Group #28355062 — subgroups

Subgroups — #28355062

Order 13851.

426 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 61 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	C₃×#13095332	C₃	1	1	yes	maximal, characteristic, complemented
2	#17987515	C₃	1	3	yes	maximal, complemented
3	#17987506	C₃	1	6	yes	maximal, complemented
4	C₃×#13095320	C₃	1	2	yes	maximal, complemented
5	C₃×C₁₉×#733	C₃	1	1	yes	maximal, characteristic, complemented
6	#13095332	—	3	3	no	—
7	#13095332	—	3	6	no	—
8	C₃×#10587025	C₃²	1	2	yes	—
9	C₃×#10587022	—	3	1	no	—
10	C₃×#10587023	—	3	1	no	—
11	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	6	yes	—
12	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	1	yes	characteristic
13	C₃×C₉×C₁₉⋊C₃	—	3	1	no	—
14	#13095320	—	3	2	no	—
15	#13095320	—	3	2	no	—
16	#13095320	—	3	2	no	—
17	#13095320	—	3	9	no	—
18	#13095315	—	3	18	no	—
19	C₃×C₁₉×3₋¹⁺²	C₃²	1	3	yes	—
20	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
21	C₁₉×#733	—	3	1	no	—
22	C₁₉×#733	—	3	1	no	—
23	C₁₉×#733	—	3	1	no	—
24	#10624757	—	19	1	no	maximal, Sylow
25	C₃³×C₁₉	C₃³	1	1	yes	characteristic
26	C₃²×C₁₉⋊C₃	—	3	1	no	—
27	C₃²×C₁₉⋊C₃	—	3	6	no	—
28	C₁₉×3₋¹⁺²	—	3	3	no	—
29	C₁₉×3₋¹⁺²	—	3	6	no	—
30	C₃×C₉×C₁₉	—	3	1	no	—
31	C₃×C₉×C₁₉	—	3	2	no	—
32	C₃×C₉×C₁₉	—	3	3	no	—
33	C₃×C₉×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
34	C₃×C₉×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
35	C₃×C₉×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
36	C₃²⋊(C₁₉⋊C₃)	—	3	2	no	—
37	C₃²⋊(C₁₉⋊C₃)	—	3	6	no	—
38	C₃²⋊(C₁₉⋊C₃)	—	3	6	no	—
39	C₃²⋊(C₁₉⋊C₃)	—	3	6	no	—
40	C₃²⋊(C₁₉⋊C₃)	H(3,𝔽₃)	1	3	yes	—
41	#10587025	—	3	2	no	—
42	#10587025	—	3	2	no	—
43	#10587025	H(3,𝔽₃)	1	6	yes	—
44	C₃×#4294	—	3	2	no	—
45	#10587023	—	3	9	no	—
46	#10587022	—	3	9	no	—
47	C₉×C₁₉⋊C₃	—	3	9	no	—
48	C₃×#737	—	19	1	no	—
49	#6911	—	19	3	no	—
50	#6909	—	19	6	no	—
51	C₃×#733	C₁₉⋊C₃	1	1	yes	characteristic, complemented
52	C₃×#732	—	19	2	no	—
53	C₃²×C₁₉	—	9	1	no	—
54	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
55	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	2	yes	—
56	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
