Group #28353508 — subgroups

Subgroups — #28353508

Order 13851.

554 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 148 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×C₉×C₁₉⋊C₃	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	2	yes	maximal, complemented
6	C₁₉×#6855	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃³×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
8	C₃²×#4294	C₃²	1	6	yes	—
9	C₃²×#4294	C₃²	1	1	yes	characteristic
10	C₃²×#4294	C₃²	1	1	yes	characteristic
11	C₃×C₉×C₁₉⋊C₃	—	3	3	no	—
12	C₃×C₉×C₁₉⋊C₃	—	3	3	no	—
13	C₃×C₉×C₁₉⋊C₃	—	3	3	no	—
14	C₃×C₉×C₁₉⋊C₃	C₉	1	9	yes	complemented
15	#13095305	—	3	9	no	—
16	#13095303	—	3	9	no	—
17	C₃²×C₉×C₁₉	C₃²	1	3	yes	—
18	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
19	#10624548	—	19	1	no	maximal, Sylow
20	C₃³×C₁₉	C₃³	1	1	yes	characteristic
21	C₃²×C₁₉⋊C₃	—	3	1	no	—
22	C₃²×C₁₉⋊C₃	—	3	1	no	—
23	C₃²×C₁₉⋊C₃	—	3	1	no	—
24	C₃²×C₁₉⋊C₃	—	3	3	no	—
25	C₃²×C₁₉⋊C₃	—	3	3	no	—
26	C₃²×C₁₉⋊C₃	C₃×C₉	1	3	yes	—
27	C₃²×C₁₉⋊C₃	3₋¹⁺²	1	9	yes	—
28	C₃×C₉×C₁₉	—	3	1	no	—
29	C₃×C₉×C₁₉	—	3	1	no	—
30	C₃×C₉×C₁₉	—	3	1	no	—
31	C₃×C₉×C₁₉	—	3	3	no	—
32	C₃×C₉×C₁₉	—	3	3	no	—
33	C₃×C₉×C₁₉	—	3	3	no	—
34	C₃×C₉×C₁₉	—	3	3	no	—
35	C₃×C₉×C₁₉	C₃×C₉	1	3	yes	—
36	C₃×#4294	—	3	1	no	—
37	C₃×#4294	—	3	1	no	—
38	C₃×#4294	—	3	1	no	—
39	C₃×#4294	—	3	6	no	—
40	C₃×#4294	—	3	6	no	—
41	C₃×#4294	—	3	6	no	—
42	C₃×#4294	—	3	6	no	—
43	C₃×#4294	C₃×C₉	1	3	yes	—
44	C₃×#4294	C₃×C₉	1	3	yes	—
45	C₃×#4294	H(3,𝔽₃)	1	3	yes	—
46	C₃×#4294	3₋¹⁺²	1	3	yes	—
47	C₃×#4294	3₋¹⁺²	1	3	yes	—
48	C₉×C₁₉⋊C₃	—	3	27	no	—
49	C₃³×C₉	—	19	1	no	—
50	C₃×C₉⋊C₉	—	19	3	no	—
51	C₃×C₃²⋊C₉	—	19	3	no	—
52	#6855	—	19	3	no	—
53	#6855	—	19	2	no	—
54	#6855	C₁₉⋊C₃	1	1	yes	characteristic, complemented
55	C₃²×C₁₉	C₃²×C₉	1	1	yes	characteristic
56	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
57	C₃²×C₁₉	C₃×3₋¹⁺²	1	1	yes	characteristic
58	C₃²×C₁₉	C₃×3₋¹⁺²	1	1	yes	characteristic
59	C₃²×C₁₉	C₃×3₋¹⁺²	1	3	yes	—
60	C₃²×C₁₉	#737	1	3	yes	—
61	C₃²×C₁₉	#737	1	3	yes	—
62	C₃×C₁₉⋊C₃	—	3	3	no	—
63	C₃×C₁₉⋊C₃	—	3	3	no	—
64	C₃×C₁₉⋊C₃	—	3	3	no	—
65	C₃×C₁₉⋊C₃	—	3	9	no	—
66	C₃×C₁₉⋊C₃	—	3	9	no	—
67	C₃×C₁₉⋊C₃	—	3	9	no	—
68	C₃×C₁₉⋊C₃	C₉⋊C₉	1	9	yes	complemented
69	C₉×C₁₉	—	3	3	no	—
70	C₉×C₁₉	—	9	3	no	—
