Group #28356725 — subgroups

Subgroups — #28356725

Order 13851.

164 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 60 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
2	C₃²×#10587020	C₃	1	1	yes	maximal, characteristic
3	#17987402	C₃	1	1	yes	maximal, characteristic
4	#13095342	—	3	2	no	—
5	#13095342	—	3	2	no	—
6	#13095342	C₉	1	6	yes	complemented
7	C₃×#10587020	—	3	1	no	—
8	C₃×#10587020	—	3	2	no	—
9	C₃×#10587020	—	3	1	no	—
10	C₃×#10587020	—	3	1	no	—
11	C₃×#10587020	C₉	1	3	yes	complemented
12	C₃²×#4294	C₃²	1	1	yes	characteristic
13	#13095305	—	3	2	no	—
14	C₃²⋊#4294	C₉	1	2	yes	—
15	C₉×#4294	—	3	1	no	—
16	C₁₉×C₃²⋊C₉	C₉	1	1	yes	characteristic
17	#10624468	—	19	1	no	maximal, Sylow
18	C₃³×C₁₉	C₃×C₉	1	1	yes	characteristic
19	C₃×C₉×C₁₉	—	3	1	no	—
20	C₃×#4294	—	3	2	no	—
21	C₃×#4294	C₃×C₉	1	1	yes	characteristic
22	C₃×#4294	C₃×C₉	1	2	yes	—
23	C₃×#4294	H(3,𝔽₃)	1	1	yes	characteristic
24	C₃×#4294	3₋¹⁺²	1	1	yes	characteristic
25	C₃×#4294	3₋¹⁺²	1	1	yes	characteristic
26	C₃×#4294	3₋¹⁺²	1	2	yes	—
27	C₃×#4294	3₋¹⁺²	1	2	yes	—
28	C₃×#4294	3₋¹⁺²	1	2	yes	—
29	#10587020	—	3	3	no	—
30	#10587020	—	3	6	no	—
31	C₃×C₂₇⋊C₃	—	19	2	no	—
32	C₃²×C₂₇	—	19	1	no	—
33	#6887	—	19	1	no	—
34	C₃²×C₁₉	—	3	1	no	—
35	C₃²×C₁₉	—	3	2	no	—
36	C₃²×C₁₉	C₉²	1	1	yes	characteristic
37	C₃²×C₁₉	C₃²⋊C₉	1	1	yes	characteristic
38	C₃²×C₁₉	C₉⋊C₉	1	1	yes	characteristic
39	C₃²×C₁₉	C₉⋊C₉	1	1	yes	characteristic
40	C₉×C₁₉	—	9	1	no	—
41	#4294	—	9	2	no	—
42	#4294	C₃²⋊C₉	1	1	yes	characteristic
43	#4294	C₃²⋊C₉	1	2	yes	—
44	#4294	C₉⋊C₉	1	2	yes	—
45	#4294	C₉⋊C₉	1	2	yes	—
46	#4294	C₉⋊C₉	1	2	yes	—
47	C₃²×C₉	—	19	1	no	—
48	C₂₇⋊C₃	—	19	6	no	—
49	C₂₇⋊C₃	—	57	2	no	—
50	C₂₇⋊C₃	—	57	2	no	—
51	C₃×C₂₇	—	19	3	no	—
52	C₃×C₂₇	—	57	1	no	—
53	C₃×C₂₇	—	57	1	no	—
54	C₃×C₂₇	—	57	1	no	—
55	C₃×C₂₇	—	57	2	no	—
56	C₉⋊C₉	—	57	2	no	—
57	C₃²⋊C₉	—	19	2	no	—
58	C₃²⋊C₉	#4296	1	1	yes	characteristic
59	C₉²	—	57	1	no	—
60	C₃×C₁₉	—	3	1	no	—
61	C₃×C₁₉	—	3	1	no	—
62	C₃×C₁₉	—	3	1	no	—
63	C₃×C₁₉	#6855	1	1	yes	characteristic
64	C₃×C₁₉	#6873	1	1	yes	characteristic
65	C₃×C₁₉	#6873	1	2	yes	—
66	C₃³	C₃×#4296	1	1	yes	characteristic
67	C₃×C₉	—	3	1	no	—
68	C₃×C₉	—	19	1	no	—
69	C₃×C₉	—	19	2	no	—
70	C₃×C₉	—	19	1	no	—
71	C₃×C₉	—	19	2	no	—
72	C₃×C₉	—	19	1	no	—
73	C₃×C₉	—	19	2	no	—
74	C₃×C₉	—	19	1	no	—
75	C₃×C₉	—	19	2	no	—
76	C₃×C₉	—	57	2	no	—
77	C₂₇	—	57	3	no	—
78	C₂₇	—	57	6	no	—
79	C₁₉	#10624468	1	1	yes	characteristic, complemented, Sylow
80	C₃²	—	3	1	no	—
81	C₃²	—	3	2	no	—
82	C₃²	C₉×#4296	1	1	yes	characteristic
83	C₃²	C₃²⋊#4296	1	1	yes	characteristic
84	C₃²	#13095338	1	1	yes	characteristic
85	C₃²	#13095339	1	1	yes	characteristic
86	C₉	—	9	1	no	—
87	C₉	—	19	1	no	—
88	C₉	—	19	2	no	—
89	C₉	—	19	2	no	—
90	C₉	—	19	2	no	—
91	C₉	—	19	2	no	—
92	C₉	—	171	2	no	—
93	C₃	—	3	1	no	—
94	C₃	—	3	1	no	—
95	C₃	—	3	1	no	—
96	C₃	(C₃×#4296)⋊C₉	1	1	yes	characteristic
97	C₃	#17987568	1	1	yes	characteristic
98	C₃	#17987569	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#13095342

