Group #28356893 — subgroups

Subgroups — #28356893

Order 13851.

332 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 90 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	C₃×#13095342	C₃	1	2	yes	maximal, complemented
2	C₃²×#10587020	C₃	1	1	yes	maximal, characteristic, complemented
3	#17987557	C₃	1	9	yes	maximal, complemented
4	C₃×#13095333	C₃	1	1	yes	maximal, characteristic
5	#13095342	—	3	6	no	—
6	C₃×#10587020	—	3	3	no	—
7	C₃×#10587020	C₃²	1	3	yes	—
8	C₃×#10587020	C₃²	1	6	yes	—
9	#13095333	—	3	3	no	—
10	#13095333	—	3	6	no	—
11	C₃×#10587025	C₉	1	1	yes	characteristic
12	C₃×#10587022	C₉	1	3	yes	—
13	C₃×#10587023	C₉	1	3	yes	—
14	C₃×C₃²⋊(C₁₉⋊C₃)	C₉	1	1	yes	characteristic
15	C₃²×#4294	C₃²	1	1	yes	characteristic
16	C₃×C₉×C₁₉⋊C₃	C₃²	1	3	yes	—
17	C₃×C₁₉×3₋¹⁺²	C₉	1	1	yes	characteristic
18	#10624684	—	19	1	no	maximal, Sylow
19	C₃³×C₁₉	C₃×C₉	1	1	yes	characteristic
20	C₃²×C₁₉⋊C₃	C₃×C₉	1	3	yes	—
21	C₁₉×3₋¹⁺²	—	3	3	no	—
22	C₃×C₉×C₁₉	C₃×C₉	1	3	yes	—
23	C₃²⋊(C₁₉⋊C₃)	—	3	3	no	—
24	#10587025	—	3	3	no	—
25	C₃×#4294	C₃×C₉	1	3	yes	—
26	C₃×#4294	C₃×C₉	1	1	yes	characteristic
27	C₃×#4294	C₃×C₉	1	1	yes	characteristic
28	C₃×#4294	H(3,𝔽₃)	1	3	yes	—
29	C₃×#4294	3₋¹⁺²	1	6	yes	—
30	C₃×#4294	C₃³	1	1	yes	characteristic
31	#10587023	—	3	9	no	—
32	#10587022	—	3	9	no	—
33	C₉×C₁₉⋊C₃	—	3	3	no	—
34	C₉×C₁₉⋊C₃	—	3	6	no	—
35	#10587020	—	3	3	no	—
36	#10587020	—	3	6	no	—
37	C₃×#737	—	19	1	no	—
38	C₃×C₂₇⋊C₃	—	19	2	no	—
39	C₃²×C₂₇	—	19	1	no	—
40	C₃²⋊C₂₇	—	19	9	no	—
41	C₃²×C₁₉	—	3	3	no	—
42	C₃²×C₁₉	C₃²⋊C₉	1	3	yes	—
43	C₃²×C₁₉	C₃²×C₉	1	1	yes	characteristic
44	C₃×C₁₉⋊C₃	—	3	3	no	—
45	C₃×C₁₉⋊C₃	—	3	9	no	—
46	C₉×C₁₉	—	3	3	no	—
47	#4294	—	3	3	no	—
48	#4294	C₃²⋊C₉	1	3	yes	—
49	#4294	C₃²⋊C₉	1	3	yes	—
50	#4294	C₃×H(3,𝔽₃)	1	1	yes	characteristic
51	#4294	C₃×3₋¹⁺²	1	2	yes	—
52	#737	—	57	3	no	—
53	#737	—	57	6	no	—
54	C₃×3₋¹⁺²	—	19	1	no	—
55	C₃×3₋¹⁺²	—	19	3	no	—
56	C₃×3₋¹⁺²	—	19	3	no	—
57	C₃×3₋¹⁺²	#4296	1	1	yes	characteristic
58	C₃×H(3,𝔽₃)	—	19	1	no	—
59	C₃²×C₉	—	19	1	no	—
60	C₃²×C₉	—	19	3	no	—
61	C₂₇⋊C₃	—	57	6	no	—
62	C₃×C₂₇	—	19	3	no	—
63	C₃×C₂₇	—	19	6	no	—
64	C₃×C₂₇	—	57	3	no	—
65	C₃×C₁₉	—	3	3	no	—
66	C₃×C₁₉	C₃×C₃²⋊C₉	1	1	yes	characteristic
67	C₃×C₁₉	#6903	1	3	yes	—
68	C₁₉⋊C₃	—	9	3	no	—
69	C₃³	—	19	3	no	—
70	C₃³	C₃×#4296	1	1	yes	characteristic
71	3₋¹⁺²	—	3	3	no	—
72	3₋¹⁺²	—	57	9	no	—
73	3₋¹⁺²	—	57	9	no	—
74	3₋¹⁺²	—	57	3	no	—
75	H(3,𝔽₃)	—	57	3	no	—
76	C₃×C₉	—	19	3	no	—
77	C₃×C₉	—	19	6	no	—
78	C₃×C₉	—	19	1	no	—
79	C₃×C₉	—	19	1	no	—
80	C₃×C₉	—	19	1	no	—
81	C₃×C₉	—	19	3	no	—
82	C₃×C₉	—	57	3	no	—
83	C₃×C₉	—	57	6	no	—
84	C₃×C₉	C₃×#4296	1	3	yes	—
85	C₂₇	—	57	3	no	—
86	C₂₇	—	57	6	no	—
87	C₁₉	#10624684	1	1	yes	characteristic, complemented, Sylow
88	C₃²	—	3	3	no	—
89	C₃²	—	57	3	no	—
90	C₃²	—	57	9	no	—
91	C₃²	C₃²⋊#4296	1	3	yes	—
92	C₃²	C₃²×#4296	1	1	yes	characteristic
93	C₉	—	3	3	no	—
94	C₉	—	19	1	no	—
95	C₉	—	19	2	no	—
96	C₉	—	19	3	no	—
97	C₉	—	19	3	no	—
98	C₉	—	57	3	no	—
99	C₃	—	3	3	no	—
100	C₃	—	171	3	no	—
101	C₃	C₃×C₃²⋊#4296	1	1	yes	characteristic
102	C₃	#17987599	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#13095342

