Group #28354574 — subgroups

Subgroups — #28354574

Order 13851.

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

normal subgroups only

This group has 184 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	#17987435	C₃	1	2	yes	maximal, complemented
3	#17987423	C₃	1	1	yes	maximal, characteristic
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	1	no	—
13	#13095305	—	3	4	no	—
14	#13095305	—	3	2	no	—
15	#13095305	—	3	2	no	—
16	C₃²⋊#4294	—	3	2	no	—
17	C₃²⋊#4294	C₃²	1	1	yes	characteristic
18	C₃²⋊#4294	C₃²	1	1	yes	characteristic
19	C₃²⋊#4294	C₃²	1	2	yes	—
20	C₃²⋊#4294	C₃²	1	2	yes	—
21	C₃²⋊#4294	C₃²	1	2	yes	—
22	C₃×C₁₉×3₋¹⁺²	C₃²	1	2	yes	—
23	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
24	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
25	#10624676	—	19	1	no	maximal, Sylow
26	C₃³×C₁₉	—	3	1	no	—
27	C₃³×C₁₉	C₃³	1	1	yes	characteristic
28	C₁₉×3₋¹⁺²	—	9	2	no	—
29	C₁₉×H(3,𝔽₃)	—	9	1	no	—
30	C₃×C₉×C₁₉	—	3	1	no	—
31	C₃×C₉×C₁₉	—	3	2	no	—
32	#10587025	—	9	3	no	—
33	C₃×#4294	—	3	1	no	—
34	C₃×#4294	—	3	1	no	—
35	C₃×#4294	—	3	2	no	—
36	C₃×#4294	—	3	2	no	—
37	C₃×#4294	—	3	2	no	—
38	C₃×#4294	—	9	1	no	—
39	C₃×#4294	H(3,𝔽₃)	1	2	yes	—
40	C₃×#4294	H(3,𝔽₃)	1	1	yes	characteristic
41	#6895	—	19	1	no	—
42	#6894	—	19	2	no	—
43	C₉⋊H(3,𝔽₃)	—	19	1	no	—
44	#6862	—	19	1	no	—
45	#6861	—	19	1	no	—
46	#6859	—	19	2	no	—
47	#6858	—	19	2	no	—
48	#6858	—	19	2	no	—
49	#6857	C₁₉⋊C₃	1	1	yes	characteristic
50	C₃²×C₁₉	—	3	1	no	—
51	C₃²×C₁₉	—	3	2	no	—
52	C₃²×C₁₉	—	3	1	no	—
53	C₃²×C₁₉	—	9	1	no	—
54	C₃²×C₁₉	—	9	2	no	—
55	C₃²×C₁₉	—	9	1	no	—
56	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
57	C₉×C₁₉	—	9	1	no	—
58	C₉×C₁₉	—	9	2	no	—
59	#4294	—	3	2	no	—
60	#4294	—	3	1	no	—
61	#4294	—	9	2	no	—
62	#4294	—	9	2	no	—
63	#4294	—	9	2	no	—
64	#4294	—	9	1	no	—
65	#4294	—	9	1	no	—
66	C₃×3₋¹⁺²	—	57	1	no	—
67	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	2	yes	—
68	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	1	yes	characteristic
69	C₃²×C₉	—	19	1	no	—
70	C₉⋊C₉	—	57	4	no	—
71	C₉⋊C₉	—	57	2	no	—
72	C₉⋊C₉	—	57	2	no	—
73	C₉⋊C₉	—	57	1	no	—
74	C₃²⋊C₉	—	19	2	no	—
75	C₃²⋊C₉	—	19	1	no	—
76	C₃²⋊C₉	—	19	1	no	—
77	C₃²⋊C₉	—	19	2	no	—
78	C₃²⋊C₉	—	19	2	no	—
79	C₃²⋊C₉	—	57	2	no	—
80	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
81	C₃×C₁₉	—	9	1	no	—
82	C₃×C₁₉	—	27	1	no	—
83	C₃×C₁₉	#6910	1	1	yes	characteristic
84	C₃×C₁₉	#6913	1	1	yes	characteristic
85	C₃×C₁₉	#6913	1	2	yes	—
86	C₃³	—	3	1	no	—
87	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
88	3₋¹⁺²	—	9	2	no	—
89	3₋¹⁺²	—	171	3	no	—
90	H(3,𝔽₃)	—	9	1	no	—
91	C₃×C₉	—	3	1	no	—
92	C₃×C₉	—	3	2	no	—
93	C₃×C₉	—	19	2	no	—
94	C₃×C₉	—	19	1	no	—
95	C₃×C₉	—	57	1	no	—
96	C₃×C₉	—	57	1	no	—
97	C₃×C₉	—	57	2	no	—
98	C₃×C₉	—	57	2	no	—
99	C₃×C₉	—	57	2	no	—
100	C₃×C₉	—	171	1	no	—
101	C₁₉	#10624676	1	1	yes	characteristic, complemented, Sylow
102	C₃²	—	3	1	no	—
103	C₃²	—	3	1	no	—
104	C₃²	—	3	2	no	—
105	C₃²	—	9	1	no	—
106	C₃²	—	9	1	no	—
107	C₃²	—	9	2	no	—
108	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
109	C₉	—	9	1	no	—
110	C₉	—	9	2	no	—
111	C₉	—	57	1	no	—
112	C₉	—	57	2	no	—
113	C₉	—	171	1	no	—
114	C₉	—	171	1	no	—
115	C₉	—	171	2	no	—
116	C₉	—	171	2	no	—
117	C₉	—	171	2	no	—
118	C₃	—	9	1	no	—
119	C₃	—	27	1	no	—
120	C₃	#17987507	1	1	yes	characteristic
121	C₃	#17987520	1	1	yes	characteristic
122	C₃	#17987521	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17987442

