Group #28355809 — subgroups

Subgroups — #28355809

Order 13851.

188 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 52 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	#17987500	C₃	1	2	yes	maximal, complemented
2	#17987405	C₃	1	1	yes	maximal, characteristic, complemented
3	#17987391	C₃	1	1	yes	maximal, characteristic, complemented
4	#17987388	C₃	1	2	yes	maximal, complemented
5	#17987386	C₃	1	1	yes	maximal, characteristic, complemented
6	#17987383	C₃	1	2	yes	maximal, complemented
7	#17987381	C₃	1	1	yes	maximal, characteristic, complemented
8	#17987378	C₃	1	2	yes	maximal, complemented
9	C₁₉×#6908	C₃	1	1	yes	maximal, characteristic, complemented
10	#13095332	—	3	2	no	—
11	#13095322	C₃²	1	2	yes	—
12	#13095322	C₃²	1	2	yes	—
13	#13095320	C₃²	1	2	yes	—
14	#13095317	—	3	2	no	—
15	#13095314	—	3	2	no	—
16	#13095307	C₃²	1	2	yes	—
17	#13095307	C₃²	1	1	yes	characteristic
18	#13095306	—	3	1	no	—
19	C₃²⋊#4294	—	3	1	no	—
20	C₉×#4294	—	3	1	no	—
21	C₁₉×#737	—	3	1	no	—
22	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
23	C₁₉×#733	C₃²	1	1	yes	characteristic
24	C₁₉×#733	C₃²	1	1	yes	characteristic
25	C₁₉×#732	C₃²	1	1	yes	characteristic
26	C₁₉×#731	—	3	1	no	—
27	C₁₉×AΣL(1,𝔽₂₇)	—	3	1	no	—
28	#10624849	—	19	1	no	maximal, Sylow
29	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
30	C₁₉×3₋¹⁺²	—	3	1	no	—
31	C₁₉×3₋¹⁺²	—	3	1	no	—
32	C₁₉×3₋¹⁺²	—	9	2	no	—
33	C₁₉×H(3,𝔽₃)	—	3	1	no	—
34	C₃×C₉×C₁₉	—	3	2	no	—
35	C₃×C₉×C₁₉	—	3	1	no	—
36	C₃×C₉×C₁₉	—	9	1	no	—
37	C₃×C₉×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
38	C₃×C₉×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
39	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
40	C₃²⋊(C₁₉⋊C₃)	—	3	2	no	—
41	#10587025	—	3	2	no	—
42	#10587025	—	3	2	no	—
43	C₃×#4294	—	3	2	no	—
44	C₃×#4294	—	3	1	no	—
45	#10587023	—	9	2	no	—
46	#10587022	—	9	2	no	—
47	C₉×C₁₉⋊C₃	—	9	2	no	—
48	#6908	—	19	2	no	—
49	#6908	C₁₉⋊C₃	1	1	yes	characteristic, complemented
50	#6887	—	19	1	no	—
51	#6883	—	19	2	no	—
52	#6883	—	19	1	no	—
53	#6882	—	19	2	no	—
54	#6882	—	19	1	no	—
55	#6881	—	19	1	no	—
56	#6881	—	19	2	no	—
57	C₃²×C₁₉	—	3	1	no	—
58	C₃²×C₁₉	—	9	1	no	—
59	C₃²×C₁₉	—	9	1	no	—
60	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
61	C₃×C₁₉⋊C₃	—	9	2	no	—
62	C₉×C₁₉	—	3	1	no	—
63	C₉×C₁₉	—	3	1	no	—
64	C₉×C₁₉	—	9	1	no	—
65	C₉×C₁₉	—	9	1	no	—
66	C₉×C₁₉	#732	1	2	yes	—
67	C₉×C₁₉	#732	1	1	yes	characteristic
68	#4294	—	9	2	no	—
69	#4294	—	9	1	no	—
70	#4294	—	9	2	no	—
71	#4294	—	9	2	no	—
72	#737	—	3	1	no	—
73	#737	—	57	2	no	—
74	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
75	#733	—	19	2	no	—
76	#733	—	19	2	no	—
77	#733	C₃×C₁₉⋊C₃	1	1	yes	characteristic
78	#733	C₃×C₁₉⋊C₃	1	1	yes	characteristic
79	#732	—	19	2	no	—
80	#732	C₃×C₁₉⋊C₃	1	1	yes	characteristic
81	#731	—	3	1	no	—
82	#731	—	57	2	no	—
83	AΣL(1,𝔽₂₇)	—	3	1	no	—
84	AΣL(1,𝔽₂₇)	—	57	2	no	—
85	C₉⋊C₉	—	19	2	no	—
86	C₉⋊C₉	—	19	1	no	—
87	C₉⋊C₉	—	57	1	no	—
88	C₃²⋊C₉	—	57	1	no	—
89	C₉²	—	57	1	no	—
90	C₃×C₁₉	—	3	1	no	—
91	C₃×C₁₉	—	9	1	no	—
92	C₃×C₁₉	—	27	1	no	—
93	C₃×C₁₉	C₃×#732	1	1	yes	characteristic
94	C₁₉⋊C₃	—	27	2	no	—
95	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
96	3₋¹⁺²	—	3	1	no	—
97	3₋¹⁺²	—	3	1	no	—
98	3₋¹⁺²	—	9	2	no	—
99	3₋¹⁺²	—	57	2	no	—
100	3₋¹⁺²	—	57	2	no	—
101	3₋¹⁺²	—	171	2	no	—
102	3₋¹⁺²	—	171	2	no	—
103	H(3,𝔽₃)	—	3	1	no	—
104	H(3,𝔽₃)	—	57	2	no	—
105	C₃×C₉	—	3	1	no	—
106	C₃×C₉	—	3	2	no	—
107	C₃×C₉	—	9	1	no	—
108	C₃×C₉	—	57	2	no	—
109	C₃×C₉	—	57	1	no	—
110	C₃×C₉	—	171	2	no	—
111	C₃×C₉	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
112	C₃×C₉	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
113	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
114	C₁₉	#10624849	1	1	yes	characteristic, complemented, Sylow
115	C₃²	—	3	1	no	—
116	C₃²	—	9	1	no	—
117	C₃²	—	9	1	no	—
118	C₃²	—	171	2	no	—
119	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
120	C₉	—	3	1	no	—
121	C₉	—	3	1	no	—
122	C₉	—	9	1	no	—
123	C₉	—	9	1	no	—
124	C₉	—	171	2	no	—
125	C₉	—	171	1	no	—
126	C₉	—	171	2	no	—
127	C₉	—	171	2	no	—
128	C₉	#13095319	1	1	yes	characteristic
129	C₉	#13095319	1	2	yes	—
130	C₃	—	513	2	no	—
131	C₃	—	3	1	no	—
132	C₃	—	9	1	no	—
133	C₃	—	27	1	no	—
134	C₃	C₃×#13095319	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17987500

