Group #28354533 — subgroups

Subgroups — #28354533

Order 13851.

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

normal subgroups only

This group has 236 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
2	#17987421	C₃	1	1	yes	maximal, characteristic
3	#17987421	C₃	1	1	yes	maximal, characteristic
4	#17987321	C₃	1	9	yes	maximal, complemented
5	C₁₉×#6891	C₃	1	1	yes	maximal, characteristic
6	C₃×#10587025	—	3	1	no	—
7	C₃×#10587025	—	3	1	no	—
8	C₃×#10587025	C₃²	1	3	yes	—
9	C₃²⋊#4294	—	3	2	no	—
10	C₃²⋊#4294	—	3	1	no	—
11	C₃²⋊#4294	—	3	2	no	—
12	C₃²⋊#4294	—	3	2	no	—
13	C₃²⋊#4294	C₃²	1	3	yes	—
14	C₃²⋊#4294	C₃²	1	3	yes	—
15	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
16	C₃×C₁₉×3₋¹⁺²	C₃²	1	3	yes	—
17	C₁₉×C₃²⋊C₉	—	3	2	no	—
18	C₁₉×C₃²⋊C₉	—	3	1	no	—
19	#10624673	—	19	1	no	maximal, Sylow
20	C₃³×C₁₉	—	9	2	no	—
21	C₃³×C₁₉	—	9	1	no	—
22	C₃³×C₁₉	—	9	1	no	—
23	C₃³×C₁₉	H(3,𝔽₃)	1	2	yes	—
24	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
25	C₃³×C₁₉	C₃³	1	1	yes	characteristic
26	C₁₉×3₋¹⁺²	—	9	3	no	—
27	C₃×C₉×C₁₉	—	3	3	no	—
28	#10587025	—	9	3	no	—
29	#10587025	—	9	3	no	—
30	#10587025	—	9	3	no	—
31	C₃×#4294	—	3	3	no	—
32	C₃×#4294	—	3	3	no	—
33	C₃×#4294	—	3	3	no	—
34	#6891	—	19	1	no	—
35	#6891	—	19	1	no	—
36	#6891	—	19	1	no	—
37	#6891	C₁₉⋊C₃	1	1	yes	characteristic
38	#6860	—	19	9	no	—
39	C₃²×C₁₉	—	3	4	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	—	3	1	no	—
46	C₃²×C₁₉	—	3	2	no	—
47	C₃²×C₁₉	—	3	2	no	—
48	C₃²×C₁₉	—	9	9	no	—
49	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
50	C₉×C₁₉	—	9	3	no	—
51	#4294	—	9	3	no	—
52	#4294	—	9	3	no	—
53	#4294	—	9	3	no	—
54	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
55	C₃×3₋¹⁺²	—	19	3	no	—
56	C₃×3₋¹⁺²	—	57	1	no	—
57	C₃×3₋¹⁺²	—	57	1	no	—
58	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	3	yes	—
59	C₃²⋊C₉	—	3	2	no	—
60	C₃²⋊C₉	—	3	1	no	—
61	C₃²⋊C₉	—	19	3	no	—
62	C₃²⋊C₉	—	19	3	no	—
63	C₃²⋊C₉	—	57	2	no	—
64	C₃²⋊C₉	—	57	2	no	—
65	C₃²⋊C₉	—	57	2	no	—
66	C₃²⋊C₉	—	57	1	no	—
67	C₃×C₁₉	—	9	1	no	—
68	C₃×C₁₉	—	9	2	no	—
69	C₃×C₁₉	—	9	1	no	—
70	C₃×C₁₉	#6909	1	1	yes	characteristic
71	C₃×C₁₉	#6909	1	1	yes	characteristic
72	C₃×C₁₉	#6912	1	2	yes	—
73	C₃³	—	9	1	no	—
74	C₃³	—	9	1	no	—
75	C₃³	—	9	2	no	—
76	C₃³	C₃²⋊(C₁₉⋊C₃)	1	2	yes	—
77	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
78	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
79	3₋¹⁺²	—	9	3	no	—
80	3₋¹⁺²	—	171	3	no	—
81	3₋¹⁺²	—	171	3	no	—
82	3₋¹⁺²	—	171	3	no	—
83	C₃×C₉	—	3	3	no	—
84	C₃×C₉	—	57	3	no	—
85	C₃×C₉	—	57	3	no	—
86	C₃×C₉	—	57	3	no	—
87	C₁₉	#10624673	1	1	yes	characteristic, complemented, Sylow
88	C₃²	—	3	1	no	—
89	C₃²	—	3	1	no	—
90	C₃²	—	3	2	no	—
91	C₃²	—	3	1	no	—
92	C₃²	—	3	1	no	—
93	C₃²	—	3	2	no	—
94	C₃²	—	3	4	no	—
95	C₃²	—	3	2	no	—
96	C₃²	—	3	2	no	—
97	C₃²	—	9	9	no	—
98	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
99	C₉	—	9	3	no	—
100	C₉	—	171	3	no	—
101	C₉	—	171	3	no	—
102	C₉	—	171	3	no	—
103	C₃	—	9	1	no	—
104	C₃	—	9	1	no	—
105	C₃	—	9	2	no	—
106	C₃	#17987505	1	1	yes	characteristic
107	C₃	#17987506	1	1	yes	characteristic
108	C₃	#17987518	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17987421

