Group #28354859 — subgroups

Subgroups — C₂₇×#10587023 #28354859

Order 13851.

220 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 106 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×C₂₇×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic
2	C₉×#10587023	C₃	1	1	yes	maximal, characteristic
3	#17987357	C₃	1	2	yes	maximal, complemented
4	#17987355	C₃	1	4	yes	maximal, complemented
5	#17987332	C₃	1	2	yes	maximal
6	C₂₇×#4294	C₃	1	2	yes	maximal, complemented
7	C₉×C₁₉×C₂₇	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃×#10587023	C₉	1	1	yes	characteristic
9	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
10	C₃×#10587018	C₃²	1	4	yes	—
11	C₃×#10587018	C₃²	1	2	yes	—
12	C₂₇×C₁₉⋊C₃	—	3	3	no	—
13	#13095307	C₉	1	2	yes	—
14	#13095305	C₉	1	4	yes	—
15	#13095303	C₉	1	2	yes	—
16	C₉×#4294	C₃²	1	2	yes	—
17	C₃×C₁₉×C₂₇	C₃²	1	2	yes	—
18	C₃×C₁₉×C₂₇	C₃²	1	1	yes	characteristic
19	C₉²×C₁₉	C₃²	1	1	yes	characteristic
20	C₂₇×3₋¹⁺²	—	19	1	no	maximal, Sylow
21	C₃²×C₁₉⋊C₃	C₃×C₉	1	1	yes	characteristic
22	C₃×C₉×C₁₉	C₃×C₉	1	2	yes	—
23	C₃×C₉×C₁₉	C₃×C₉	1	1	yes	characteristic
24	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
25	C₃×#4294	C₃×C₉	1	4	yes	—
26	C₃×#4294	C₃×C₉	1	2	yes	—
27	C₃×#4294	C₃×C₉	1	2	yes	—
28	#10587023	C₂₇	1	9	yes	direct product factor
29	C₉×C₁₉⋊C₃	—	3	1	no	—
30	C₉×C₁₉⋊C₃	—	3	2	no	—
31	C₁₉×C₂₇	—	3	2	no	—
32	C₁₉×C₂₇	3₋¹⁺²	1	3	yes	complemented
33	#10587018	—	3	2	no	—
34	#10587018	—	3	4	no	—
35	C₃²×C₂₇	—	19	1	no	—
36	C₉×3₋¹⁺²	—	19	1	no	—
37	C₉⋊C₂₇	—	19	4	no	—
38	C₉⋊C₂₇	—	19	2	no	—
39	C₃²⋊C₂₇	—	19	2	no	—
40	C₉×C₂₇	—	19	2	no	—
41	C₉×C₂₇	C₁₉⋊C₃	1	1	yes	characteristic, complemented
42	C₃²×C₁₉	C₃²×C₉	1	1	yes	characteristic
43	C₃×C₁₉⋊C₃	—	3	1	no	—
44	C₃×C₁₉⋊C₃	—	3	2	no	—
45	C₃×C₁₉⋊C₃	C₃×C₂₇	1	3	yes	—
46	C₉×C₁₉	—	3	2	no	—
47	C₉×C₁₉	C₃×C₂₇	1	3	yes	complemented
48	C₉×C₁₉	C₃×3₋¹⁺²	1	1	yes	characteristic
49	C₉×C₁₉	#737	1	2	yes	—
50	#4294	—	3	2	no	—
51	#4294	—	3	4	no	—
52	#4294	C₃×C₂₇	1	6	yes	complemented
53	C₃×3₋¹⁺²	—	19	1	no	—
54	C₃²×C₉	—	19	1	no	—
55	C₃×C₂₇	—	19	4	no	—
56	C₃×C₂₇	—	19	2	no	—
57	C₃×C₂₇	—	57	3	no	—
58	C₃×C₂₇	C₃×C₁₉⋊C₃	1	2	yes	—
59	C₃×C₂₇	C₃×C₁₉⋊C₃	1	1	yes	characteristic
60	C₉⋊C₉	—	19	4	no	—
61	C₉⋊C₉	—	19	2	no	—
62	C₃²⋊C₉	—	19	2	no	—
63	C₉²	—	19	2	no	—
64	C₉²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
65	C₃×C₁₉	C₉×3₋¹⁺²	1	1	yes	characteristic
66	C₃×C₁₉	C₃²×C₂₇	1	1	yes	characteristic
67	C₃×C₁₉	#6903	1	2	yes	—
68	C₁₉⋊C₃	—	3	3	no	—
69	C₃³	—	19	1	no	—
70	3₋¹⁺²	—	19	9	no	—
71	C₃×C₉	—	19	2	no	—
72	C₃×C₉	—	19	4	no	—
73	C₃×C₉	—	19	2	no	—
74	C₃×C₉	—	57	1	no	—
75	C₃×C₉	—	57	2	no	—
76	C₃×C₉	C₉×C₁₉⋊C₃	1	2	yes	—
77	C₃×C₉	C₉×C₁₉⋊C₃	1	1	yes	characteristic
78	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
79	C₂₇	—	3	2	no	—
80	C₂₇	—	57	2	no	—
81	C₂₇	—	57	4	no	—
82	C₂₇	#10587023	1	3	yes	direct product factor
83	C₁₉	C₂₇×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
84	C₃²	—	19	3	no	—
85	C₃²	—	57	1	no	—
86	C₃²	—	57	2	no	—
87	C₃²	C₃×C₉×C₁₉⋊C₃	1	1	yes	characteristic
88	C₉	—	3	2	no	—
89	C₉	—	19	6	no	—
90	C₉	—	57	2	no	—
91	C₉	—	57	4	no	—
92	C₉	C₂₇×C₁₉⋊C₃	1	3	yes	complemented
93	C₉	C₃×#10587023	1	1	yes	characteristic
94	C₉	#13095332	1	2	yes	—
95	C₃	—	57	3	no	—
96	C₃	C₉×#10587023	1	1	yes	characteristic
97	C₃	C₃×C₂₇×C₁₉⋊C₃	1	1	yes	characteristic
98	C₃	#17987479	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₂₇×C₁₉⋊C₃

