Group #41008541 — subgroups

Subgroups — #41008541

Order 26973.

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

normal subgroups only

This group has 218 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×#15688543	C₃	1	1	yes	maximal, characteristic
2	C₃×#15688542	C₃	1	1	yes	maximal, characteristic
3	C₃×(C₃₇⋊C₃)⋊C₂₇	C₃	1	1	yes	maximal, characteristic
4	#22582705	C₃	1	6	yes	maximal, complemented
5	#22582703	C₃	1	3	yes	maximal, complemented
6	#22582537	C₃	1	1	yes	maximal, characteristic
7	#15688543	—	3	3	no	—
8	#15688542	—	3	3	no	—
9	(C₃₇⋊C₃)⋊C₂₇	—	3	3	no	—
10	C₃×#11758603	C₃²	1	1	yes	characteristic
11	C₃×#11758603	C₃²	1	3	yes	—
12	C₃×#11758603	C₃²	1	3	yes	—
13	C₃×#11758603	C₃²	1	2	yes	—
14	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
15	#15688506	C₉	1	1	yes	characteristic
16	#15688505	C₉	1	1	yes	characteristic
17	#15688504	C₉	1	3	yes	—
18	#15688502	C₉	1	3	yes	—
19	C₉×#66524	C₃²	1	3	yes	—
20	C₃₇×C₉⋊C₉	C₉	1	1	yes	characteristic
21	C₃²×C₃₇⋊C₃	C₃×C₉	1	1	yes	characteristic
22	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
23	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
24	C₃×#66524	C₃×C₉	1	3	yes	—
25	C₃×#66524	C₃×C₉	1	3	yes	—
26	C₃×#66524	C₃×C₉	1	1	yes	characteristic
27	C₃×#66524	C₃³	1	1	yes	characteristic
28	C₉×C₃₇⋊C₃	3₋¹⁺²	1	3	yes	—
29	C₉×C₃₇⋊C₃	3₋¹⁺²	1	3	yes	—
30	C₉×C₃₇⋊C₃	3₋¹⁺²	1	3	yes	—
31	#11758603	—	3	3	no	—
32	#11758603	—	3	1	no	—
33	#11758603	—	3	2	no	—
34	#11758603	—	3	3	no	—
35	#10624707	—	37	1	no	maximal, Sylow
36	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
37	C₃×C₃₇⋊C₃	—	3	3	no	—
38	C₃×C₃₇⋊C₃	C₉⋊C₉	1	3	yes	—
39	C₉×C₃₇	—	3	3	no	—
40	C₉×C₃₇	C₉⋊C₉	1	3	yes	—
41	#66524	—	3	3	no	—
42	#66524	—	3	3	no	—
43	#66524	C₉⋊C₉	1	3	yes	—
44	#66524	C₃×3₋¹⁺²	1	1	yes	characteristic
45	#66524	C₃×3₋¹⁺²	1	1	yes	characteristic
46	#66524	C₃×3₋¹⁺²	1	1	yes	characteristic
47	C₃×C₂₇⋊C₃	—	37	1	no	—
48	C₃×C₂₇⋊C₃	—	37	1	no	—
49	C₃²×C₂₇	—	37	1	no	—
50	#6887	—	37	1	no	—
51	C₉⋊C₂₇	—	37	3	no	—
52	C₉⋊C₂₇	—	37	6	no	—
53	C₃×C₃₇	C₃×C₉⋊C₉	1	1	yes	characteristic
54	C₃×C₃₇	#6903	1	3	yes	—
55	C₃₇⋊C₃	—	3	3	no	—
56	C₃²×C₉	—	37	1	no	—
57	C₂₇⋊C₃	—	111	3	no	—
58	C₂₇⋊C₃	—	111	3	no	—
59	C₃×C₂₇	—	37	3	no	—
60	C₃×C₂₇	—	37	1	no	—
61	C₃×C₂₇	—	37	3	no	—
62	C₃×C₂₇	—	37	2	no	—
63	C₃×C₂₇	—	111	3	no	—
64	C₉⋊C₉	—	37	3	no	—
65	C₉⋊C₉	—	37	1	no	—
66	C₉⋊C₉	—	37	1	no	—
67	C₉⋊C₉	#66526	1	1	yes	characteristic
68	C₃²⋊C₉	—	37	3	no	—
69	C₉²	—	37	3	no	—
70	C₃₇	#10624707	1	1	yes	characteristic, complemented, Sylow
71	C₃³	—	37	1	no	—
72	C₃×C₉	—	37	1	no	—
73	C₃×C₉	—	37	1	no	—
74	C₃×C₉	—	37	3	no	—
75	C₃×C₉	—	37	3	no	—
76	C₃×C₉	—	37	3	no	—
77	C₃×C₉	—	37	3	no	—
78	C₃×C₉	—	37	3	no	—
79	C₃×C₉	C₃×#66526	1	3	yes	—
80	C₃×C₉	C₃×#66526	1	1	yes	characteristic
81	C₂₇	—	111	3	no	—
82	C₂₇	—	111	1	no	—
83	C₂₇	—	111	2	no	—
84	C₂₇	—	111	3	no	—
85	C₃²	—	37	3	no	—
86	C₃²	—	111	3	no	—
87	C₃²	C₃²×#66526	1	1	yes	characteristic
88	C₉	—	3	3	no	—
89	C₉	—	37	1	no	—
90	C₉	—	37	1	no	—
91	C₉	—	37	1	no	—
92	C₉	—	37	3	no	—
93	C₉	—	111	3	no	—
94	C₉	—	111	3	no	—
95	C₉	#15688537	1	3	yes	—
96	C₃	—	111	3	no	—
97	C₃	C₃×#15688537	1	1	yes	characteristic
98	C₃	#22582732	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688543

