Group #28354462 — subgroups

Subgroups — #28354462

Order 13851.

320 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 52 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	#17987421	C₃	1	1	yes	maximal, characteristic, complemented
2	#17987417	C₃	1	2	yes	maximal, complemented
3	#17987310	C₃	1	9	yes	maximal, complemented
4	C₁₉×#6891	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×#10587025	—	3	1	no	—
6	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	2	no	—
7	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	2	no	—
8	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	2	no	—
9	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	6	yes	—
10	C₃²⋊#4294	—	3	2	no	—
11	C₃²⋊#4294	C₃²	1	3	yes	—
12	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
13	C₃×C₁₉×3₋¹⁺²	—	3	1	no	—
14	C₁₉×C₃²⋊C₉	—	3	2	no	—
15	C₁₉×C₃²⋊C₉	C₃²	1	3	yes	—
16	#10624665	—	19	1	no	maximal, Sylow
17	C₃³×C₁₉	—	9	2	no	—
18	C₃³×C₁₉	—	9	1	no	—
19	C₃³×C₁₉	—	9	1	no	—
20	C₃³×C₁₉	H(3,𝔽₃)	1	2	yes	—
21	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
22	C₃³×C₁₉	C₃³	1	1	yes	characteristic
23	C₃²×C₁₉⋊C₃	—	3	6	no	—
24	C₁₉×3₋¹⁺²	—	9	3	no	—
25	C₃×C₉×C₁₉	—	3	3	no	—
26	C₃²⋊(C₁₉⋊C₃)	—	9	6	no	—
27	C₃²⋊(C₁₉⋊C₃)	—	9	6	no	—
28	C₃²⋊(C₁₉⋊C₃)	—	9	6	no	—
29	C₃²⋊(C₁₉⋊C₃)	—	9	6	no	—
30	#10587025	—	9	3	no	—
31	C₃×#4294	—	3	3	no	—
32	#6891	—	19	1	no	—
33	#6891	C₁₉⋊C₃	1	1	yes	characteristic, complemented
34	C₃²⋊H(3,𝔽₃)	—	19	2	no	—
35	#6856	—	19	9	no	—
36	C₃²×C₁₉	—	3	2	no	—
37	C₃²×C₁₉	—	3	2	no	—
38	C₃²×C₁₉	—	3	4	no	—
39	C₃²×C₁₉	—	3	1	no	—
40	C₃²×C₁₉	—	3	2	no	—
41	C₃²×C₁₉	—	3	2	no	—
42	C₃²×C₁₉	—	3	1	no	—
43	C₃²×C₁₉	—	3	1	no	—
44	C₃²×C₁₉	—	3	1	no	—
45	C₃²×C₁₉	—	9	9	no	—
46	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
47	C₃×C₁₉⋊C₃	—	9	6	no	—
48	C₃×C₁₉⋊C₃	—	9	6	no	—
49	C₃×C₁₉⋊C₃	—	9	6	no	—
50	C₃×C₁₉⋊C₃	—	9	6	no	—
51	C₉×C₁₉	—	9	3	no	—
52	#4294	—	9	3	no	—
53	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
54	C₃×3₋¹⁺²	—	3	1	no	—
55	C₃×3₋¹⁺²	—	57	1	no	—
56	C₃×H(3,𝔽₃)	—	19	6	no	—
57	C₃×H(3,𝔽₃)	—	57	2	no	—
58	C₃×H(3,𝔽₃)	—	57	2	no	—
59	C₃×H(3,𝔽₃)	—	57	2	no	—
60	C₃²⋊C₉	—	3	2	no	—
61	C₃²⋊C₉	—	19	3	no	—
62	C₃²⋊C₉	—	57	2	no	—
63	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	—
64	C₃×C₁₉	—	9	1	no	—
65	C₃×C₁₉	—	9	1	no	—
66	C₃×C₁₉	—	9	2	no	—
67	C₃×C₁₉	#6909	1	2	yes	—
68	C₃×C₁₉	#6911	1	1	yes	characteristic
69	C₃×C₁₉	#6911	1	1	yes	characteristic
70	C₁₉⋊C₃	—	27	6	no	—
71	C₃³	—	9	2	no	—
72	C₃³	—	9	1	no	—
73	C₃³	—	9	1	no	—
74	C₃³	—	57	6	no	—
75	C₃³	C₃²⋊(C₁₉⋊C₃)	1	2	yes	—
76	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
77	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
78	3₋¹⁺²	—	9	3	no	—
79	3₋¹⁺²	—	171	3	no	—
80	H(3,𝔽₃)	—	171	6	no	—
81	H(3,𝔽₃)	—	171	6	no	—
82	H(3,𝔽₃)	—	171	6	no	—
83	H(3,𝔽₃)	—	171	6	no	—
84	C₃×C₉	—	3	3	no	—
85	C₃×C₉	—	57	3	no	—
86	C₁₉	#10624665	1	1	yes	characteristic, complemented, Sylow
87	C₃²	—	3	2	no	—
88	C₃²	—	3	2	no	—
89	C₃²	—	3	4	no	—
90	C₃²	—	3	2	no	—
91	C₃²	—	3	1	no	—
92	C₃²	—	3	1	no	—
93	C₃²	—	3	2	no	—
94	C₃²	—	3	1	no	—
95	C₃²	—	3	1	no	—
96	C₃²	—	9	9	no	—
97	C₃²	—	171	6	no	—
98	C₃²	—	171	6	no	—
99	C₃²	—	171	6	no	—
100	C₃²	—	171	6	no	—
101	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
102	C₉	—	9	3	no	—
103	C₉	—	171	3	no	—
104	C₃	—	513	6	no	—
105	C₃	—	9	1	no	—
106	C₃	—	9	1	no	—
107	C₃	—	9	2	no	—
108	C₃	#17987506	1	2	yes	—
109	C₃	#17987514	1	1	yes	characteristic
110	C₃	#17987515	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17987421

