Group #28354849 — subgroups

Subgroups — #28354849

Order 13851.

216 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 73 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	#17987497	C₃	1	1	yes	maximal, characteristic
2	#17987481	C₃	1	2	yes	maximal, complemented
3	#17987480	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×#13095310	C₃	1	1	yes	maximal, characteristic, complemented
5	#17987353	C₃	1	1	yes	maximal, characteristic, complemented
6	#17987352	C₃	1	2	yes	maximal, complemented
7	#17987351	C₃	1	2	yes	maximal, complemented
8	#17987340	C₃	1	2	yes	maximal, complemented
9	C₁₉×#6872	C₃	1	1	yes	maximal, characteristic, complemented
10	#13095332	C₃²	1	2	yes	—
11	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
12	#13095321	C₉	1	2	yes	—
13	#13095319	C₉	1	1	yes	characteristic
14	#13095318	C₉	1	2	yes	—
15	#13095316	C₉	1	1	yes	characteristic
16	#13095315	C₉	1	2	yes	—
17	#13095313	C₉	1	1	yes	characteristic
18	#13095312	—	3	2	no	—
19	#13095312	C₃²	1	2	yes	—
20	#13095312	C₃²	1	2	yes	—
21	#13095311	—	3	2	no	—
22	#13095310	—	3	2	no	—
23	#13095310	—	3	1	no	—
24	#13095310	—	3	2	no	—
25	C₃×#10587018	—	3	1	no	—
26	C₃×#10587018	C₃²	1	1	yes	characteristic
27	C₃×#10587018	C₃²	1	1	yes	characteristic
28	C₂₇×C₁₉⋊C₃	—	3	2	no	—
29	C₁₉×#737	C₃²	1	1	yes	characteristic
30	C₁₉×C₂₇⋊C₃	C₃²	1	2	yes	—
31	C₃×C₁₉×C₂₇	C₃²	1	1	yes	characteristic
32	#10624725	—	19	1	no	maximal, Sylow
33	C₃²×C₁₉⋊C₃	C₃×C₉	1	1	yes	characteristic
34	C₁₉×3₋¹⁺²	—	3	2	no	—
35	C₁₉×3₋¹⁺²	C₃×C₉	1	2	yes	—
36	C₁₉×H(3,𝔽₃)	C₃×C₉	1	1	yes	characteristic
37	C₃×C₉×C₁₉	—	3	1	no	—
38	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
39	C₃²⋊(C₁₉⋊C₃)	C₃×C₉	1	2	yes	—
40	#10587025	C₃×C₉	1	2	yes	—
41	#10587025	C₃×C₉	1	2	yes	—
42	C₃×#4294	C₃×C₉	1	1	yes	characteristic
43	C₃×#4294	C₃×C₉	1	1	yes	characteristic
44	#10587023	—	3	2	no	—
45	#10587022	—	3	2	no	—
46	C₉×C₁₉⋊C₃	—	3	2	no	—
47	C₉×C₁₉⋊C₃	—	3	1	no	—
48	C₉×C₁₉⋊C₃	—	3	2	no	—
49	C₁₉×C₂₇	—	3	1	no	—
50	C₁₉×C₂₇	—	3	2	no	—
51	#10587018	—	3	2	no	—
52	#10587018	—	3	2	no	—
53	#10587018	—	3	1	no	—
54	#10587018	H(3,𝔽₃)	1	2	yes	—
55	#10587018	H(3,𝔽₃)	1	1	yes	characteristic
56	#6908	—	19	1	no	—
57	#6903	—	19	1	no	—
58	#6903	—	19	2	no	—
59	C₃×C₂₇⋊C₃	—	19	1	no	—
60	#6873	—	19	1	no	—
61	#6873	—	19	2	no	—
62	#6872	—	19	2	no	—
63	#6872	C₁₉⋊C₃	1	1	yes	characteristic, complemented
64	#6869	—	19	2	no	—
65	C₃²×C₁₉	—	3	1	no	—
66	C₃²×C₁₉	C₃²×C₉	1	1	yes	characteristic
67	C₃×C₁₉⋊C₃	—	3	1	no	—
68	C₃×C₁₉⋊C₃	—	3	2	no	—
69	C₃×C₁₉⋊C₃	—	9	1	no	—
70	C₉×C₁₉	—	3	2	no	—
71	C₉×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
72	C₉×C₁₉	#737	1	2	yes	—
73	#4294	—	3	1	no	—
74	#4294	—	3	1	no	—
75	#4294	—	3	2	no	—
76	#4294	—	3	2	no	—
77	#737	—	19	2	no	—
78	#737	C₃×C₁₉⋊C₃	1	1	yes	characteristic
79	C₃²×C₉	—	19	1	no	—
80	#733	—	19	2	no	—
81	#732	—	19	1	no	—
82	#731	—	19	2	no	—
83	#731	—	19	1	no	—
84	AΣL(1,𝔽₂₇)	—	19	2	no	—
85	AΣL(1,𝔽₂₇)	—	19	1	no	—
86	C₂₇⋊C₃	—	19	2	no	—
87	C₂₇⋊C₃	—	19	2	no	—
88	C₂₇⋊C₃	—	57	2	no	—
89	C₂₇⋊C₃	—	57	1	no	—
90	C₂₇⋊C₃	—	57	2	no	—
91	C₂₇⋊C₃	—	57	2	no	—
92	C₂₇⋊C₃	—	57	2	no	—
93	C₂₇⋊C₃	C₃×C₁₉⋊C₃	1	2	yes	—
94	C₃×C₂₇	—	19	1	no	—
95	C₃×C₂₇	—	19	1	no	—
96	C₃×C₂₇	—	57	2	no	—
97	C₃×C₂₇	—	57	1	no	—
98	C₃×C₂₇	C₃×C₁₉⋊C₃	1	1	yes	characteristic
99	C₃×C₁₉	—	3	1	no	—
100	C₃×C₁₉	—	9	1	no	—
101	C₃×C₁₉	C₉×H(3,𝔽₃)	1	1	yes	characteristic
102	C₁₉⋊C₃	—	9	2	no	—
103	C₁₉⋊C₃	—	9	1	no	—
104	C₃³	—	19	1	no	—
105	3₋¹⁺²	—	3	2	no	—
106	3₋¹⁺²	—	19	2	no	—
107	3₋¹⁺²	—	19	2	no	—
108	3₋¹⁺²	—	57	2	no	—
109	3₋¹⁺²	—	57	2	no	—
110	3₋¹⁺²	C₉×C₁₉⋊C₃	1	2	yes	—
111	H(3,𝔽₃)	—	19	2	no	—
112	H(3,𝔽₃)	C₉×C₁₉⋊C₃	1	1	yes	characteristic
113	C₃×C₉	—	3	1	no	—
114	C₃×C₉	—	19	1	no	—
115	C₃×C₉	—	19	1	no	—
116	C₃×C₉	—	57	2	no	—
117	C₃×C₉	—	57	1	no	—
118	C₃×C₉	—	57	2	no	—
119	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
120	C₂₇	—	3	2	no	—
121	C₂₇	—	3	1	no	—
122	C₂₇	—	19	2	no	—
123	C₂₇	—	19	1	no	—
124	C₂₇	—	57	1	no	—
125	C₂₇	—	57	2	no	—
126	C₂₇	—	57	2	no	—
127	C₁₉	#10624725	1	1	yes	characteristic, complemented, Sylow
128	C₃²	—	3	1	no	—
129	C₃²	—	57	1	no	—
130	C₃²	—	57	2	no	—
131	C₃²	—	171	1	no	—
132	C₃²	C₃×C₉×C₁₉⋊C₃	1	1	yes	characteristic
133	C₉	—	3	2	no	—
134	C₉	—	57	1	no	—
135	C₉	—	57	1	no	—
136	C₉	—	57	2	no	—
137	C₉	—	57	2	no	—
138	C₉	H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
139	C₉	#13095333	1	2	yes	—
140	C₃	—	3	1	no	—
141	C₃	—	9	1	no	—
142	C₃	—	171	1	no	—
143	C₃	—	171	2	no	—
144	C₃	#17987406	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17987497