C₃

yes

maximal, characteristic, complemented

#17987435

C₃

yes

maximal, complemented

#17987423

C₃

yes

maximal, characteristic

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

characteristic

C₃²⋊#4294

C₃²

yes

characteristic

C₃²⋊#4294

C₃²

yes

—

C₃²⋊#4294

C₃²

yes

—

C₃²⋊#4294

C₃²

yes

—

C₃×C₁₉×3₋¹⁺²

C₃²

yes

—

C₃×C₁₉×H(3,𝔽₃)

C₃²

yes

characteristic

C₁₉×C₃²⋊C₉

C₃²

yes

characteristic

#10624676

—

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)

—

—

—

—

—

—

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

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

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

—

C₃×C₉

—

C₃×C₉

—

100

C₃×C₉

—

171

—

101

C₁₉

#10624676

yes

characteristic, complemented, Sylow

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

H(3,𝔽₃)×C₁₉⋊C₃

yes

characteristic

109

C₉

—

110

C₉

—

111

C₉

—

112

C₉

—

113

C₉

—

171

—

114

C₉

—

171

—

115

C₉

—

171

—

116

C₉

—

171

—

117

C₉

—

171

—

118

C₃

—

119

C₃

—

120

C₃

#17987507

yes

characteristic

121

C₃

#17987520

yes

characteristic

122

C₃

#17987521

yes

—

	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	#10624676
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
2. #17987435	C₃	semidirect
4. C₁₉⋊#6862	C₃	semidirect
5. #17987324	C₃	semidirect
7. #17987314	C₃	semidirect
8. #17987314	C₃	semidirect
101. C₁₉	#10624676	semidirect

H ≅ G/N

type

1. #17987442

C₃

semidirect

2. #17987435

C₃

semidirect

4. C₁₉⋊#6862

C₃

semidirect

5. #17987324

C₃

semidirect

7. #17987314

C₃

semidirect

8. #17987314

C₃

semidirect

101. C₁₉

#10624676

semidirect

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

subgroup	index
#17987442	3
#17987435	3
#17987423	3
C₁₉⋊#6862	3
#17987324	3
#17987318	3
#17987314	3
C₁₉×#6857	3
#10624676	19

subgroup

index

C₁₉⋊#6862

C₁₉×#6857

Subgroups — #28354574

Conjugacy classes of subgroups

Complements

Maximal subgroups