Group #28354058 — subgroups

Subgroups — #28354058

Order 13851.

200 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 49 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	#17987442	C₃	1	1	yes	maximal, characteristic, complemented
2	#17987423	C₃	1	1	yes	maximal, characteristic
3	3₋¹⁺²×#4294	C₃	1	2	yes	maximal, complemented
4	C₁₉⋊#6862	C₃	1	1	yes	maximal, characteristic, complemented
5	#17987324	C₃	1	1	yes	maximal, characteristic, complemented
6	#17987318	C₃	1	2	yes	maximal
7	#17987314	C₃	1	2	yes	maximal, complemented
8	#17987314	C₃	1	2	yes	maximal, complemented
9	C₁₉×#6857	C₃	1	1	yes	maximal, characteristic
10	C₃×#10587025	—	3	1	no	—
11	C₃²×#4294	C₃²	1	1	yes	characteristic
12	#13095307	—	3	2	no	—
13	#13095307	—	3	1	no	—
14	#13095306	—	3	2	no	—
15	#13095305	—	3	2	no	—
16	C₃²⋊#4294	—	3	2	no	—
17	C₃²⋊#4294	C₃²	1	2	yes	—
18	C₃²⋊#4294	C₃²	1	2	yes	—
19	C₃²⋊#4294	C₃²	1	1	yes	characteristic
20	C₃²⋊#4294	C₃²	1	1	yes	characteristic
21	C₃²⋊#4294	C₃²	1	2	yes	—
22	C₉×#4294	—	3	2	no	—
23	C₃×C₁₉×3₋¹⁺²	C₃²	1	2	yes	—
24	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
25	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
26	#10624625	—	19	1	no	maximal, Sylow
27	C₃³×C₁₉	—	3	1	no	—
28	C₃³×C₁₉	C₃³	1	1	yes	characteristic
29	C₁₉×3₋¹⁺²	—	3	6	no	—
30	C₁₉×H(3,𝔽₃)	—	9	1	no	—
31	C₃×C₉×C₁₉	—	3	2	no	—
32	C₃×C₉×C₁₉	—	3	1	no	—
33	#10587025	—	9	3	no	—
34	C₃×#4294	—	3	2	no	—
35	C₃×#4294	—	3	2	no	—
36	C₃×#4294	—	3	1	no	—
37	C₃×#4294	—	3	1	no	—
38	C₃×#4294	—	3	2	no	—
39	C₃×#4294	—	9	1	no	—
40	C₃×#4294	H(3,𝔽₃)	1	2	yes	—
41	C₃×#4294	H(3,𝔽₃)	1	1	yes	characteristic
42	#6895	—	19	1	no	—
43	C₉⋊H(3,𝔽₃)	—	19	1	no	—
44	C₉×3₋¹⁺²	—	19	2	no	—
45	#6862	—	19	1	no	—
46	#6861	—	19	1	no	—
47	#6859	—	19	2	no	—
48	#6858	—	19	2	no	—
49	#6858	—	19	2	no	—
50	#6857	C₁₉⋊C₃	1	1	yes	characteristic
51	C₃²×C₁₉	—	3	1	no	—
52	C₃²×C₁₉	—	3	2	no	—
53	C₃²×C₁₉	—	3	1	no	—
54	C₃²×C₁₉	—	9	1	no	—
55	C₃²×C₁₉	—	9	2	no	—
56	C₃²×C₁₉	—	9	1	no	—
57	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
58	C₉×C₁₉	—	3	6	no	—
59	C₉×C₁₉	—	9	1	no	—
60	#4294	—	3	2	no	—
61	#4294	—	3	1	no	—
62	#4294	—	9	2	no	—
63	#4294	—	9	2	no	—
64	#4294	—	9	2	no	—
65	#4294	—	9	1	no	—
66	#4294	—	9	1	no	—
67	C₃×3₋¹⁺²	—	57	1	no	—
68	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	2	yes	—
69	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	1	yes	characteristic
70	C₃²×C₉	—	19	1	no	—
71	C₉⋊C₉	—	57	2	no	—
72	C₉⋊C₉	—	57	2	no	—
73	C₉⋊C₉	—	57	2	no	—
74	C₉⋊C₉	—	57	1	no	—
75	C₃²⋊C₉	—	19	2	no	—
76	C₃²⋊C₉	—	19	1	no	—
77	C₃²⋊C₉	—	19	1	no	—
78	C₃²⋊C₉	—	19	2	no	—
79	C₃²⋊C₉	—	19	2	no	—
80	C₃²⋊C₉	—	57	2	no	—
81	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
82	C₉²	—	57	2	no	—
83	C₃×C₁₉	—	9	1	no	—
84	C₃×C₁₉	—	27	1	no	—
85	C₃×C₁₉	#6908	1	2	yes	—
86	C₃×C₁₉	#6910	1	1	yes	characteristic
87	C₃×C₁₉	#6913	1	1	yes	characteristic
88	C₃³	—	3	1	no	—
89	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
90	3₋¹⁺²	—	3	6	no	—
91	3₋¹⁺²	—	171	3	no	—
92	H(3,𝔽₃)	—	9	1	no	—
93	C₃×C₉	—	3	2	no	—
94	C₃×C₉	—	3	1	no	—
95	C₃×C₉	—	19	2	no	—
96	C₃×C₉	—	19	1	no	—
97	C₃×C₉	—	57	2	no	—
98	C₃×C₉	—	57	2	no	—
99	C₃×C₉	—	57	1	no	—
100	C₃×C₉	—	57	1	no	—
101	C₃×C₉	—	57	2	no	—
102	C₃×C₉	—	171	1	no	—
103	C₁₉	#10624625	1	1	yes	characteristic, complemented, Sylow
104	C₃²	—	3	2	no	—
105	C₃²	—	3	1	no	—
106	C₃²	—	3	1	no	—
107	C₃²	—	9	2	no	—
108	C₃²	—	9	1	no	—
109	C₃²	—	9	1	no	—
110	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
111	C₉	—	3	6	no	—
112	C₉	—	9	1	no	—
113	C₉	—	57	1	no	—
114	C₉	—	57	2	no	—
115	C₉	—	171	2	no	—
116	C₉	—	171	2	no	—
117	C₉	—	171	1	no	—
118	C₉	—	171	1	no	—
119	C₉	—	171	2	no	—
120	C₃	—	9	1	no	—
121	C₃	—	27	1	no	—
122	C₃	#17987496	1	2	yes	—
123	C₃	#17987507	1	1	yes	characteristic
124	C₃	#17987521	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17987442

