Group #28356060 — subgroups

Subgroups — #28356060

Order 13851.

1058 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 274 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×#13095333	C₃	1	3	yes	maximal, complemented
2	C₃²×#10587025	C₃	1	1	yes	maximal, characteristic, complemented
3	#17987432	C₃	1	6	yes	maximal, complemented
4	#17987424	C₃	1	27	yes	maximal, complemented
5	#17987421	C₃	1	1	yes	maximal, characteristic, complemented
6	#17987417	C₃	1	1	yes	maximal, characteristic
7	C₁₉×#6891	C₃	1	1	yes	maximal, characteristic, complemented
8	#13095333	—	3	27	no	—
9	C₃×#10587025	—	3	9	no	—
10	C₃×#10587025	C₃²	1	3	yes	—
11	C₃×#10587025	C₃²	1	9	yes	—
12	C₃×#10587022	C₃²	1	9	yes	complemented
13	C₃×#10587023	C₃²	1	9	yes	complemented
14	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	3	yes	—
15	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	9	yes	—
16	C₃²×#4294	C₃²	1	2	yes	—
17	C₃²×#4294	C₃²	1	1	yes	characteristic
18	C₃×C₉×C₁₉⋊C₃	C₃²	1	9	yes	complemented
19	C₃²⋊#4294	C₃²	1	9	yes	—
20	#13095303	C₃²	1	54	yes	complemented
21	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
22	C₃×C₁₉×3₋¹⁺²	C₃²	1	3	yes	—
23	C₁₉×C₃²⋊C₉	C₃²	1	9	yes	—
24	#10624882	—	19	1	no	maximal, Sylow
25	C₃³×C₁₉	—	3	3	no	—
26	C₃³×C₁₉	—	9	3	no	—
27	C₃³×C₁₉	C₃³	1	3	yes	—
28	C₃³×C₁₉	C₃³	1	1	yes	characteristic
29	C₃²×C₁₉⋊C₃	C₃³	1	9	yes	—
30	C₁₉×3₋¹⁺²	—	3	9	no	—
31	C₃×C₉×C₁₉	C₃³	1	9	yes	—
32	C₃²⋊(C₁₉⋊C₃)	—	3	9	no	—
33	C₃²⋊(C₁₉⋊C₃)	—	9	9	no	—
34	#10587025	—	3	9	no	—
35	#10587025	—	3	27	no	—
36	C₃×#4294	—	3	6	no	—
37	C₃×#4294	H(3,𝔽₃)	1	9	yes	complemented
38	C₃×#4294	C₃³	1	6	yes	—
39	C₃×#4294	C₃³	1	9	yes	—
40	C₃×#4294	C₃³	1	3	yes	—
41	#10587023	—	3	27	no	—
42	#10587022	—	3	27	no	—
43	C₉×C₁₉⋊C₃	—	3	27	no	—
44	C₃×#737	—	19	3	no	—
45	C₃²×3₋¹⁺²	—	19	1	no	—
46	#6893	—	19	6	no	—
47	C₉⋊H(3,𝔽₃)	—	19	27	no	—
48	#6891	—	19	1	no	—
49	#6891	C₁₉⋊C₃	1	1	yes	characteristic, complemented
50	C₃²⋊H(3,𝔽₃)	—	19	1	no	—
51	C₃²×C₁₉	—	3	3	no	—
52	C₃²×C₁₉	—	3	3	no	—
53	C₃²×C₁₉	—	3	3	no	—
54	C₃²×C₁₉	—	3	6	no	—
55	C₃²×C₁₉	—	9	9	no	—
56	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	3	yes	—
57	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
58	C₃×C₁₉⋊C₃	—	3	9	no	—
59	C₃×C₁₉⋊C₃	—	3	27	no	—
60	C₉×C₁₉	—	3	9	no	—
61	#4294	—	3	9	no	—
62	#4294	—	3	6	no	—
63	#4294	C₃×H(3,𝔽₃)	1	9	yes	—
64	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
65	#737	—	57	27	no	—
66	C₃×3₋¹⁺²	—	19	9	no	—
67	C₃×3₋¹⁺²	—	19	3	no	—
68	C₃×3₋¹⁺²	—	19	9	no	—
69	C₃×3₋¹⁺²	—	19	9	no	—
70	C₃×3₋¹⁺²	—	57	9	no	—
71	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	3	yes	—
72	C₃×H(3,𝔽₃)	—	19	3	no	—
73	C₃×H(3,𝔽₃)	—	19	9	no	—
74	C₃²×C₉	—	19	1	no	—
75	C₃²×C₉	—	19	2	no	—
76	C₃²×C₉	—	19	9	no	—
77	C₃²⋊C₉	—	19	9	no	—
78	C₃²⋊C₉	—	19	54	no	—
79	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	9	yes	—
80	C₃×C₁₉	—	3	3	no	—
81	C₃×C₁₉	—	9	3	no	—
82	C₃×C₁₉	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
83	C₃×C₁₉	3₋¹⁺⁴	1	3	yes	—
84	C₁₉⋊C₃	—	9	9	no	—
85	C₃³	—	3	3	no	—
86	C₃³	—	9	3	no	—
87	C₃³	—	19	9	no	—
88	C₃³	C₃²×C₁₉⋊C₃	1	3	yes	—
89	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
90	3₋¹⁺²	—	3	9	no	—
91	3₋¹⁺²	—	57	27	no	—
92	3₋¹⁺²	—	57	27	no	—
93	3₋¹⁺²	—	57	27	no	—
94	3₋¹⁺²	—	57	9	no	—
95	H(3,𝔽₃)	—	57	9	no	—
96	H(3,𝔽₃)	—	171	9	no	—
97	C₃×C₉	—	19	9	no	—
98	C₃×C₉	—	19	3	no	—
99	C₃×C₉	—	19	6	no	—
100	C₃×C₉	—	19	9	no	—
101	C₃×C₉	—	57	27	no	—
102	C₃×C₉	—	57	6	no	—
103	C₃×C₉	C₃²×C₁₉⋊C₃	1	9	yes	—
104	C₁₉	#10624882	1	1	yes	characteristic, complemented, Sylow
105	C₃²	—	3	3	no	—
106	C₃²	—	3	3	no	—
107	C₃²	—	3	3	no	—
108	C₃²	—	3	6	no	—
109	C₃²	—	9	9	no	—
110	C₃²	—	57	9	no	—
111	C₃²	—	57	27	no	—
112	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	3	yes	—
113	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
114	C₉	—	3	9	no	—
115	C₉	—	19	9	no	—
116	C₉	—	57	9	no	—
117	C₉	—	57	6	no	—
118	C₃	—	3	3	no	—
119	C₃	—	9	3	no	—
120	C₃	—	171	9	no	—
121	C₃	C₃×H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
122	C₃	#17987546	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#13095333