C₃

yes

maximal, complemented

C₃²×#10587020

C₃

yes

maximal, characteristic, complemented

#17987557

C₃

yes

maximal, complemented

C₃×#13095333

C₃

yes

maximal, characteristic

#13095342

—

C₃×#10587020

—

C₃×#10587020

C₃²

yes

—

C₃×#10587020

C₃²

yes

—

#13095333

—

#13095333

—

C₃×#10587025

C₉

yes

characteristic

C₃×#10587022

C₉

yes

—

C₃×#10587023

C₉

yes

—

C₃×C₃²⋊(C₁₉⋊C₃)

C₉

yes

characteristic

C₃²×#4294

C₃²

yes

characteristic

C₃×C₉×C₁₉⋊C₃

C₃²

yes

—

C₃×C₁₉×3₋¹⁺²

C₉

yes

characteristic

#10624684

—

maximal, Sylow

C₃³×C₁₉

C₃×C₉

yes

characteristic

C₃²×C₁₉⋊C₃

C₃×C₉

yes

—

C₁₉×3₋¹⁺²

—

C₃×C₉×C₁₉

C₃×C₉

yes

—

C₃²⋊(C₁₉⋊C₃)

—

#10587025

—

C₃×#4294

C₃×C₉

yes

—

C₃×#4294

C₃×C₉

yes

characteristic

C₃×#4294

C₃×C₉

yes

characteristic

C₃×#4294

H(3,𝔽₃)

yes

—

C₃×#4294

3₋¹⁺²

yes

—

C₃×#4294

C₃³

yes

characteristic

#10587023

—

#10587022

—

C₉×C₁₉⋊C₃

—

C₉×C₁₉⋊C₃

—

#10587020

—

#10587020

—

C₃×#737

—

C₃×C₂₇⋊C₃

—

C₃²×C₂₇

—

C₃²⋊C₂₇

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃²⋊C₉

yes

—

C₃²×C₁₉

C₃²×C₉

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₉×C₁₉

—

#4294

—

#4294

C₃²⋊C₉

yes

—

#4294

C₃²⋊C₉

yes

—

#4294

C₃×H(3,𝔽₃)

yes

characteristic

#4294

C₃×3₋¹⁺²

yes

—

#737

—

#737

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

#4296

yes

characteristic

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

C₂₇⋊C₃

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₁₉

—

C₃×C₁₉

C₃×C₃²⋊C₉

yes

characteristic

C₃×C₁₉

#6903

yes

—

C₁₉⋊C₃

—

C₃³

—

C₃³

C₃×#4296

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃×#4296

yes

—

C₂₇

—

C₂₇

—

C₁₉

#10624684

yes

characteristic, complemented, Sylow

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃²⋊#4296

yes

—

C₃²

C₃²×#4296

yes

characteristic

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

C₃

—

100

C₃

—

171

—

101

C₃

C₃×C₃²⋊#4296

yes

characteristic

102

C₃

#17987599

yes

—

	subgroup	G/H
center, Frattini	C₃²	C₃²×#4296
commutator, socle	C₃²×C₁₉	C₃²×C₉
Fitting	C₃×C₁₉×3₋¹⁺²	C₉
prime radical	C₃×C₃²⋊(C₁₉⋊C₃)	C₉
cosocle	C₃×#4294	C₃³
hypercenter	C₃×3₋¹⁺²	#4296
hypocenter	C₁₉	#10624684
LCS	G ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃×3₋¹⁺²

subgroup

G/H

center, Frattini

C₃²

C₃²×#4296

commutator, socle

C₃²×C₁₉

C₃²×C₉

Fitting

C₃×C₁₉×3₋¹⁺²

C₉

prime radical

C₃×C₃²⋊(C₁₉⋊C₃)

C₉

C₃³

C₃×3₋¹⁺²

C₁₉

G ▷ C₃²×C₁₉ ▷ C₁₉

C₁ ◁ C₃² ◁ C₃×3₋¹⁺²

N	H ≅ G/N	type
1. C₃×#13095342	C₃	semidirect
2. C₃²×#10587020	C₃	semidirect
3. #17987557	C₃	semidirect
87. C₁₉	#10624684	semidirect

H ≅ G/N

type

1. C₃×#13095342

C₃

semidirect

2. C₃²×#10587020

C₃

semidirect

3. #17987557

C₃

semidirect

87. C₁₉

#10624684

semidirect

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

subgroup	index
C₃×#13095342	3
C₃²×#10587020	3
#17987557	3
C₃×#13095333	3
#10624684	19

subgroup

index

Subgroups — #28356893

Conjugacy classes of subgroups

Complements

Maximal subgroups