C₃

yes

maximal, characteristic

#17987421

C₃

yes

maximal, characteristic

#17987421

C₃

yes

maximal, characteristic

#17987321

C₃

yes

maximal, complemented

C₁₉×#6891

C₃

yes

maximal, characteristic

C₃×#10587025

—

C₃×#10587025

—

C₃×#10587025

C₃²

yes

—

C₃²⋊#4294

—

C₃²⋊#4294

—

C₃²⋊#4294

—

C₃²⋊#4294

—

C₃²⋊#4294

C₃²

yes

—

C₃²⋊#4294

C₃²

yes

—

C₃⁴×C₁₉

C₃²

yes

characteristic

C₃×C₁₉×3₋¹⁺²

C₃²

yes

—

C₁₉×C₃²⋊C₉

—

C₁₉×C₃²⋊C₉

—

#10624673

—

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

C₁₉×3₋¹⁺²

—

C₃×C₉×C₁₉

—

—

—

—

—

—

—

—

—

—

C₁₉⋊C₃

yes

characteristic

#6860

—

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

C₉×C₁₉

—

#4294

—

#4294

—

#4294

—

C₃⁴

C₃×C₁₉⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₁₉⋊C₃

yes

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

yes

C₃×C₁₉

yes

C₃×C₁₉

yes

—

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

—

3₋¹⁺²

—

171

—

3₋¹⁺²

—

171

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₁₉

#10624673

yes

characteristic, complemented, Sylow

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

C₉

—

100

C₉

—

171

—

101

C₉

—

171

—

102

C₉

—

171

—

103

C₃

—

104

C₃

—

105

C₃

—

106

C₃

#17987505

yes

characteristic

107

C₃

#17987506

yes

characteristic

108

C₃

#17987518

yes

—

	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)
prime radical	C₃⁴×C₁₉	C₃²
hypercenter	#6891	C₁₉⋊C₃
hypocenter	C₁₉	#10624673
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)

prime radical

C₃⁴×C₁₉

C₃²

C₁₉⋊C₃

C₁₉

G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ #6891

N	H ≅ G/N	type
4. #17987321	C₃	semidirect
87. C₁₉	#10624673	semidirect

H ≅ G/N

type

4. #17987321

C₃

semidirect

87. C₁₉

#10624673

semidirect

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

subgroup	index
#17987421	3
#17987321	3
C₁₉×#6891	3
#10624673	19

subgroup

index

#17987421

#17987321

C₁₉×#6891

#10624673

Subgroups — #28354533

Conjugacy classes of subgroups

Complements

Maximal subgroups