C₃

yes

maximal, complemented

C₃²×#10587025

C₃

yes

maximal, characteristic, complemented

#17987432

C₃

yes

maximal, complemented

#17987424

C₃

yes

maximal, complemented

#17987421

C₃

yes

maximal, characteristic, complemented

#17987417

C₃

yes

maximal, characteristic

C₁₉×#6891

C₃

yes

maximal, characteristic, complemented

#13095333

—

C₃×#10587025

—

C₃×#10587025

C₃²

yes

—

C₃×#10587025

C₃²

yes

—

C₃×#10587022

C₃²

yes

complemented

C₃×#10587023

C₃²

yes

complemented

C₃×C₃²⋊(C₁₉⋊C₃)

C₃²

yes

—

C₃×C₃²⋊(C₁₉⋊C₃)

C₃²

yes

—

C₃²×#4294

C₃²

yes

—

C₃²×#4294

C₃²

yes

characteristic

C₃×C₉×C₁₉⋊C₃

C₃²

yes

complemented

C₃²⋊#4294

C₃²

yes

—

#13095303

C₃²

yes

complemented

C₃⁴×C₁₉

C₃²

yes

characteristic

C₃×C₁₉×3₋¹⁺²

C₃²

yes

—

C₁₉×C₃²⋊C₉

C₃²

yes

—

#10624882

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

C₃³

yes

—

C₃³×C₁₉

C₃³

yes

characteristic

C₃²×C₁₉⋊C₃

C₃³

yes

—

C₁₉×3₋¹⁺²

—

C₃×C₉×C₁₉

C₃³

yes

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

—

—

—

yes

C₃³

yes

—

C₃×#4294

C₃³

yes

—

C₃×#4294

C₃³

yes

—

#10587023

—

#10587022

—

C₉×C₁₉⋊C₃

—

C₃×#737

—

C₃²×3₋¹⁺²

—

#6893

—

C₉⋊H(3,𝔽₃)