C₃

yes

maximal, complemented

#17987405

C₃

yes

maximal, characteristic, complemented

#17987391

C₃

yes

maximal, characteristic, complemented

#17987388

C₃

yes

maximal, complemented

#17987386

C₃

yes

maximal, characteristic, complemented

#17987383

C₃

yes

maximal, complemented

#17987381

C₃

yes

maximal, characteristic, complemented

#17987378

C₃

yes

maximal, complemented

C₁₉×#6908

C₃

yes

maximal, characteristic, complemented

#13095332

—

#13095322

C₃²

yes

—

#13095322

C₃²

yes

—

#13095320

C₃²

yes

—

#13095317

—

#13095314

—

#13095307

C₃²

yes

—

#13095307

C₃²

yes

characteristic

#13095306

—

C₃²⋊#4294

—

C₉×#4294

—

C₁₉×#737

—

C₃²×C₉×C₁₉

C₃²

yes

characteristic

C₁₉×#733

C₃²

yes

characteristic

C₁₉×#733

C₃²

yes

characteristic

C₁₉×#732

C₃²

yes

characteristic

C₁₉×#731

—

C₁₉×AΣL(1,𝔽₂₇)

—

—

C₃³×C₁₉

yes

C₁₉×3₋¹⁺²

—

C₁₉×3₋¹⁺²

—

C₁₉×3₋¹⁺²

—

C₁₉×H(3,𝔽₃)

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

yes

C₃×C₉×C₁₉

yes

C₃×C₉×C₁₉

C₃³

yes

characteristic

C₃²⋊(C₁₉⋊C₃)

