Group #28355825 — subgroups

Subgroups — #28355825

Order 13851.

200 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 46 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	#17987528	C₃	1	2	yes	maximal, complemented
2	#17987515	C₃	1	1	yes	maximal, characteristic, complemented
3	#17987390	C₃	1	2	yes	maximal, complemented
4	#17987380	C₃	1	2	yes	maximal, complemented
5	#17987380	C₃	1	2	yes	maximal, complemented
6	#17987374	C₃	1	1	yes	maximal, characteristic, complemented
7	#17987370	C₃	1	1	yes	maximal, characteristic, complemented
8	#17987369	C₃	1	1	yes	maximal, characteristic, complemented
9	C₁₉×#6900	C₃	1	1	yes	maximal, characteristic, complemented
10	#13095332	—	3	2	no	—
11	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	1	no	—
12	#13095322	C₃²	1	2	yes	complemented
13	#13095322	C₃²	1	2	yes	complemented
14	#13095320	—	3	2	no	—
15	#13095320	C₃²	1	1	yes	characteristic
16	#13095320	C₃²	1	1	yes	characteristic
17	#13095320	C₃²	1	1	yes	characteristic
18	#13095320	C₃²	1	2	yes	complemented
19	#13095317	—	3	2	no	—
20	#13095314	—	3	2	no	—
21	C₃×C₁₉×3₋¹⁺²	—	3	1	no	—
22	C₃×C₁₉×3₋¹⁺²	C₃²	1	1	yes	characteristic
23	C₁₉×C₉⋊C₉	—	3	2	no	—
24	C₁₉×C₉⋊C₉	C₃²	1	2	yes	complemented
25	C₉²×C₁₉	C₃²	1	1	yes	characteristic, complemented
26	#10624851	—	19	1	no	maximal, Sylow
27	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
28	C₃²×C₁₉⋊C₃	—	9	1	no	—
29	C₁₉×3₋¹⁺²	—	9	1	no	—
30	C₁₉×3₋¹⁺²	—	9	3	no	—
31	C₃×C₉×C₁₉	—	3	1	no	—
32	C₃×C₉×C₁₉	—	3	2	no	—
33	C₃×C₉×C₁₉	H(3,𝔽₃)	1	2	yes	—
34	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
35	C₃²⋊(C₁₉⋊C₃)	—	3	2	no	—
36	C₃²⋊(C₁₉⋊C₃)	—	3	1	no	—
37	C₃²⋊(C₁₉⋊C₃)	—	3	1	no	—
38	C₃²⋊(C₁₉⋊C₃)	—	3	1	no	—
39	C₃²⋊(C₁₉⋊C₃)	—	9	2	no	—
40	#10587025	—	3	2	no	—
41	#10587025	—	3	2	no	—
42	#10587023	—	9	2	no	—
43	#10587022	—	9	2	no	—
44	C₉×C₁₉⋊C₃	—	9	2	no	—
45	#6913	—	19	2	no	—
46	#6911	—	19	1	no	—
47	#6900	C₁₉⋊C₃	1	1	yes	characteristic, complemented
48	#6883	—	19	2	no	—
49	#6881	—	19	2	no	—
50	#6881	—	19	2	no	—
51	#6879	—	19	1	no	—
52	#6878	—	19	1	no	—
53	#6878	—	19	1	no	—
54	C₃²×C₁₉	—	3	1	no	—
55	C₃²×C₁₉	—	9	1	no	—
56	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
57	C₃×C₁₉⋊C₃	—	9	2	no	—
58	C₃×C₁₉⋊C₃	—	9	1	no	—
59	C₃×C₁₉⋊C₃	—	9	1	no	—
60	C₃×C₁₉⋊C₃	—	9	1	no	—
61	C₃×C₁₉⋊C₃	—	27	3	no	—
62	C₉×C₁₉	—	3	3	no	—
63	C₉×C₁₉	—	3	2	no	—
64	C₉×C₁₉	—	3	1	no	—
65	C₉×C₁₉	—	9	2	no	—
66	#4294	—	9	2	no	—
67	#4294	—	9	2	no	—
68	#737	—	57	2	no	—
69	C₃×3₋¹⁺²	—	3	1	no	—
70	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
71	C₃×H(3,𝔽₃)	—	57	1	no	—
72	#733	—	19	2	no	—
73	#733	—	19	2	no	—
74	#732	—	19	2	no	—
75	#732	—	19	1	no	—
76	#732	—	19	1	no	—
77	#732	—	19	1	no	—
78	#732	—	57	2	no	—
79	#731	—	57	2	no	—
80	AΣL(1,𝔽₂₇)	—	57	2	no	—
81	C₉⋊C₉	—	3	2	no	—
82	C₉⋊C₉	C₃×C₁₉⋊C₃	1	2	yes	complemented
83	C₉²	C₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented
84	C₃×C₁₉	—	3	1	no	—
85	C₃×C₁₉	—	9	1	no	—
86	C₃×C₁₉	#6911	1	1	yes	characteristic
87	C₁₉⋊C₃	—	27	2	no	—
88	C₁₉⋊C₃	—	27	1	no	—
89	C₁₉⋊C₃	—	27	1	no	—
90	C₁₉⋊C₃	—	27	1	no	—
91	C₃³	—	171	1	no	—
92	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
93	3₋¹⁺²	—	9	1	no	—
94	3₋¹⁺²	—	9	3	no	—
95	3₋¹⁺²	—	57	2	no	—
96	3₋¹⁺²	—	57	2	no	—
97	3₋¹⁺²	—	171	2	no	—
98	3₋¹⁺²	—	171	2	no	—
99	H(3,𝔽₃)	—	57	2	no	—
100	H(3,𝔽₃)	—	57	1	no	—
101	H(3,𝔽₃)	—	57	1	no	—
102	H(3,𝔽₃)	—	57	1	no	—
103	H(3,𝔽₃)	—	171	2	no	—
104	C₃×C₉	—	3	1	no	—
105	C₃×C₉	—	3	2	no	—
106	C₃×C₉	—	171	2	no	—
107	C₃×C₉	C₃²⋊(C₁₉⋊C₃)	1	2	yes	—
108	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
109	C₁₉	#10624851	1	1	yes	characteristic, complemented, Sylow
110	C₃²	—	513	3	no	—
111	C₃²	—	3	1	no	—
112	C₃²	—	9	1	no	—
113	C₃²	—	171	2	no	—
114	C₃²	—	171	1	no	—
115	C₃²	—	171	1	no	—
116	C₃²	—	171	1	no	—
117	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
118	C₉	—	3	3	no	—
119	C₉	—	3	2	no	—
120	C₉	—	3	1	no	—
121	C₉	—	9	2	no	—
122	C₉	—	171	2	no	—
123	C₉	—	171	2	no	—
124	C₃	—	513	1	no	—
125	C₃	—	513	1	no	—
126	C₃	—	513	1	no	—
127	C₃	—	513	2	no	—
128	C₃	—	3	1	no	—
129	C₃	—	9	1	no	—
130	C₃	#17987515	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17987528

