Group #28354531 — subgroups

Subgroups — #28354531

Order 13851.

236 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 49 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	#17987420	C₃	1	1	yes	maximal, characteristic
2	#17987419	C₃	1	1	yes	maximal, characteristic
3	#17987418	C₃	1	1	yes	maximal, characteristic
4	#17987418	C₃	1	1	yes	maximal, characteristic
5	#17987322	C₃	1	2	yes	maximal, complemented
6	#17987322	C₃	1	2	yes	maximal, complemented
7	#17987321	C₃	1	2	yes	maximal, complemented
8	#17987321	C₃	1	2	yes	maximal, complemented
9	C₁₉×#6860	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃³×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
11	C₃×#10587025	C₃²	1	2	yes	—
12	C₃×#10587025	C₃²	1	2	yes	—
13	C₃×#10587022	—	3	1	no	—
14	C₃×#10587023	—	3	1	no	—
15	C₃²⋊#4294	C₃²	1	2	yes	—
16	C₃²⋊#4294	C₃²	1	2	yes	—
17	#13095303	—	3	2	no	—
18	#13095303	—	3	2	no	—
19	#13095303	—	3	1	no	—
20	#13095303	—	3	1	no	—
21	#13095303	—	3	2	no	—
22	#13095303	—	3	2	no	—
23	C₃×C₁₉×3₋¹⁺²	C₃²	1	1	yes	characteristic
24	C₃×C₁₉×3₋¹⁺²	C₃²	1	1	yes	characteristic
25	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
26	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
27	#10624673	—	19	1	no	maximal, Sylow
28	C₃³×C₁₉	C₃³	1	1	yes	characteristic
29	C₃²×C₁₉⋊C₃	—	9	1	no	—
30	C₃²×C₁₉⋊C₃	—	9	1	no	—
31	C₃²×C₁₉⋊C₃	—	9	1	no	—
32	C₃²×C₁₉⋊C₃	—	9	1	no	—
33	C₃²×C₁₉⋊C₃	H(3,𝔽₃)	1	2	yes	—
34	C₃²×C₁₉⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
35	C₁₉×3₋¹⁺²	—	9	1	no	—
36	C₁₉×3₋¹⁺²	—	9	1	no	—
37	C₃×C₉×C₁₉	—	3	1	no	—
38	C₃×C₉×C₁₉	—	3	1	no	—
39	C₃×C₉×C₁₉	—	3	1	no	—
40	C₃×C₉×C₁₉	—	3	1	no	—
41	#10587025	—	9	2	no	—
42	#10587025	—	9	2	no	—
43	C₃×#4294	—	3	2	no	—
44	C₃×#4294	—	3	2	no	—
45	C₃×#4294	—	3	2	no	—
46	C₃×#4294	—	3	2	no	—
47	#10587023	—	9	3	no	—
48	#10587022	—	9	3	no	—
49	#6891	—	19	1	no	—
50	#6891	—	19	1	no	—
51	#6891	—	19	1	no	—
52	#6891	—	19	1	no	—
53	#6860	—	19	2	no	—
54	#6860	—	19	2	no	—
55	#6860	—	19	2	no	—
56	#6860	—	19	2	no	—
57	#6860	C₁₉⋊C₃	1	1	yes	characteristic, complemented
58	C₃²×C₁₉	—	3	1	no	—
59	C₃²×C₁₉	—	3	1	no	—
60	C₃²×C₁₉	—	3	1	no	—
61	C₃²×C₁₉	—	3	1	no	—
62	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
63	C₃×C₁₉⋊C₃	—	3	2	no	—
64	C₃×C₁₉⋊C₃	—	3	1	no	—
65	C₃×C₁₉⋊C₃	—	3	2	no	—
66	C₃×C₁₉⋊C₃	—	3	1	no	—
67	C₃×C₁₉⋊C₃	—	3	1	no	—
68	C₃×C₁₉⋊C₃	—	3	1	no	—
69	C₃×C₁₉⋊C₃	—	3	2	no	—
70	C₃×C₁₉⋊C₃	—	3	2	no	—
71	C₃×C₁₉⋊C₃	—	9	9	no	—
72	C₉×C₁₉	—	9	1	no	—
73	C₉×C₁₉	—	9	1	no	—
74	C₉×C₁₉	—	9	1	no	—
75	C₉×C₁₉	—	9	1	no	—
76	#4294	—	9	2	no	—
77	#4294	—	9	2	no	—
78	#4294	—	9	2	no	—
79	#4294	—	9	2	no	—
80	C₃⁴	—	19	1	no	—
81	C₃×3₋¹⁺²	—	19	2	no	—
82	C₃×3₋¹⁺²	—	19	2	no	—
83	C₃×3₋¹⁺²	—	57	1	no	—
84	C₃×3₋¹⁺²	—	57	1	no	—
85	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
86	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
87	C₃²⋊C₉	—	19	2	no	—
88	C₃²⋊C₉	—	19	2	no	—
89	C₃²⋊C₉	—	57	2	no	—
90	C₃²⋊C₉	—	57	2	no	—
91	C₃²⋊C₉	—	57	1	no	—
92	C₃²⋊C₉	—	57	1	no	—
93	C₃²⋊C₉	—	57	2	no	—
94	C₃²⋊C₉	—	57	2	no	—
95	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
96	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
97	C₃×C₁₉	—	9	1	no	—
98	C₃×C₁₉	#6909	1	1	yes	characteristic
99	C₃×C₁₉	#6909	1	1	yes	characteristic
100	C₃×C₁₉	#6912	1	1	yes	characteristic
101	C₃×C₁₉	#6912	1	1	yes	characteristic
102	C₁₉⋊C₃	—	9	1	no	—
103	C₁₉⋊C₃	—	9	2	no	—
104	C₃³	—	19	2	no	—
105	C₃³	—	19	1	no	—
106	C₃³	—	171	1	no	—
107	C₃³	—	171	1	no	—
108	C₃³	—	171	1	no	—
109	C₃³	—	171	1	no	—
110	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
111	3₋¹⁺²	—	9	1	no	—
112	3₋¹⁺²	—	9	1	no	—
113	3₋¹⁺²	—	171	3	no	—
114	3₋¹⁺²	—	171	3	no	—
115	3₋¹⁺²	—	171	2	no	—
116	3₋¹⁺²	—	171	2	no	—
117	C₃×C₉	—	3	1	no	—
118	C₃×C₉	—	3	1	no	—
119	C₃×C₉	—	3	1	no	—
120	C₃×C₉	—	3	1	no	—
121	C₃×C₉	—	57	2	no	—
122	C₃×C₉	—	57	2	no	—
123	C₃×C₉	—	57	2	no	—
124	C₃×C₉	—	57	2	no	—
125	C₁₉	#10624673	1	1	yes	characteristic, complemented, Sylow
126	C₃²	—	3	1	no	—
127	C₃²	—	3	1	no	—
128	C₃²	—	3	1	no	—
129	C₃²	—	3	1	no	—
130	C₃²	—	57	1	no	—
131	C₃²	—	57	2	no	—
132	C₃²	—	57	1	no	—
133	C₃²	—	57	2	no	—
134	C₃²	—	57	2	no	—
135	C₃²	—	57	1	no	—
136	C₃²	—	57	2	no	—
137	C₃²	—	57	1	no	—
138	C₃²	—	171	9	no	—
139	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
140	C₉	—	9	1	no	—
141	C₉	—	9	1	no	—
142	C₉	—	9	1	no	—
143	C₉	—	9	1	no	—
144	C₉	—	171	2	no	—
145	C₉	—	171	2	no	—
146	C₉	—	171	2	no	—
147	C₉	—	171	2	no	—
148	C₃	—	9	1	no	—
149	C₃	—	171	1	no	—
150	C₃	—	171	2	no	—
151	C₃	#17987502	1	1	yes	characteristic
152	C₃	#17987503	1	1	yes	characteristic
153	C₃	#17987516	1	1	yes	characteristic
154	C₃	#17987517	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17987420

