₃³

yes

characteristic

C₃×#4294

C₃³

yes

characteristic

C₃×#4294

C₃³

yes

characteristic

#10587023

—

#10587022

—

C₉×C₁₉⋊C₃

—

C₉×C₁₉⋊C₃

—

—

—

—

—

—

C₁₉⋊C₃

yes

characteristic, complemented

#6910

—

#6909

—

#6908

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

—

C₃²×C₁₉

C₃⁴

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₉×C₁₉

—

#4294

—

#4294

—

#4294

C₃×H(3,𝔽₃)

yes

characteristic

#4294

C₃×H(3,𝔽₃)

yes

—

#4294

C₃×H(3,𝔽₃)

yes

—

#4294

C₃×H(3,𝔽₃)

yes

—

—

—

—

—

—

—

—

—

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×C₁₉⋊C₃

yes

characteristic

C₃×H(3,𝔽₃)

—

100

C₃×H(3,𝔽₃)

C₃×C₁₉⋊C₃

yes

—

101

C₃²×C₉

—

102

C₃²×C₉

—

103

C₃²×C₉

—

104

#733

—

105

#732

C₃×C₁₉⋊C₃

yes

—

106

#731

—

107

AΣL(1,𝔽₂₇)

—

108

C₃×C₁₉

—

109

C₃×C₁₉

—

110

C₃×C₁₉

—

111

C₃×C₁₉

C₃²×H(3,𝔽₃)

yes

characteristic

112

C₁₉⋊C₃

—

113

C₃³

—

114

C₃³

—

115

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

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

H(3,𝔽₃)

—

126

H(3,𝔽₃)

—

127

H(3,𝔽₃)

C₃²×C₁₉⋊C₃

yes

—

128

C₃×C₉

—

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

C₃²×C₁₉⋊C₃

yes

—

141

C₁₉

#10624929

yes

characteristic, complemented, Sylow

142

C₃²

—

143

C₃²

—

144

C₃²

—

145

C₃²

—

146

C₃²

—

171

—

147

C₃²

H(3,𝔽₃)×C₁₉⋊C₃

yes

—

148

C₃²

C₃³×C₁₉⋊C₃

yes

characteristic

149

C₉

—

150

C₉

—

151

C₉

—

152

C₉

—

153

C₉

—

154

C₉

—

155

C₉

—

156

C₃

—

157

C₃

—

158

C₃

—

159

C₃

—

171

—

160

C₃

C₃×H(3,𝔽₃)×C₁₉⋊C₃

yes

characteristic

	subgroup	G/H
center	C₃	C₃×H(3,𝔽₃)×C₁₉⋊C₃
commutator, cosocle	C₃²×C₁₉	C₃⁴
Frattini	C₃²	C₃³×C₁₉⋊C₃
Fitting	C₁₉×#6911	C₃
socle	C₃×C₁₉	C₃²×H(3,𝔽₃)
hypercenter	#6911	C₁₉⋊C₃
hypocenter	C₁₉	#10624929
LCS	G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ #6911

subgroup

G/H

center

C₃

C₃×H(3,𝔽₃)×C₁₉⋊C₃

commutator, cosocle

C₃²×C₁₉

C₃⁴

Frattini

C₃²

C₃³×C₁₉⋊C₃

Fitting

C₁₉×#6911

C₃

socle

C₃×C₁₉

C₃²×H(3,𝔽₃)

C₁₉⋊C₃

C₁₉

G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃ ◁ C₃³ ◁ #6911

N	H ≅ G/N	type
1. C₃×#13095333	C₃	semidirect
2. C₃×#13095331	C₃	semidirect
3. #17987525	C₃	semidirect
4. #17987524	C₃	semidirect
5. #17987519	C₃	semidirect
6. #17987509	C₃	semidirect
7. #17987504	C₃	semidirect
8. #17987498	C₃	semidirect
9. C₁₉×#6911	C₃	semidirect
15. #13095331	C₃²	semidirect
16. #13095331	C₃²	semidirect
17. #13095331	C₃²	semidirect
21. C₃×#10587022	C₃²	semidirect
22. C₃×#10587023	C₃²	semidirect
26. C₃×C₉×C₁₉⋊C₃	C₃²	semidirect
27. #13095322	C₃²	semidirect
28. #13095316	C₃²	semidirect
29. #13095313	C₃²	semidirect
67. #6911	C₁₉⋊C₃	semidirect
141. C₁₉	#10624929	semidirect

H ≅ G/N

type

1. C₃×#13095333

C₃

semidirect

2. C₃×#13095331

C₃

semidirect

3. #17987525

C₃

semidirect

4. #17987524

C₃

semidirect

5. #17987519

C₃

semidirect

6. #17987509

C₃

semidirect

7. #17987504

C₃

semidirect

8. #17987498

C₃

semidirect

9. C₁₉×#6911

C₃

semidirect

15. #13095331

C₃²

semidirect

16. #13095331

C₃²

semidirect

17. #13095331

C₃²

semidirect

21. C₃×#10587022

C₃²

semidirect

22. C₃×#10587023

C₃²

semidirect

26. C₃×C₉×C₁₉⋊C₃

C₃²

semidirect

27. #13095322

C₃²

semidirect

28. #13095316

C₃²

semidirect

29. #13095313

C₃²

semidirect

67. #6911

C₁₉⋊C₃

semidirect

141. C₁₉

#10624929

semidirect

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

subgroup	index
C₃×#13095333	3
C₃×#13095331	3
#17987525	3
#17987524	3
#17987519	3
#17987509	3
#17987504	3
#17987498	3
C₁₉×#6911	3
#10624929	19

subgroup

index

C₁₉×#6911

Subgroups — #28356595

Conjugacy classes of subgroups

Complements

Maximal subgroups