C₃

yes

maximal, characteristic, complemented

#17987423

C₃

yes

maximal, characteristic

3₋¹⁺²×#4294

C₃

yes

maximal, complemented

C₁₉⋊#6862

C₃

yes

maximal, characteristic, complemented

#17987324

C₃

yes

maximal, characteristic, complemented

#17987318

C₃

yes

maximal

#17987314

C₃

yes

maximal, complemented

#17987314

C₃

yes

maximal, complemented

C₁₉×#6857

C₃

yes

maximal, characteristic

C₃×#10587025

—

C₃²×#4294

C₃²

yes

—

—

—

—

C₃²⋊#4294

—

C₃²⋊#4294

C₃²

yes

—

C₃²⋊#4294

C₃²

yes

—

C₃²⋊#4294

C₃²

yes

characteristic

C₃²⋊#4294

C₃²

yes

characteristic

C₃²⋊#4294

C₃²

yes

—

C₉×#4294

—

C₃×C₁₉×3₋¹⁺²

C₃²

yes

—

C₃×C₁₉×H(3,𝔽₃)

C₃²

yes

characteristic

C₁₉×C₃²⋊C₉

C₃²

yes

characteristic

#10624625

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

C₃³

yes

characteristic

C₁₉×3₋¹⁺²

—

C₁₉×H(3,𝔽₃)

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

—

—

—

—

—

—

—

yes

—

yes

—

C₉⋊H(3,𝔽₃)

—

C₉×3₋¹⁺²

—

—

—

—

—

—

C₁₉⋊C₃

yes

characteristic

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₁₉

—

C₉×C₁₉

—

—

—

—

—

—

—

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₁₉⋊C₃

yes

—

C₃×H(3,𝔽₃)

C₃×C₁₉⋊C₃

yes

characteristic

C₃²×C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₁₉⋊C₃

yes

characteristic

C₉²

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

#6908

yes

—

C₃×C₁₉

yes

C₃×C₁₉

yes

C₃³

—

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

171

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

100

C₃×C₉

—

101

C₃×C₉

—

102

C₃×C₉

—

171

—

103

C₁₉

#10624625

yes

characteristic, complemented, Sylow

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

H(3,𝔽₃)×C₁₉⋊C₃

yes

characteristic

111

C₉

—

112

C₉

—

113

C₉

—

114

C₉

—

115

C₉

—

171

—

116

C₉

—

171

—

117

C₉

—

171

—

118

C₉

—

171

—

119

C₉

—

171

—

120

C₃

—

121

C₃

—

122

C₃

#17987496

yes

—

123

C₃

#17987507

yes

characteristic

124

C₃

#17987521

yes

characteristic

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₁₉⋊C₃
commutator, cosocle	C₃³×C₁₉	C₃³
Frattini	C₃³	C₃²×C₁₉⋊C₃
Fitting	C₁₉×#6857	C₃
socle	C₃²×C₁₉	C₃×H(3,𝔽₃)
prime radical	C₃×C₁₉×H(3,𝔽₃)	C₃²
hypercenter	#6857	C₁₉⋊C₃
hypocenter	C₁₉	#10624625
LCS	G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ #6857

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₁₉⋊C₃

commutator, cosocle

C₃³×C₁₉

C₃³

Frattini

C₃³

C₃²×C₁₉⋊C₃

Fitting

C₁₉×#6857

C₃

socle

C₃²×C₁₉

C₃×H(3,𝔽₃)

prime radical

C₃×C₁₉×H(3,𝔽₃)

C₃²

C₁₉⋊C₃

C₁₉

G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ #6857

N	H ≅ G/N	type
1. #17987442	C₃	semidirect
3. 3₋¹⁺²×#4294	C₃	semidirect
4. C₁₉⋊#6862	C₃	semidirect
5. #17987324	C₃	semidirect
7. #17987314	C₃	semidirect
8. #17987314	C₃	semidirect
103. C₁₉	#10624625	semidirect

H ≅ G/N

type

1. #17987442

C₃

semidirect

3. 3₋¹⁺²×#4294

C₃

semidirect

4. C₁₉⋊#6862

C₃

semidirect

5. #17987324

C₃

semidirect

7. #17987314

C₃

semidirect

8. #17987314

C₃

semidirect

103. C₁₉

#10624625

semidirect

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

subgroup	index
#17987442	3
#17987423	3
3₋¹⁺²×#4294	3
C₁₉⋊#6862	3
#17987324	3
#17987318	3
#17987314	3
C₁₉×#6857	3
#10624625	19

subgroup

index

#17987442

#17987423

3₋¹⁺²×#4294

C₁₉⋊#6862

C₁₉×#6857

Subgroups — #28354058

Conjugacy classes of subgroups

Complements

Maximal subgroups