—

#6891

—

#6891

C₁₉⋊C₃

yes

characteristic, complemented

C₃²⋊H(3,𝔽₃)

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

—

C₃²×C₁₉

C₃⁴

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₉×C₁₉

—

#4294

—

#4294

—

#4294

C₃×H(3,𝔽₃)

yes

—

C₃⁴

C₃×C₁₉⋊C₃

yes

characteristic

#737

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

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

—

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₁₉⋊C₃

yes

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

C₃²×H(3,𝔽₃)

yes

characteristic

C₃×C₁₉

3₋¹⁺⁴

yes

—

C₁₉⋊C₃

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²×C₁₉⋊C₃

yes

—

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

171

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

100

C₃×C₉

—

101

C₃×C₉

—

102

C₃×C₉

—

103

C₃×C₉

C₃²×C₁₉⋊C₃

yes

—

104

C₁₉

#10624882

yes

characteristic, complemented, Sylow

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

—

111

C₃²

—

112

C₃²

H(3,𝔽₃)×C₁₉⋊C₃

yes

—

113

C₃²

C₃³×C₁₉⋊C₃

yes

characteristic

114

C₉

—

115

C₉

—

116

C₉

—

117

C₉

—

118

C₃

—

119

C₃

—

120

C₃

—

171

—

121

C₃

C₃×H(3,𝔽₃)×C₁₉⋊C₃

yes

characteristic

122

C₃

#17987546

yes

—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₁₉⋊C₃
commutator, socle, cosocle	C₃²×C₁₉	C₃⁴
Fitting	C₁₉×#6891	C₃
prime radical	#17987417	C₃
hypercenter	#6891	C₁₉⋊C₃
hypocenter	C₁₉	#10624882
LCS	G ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ #6891

subgroup

G/H

center, Frattini

C₃²

C₃³×C₁₉⋊C₃

commutator, socle, cosocle

C₃²×C₁₉

C₃⁴

Fitting

C₁₉×#6891

C₃

C₃

C₁₉⋊C₃

C₁₉

G ▷ C₃²×C₁₉ ▷ C₁₉

C₁ ◁ C₃² ◁ #6891

N	H ≅ G/N	type
1. C₃×#13095333	C₃	semidirect
2. C₃²×#10587025	C₃	semidirect
3. #17987432	C₃	semidirect
4. #17987424	C₃	semidirect
5. #17987421	C₃	semidirect
7. C₁₉×#6891	C₃	semidirect
12. C₃×#10587022	C₃²	semidirect
13. C₃×#10587023	C₃²	semidirect
18. C₃×C₉×C₁₉⋊C₃	C₃²	semidirect
20. #13095303	C₃²	semidirect
37. C₃×#4294	H(3,𝔽₃)	semidirect
49. #6891	C₁₉⋊C₃	semidirect
104. C₁₉	#10624882	semidirect

H ≅ G/N

type

1. C₃×#13095333

C₃

semidirect

2. C₃²×#10587025

C₃

semidirect

3. #17987432

C₃

semidirect

4. #17987424

C₃

semidirect

5. #17987421

C₃

semidirect

7. C₁₉×#6891

C₃

semidirect

12. C₃×#10587022

C₃²

semidirect

13. C₃×#10587023

C₃²

semidirect

18. C₃×C₉×C₁₉⋊C₃

C₃²

semidirect

20. #13095303

C₃²

semidirect

37. C₃×#4294

H(3,𝔽₃)

semidirect

49. #6891

C₁₉⋊C₃

semidirect

104. C₁₉

#10624882

semidirect

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

subgroup	index
C₃×#13095333	3
C₃²×#10587025	3
#17987432	3
#17987424	3
#17987421	3
#17987417	3
C₁₉×#6891	3
#10624882	19

subgroup

index

C₁₉×#6891

Subgroups — #28356060

Conjugacy classes of subgroups

Complements

Maximal subgroups