C₃

yes

maximal, characteristic

C₃×#15688542

C₃

yes

maximal, characteristic

C₃×(C₃₇⋊C₃)⋊C₂₇

C₃

yes

maximal, characteristic

#22582705

C₃

yes

maximal, complemented

#22582703

C₃

yes

maximal, complemented

#22582537

C₃

yes

maximal, characteristic

#15688543

—

#15688542

—

(C₃₇⋊C₃)⋊C₂₇

—

C₃×#11758603

C₃²

yes

characteristic

C₃×#11758603

C₃²

yes

—

C₃×#11758603

C₃²

yes

—

C₃×#11758603

C₃²

yes

—

C₃×C₉×C₃₇⋊C₃

C₃²

yes

characteristic

#15688506

C₉

yes

characteristic

#15688505

C₉

yes

characteristic

#15688504

C₉

yes

—

#15688502

C₉

yes

—

C₉×#66524

C₃²

yes

—

C₃₇×C₉⋊C₉

C₉

yes

characteristic

C₃²×C₃₇⋊C₃

C₃×C₉

yes

characteristic

C₃×C₉×C₃₇

C₃×C₉

yes

—

C₃×C₉×C₃₇

C₃×C₉

yes

characteristic

C₃×#66524

C₃×C₉

yes

—

C₃×#66524

C₃×C₉

yes

—

C₃×#66524

C₃×C₉

yes

characteristic

C₃×#66524

C₃³

yes

characteristic

C₉×C₃₇⋊C₃

3₋¹⁺²

yes

—

C₉×C₃₇⋊C₃

3₋¹⁺²

yes

—

C₉×C₃₇⋊C₃

3₋¹⁺²

yes

—

—

—

—

—

—

C₃²×C₃₇

C₃²×C₉

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

C₉⋊C₉

yes

—

C₉×C₃₇

—

C₉×C₃₇

C₉⋊C₉

yes

—

#66524

—

#66524

—

#66524

C₉⋊C₉

yes

—

#66524

C₃×3₋¹⁺²

yes

characteristic

#66524

C₃×3₋¹⁺²

yes

characteristic

#66524

C₃×3₋¹⁺²

yes

characteristic

C₃×C₂₇⋊C₃

—

C₃×C₂₇⋊C₃

—

C₃²×C₂₇

—

#6887

—

C₉⋊C₂₇

—

C₉⋊C₂₇

—

C₃×C₃₇

C₃×C₉⋊C₉

yes

characteristic

C₃×C₃₇

#6903

yes

—

C₃₇⋊C₃

—

C₃²×C₉

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

—

111

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

111

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

#66526

yes

characteristic

C₃²⋊C₉

—

C₉²

—

C₃₇

#10624707

yes

characteristic, complemented, Sylow

C₃³

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃×#66526

yes

—

C₃×C₉

C₃×#66526

yes

characteristic

C₂₇

—

111

—

C₂₇

—

111

—

C₂₇

—

111

—

C₂₇

—

111

—

C₃²

—

C₃²

—

111

—

C₃²

C₃²×#66526

yes

characteristic

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

111

—

C₉

—

111

—

C₉

#15688537

yes

—

C₃

—

111

—

C₃

C₃×#15688537

yes

characteristic

C₃

#22582732

yes

—

	subgroup	G/H
center, Frattini	C₃²	C₃²×#66526
commutator, socle	C₃²×C₃₇	C₃²×C₉
Fitting	C₃₇×C₉⋊C₉	C₉
prime radical	C₃²×C₃₇⋊C₃	C₃×C₉
cosocle	C₃×#66524	C₃³
hypercenter	C₉⋊C₉	#66526
hypocenter	C₃₇	#10624707
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₉⋊C₉

subgroup

G/H

center, Frattini

C₃²

C₃²×#66526

commutator, socle

C₃²×C₃₇

C₃²×C₉

Fitting

C₃₇×C₉⋊C₉

C₉

prime radical

C₃²×C₃₇⋊C₃

C₃×C₉

C₃³

C₉⋊C₉

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃₇

C₁ ◁ C₃² ◁ C₉⋊C₉

N	H ≅ G/N	type
4. #22582705	C₃	semidirect
5. #22582703	C₃	semidirect
70. C₃₇	#10624707	semidirect

H ≅ G/N

type

4. #22582705

C₃

semidirect

5. #22582703

C₃

semidirect

70. C₃₇

#10624707

semidirect

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

subgroup	index
C₃×#15688543	3
C₃×#15688542	3
C₃×(C₃₇⋊C₃)⋊C₂₇	3
#22582705	3
#22582703	3
#22582537	3
#10624707	37

subgroup

index

C₃×(C₃₇⋊C₃)⋊C₂₇

Subgroups — #41008541

Conjugacy classes of subgroups

Complements

Maximal subgroups