C₃

yes

maximal

C₃²×#10587020

C₃

yes

maximal, characteristic

#17987402

C₃

yes

maximal, characteristic

#13095342

—

#13095342

—

#13095342

C₉

yes

—

—

—

—

C₉

yes

complemented

C₃²×#4294

C₃²

yes

characteristic

#13095305

—

C₃²⋊#4294

C₉

yes

—

C₉×#4294

—

C₁₉×C₃²⋊C₉

C₉

yes

characteristic

#10624468

—

maximal, Sylow

C₃³×C₁₉

C₃×C₉

yes

characteristic

C₃×C₉×C₁₉

—

C₃×#4294

—

C₃×#4294

C₃×C₉

yes

characteristic

C₃×#4294

C₃×C₉

yes

—

yes

3₋¹⁺²

yes

characteristic

C₃×#4294

3₋¹⁺²

yes

characteristic

C₃×#4294

3₋¹⁺²

yes

—

C₃×#4294

3₋¹⁺²

yes

—

C₃×#4294

3₋¹⁺²

yes

—

#10587020

—

#10587020

—

C₃×C₂₇⋊C₃

—

C₃²×C₂₇

—

#6887

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₉²

yes

characteristic

C₃²×C₁₉

C₃²⋊C₉

yes

characteristic

C₃²×C₁₉

C₉⋊C₉

yes

characteristic

C₃²×C₁₉

C₉⋊C₉

yes

characteristic

C₉×C₁₉

—

#4294

—

#4294

C₃²⋊C₉

yes

characteristic

#4294

C₃²⋊C₉

yes

—

#4294

C₉⋊C₉

yes

—

#4294

C₉⋊C₉

yes

—

#4294

C₉⋊C₉

yes

—

C₃²×C₉

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

#4296

yes

characteristic

C₉²

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

yes

C₃×C₁₉

yes

C₃×C₁₉

yes

—

C₃³

C₃×#4296

yes

characteristic

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₂₇

—

C₂₇

—

C₁₉

#10624468

yes

characteristic, complemented, Sylow

C₃²

—

C₃²

—

C₃²

C₉×#4296

yes

characteristic

C₃²

C₃²⋊#4296

yes

C₃²

yes

C₃²

yes

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

171

—

C₃

—

C₃

—

C₃

—

C₃

yes

C₃

yes

C₃

yes

—

	subgroup	G/H
center	C₃²	C₃²⋊#4296
commutator	C₃²×C₁₉	C₉²
Frattini	C₃³	C₃×#4296
Fitting	C₁₉×C₃²⋊C₉	C₉
socle	C₃²×C₁₉	C₃²⋊C₉
prime radical	C₃³×C₁₉	C₃×C₉
cosocle	C₃²×#4294	C₃²
hypercenter	C₃²⋊C₉	#4296
hypocenter	C₁₉	#10624468
LCS	G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²⋊C₉

subgroup

G/H

center

C₃²

C₃²⋊#4296

commutator

C₃²×C₁₉

C₉²

Frattini

C₃³

C₃×#4296

Fitting

C₁₉×C₃²⋊C₉

C₉

socle

C₃²×C₁₉

C₃²⋊C₉

prime radical

C₃³×C₁₉

C₃×C₉

cosocle

C₃²×#4294

C₃²

C₃²⋊C₉

C₁₉

G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ C₃²⋊C₉

N	H ≅ G/N	type
6. #13095342	C₉	semidirect
11. C₃×#10587020	C₉	semidirect
79. C₁₉	#10624468	semidirect

H ≅ G/N

type

6. #13095342

C₉

semidirect

11. C₃×#10587020

C₉

semidirect

79. C₁₉

#10624468

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
#17987402	3
#10624468	19

subgroup

index

Subgroups — #28356725

Conjugacy classes of subgroups

Complements

Maximal subgroups