C₃

yes

maximal, complemented

#17987515

C₃

yes

maximal, characteristic, complemented

#17987390

C₃

yes

maximal, complemented

#17987380

C₃

yes

maximal, complemented

#17987380

C₃

yes

maximal, complemented

#17987374

C₃

yes

maximal, characteristic, complemented

#17987370

C₃

yes

maximal, characteristic, complemented

#17987369

C₃

yes

maximal, characteristic, complemented

C₁₉×#6900

C₃

yes

maximal, characteristic, complemented

#13095332

—

C₃×C₃²⋊(C₁₉⋊C₃)

—

#13095322

C₃²

yes

complemented

#13095322

C₃²

yes

complemented

#13095320

—

#13095320

C₃²

yes

characteristic

#13095320

C₃²

yes

characteristic

#13095320

C₃²

yes

characteristic

#13095320

C₃²

yes

complemented

#13095317

—

#13095314

—

C₃×C₁₉×3₋¹⁺²

—

C₃×C₁₉×3₋¹⁺²

C₃²

yes

characteristic

C₁₉×C₉⋊C₉

—

C₁₉×C₉⋊C₉

C₃²

yes

complemented

C₉²×C₁₉

C₃²

yes

characteristic, complemented

—

C₃³×C₁₉

yes

C₃²×C₁₉⋊C₃

—

C₁₉×3₋¹⁺²

—

C₁₉×3₋¹⁺²

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

H(3,𝔽₃)

yes

—

C₃×C₉×C₁₉

C₃³

yes

characteristic

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

—

—

—

—

C₉×C₁₉⋊C₃

—

#6913

—

#6911

—

#6900

C₁₉⋊C₃

yes

characteristic, complemented

—

—

—

—

—

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₉×C₁₉

—

C₉×C₁₉

—

C₉×C₁₉

—

C₉×C₁₉

—

#4294

—

#4294

—

#737

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₁₉⋊C₃

yes

characteristic

C₃×H(3,𝔽₃)

—

—

—

—

—

—

—

—

—

AΣL(1,𝔽₂₇)