71	#4294	—	3	3	no	—
72	#4294	—	9	6	no	—
73	#4294	C₃²⋊C₉	1	9	yes	complemented
74	C₃⁴	—	19	1	no	—
75	C₃²×C₉	—	19	1	no	—
76	C₃²×C₉	—	19	1	no	—
77	C₃²×C₉	—	19	9	no	—
78	C₃²×C₉	—	19	6	no	—
79	C₃²×C₉	—	57	3	no	—
80	C₃²×C₉	—	57	3	no	—
81	C₃²×C₉	—	57	3	no	—
82	C₃²×C₉	C₃×C₁₉⋊C₃	1	3	yes	—
83	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
84	C₉⋊C₉	—	57	9	no	—
85	C₃²⋊C₉	—	57	9	no	—
86	C₃×C₁₉	C₃×C₃²⋊C₉	1	1	yes	characteristic
87	C₃×C₁₉	C₃×C₉⋊C₉	1	1	yes	characteristic
88	C₃×C₁₉	#6887	1	1	yes	characteristic
89	C₃×C₁₉	#6887	1	1	yes	characteristic
90	C₃×C₁₉	C₉⋊H(3,𝔽₃)	1	3	yes	—
91	C₃×C₁₉	#6894	1	3	yes	—
92	C₃×C₁₉	#6894	1	3	yes	—
93	C₁₉⋊C₃	—	3	9	no	—
94	C₃³	—	19	3	no	—
95	C₃³	—	19	9	no	—
96	C₃³	—	57	1	no	—
97	C₃³	—	57	1	no	—
98	C₃³	—	57	1	no	—
99	C₃³	—	57	3	no	—
100	C₃³	—	57	3	no	—
101	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
102	C₃×C₉	—	3	1	no	—
103	C₃×C₉	—	3	1	no	—
104	C₃×C₉	—	3	1	no	—
105	C₃×C₉	—	3	3	no	—
106	C₃×C₉	—	3	3	no	—
107	C₃×C₉	—	3	3	no	—
108	C₃×C₉	—	3	3	no	—
109	C₃×C₉	—	19	3	no	—
110	C₃×C₉	—	19	3	no	—
111	C₃×C₉	—	19	3	no	—
112	C₃×C₉	—	19	3	no	—
113	C₃×C₉	—	19	3	no	—
114	C₃×C₉	—	57	27	no	—
115	C₃×C₉	—	57	1	no	—
116	C₃×C₉	—	57	1	no	—
117	C₃×C₉	—	57	1	no	—
118	C₃×C₉	—	57	6	no	—
119	C₃×C₉	—	57	6	no	—
120	C₃×C₉	—	57	6	no	—
121	C₃×C₉	—	57	6	no	—
122	C₃×C₉	C₉×C₁₉⋊C₃	1	3	yes	—
123	C₁₉	#10624548	1	1	yes	characteristic, complemented, Sylow
124	C₃²	—	19	9	no	—
125	C₃²	—	57	3	no	—
126	C₃²	—	57	3	no	—
127	C₃²	—	57	3	no	—
128	C₃²	—	57	9	no	—
129	C₃²	—	57	9	no	—
130	C₃²	—	57	9	no	—
131	C₃²	C₃×C₉×C₁₉⋊C₃	1	1	yes	characteristic
132	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
133	C₃²	3₋¹⁺²×C₁₉⋊C₃	1	1	yes	characteristic
134	C₃²	3₋¹⁺²×C₁₉⋊C₃	1	1	yes	characteristic
135	C₃²	3₋¹⁺²×C₁₉⋊C₃	1	3	yes	—
136	C₃²	#13095332	1	3	yes	—
137	C₃²	#13095333	1	3	yes	—
138	C₉	—	3	3	no	—
139	C₉	—	9	3	no	—
140	C₉	—	19	9	no	—
141	C₉	—	57	3	no	—
142	C₉	—	171	6	no	—
143	C₃	—	57	9	no	—
144	C₃	C₁₉⋊C₃×C₃²⋊C₉	1	1	yes	characteristic
145	C₃	C₁₉⋊C₃×C₉⋊C₉	1	1	yes	characteristic
146	C₃	#17987403	1	1	yes	characteristic
147	C₃	#17987404	1	1	yes	characteristic
148	C₃	#17987424	1	3	yes	—
149	C₃	#17987436	1	3	yes	—
150	C₃	#17987437	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₉×C₁₉⋊C₃