C₃

yes

maximal, characteristic, complemented

#17987417

C₃

yes

maximal, complemented

#17987310

C₃

yes

maximal, complemented

C₁₉×#6891

C₃

yes

maximal, characteristic, complemented

C₃×#10587025

—

C₃×C₃²⋊(C₁₉⋊C₃)

—

C₃×C₃²⋊(C₁₉⋊C₃)

—

C₃×C₃²⋊(C₁₉⋊C₃)

—

C₃×C₃²⋊(C₁₉⋊C₃)

C₃²

yes

—

C₃²⋊#4294

—

C₃²⋊#4294

C₃²

yes

—

C₃⁴×C₁₉

C₃²

yes

characteristic

C₃×C₁₉×3₋¹⁺²

—

C₁₉×C₃²⋊C₉

—

C₁₉×C₃²⋊C₉

C₃²

yes

—

#10624665

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

H(3,𝔽₃)

yes

—

C₃³×C₁₉

H(3,𝔽₃)

yes

characteristic

C₃³×C₁₉

C₃³

yes

characteristic

C₃²×C₁₉⋊C₃

—

C₁₉×3₋¹⁺²

—

C₃×C₉×C₁₉

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

—

—

—

C₁₉⋊C₃

yes

characteristic, complemented

C₃²⋊H(3,𝔽₃)

—

#6856

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₉×C₁₉

—

#4294

—

C₃⁴

C₃×C₁₉⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₁₉⋊C₃

yes

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

#6909

yes

—

C₃×C₁₉

yes

C₃×C₁₉

yes

C₁₉⋊C₃

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

—

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

171

—

H(3,𝔽₃)

—

171

—

H(3,𝔽₃)

—

171

—

H(3,𝔽₃)

—

171

—

H(3,𝔽₃)

—

171

—

C₃×C₉

—

C₃×C₉

—

C₁₉

#10624665

yes

characteristic, complemented, Sylow

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

171

—

C₃²

—

171

—

C₃²

—

171

—

100

C₃²

—

171

—

101

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

102

C₉

—

103

C₉

—

171

—

104

C₃

—

513

—

105

C₃

—

106

C₃

—

107

C₃

—

108

C₃

#17987506

yes

—

109

C₃

#17987514

yes

characteristic

110

C₃

#17987515

yes

characteristic

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)
commutator, cosocle	C₃³×C₁₉	C₃³
Frattini	C₃³	C₃²×C₁₉⋊C₃
Fitting	C₁₉×#6891	C₃
socle	C₃²×C₁₉	C₃×H(3,𝔽₃)
hypercenter	#6891	C₁₉⋊C₃
hypocenter	C₁₉	#10624665
LCS	G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ #6891

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

commutator, cosocle

C₃³×C₁₉

C₃³

Frattini

C₃³

C₃²×C₁₉⋊C₃

Fitting

C₁₉×#6891

C₃

socle

C₃²×C₁₉

C₃×H(3,𝔽₃)

C₁₉⋊C₃

C₁₉

G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ #6891

N	H ≅ G/N	type
1. #17987421	C₃	semidirect
2. #17987417	C₃	semidirect
3. #17987310	C₃	semidirect
4. C₁₉×#6891	C₃	semidirect
33. #6891	C₁₉⋊C₃	semidirect
86. C₁₉	#10624665	semidirect

H ≅ G/N

type

1. #17987421

C₃

semidirect

2. #17987417

C₃

semidirect

3. #17987310

C₃

semidirect

4. C₁₉×#6891

C₃

semidirect

33. #6891

C₁₉⋊C₃

semidirect

86. C₁₉

#10624665

semidirect

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

subgroup	index
#17987421	3
#17987417	3
#17987310	3
C₁₉×#6891	3
#10624665	19

subgroup

index

C₁₉×#6891

Subgroups — #28354462

Conjugacy classes of subgroups

Complements

Maximal subgroups