C₃

yes

maximal, characteristic

#17987481

C₃

yes

maximal, complemented

#17987480

C₃

yes

maximal, characteristic, complemented

C₃×#13095310

C₃

yes

maximal, characteristic, complemented

#17987353

C₃

yes

maximal, characteristic, complemented

#17987352

C₃

yes

maximal, complemented

#17987351

C₃

yes

maximal, complemented

#17987340

C₃

yes

maximal, complemented

C₁₉×#6872

C₃

yes

maximal, characteristic, complemented

#13095332

C₃²

yes

—

C₃×C₉×C₁₉⋊C₃

C₃²

yes

characteristic

#13095321

C₉

yes

—

#13095319

C₉

yes

characteristic

#13095318

C₉

yes

—

#13095316

C₉

yes

characteristic

#13095315

C₉

yes

—

#13095313

C₉

yes

characteristic

#13095312

—

#13095312

C₃²

yes

—

#13095312

C₃²

yes

—

—

—

—

—

—

C₃²

yes

characteristic

C₃×#10587018

C₃²

yes

characteristic

C₂₇×C₁₉⋊C₃

—

C₁₉×#737

C₃²

yes

characteristic

C₁₉×C₂₇⋊C₃

C₃²

yes

—

C₃×C₁₉×C₂₇

C₃²

yes

characteristic

#10624725

—

maximal, Sylow

C₃²×C₁₉⋊C₃

C₃×C₉

yes

characteristic

C₁₉×3₋¹⁺²

—

C₁₉×3₋¹⁺²

C₃×C₉

yes

—

C₁₉×H(3,𝔽₃)

C₃×C₉

yes

characteristic

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

C₃³

yes

characteristic

C₃²⋊(C₁₉⋊C₃)