—

—

—

—

—

—

—

C₉×C₁₉⋊C₃

—

#6908

—

#6908

C₁₉⋊C₃

yes

characteristic, complemented

—

—

—

—

—

—

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₁₉⋊C₃

—

C₉×C₁₉

—

C₉×C₁₉

—

C₉×C₁₉

—

C₉×C₁₉

—

C₉×C₁₉

#732

yes

—

C₉×C₁₉

yes

—

—

—

—

—

—

C₃²×C₉

C₃×C₁₉⋊C₃

yes

—

—

C₃×C₁₉⋊C₃

yes

characteristic

#733

C₃×C₁₉⋊C₃

yes

characteristic

#732

—

#732

C₃×C₁₉⋊C₃

yes

characteristic

#731

—

#731

—

AΣL(1,𝔽₂₇)

—

AΣL(1,𝔽₂₇)

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₉²

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

C₃×#732

yes

characteristic

C₁₉⋊C₃

—

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

100

3₋¹⁺²

—

101

3₋¹⁺²

—

171

—

102

3₋¹⁺²

—

171

—

103

H(3,𝔽₃)

—

104

H(3,𝔽₃)

—

105

C₃×C₉

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

—

110

C₃×C₉

—

171

—

111

C₃×C₉

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

112

C₃×C₉

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

113

C₃×C₉

C₃²×C₁₉⋊C₃

yes

characteristic

114

C₁₉

#10624849

yes

characteristic, complemented, Sylow

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

—

171

—

119

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

120

C₉

—

121

C₉

—

122

C₉

—

123

C₉

—

124

C₉

—

171

—

125

C₉

—

171

—

126

C₉

—

171

—

127

C₉

—

171

—

128

C₉

#13095319

yes

characteristic

129

C₉

#13095319

yes

—

130

C₃

—

513

—

131

C₃

—

132

C₃

—

133

C₃

—

134

C₃

C₃×#13095319

yes

characteristic

	subgroup	G/H
center	C₃	C₃×#13095319
commutator, cosocle	C₃×C₉×C₁₉	C₃³
Frattini	C₃×C₉	C₃²×C₁₉⋊C₃
Fitting	C₁₉×#6908	C₃
socle	C₃×C₁₉	C₃×#732
hypercenter	#6908	C₁₉⋊C₃
hypocenter	C₁₉	#10624849
LCS	G ▷ C₃×C₉×C₁₉ ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ #6908

subgroup

G/H

center

C₃

C₃×#13095319

commutator, cosocle

C₃×C₉×C₁₉

C₃³

Frattini

C₃×C₉

C₃²×C₁₉⋊C₃

Fitting

C₁₉×#6908

C₃

C₃×C₁₉

C₁₉⋊C₃

C₁₉

G ▷ C₃×C₉×C₁₉ ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ #6908

N	H ≅ G/N	type
1. #17987500	C₃	semidirect
2. #17987405	C₃	semidirect
3. #17987391	C₃	semidirect
4. #17987388	C₃	semidirect
5. #17987386	C₃	semidirect
6. #17987383	C₃	semidirect
7. #17987381	C₃	semidirect
8. #17987378	C₃	semidirect
9. C₁₉×#6908	C₃	semidirect
49. #6908	C₁₉⋊C₃	semidirect
114. C₁₉	#10624849	semidirect

H ≅ G/N

type

1. #17987500

C₃

semidirect

2. #17987405

C₃

semidirect

3. #17987391

C₃

semidirect

4. #17987388

C₃

semidirect

5. #17987386

C₃

semidirect

6. #17987383

C₃

semidirect

7. #17987381

C₃

semidirect

8. #17987378

C₃

semidirect

9. C₁₉×#6908

C₃

semidirect

49. #6908

C₁₉⋊C₃

semidirect

114. C₁₉

#10624849

semidirect

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

subgroup	index
#17987500	3
#17987405	3
#17987391	3
#17987388	3
#17987386	3
#17987383	3
#17987381	3
#17987378	3
C₁₉×#6908	3
#10624849	19

subgroup

index

C₁₉×#6908

Subgroups — #28355809

Conjugacy classes of subgroups

Complements

Maximal subgroups