C₃

yes

maximal, characteristic

C₃×#13095305

C₃

yes

maximal, complemented

C₃×#13095303

C₃

yes

maximal

#17987308

C₃

yes

maximal, complemented

#17987308

C₃

yes

maximal, complemented

C₁₉×#6855

C₃

yes

maximal, characteristic, complemented

C₃³×C₁₉⋊C₃

C₃²

yes

characteristic

C₃²×#4294

C₃²

yes

—

C₃²×#4294

C₃²

yes

characteristic

C₃²×#4294

C₃²

yes

characteristic

C₃×C₉×C₁₉⋊C₃

—

C₃×C₉×C₁₉⋊C₃

—

C₃×C₉×C₁₉⋊C₃

—

C₃×C₉×C₁₉⋊C₃

C₉

yes

complemented

#13095305

—

#13095303

—

C₃²×C₉×C₁₉

C₃²

yes

—

C₃²×C₉×C₁₉

C₃²

yes

characteristic

#10624548

—

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

C₃×C₉

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

—

—

—

—

—

—

—

—

C₃×C₉

yes

—

C₃×#4294

C₃×C₉

yes

—

C₃×#4294

H(3,𝔽₃)

yes

—

C₃×#4294

3₋¹⁺²

yes

—

C₃×#4294

3₋¹⁺²

yes

—

C₉×C₁₉⋊C₃

—

C₃³×C₉

—

C₃×C₉⋊C₉

—

C₃×C₃²⋊C₉

—

#6855

—

#6855

—

#6855

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

characteristic

C₃²×C₁₉

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₁₉

C₃×3₋¹⁺²

yes

—

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⋊C₉

yes

complemented

C₉×C₁₉

—

C₉×C₁₉

—

#4294

—

#4294

—

#4294

C₃²⋊C₉

yes

complemented

C₃⁴

—

C₃²×C₉

—

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

C₉⋊C₉

—

C₃²⋊C₉

—

C₃×C₁₉

C₃×C₃²⋊C₉

yes

characteristic

C₃×C₁₉

C₃×C₉⋊C₉

yes

C₃×C₁₉

yes

C₃×C₁₉

yes

C₃×C₁₉

C₉⋊H(3,𝔽₃)

yes

—

C₃×C₁₉

#6894

yes

—

C₃×C₁₉

#6894

yes

—

C₁₉⋊C₃

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

100

C₃³

—

101

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

102

C₃×C₉

—

103

C₃×C₉

—

104

C₃×C₉

—

105

C₃×C₉

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

—

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

C₉×C₁₉⋊C₃

yes

—

123

C₁₉

#10624548

yes

characteristic, complemented, Sylow

124

C₃²

—

125

C₃²

—

126

C₃²

—

127

C₃²

—

128

C₃²

—

129

C₃²

—

130

C₃²

—

131

C₃²

C₃×C₉×C₁₉⋊C₃

yes

characteristic

132

C₃²

H(3,𝔽₃)×C₁₉⋊C₃

yes

characteristic

133

C₃²

3₋¹⁺²×C₁₉⋊C₃

yes

characteristic

134

C₃²

3₋¹⁺²×C₁₉⋊C₃

yes

characteristic

135

C₃²

3₋¹⁺²×C₁₉⋊C₃

yes

—

136

C₃²

#13095332

yes

—

137

C₃²

#13095333

yes

—

138

C₉

—

139

C₉

—

140

C₉

—

141

C₉

—

142

C₉

—

171

—

143

C₃

—

144

C₃

C₁₉⋊C₃×C₃²⋊C₉

yes

characteristic

145

C₃

C₁₉⋊C₃×C₉⋊C₉

yes

characteristic

146

C₃

#17987403

yes

characteristic

147

C₃

#17987404

yes

characteristic

148

C₃

#17987424

yes

—

149

C₃

#17987436

yes

—

150

C₃

#17987437

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	#10624548
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
2. C₃×#13095305	C₃	semidirect
4. #17987308	C₃	semidirect
5. #17987308	C₃	semidirect
6. C₁₉×#6855	C₃	semidirect
14. C₃×C₉×C₁₉⋊C₃	C₉	semidirect
54. #6855	C₁₉⋊C₃	semidirect
68. C₃×C₁₉⋊C₃	C₉⋊C₉	semidirect
73. #4294	C₃²⋊C₉	semidirect
123. C₁₉	#10624548	semidirect

H ≅ G/N

type

2. C₃×#13095305

C₃

semidirect

4. #17987308

C₃

semidirect

5. #17987308

C₃

semidirect

6. C₁₉×#6855

C₃

semidirect

14. C₃×C₉×C₁₉⋊C₃

C₉

semidirect

54. #6855

C₁₉⋊C₃

semidirect

68. C₃×C₁₉⋊C₃

C₉⋊C₉

semidirect

73. #4294

C₃²⋊C₉

semidirect

123. C₁₉

#10624548

semidirect

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

subgroup	index
C₃²×C₉×C₁₉⋊C₃	3
C₃×#13095305	3
C₃×#13095303	3
#17987308	3
C₁₉×#6855	3
#10624548	19

subgroup

index

C₃²×C₉×C₁₉⋊C₃

C₁₉×#6855

Subgroups — #28353508

Conjugacy classes of subgroups

Complements

Maximal subgroups