Group #28353748 — subgroups

Subgroups — #28353748

Order 13851.

356 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 64 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×#10587025	C₃	1	1	yes	maximal, characteristic
2	#17987421	C₃	1	1	yes	maximal, characteristic
3	C₃×C₃²⋊#4294	C₃	1	1	yes	maximal, characteristic
4	#17987317	C₃	1	9	yes	maximal, complemented
5	C₁₉×#6891	C₃	1	1	yes	maximal, characteristic
6	C₃×#10587025	—	3	1	no	—
7	C₃×#10587025	—	3	1	no	—
8	C₃×#10587025	—	3	9	no	—
9	C₃×#10587025	—	3	1	no	—
10	C₃×#10587025	C₃²	1	3	yes	—
11	C₃²×#4294	—	3	1	no	—
12	C₃²×#4294	—	3	1	no	—
13	C₃²⋊#4294	—	3	1	no	—
14	C₃²⋊#4294	—	3	1	no	—
15	C₃²⋊#4294	—	3	1	no	—
16	C₃²⋊#4294	—	3	1	no	—
17	C₃²⋊#4294	C₃²	1	3	yes	—
18	C₃²⋊#4294	C₃²	1	3	yes	—
19	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
20	C₃×C₁₉×3₋¹⁺²	—	3	1	no	—
21	C₁₉×C₃²⋊C₉	—	3	1	no	—
22	C₁₉×C₃²⋊C₉	—	3	1	no	—
23	C₁₉×C₃²⋊C₉	C₃²	1	3	yes	—
24	#10624592	—	19	1	no	maximal, Sylow
25	C₃³×C₁₉	—	3	3	no	—
26	C₃³×C₁₉	—	3	3	no	—
27	C₃³×C₁₉	—	9	1	no	—
28	C₃³×C₁₉	—	9	1	no	—
29	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
30	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
31	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
32	C₃³×C₁₉	C₃³	1	1	yes	characteristic
33	C₁₉×3₋¹⁺²	—	9	3	no	—
34	C₃×C₉×C₁₉	—	3	3	no	—
35	#10587025	—	3	9	no	—
36	#10587025	—	3	9	no	—
37	#10587025	—	3	9	no	—
38	#10587025	—	9	3	no	—
39	C₃×#4294	—	3	9	no	—
40	C₃×#4294	—	3	3	no	—
41	C₃×#4294	—	3	3	no	—
42	C₃×#4294	—	3	3	no	—
43	C₃×#4294	—	9	3	no	—
44	C₃²×3₋¹⁺²	—	19	1	no	—
45	#6891	—	19	1	no	—
46	#6891	C₁₉⋊C₃	1	1	yes	characteristic
47	C₃×C₃²⋊C₉	—	19	1	no	—
48	#6858	—	19	9	no	—
49	C₃²×C₁₉	—	3	1	no	—
50	C₃²×C₁₉	—	3	1	no	—
51	C₃²×C₁₉	—	3	1	no	—
52	C₃²×C₁₉	—	3	1	no	—
53	C₃²×C₁₉	—	3	1	no	—
54	C₃²×C₁₉	—	3	1	no	—
55	C₃²×C₁₉	—	3	1	no	—
56	C₃²×C₁₉	—	3	1	no	—
57	C₃²×C₁₉	—	3	1	no	—
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	—	3	1	no	—
63	C₃²×C₁₉	—	9	9	no	—
64	C₃²×C₁₉	AΣL(1,𝔽₂₇)	1	3	yes	—
65	C₃²×C₁₉	#731	1	3	yes	—
66	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
67	C₉×C₁₉	—	9	3	no	—
68	#4294	—	3	9	no	—
69	#4294	—	9	3	no	—
70	#4294	—	9	3	no	—