57	C₃×C₁₉⋊C₃	—	3	3	no	—
58	C₃×C₁₉⋊C₃	—	9	3	no	—
59	C₃×C₁₉⋊C₃	—	9	6	no	—
60	C₃×C₁₉⋊C₃	—	27	6	no	—
61	C₉×C₁₉	—	3	1	no	—
62	C₉×C₁₉	—	3	1	no	—
63	C₉×C₁₉	—	3	1	no	—
64	C₉×C₁₉	—	9	3	no	—
65	#4294	—	3	6	no	—
66	#737	—	57	3	no	—
67	#737	—	57	6	no	—
68	C₃×3₋¹⁺²	—	19	2	no	—
69	C₃×3₋¹⁺²	—	57	1	no	—
70	C₃×3₋¹⁺²	—	57	1	no	—
71	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	3	yes	—
72	C₃×H(3,𝔽₃)	—	19	1	no	—
73	C₃×H(3,𝔽₃)	—	19	6	no	—
74	C₃²×C₉	—	57	1	no	—
75	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
76	#733	—	3	1	no	—
77	#733	—	3	1	no	—
78	#733	—	3	1	no	—
79	#732	—	57	9	no	—
80	#732	—	57	2	no	—
81	#732	—	57	2	no	—
82	#732	—	57	2	no	—
83	AΣL(1,𝔽₂₇)	—	57	18	no	—
84	C₃×C₁₉	—	3	1	no	—
85	C₃×C₁₉	—	3	2	no	—
86	C₃×C₁₉	—	3	1	no	—
87	C₃×C₁₉	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
88	C₁₉⋊C₃	—	9	3	no	—
89	C₁₉⋊C₃	—	27	6	no	—
90	C₃³	—	57	6	no	—
91	C₃³	—	57	1	no	—
92	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
93	3₋¹⁺²	—	3	6	no	—
94	3₋¹⁺²	—	3	3	no	—
95	3₋¹⁺²	—	19	6	no	—
96	3₋¹⁺²	—	57	9	no	—
97	3₋¹⁺²	—	57	9	no	—
98	3₋¹⁺²	—	57	2	no	—
99	3₋¹⁺²	—	57	2	no	—
100	H(3,𝔽₃)	—	19	3	no	—
101	H(3,𝔽₃)	—	57	2	no	—
102	H(3,𝔽₃)	—	57	6	no	—
103	H(3,𝔽₃)	—	57	6	no	—
104	H(3,𝔽₃)	—	57	6	no	—
105	C₃×C₉	—	3	1	no	—
106	C₃×C₉	—	3	2	no	—
107	C₃×C₉	—	3	3	no	—
108	C₃×C₉	—	57	9	no	—
109	C₃×C₉	—	57	2	no	—
110	C₃×C₉	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
111	C₃×C₉	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
112	C₃×C₉	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
113	C₁₉	#10624757	1	1	yes	characteristic, complemented, Sylow
114	C₃²	—	513	6	no	—
115	C₃²	—	9	1	no	—
116	C₃²	—	57	3	no	—
117	C₃²	—	171	6	no	—
118	C₃²	—	171	3	no	—
119	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
120	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	2	yes	—
121	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
122	C₉	—	3	1	no	—
123	C₉	—	3	1	no	—
124	C₉	—	3	1	no	—
125	C₉	—	9	3	no	—
126	C₉	—	57	6	no	—
127	C₃	—	513	6	no	—
128	C₃	—	3	2	no	—
129	C₃	—	3	1	no	—
130	C₃	—	3	1	no	—
131	C₃	—	171	3	no	—
132	C₃	(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#13095332