C₃

yes

maximal, characteristic

C₉×#10587023

C₃

yes

maximal, characteristic

#17987357

C₃

yes

maximal, complemented

#17987355

C₃

yes

maximal, complemented

#17987332

C₃

yes

maximal

C₂₇×#4294

C₃

yes

maximal, complemented

C₉×C₁₉×C₂₇

C₃

yes

maximal, characteristic, complemented

C₃×#10587023

C₉

yes

characteristic

C₃×C₉×C₁₉⋊C₃

C₃²

yes

characteristic

C₃×#10587018

C₃²

yes

—

C₃×#10587018

C₃²

yes

—

C₂₇×C₁₉⋊C₃

—

#13095307

C₉

yes

—

#13095305

C₉

yes

—

#13095303

C₉

yes

—

C₉×#4294

C₃²

yes

—

C₃×C₁₉×C₂₇

C₃²

yes

—

C₃×C₁₉×C₂₇

C₃²

yes

characteristic

C₉²×C₁₉

C₃²

yes

characteristic

C₂₇×3₋¹⁺²

—

maximal, Sylow

C₃²×C₁₉⋊C₃

C₃×C₉

yes

characteristic

C₃×C₉×C₁₉

C₃×C₉

yes

—

C₃×C₉×C₁₉

C₃×C₉

yes

characteristic

C₃×C₉×C₁₉

C₃³

yes

characteristic

C₃×#4294

C₃×C₉

yes

—

C₃×#4294

C₃×C₉

yes

—

C₃×#4294

C₃×C₉

yes

—

#10587023

C₂₇

yes

direct product factor

C₉×C₁₉⋊C₃

—

C₉×C₁₉⋊C₃

—

C₁₉×C₂₇

—

C₁₉×C₂₇

3₋¹⁺²

yes

complemented

#10587018

—

#10587018

—

C₃²×C₂₇

—

C₉×3₋¹⁺²

—

C₉⋊C₂₇

—

C₉⋊C₂₇

—

C₃²⋊C₂₇

—

C₉×C₂₇

—

C₉×C₂₇

C₁₉⋊C₃

yes

characteristic, complemented

C₃²×C₁₉

C₃²×C₉

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

C₃×C₂₇

yes

—

C₉×C₁₉

—

C₉×C₁₉

C₃×C₂₇

yes

complemented

C₉×C₁₉

C₃×3₋¹⁺²

yes

characteristic

C₉×C₁₉

#737

yes

—

#4294

—

#4294

—

#4294

C₃×C₂₇

yes

complemented

C₃×3₋¹⁺²

—

C₃²×C₉

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

C₃×C₁₉⋊C₃

yes

—

C₃×C₂₇

C₃×C₁₉⋊C₃

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₉²

—

C₉²

C₃×C₁₉⋊C₃

yes

characteristic

C₃×C₁₉

C₉×3₋¹⁺²

yes

characteristic

C₃×C₁₉

C₃²×C₂₇

yes

characteristic

C₃×C₁₉

#6903

yes

—

C₁₉⋊C₃

—

C₃³

—

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₉×C₁₉⋊C₃

yes

—

C₃×C₉

C₉×C₁₉⋊C₃

yes

characteristic

C₃×C₉

C₃²×C₁₉⋊C₃

yes

characteristic

C₂₇

—

C₂₇

—

C₂₇

—

C₂₇

#10587023

yes

direct product factor

C₁₉

C₂₇×3₋¹⁺²

yes

characteristic, complemented, Sylow

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃×C₉×C₁₉⋊C₃