71	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
72	C₃×3₋¹⁺²	—	3	1	no	—
73	C₃×3₋¹⁺²	—	19	3	no	—
74	C₃×3₋¹⁺²	—	57	1	no	—
75	C₃×3₋¹⁺²	—	57	1	no	—
76	C₃×3₋¹⁺²	—	57	9	no	—
77	C₃×3₋¹⁺²	—	57	1	no	—
78	C₃²×C₉	—	57	1	no	—
79	C₃²×C₉	—	57	1	no	—
80	C₃²⋊C₉	—	3	1	no	—
81	C₃²⋊C₉	—	3	1	no	—
82	C₃²⋊C₉	—	19	3	no	—
83	C₃²⋊C₉	—	19	3	no	—
84	C₃²⋊C₉	—	57	1	no	—
85	C₃²⋊C₉	—	57	1	no	—
86	C₃²⋊C₉	—	57	1	no	—
87	C₃²⋊C₉	—	57	1	no	—
88	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	—
89	C₃×C₁₉	—	3	3	no	—
90	C₃×C₁₉	—	3	3	no	—
91	C₃×C₁₉	—	9	1	no	—
92	C₃×C₁₉	—	9	1	no	—
93	C₃×C₁₉	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
94	C₃×C₁₉	C₃×#731	1	1	yes	characteristic
95	C₃×C₁₉	#6910	1	1	yes	characteristic
96	C₃×C₁₉	#6912	1	1	yes	characteristic
97	C₃³	—	3	3	no	—
98	C₃³	—	3	3	no	—
99	C₃³	—	9	1	no	—
100	C₃³	—	9	1	no	—
101	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
102	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
103	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
104	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
105	3₋¹⁺²	—	9	3	no	—
106	3₋¹⁺²	—	57	9	no	—
107	3₋¹⁺²	—	57	9	no	—
108	3₋¹⁺²	—	57	9	no	—
109	3₋¹⁺²	—	171	3	no	—
110	C₃×C₉	—	3	3	no	—
111	C₃×C₉	—	57	9	no	—
112	C₃×C₉	—	57	3	no	—
113	C₃×C₉	—	57	3	no	—
114	C₃×C₉	—	57	3	no	—
115	C₃×C₉	—	171	3	no	—
116	C₁₉	#10624592	1	1	yes	characteristic, complemented, Sylow
117	C₃²	—	3	1	no	—
118	C₃²	—	3	1	no	—
119	C₃²	—	3	1	no	—
120	C₃²	—	3	1	no	—
121	C₃²	—	3	1	no	—
122	C₃²	—	3	1	no	—
123	C₃²	—	3	1	no	—
124	C₃²	—	3	1	no	—
125	C₃²	—	3	1	no	—
126	C₃²	—	3	1	no	—
127	C₃²	—	3	1	no	—
128	C₃²	—	3	1	no	—
129	C₃²	—	3	1	no	—
130	C₃²	—	3	1	no	—
131	C₃²	—	9	9	no	—
132	C₃²	#13095315	1	3	yes	—
133	C₃²	#13095316	1	3	yes	—
134	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
135	C₉	—	9	3	no	—
136	C₉	—	57	9	no	—
137	C₉	—	171	3	no	—
138	C₉	—	171	3	no	—
139	C₃	—	3	3	no	—
140	C₃	—	3	3	no	—
141	C₃	—	9	1	no	—
142	C₃	—	9	1	no	—
143	C₃	C₃×#13095315	1	1	yes	characteristic
144	C₃	C₃×#13095316	1	1	yes	characteristic
145	C₃	#17987510	1	1	yes	characteristic
146	C₃	#17987518	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#10587025