—

C₉⋊C₉

—

C₉⋊C₉

C₃×C₁₉⋊C₃

yes

complemented

C₉²

C₃×C₁₉⋊C₃

yes

characteristic, complemented

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

#6911

yes

characteristic

C₁₉⋊C₃

—

C₁₉⋊C₃

—

C₁₉⋊C₃

—

C₁₉⋊C₃

—

C₃³

—

171

—

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

171

—

3₋¹⁺²

—

171

—

H(3,𝔽₃)

—

100

H(3,𝔽₃)

—

101

H(3,𝔽₃)

—

102

H(3,𝔽₃)

—

103

H(3,𝔽₃)

—

171

—

104

C₃×C₉

—

105

C₃×C₉

—

106

C₃×C₉

—

171

—

107

C₃×C₉

C₃²⋊(C₁₉⋊C₃)

yes

—

108

C₃×C₉

C₃²×C₁₉⋊C₃

yes

characteristic

109

C₁₉

#10624851

yes

characteristic, complemented, Sylow

110

C₃²

—

513

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

171

—

114

C₃²

—

171

—

115

C₃²

—

171

—

116

C₃²

—

171

—

117

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

118

C₉

—

119

C₉

—

120

C₉

—

121

C₉

—

122

C₉

—

171

—

123

C₉

—

171

—

124

C₃

—

513

—

125

C₃

—

513

—

126

C₃

—

513

—

127

C₃

—

513

—

128

C₃

—

129

C₃

—

130

C₃

#17987515

yes

characteristic

	subgroup	G/H
center	C₃	#17987515
commutator, cosocle	C₃×C₉×C₁₉	C₃³
Frattini	C₃×C₉	C₃²×C₁₉⋊C₃
Fitting	C₁₉×#6900	C₃
socle	C₃×C₁₉	#6911
hypercenter	#6900	C₁₉⋊C₃
hypocenter	C₁₉	#10624851
LCS	G ▷ C₃×C₉×C₁₉ ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×3₋¹⁺² ◁ #6900

subgroup

G/H

center

C₃

#17987515

commutator, cosocle

C₃×C₉×C₁₉

C₃³

Frattini

C₃×C₉

C₃²×C₁₉⋊C₃

Fitting

C₁₉×#6900

C₃

C₃×C₁₉

C₁₉⋊C₃

C₁₉

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

UCS

C₁ ◁ C₃ ◁ C₃² ◁ C₃×3₋¹⁺² ◁ #6900

N	H ≅ G/N	type
1. #17987528	C₃	semidirect
2. #17987515	C₃	semidirect
3. #17987390	C₃	semidirect
4. #17987380	C₃	semidirect
5. #17987380	C₃	semidirect
6. #17987374	C₃	semidirect
7. #17987370	C₃	semidirect
8. #17987369	C₃	semidirect
9. C₁₉×#6900	C₃	semidirect
12. #13095322	C₃²	semidirect
13. #13095322	C₃²	semidirect
18. #13095320	C₃²	semidirect
24. C₁₉×C₉⋊C₉	C₃²	semidirect
25. C₉²×C₁₉	C₃²	semidirect
47. #6900	C₁₉⋊C₃	semidirect
82. C₉⋊C₉	C₃×C₁₉⋊C₃	semidirect
83. C₉²	C₃×C₁₉⋊C₃	semidirect
109. C₁₉	#10624851	semidirect

H ≅ G/N

type

1. #17987528

C₃

semidirect

2. #17987515

C₃

semidirect

3. #17987390

C₃

semidirect

4. #17987380

C₃

semidirect

5. #17987380

C₃

semidirect

6. #17987374

C₃

semidirect

7. #17987370

C₃

semidirect

8. #17987369

C₃

semidirect

9. C₁₉×#6900

C₃

semidirect

12. #13095322

C₃²

semidirect

13. #13095322

C₃²

semidirect

18. #13095320

C₃²

semidirect

24. C₁₉×C₉⋊C₉

C₃²

semidirect

25. C₉²×C₁₉

C₃²

semidirect

47. #6900

C₁₉⋊C₃

semidirect

82. C₉⋊C₉

C₃×C₁₉⋊C₃

semidirect

83. C₉²

C₃×C₁₉⋊C₃

semidirect

109. C₁₉

#10624851

semidirect

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

subgroup	index
#17987528	3
#17987515	3
#17987390	3
#17987380	3
#17987374	3
#17987370	3
#17987369	3
C₁₉×#6900	3
#10624851	19

subgroup

index

C₁₉×#6900

Subgroups — #28355825

Conjugacy classes of subgroups

Complements

Maximal subgroups