yes

characteristic

C₉

—

C₉

—

C₉

—

C₉

—

C₉

C₂₇×C₁₉⋊C₃

yes

C₉

yes

C₉

yes

—

C₃

—

C₃

C₉×#10587023

yes

characteristic

C₃

C₃×C₂₇×C₁₉⋊C₃

yes

characteristic

C₃

#17987479

yes

—

	subgroup	G/H
center	C₃×C₂₇	C₃×C₁₉⋊C₃
commutator	C₃×C₁₉	C₃²×C₂₇
Frattini	C₃×C₉	C₃²×C₁₉⋊C₃
Fitting	C₉×C₁₉×C₂₇	C₃
socle	C₃²×C₁₉	C₃²×C₉
prime radical	C₃²×C₁₉⋊C₃	C₃×C₉
cosocle	C₃×C₉×C₁₉	C₃³
hypercenter	C₉×C₂₇	C₁₉⋊C₃
hypocenter	C₁₉	C₂₇×3₋¹⁺²
LCS	G ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃×C₂₇ ◁ C₉×C₂₇

subgroup

G/H

center

C₃×C₂₇

C₃×C₁₉⋊C₃

commutator

C₃×C₁₉

C₃²×C₂₇

Frattini

C₃×C₉

C₃²×C₁₉⋊C₃

Fitting

C₉×C₁₉×C₂₇

C₃

socle

C₃²×C₁₉

C₃²×C₉

prime radical

C₃²×C₁₉⋊C₃

C₃×C₉

cosocle

C₃×C₉×C₁₉

C₃³

hypercenter

C₉×C₂₇

C₁₉⋊C₃

hypocenter

C₁₉

C₂₇×3₋¹⁺²

LCS

G ▷ C₃×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃×C₂₇ ◁ C₉×C₂₇

N	H ≅ G/N	type
3. #17987357	C₃	semidirect
4. #17987355	C₃	semidirect
6. C₂₇×#4294	C₃	semidirect
7. C₉×C₁₉×C₂₇	C₃	semidirect
28. #10587023	C₂₇	direct (N × H)
32. C₁₉×C₂₇	3₋¹⁺²	semidirect
41. C₉×C₂₇	C₁₉⋊C₃	semidirect
47. C₉×C₁₉	C₃×C₂₇	semidirect
52. #4294	C₃×C₂₇	semidirect
82. C₂₇	#10587023	direct (N × H)
83. C₁₉	C₂₇×3₋¹⁺²	semidirect
92. C₉	C₂₇×C₁₉⋊C₃	semidirect

H ≅ G/N

type

3. #17987357

C₃

semidirect

4. #17987355

C₃

semidirect

6. C₂₇×#4294

C₃

semidirect

7. C₉×C₁₉×C₂₇

C₃

semidirect

28. #10587023

C₂₇

direct (N × H)

32. C₁₉×C₂₇

3₋¹⁺²

semidirect

41. C₉×C₂₇

C₁₉⋊C₃

semidirect

47. C₉×C₁₉

C₃×C₂₇

semidirect

52. #4294

C₃×C₂₇

semidirect

82. C₂₇

#10587023

direct (N × H)

83. C₁₉

C₂₇×3₋¹⁺²

semidirect

92. C₉

C₂₇×C₁₉⋊C₃

semidirect

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

subgroup	index
C₃×C₂₇×C₁₉⋊C₃	3
C₉×#10587023	3
#17987357	3
#17987355	3
#17987332	3
C₂₇×#4294	3
C₉×C₁₉×C₂₇	3
C₂₇×3₋¹⁺²	19

subgroup

index

C₃×C₂₇×C₁₉⋊C₃

C₂₇×#4294

C₉×C₁₉×C₂₇

C₂₇×3₋¹⁺²

Subgroups — C₂₇×#10587023 #28354859

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C27×#10587023 #28354859

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₇×#10587023 #28354859