C₃

yes

maximal, characteristic

#17987421

C₃

yes

maximal, characteristic

C₃×C₃²⋊#4294

C₃

yes

maximal, characteristic

#17987317

C₃

yes

maximal, complemented

C₁₉×#6891

C₃

yes

maximal, characteristic

—

—

—

—

C₃²

yes

—

C₃²×#4294

—

C₃²×#4294

—

C₃²⋊#4294

—

C₃²⋊#4294

—

C₃²⋊#4294

—

C₃²⋊#4294

—

C₃²⋊#4294

C₃²

yes

—

C₃²⋊#4294

C₃²

yes

—

C₃⁴×C₁₉

C₃²

yes

characteristic

C₃×C₁₉×3₋¹⁺²

—

C₁₉×C₃²⋊C₉

—

C₁₉×C₃²⋊C₉

—

C₁₉×C₃²⋊C₉

C₃²

yes

—

#10624592

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

yes

C₃³×C₁₉

yes

C₃³×C₁₉

yes

C₃³×C₁₉

C₃³

yes

characteristic

C₁₉×3₋¹⁺²

—

C₃×C₉×C₁₉

—

—

—

—

—

—

—

—

—

—

C₃²×3₋¹⁺²

—

#6891

—

#6891

C₁₉⋊C₃

yes

characteristic

C₃×C₃²⋊C₉

—

#6858

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₁₉

#731

yes

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₁₉

—

#4294

—

#4294

—

#4294

—

C₃⁴

C₃×C₁₉⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

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

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₁₉⋊C₃

yes

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

C₃×AΣL(1,𝔽₂₇)

yes

C₃×C₁₉

yes

C₃×C₁₉

yes

C₃×C₁₉

yes

C₃³

—

C₃³

—

C₃³

—

100

C₃³

—

101

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

102

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

103

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

104

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

105

3₋¹⁺²

—

106

3₋¹⁺²

—

107

3₋¹⁺²

—

108

3₋¹⁺²

—

109

3₋¹⁺²

—

171

—

110

C₃×C₉

—

111

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

—

114

C₃×C₉

—

115

C₃×C₉

—

171

—

116

C₁₉

#10624592

yes

characteristic, complemented, Sylow

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

—

125

C₃²

—

126

C₃²

—

127

C₃²

—

128

C₃²

—

129

C₃²

—

130

C₃²

—

131

C₃²

—

132

C₃²

#13095315

yes

—

133

C₃²

#13095316

yes

—

134

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

135

C₉

—

136

C₉

—

137

C₉

—

171

—

138

C₉

—

171

—

139

C₃

—

140

C₃

—

141

C₃

—

142

C₃

—

143

C₃

C₃×#13095315

yes

characteristic

144

C₃

C₃×#13095316

yes

characteristic

145

C₃

#17987510

yes

characteristic

146

C₃

#17987518

yes

characteristic

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)
commutator, cosocle	C₃³×C₁₉	C₃³
Frattini	C₃³	C₃²×C₁₉⋊C₃
Fitting	C₁₉×#6891	C₃
socle	C₃²×C₁₉	C₃×H(3,𝔽₃)
prime radical	C₃⁴×C₁₉	C₃²
hypercenter	#6891	C₁₉⋊C₃
hypocenter	C₁₉	#10624592
LCS	G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ #6891

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

commutator, cosocle

C₃³×C₁₉

C₃³

Frattini

C₃³

C₃²×C₁₉⋊C₃

Fitting

C₁₉×#6891

C₃

socle

C₃²×C₁₉

C₃×H(3,𝔽₃)

prime radical

C₃⁴×C₁₉

C₃²

C₁₉⋊C₃

C₁₉

G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ #6891

N	H ≅ G/N	type
4. #17987317	C₃	semidirect
116. C₁₉	#10624592	semidirect

H ≅ G/N

type

4. #17987317

C₃

semidirect

116. C₁₉

#10624592

semidirect

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

subgroup	index
C₃²×#10587025	3
#17987421	3
C₃×C₃²⋊#4294	3
#17987317	3
C₁₉×#6891	3
#10624592	19

subgroup

index

C₃×C₃²⋊#4294

C₁₉×#6891

Subgroups — #28353748

Conjugacy classes of subgroups

Complements

Maximal subgroups