C₃

yes

maximal, characteristic, complemented

#17987515

C₃

yes

maximal, complemented

#17987506

C₃

yes

maximal, complemented

C₃×#13095320

C₃

yes

maximal, complemented

C₃×C₁₉×#733

C₃

yes

maximal, characteristic, complemented

#13095332

—

#13095332

—

C₃×#10587025

C₃²

yes

—

C₃×#10587022

—

C₃×#10587023

—

C₃×C₃²⋊(C₁₉⋊C₃)

C₃²

yes

—

C₃×C₃²⋊(C₁₉⋊C₃)

C₃²

yes

characteristic

C₃×C₉×C₁₉⋊C₃

—

—

—

—

—

—

C₃×C₁₉×3₋¹⁺²

C₃²

yes

—

C₃²×C₉×C₁₉

C₃²

yes

characteristic

C₁₉×#733

—

C₁₉×#733

—

C₁₉×#733

—

#10624757

—

maximal, Sylow

C₃³×C₁₉

C₃³

yes

characteristic

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₁₉×3₋¹⁺²

—

C₁₉×3₋¹⁺²

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

yes

C₃×C₉×C₁₉

yes

C₃×C₉×C₁₉

yes

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

H(3,𝔽₃)

yes

—

—

—

yes

—

C₃×#4294

—

#10587023

—

#10587022

—

C₉×C₁₉⋊C₃

—

—

—

—

C₁₉⋊C₃

yes

characteristic, complemented

C₃×#732

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₉×C₁₉

—

C₉×C₁₉

—

C₉×C₁₉

—

C₉×C₁₉

—

#4294

—

#737

—

#737

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₁₉⋊C₃

yes

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

C₃×C₁₉⋊C₃

yes

—

—

—

—

—

—

—

AΣL(1,𝔽₂₇)

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

C₃²⋊H(3,𝔽₃)

yes

characteristic

C₁₉⋊C₃

—

C₁₉⋊C₃

—

C₃³

—

C₃³

—

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

100

H(3,𝔽₃)

—

101

H(3,𝔽₃)

—

102

H(3,𝔽₃)

—

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

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

111

C₃×C₉

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

112

C₃×C₉

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

113

C₁₉

#10624757

yes

characteristic, complemented, Sylow

114

C₃²

—

513

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

171

—

118

C₃²

—

171

—

119

C₃²

H(3,𝔽₃)×C₁₉⋊C₃

yes

characteristic

120

C₃²

H(3,𝔽₃)×C₁₉⋊C₃

yes

—

121

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

122

C₉

—

123

C₉

—

124

C₉

—

125

C₉

—

126

C₉

—

127

C₃

—

513

—

128

C₃

—

129

C₃

—

130

C₃

—

131

C₃

—

171

—

132

C₃

(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)

yes

characteristic

	subgroup	G/H
center	C₃	(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)
commutator, cosocle	C₃³×C₁₉	C₃³
Frattini	C₃³	C₃²×C₁₉⋊C₃
Fitting	C₃×C₁₉×#733	C₃
socle	C₃×C₁₉	C₃²⋊H(3,𝔽₃)
hypercenter	C₃×#733	C₁₉⋊C₃
hypocenter	C₁₉	#10624757
LCS	G ▷ C₃³×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×#733

subgroup

G/H

center

C₃

(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)

commutator, cosocle

C₃³×C₁₉

C₃³

Frattini

C₃³

C₃²×C₁₉⋊C₃

Fitting

C₃×C₁₉×#733

C₃

socle

C₃×C₁₉

C₃²⋊H(3,𝔽₃)

C₁₉⋊C₃

C₁₉

G ▷ C₃³×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃ ◁ C₃³ ◁ C₃×#733

N	H ≅ G/N	type
1. C₃×#13095332	C₃	semidirect
2. #17987515	C₃	semidirect
3. #17987506	C₃	semidirect
4. C₃×#13095320	C₃	semidirect
5. C₃×C₁₉×#733	C₃	semidirect
51. C₃×#733	C₁₉⋊C₃	semidirect
113. C₁₉	#10624757	semidirect

H ≅ G/N

type

1. C₃×#13095332

C₃

semidirect

2. #17987515

C₃

semidirect

3. #17987506

C₃

semidirect

4. C₃×#13095320

C₃

semidirect

5. C₃×C₁₉×#733

C₃

semidirect

51. C₃×#733

C₁₉⋊C₃

semidirect

113. C₁₉

#10624757

semidirect

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

subgroup	index
C₃×#13095332	3
#17987515	3
#17987506	3
C₃×#13095320	3
C₃×C₁₉×#733	3
#10624757	19

subgroup

index

C₃×C₁₉×#733

Subgroups — #28355062

Conjugacy classes of subgroups

Complements

Maximal subgroups