C₃×C₉

yes

—

#10587025

C₃×C₉

yes

—

#10587025

C₃×C₉

yes

—

C₃×#4294

C₃×C₉

yes

characteristic

C₃×#4294

C₃×C₉

yes

characteristic

#10587023

—

#10587022

—

C₉×C₁₉⋊C₃

—

C₉×C₁₉⋊C₃

—

C₉×C₁₉⋊C₃

—

C₁₉×C₂₇

—

C₁₉×C₂₇

—

—

—

—

yes

—

yes

—

—

—

C₃×C₂₇⋊C₃

—

—

—

—

C₁₉⋊C₃

yes

characteristic, complemented

#6869

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃²×C₉

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₉×C₁₉

—

C₉×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₁₉

#737

yes

—

—

—

—

—

—

C₃×C₁₉⋊C₃

yes

characteristic

C₃²×C₉

—

—

—

—

—

AΣL(1,𝔽₂₇)

—

AΣL(1,𝔽₂₇)

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

C₃×C₁₉⋊C₃

yes

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

C₃×C₁₉⋊C₃

yes

characteristic

C₃×C₁₉

—

100

C₃×C₁₉

—

101

C₃×C₁₉

C₉×H(3,𝔽₃)

yes

characteristic

102

C₁₉⋊C₃

—

103

C₁₉⋊C₃

—

104

C₃³

—

105

3₋¹⁺²

—

106

3₋¹⁺²

—

107

3₋¹⁺²

—

108

3₋¹⁺²

—

109

3₋¹⁺²

—

110

3₋¹⁺²

C₉×C₁₉⋊C₃

yes

—

111

H(3,𝔽₃)

—

112

H(3,𝔽₃)

C₉×C₁₉⋊C₃

yes

characteristic

113

C₃×C₉

—

114

C₃×C₉

—

115

C₃×C₉

—

116

C₃×C₉

—

117

C₃×C₉

—

118

C₃×C₉

—

119

C₃×C₉

C₃²×C₁₉⋊C₃

yes

characteristic

120

C₂₇

—

121

C₂₇

—

122

C₂₇

—

123

C₂₇

—

124

C₂₇

—

125

C₂₇

—

126

C₂₇

—

127

C₁₉

#10624725

yes

characteristic, complemented, Sylow

128

C₃²

—

129

C₃²

—

130

C₃²

—

131

C₃²

—

171

—

132

C₃²

C₃×C₉×C₁₉⋊C₃

yes

characteristic

133

C₉

—

134

C₉

—

135

C₉

—

136

C₉

—

137

C₉

—

138

C₉

H(3,𝔽₃)×C₁₉⋊C₃

yes

characteristic

139

C₉

#13095333

yes

—

140

C₃

—

141

C₃

—

142

C₃

—

171

—

143

C₃

—

171

—

144

C₃

#17987406

yes

characteristic

	subgroup	G/H
center	C₉	H(3,𝔽₃)×C₁₉⋊C₃
commutator	C₃²×C₁₉	C₃²×C₉
Frattini	C₃×C₉	C₃²×C₁₉⋊C₃
Fitting	C₁₉×#6872	C₃
socle	C₃×C₁₉	C₉×H(3,𝔽₃)
prime radical	#17987497	C₃
cosocle	C₃×C₉×C₁₉	C₃³
hypercenter	#6872	C₁₉⋊C₃
hypocenter	C₁₉	#10624725
LCS	G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₉ ◁ C₃×C₉ ◁ #6872

subgroup

G/H

center

C₉

H(3,𝔽₃)×C₁₉⋊C₃

commutator

C₃²×C₁₉

C₃²×C₉

Frattini

C₃×C₉

C₃²×C₁₉⋊C₃

Fitting

C₁₉×#6872

C₃

socle

C₃×C₁₉

C₉×H(3,𝔽₃)

prime radical

#17987497

C₃

cosocle

C₃×C₉×C₁₉

C₃³

C₁₉⋊C₃

C₁₉

G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₉ ◁ C₃×C₉ ◁ #6872

N	H ≅ G/N	type
2. #17987481	C₃	semidirect
3. #17987480	C₃	semidirect
4. C₃×#13095310	C₃	semidirect
5. #17987353	C₃	semidirect
6. #17987352	C₃	semidirect
7. #17987351	C₃	semidirect
8. #17987340	C₃	semidirect
9. C₁₉×#6872	C₃	semidirect
63. #6872	C₁₉⋊C₃	semidirect
127. C₁₉	#10624725	semidirect

H ≅ G/N

type

2. #17987481

C₃

semidirect

3. #17987480

C₃

semidirect

4. C₃×#13095310

C₃

semidirect

5. #17987353

C₃

semidirect

6. #17987352

C₃

semidirect

7. #17987351

C₃

semidirect

8. #17987340

C₃

semidirect

9. C₁₉×#6872

C₃

semidirect

63. #6872

C₁₉⋊C₃

semidirect

127. C₁₉

#10624725

semidirect

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

subgroup	index
#17987497	3
#17987481	3
#17987480	3
C₃×#13095310	3
#17987353	3
#17987352	3
#17987351	3
#17987340	3
C₁₉×#6872	3
#10624725	19

subgroup

index

C₁₉×#6872

Subgroups — #28354849

Conjugacy classes of subgroups

Complements

Maximal subgroups