C₃

yes

maximal, characteristic

#17987419

C₃

yes

maximal, characteristic

#17987418

C₃

yes

maximal, characteristic

#17987418

C₃

yes

maximal, characteristic

#17987322

C₃

yes

maximal, complemented

#17987322

C₃

yes

maximal, complemented

#17987321

C₃

yes

maximal, complemented

#17987321

C₃

yes

maximal, complemented

C₁₉×#6860

C₃

yes

maximal, characteristic, complemented

C₃³×C₁₉⋊C₃

C₃²

yes

characteristic

C₃×#10587025

C₃²

yes

—

C₃×#10587025

C₃²

yes

—

C₃×#10587022

—

C₃×#10587023

—

C₃²⋊#4294

C₃²

yes

—

C₃²⋊#4294

C₃²

yes

—

—

—

—

—

—

—

C₃×C₁₉×3₋¹⁺²

C₃²

yes

characteristic

C₃×C₁₉×3₋¹⁺²

C₃²

yes

characteristic

C₁₉×C₃²⋊C₉

C₃²

yes

characteristic

C₁₉×C₃²⋊C₉

C₃²

yes

characteristic

#10624673

—

maximal, Sylow

C₃³×C₁₉

C₃³

yes

characteristic

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

H(3,𝔽₃)

yes

—

C₃²×C₁₉⋊C₃

H(3,𝔽₃)

yes

characteristic

C₁₉×3₋¹⁺²

—

C₁₉×3₋¹⁺²

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

C₁₉⋊C₃

yes

characteristic, complemented

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

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

—

C₉×C₁₉

—

C₉×C₁₉

—

—

—

—

—

C₃⁴

—

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

C₃×3₋¹⁺²

C₃×C₁₉⋊C₃

yes

characteristic

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₁₉⋊C₃

yes

characteristic

C₃²⋊C₉

C₃×C₁₉⋊C₃

yes

characteristic

C₃×C₁₉

—

C₃×C₁₉

yes

C₃×C₁₉

yes

100

C₃×C₁₉

#6912

yes

characteristic

101

C₃×C₁₉

#6912

yes

characteristic

102

C₁₉⋊C₃

—

103

C₁₉⋊C₃

—

104

C₃³

—

105

C₃³

—

106

C₃³

—

171

—

107

C₃³

—

171

—

108

C₃³

—

171

—

109

C₃³

—

171

—

110

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

111

3₋¹⁺²

—

112

3₋¹⁺²

—

113

3₋¹⁺²

—

171

—

114

3₋¹⁺²

—

171

—

115

3₋¹⁺²

—

171

—

116

3₋¹⁺²

—

171

—

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

C₁₉

#10624673

yes

characteristic, complemented, Sylow

126

C₃²

—

127

C₃²

—

128

C₃²

—

129

C₃²

—

130

C₃²

—

131

C₃²

—

132

C₃²

—

133

C₃²

—

134

C₃²

—

135

C₃²

—

136

C₃²

—

137

C₃²

—

138

C₃²

—

171

—

139

C₃²

H(3,𝔽₃)×C₁₉⋊C₃

yes

characteristic

140

C₉

—

141

C₉

—

142

C₉

—

143

C₉

—

144

C₉

—

171

—

145

C₉

—

171

—

146

C₉

—

171

—

147

C₉

—

171

—

148

C₃

—

149

C₃

—

171

—

150

C₃

—

171

—

151

C₃

#17987502

yes

characteristic

152

C₃

#17987503

yes

characteristic

153

C₃

#17987516

yes

characteristic

154

C₃

#17987517

yes

characteristic

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₁₉⋊C₃
commutator, cosocle	C₃³×C₁₉	C₃³
Frattini	C₃³	C₃²×C₁₉⋊C₃
Fitting	C₁₉×#6860	C₃
socle	C₃²×C₁₉	C₃×H(3,𝔽₃)
prime radical	C₃³×C₁₉⋊C₃	C₃²
hypercenter	#6860	C₁₉⋊C₃
hypocenter	C₁₉	#10624673
LCS	G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ #6860

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₁₉⋊C₃

commutator, cosocle

C₃³×C₁₉

C₃³

Frattini

C₃³

C₃²×C₁₉⋊C₃

Fitting

C₁₉×#6860

C₃

socle

C₃²×C₁₉

C₃×H(3,𝔽₃)

prime radical

C₃³×C₁₉⋊C₃

C₃²

C₁₉⋊C₃

C₁₉

G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ #6860

N	H ≅ G/N	type
5. #17987322	C₃	semidirect
6. #17987322	C₃	semidirect
7. #17987321	C₃	semidirect
8. #17987321	C₃	semidirect
9. C₁₉×#6860	C₃	semidirect
57. #6860	C₁₉⋊C₃	semidirect
125. C₁₉	#10624673	semidirect

H ≅ G/N

type

5. #17987322

C₃

semidirect

6. #17987322

C₃

semidirect

7. #17987321

C₃

semidirect

8. #17987321

C₃

semidirect

9. C₁₉×#6860

C₃

semidirect

57. #6860

C₁₉⋊C₃

semidirect

125. C₁₉

#10624673

semidirect

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

subgroup	index
#17987420	3
#17987419	3
#17987418	3
#17987322	3
#17987321	3
C₁₉×#6860	3
#10624673	19

subgroup

index

C₁₉×#6860

Subgroups — #28354531

Conjugacy classes of subgroups

Complements

Maximal subgroups