Group #41006465 — subgroups

Subgroups — #41006465

Order 26973.

192 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 85 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×#15688510	C₃	1	1	yes	maximal, characteristic
2	C₉×#11758605	C₃	1	1	yes	maximal, characteristic
3	#22582490	C₃	1	1	yes	maximal, characteristic, complemented
4	#22582488	C₃	1	2	yes	maximal, complemented
5	#22582488	C₃	1	2	yes	maximal, complemented
6	#22582465	C₃	1	2	yes	maximal
7	#22582463	C₃	1	2	yes	maximal, complemented
8	C₉×#11758601	C₃	1	1	yes	maximal, characteristic, complemented
9	C₃₇×C₉⋊C₂₇	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃×#11758605	C₉	1	1	yes	characteristic
11	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
12	#15688510	—	3	3	no	—
13	C₃×#11758601	C₃²	1	1	yes	characteristic
14	C₃×#11758601	C₃²	1	2	yes	—
15	C₃×#11758601	C₃²	1	2	yes	—
16	C₃×#11758601	C₃²	1	1	yes	characteristic
17	#15688505	C₉	1	2	yes	—
18	#15688504	C₉	1	4	yes	—
19	#15688502	C₉	1	2	yes	—
20	C₉×#66524	C₃²	1	2	yes	—
21	C₃×C₂₇×C₃₇	C₃²	1	1	yes	characteristic
22	C₃×C₂₇×C₃₇	C₃²	1	2	yes	—
23	C₉²×C₃₇	C₃²	1	1	yes	characteristic
24	C₃²×C₃₇⋊C₃	C₃×C₉	1	1	yes	characteristic
25	C₃×C₉×C₃₇	C₃×C₉	1	2	yes	—
26	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
27	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
28	C₃×#66524	C₃×C₉	1	4	yes	—
29	C₃×#66524	C₃×C₉	1	2	yes	—
30	C₃×#66524	C₃×C₉	1	2	yes	—
31	#11758605	—	3	3	no	—
32	C₉×C₃₇⋊C₃	—	3	1	no	—
33	C₉×C₃₇⋊C₃	—	3	2	no	—
34	C₂₇×C₃₇	—	3	1	no	—
35	C₂₇×C₃₇	—	3	2	no	—
36	#11758601	—	3	1	no	—
37	#11758601	—	3	2	no	—
38	#11758601	—	3	2	no	—
39	#11758601	3₋¹⁺²	1	3	yes	complemented
40	#10624728	—	37	1	no	maximal, Sylow
41	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
42	C₃×C₃₇⋊C₃	—	3	1	no	—
43	C₃×C₃₇⋊C₃	—	3	1	no	—
44	C₃×C₃₇⋊C₃	—	3	2	no	—
45	C₉×C₃₇	—	3	2	no	—
46	C₉×C₃₇	C₂₇⋊C₃	1	3	yes	complemented
47	C₉×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
48	C₉×C₃₇	#737	1	2	yes	—
49	#66524	—	3	2	no	—
50	#66524	—	3	2	no	—
51	#66524	—	3	4	no	—
52	C₃×C₂₇⋊C₃	—	37	1	no	—
53	C₉×3₋¹⁺²	—	37	1	no	—
54	C₉⋊C₂₇	—	37	2	no	—
55	C₉⋊C₂₇	—	37	2	no	—
56	C₉⋊C₂₇	—	37	1	no	—
57	C₉⋊C₂₇	C₃₇⋊C₃	1	1	yes	characteristic, complemented
58	C₃²⋊C₂₇	—	37	2	no	—
59	#6864	—	37	2	no	—
60	C₉×C₂₇	—	37	1	no	—
61	C₃×C₃₇	C₉×3₋¹⁺²	1	1	yes	characteristic
62	C₃×C₃₇	C₃×C₂₇⋊C₃	1	1	yes	characteristic
63	C₃×C₃₇	#6903	1	2	yes	—
64	C₃₇⋊C₃	—	9	1	no	—
65	C₃×3₋¹⁺²	—	37	1	no	—
66	C₃²×C₉	—	37	1	no	—
67	C₂₇⋊C₃	—	111	3	no	—
68	C₃×C₂₇	—	37	1	no	—
69	C₃×C₂₇	—	37	2	no	—
70	C₃×C₂₇	—	37	2	no	—
71	C₃×C₂₇	—	37	1	no	—
72	C₃×C₂₇	C₃×C₃₇⋊C₃	1	2	yes	—
73	C₃×C₂₇	C₃×C₃₇⋊C₃	1	1	yes	characteristic
74	C₉⋊C₉	—	37	4	no	—
75	C₉⋊C₉	—	37	2	no	—
76	C₃²⋊C₉	—	37	2	no	—
77	C₉²	—	37	2	no	—
78	C₉²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
79	C₃₇	#10624728	1	1	yes	characteristic, complemented, Sylow
80	C₃³	—	37	1	no	—
81	3₋¹⁺²	—	111	3	no	—
82	C₃×C₉	—	37	4	no	—
83	C₃×C₉	—	37	2	no	—
84	C₃×C₉	—	37	2	no	—
85	C₃×C₉	—	111	1	no	—
86	C₃×C₉	—	111	2	no	—
87	C₃×C₉	C₉×C₃₇⋊C₃	1	2	yes	—
88	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
89	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
90	C₂₇	—	3	1	no	—
91	C₂₇	—	3	2	no	—
92	C₂₇	—	37	3	no	—
93	C₂₇	—	111	1	no	—
94	C₂₇	—	111	2	no	—
95	C₂₇	—	111	2	no	—
96	C₃²	—	111	1	no	—
97	C₃²	—	111	1	no	—
98	C₃²	—	111	2	no	—
99	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
100	C₉	—	3	2	no	—
101	C₉	—	111	2	no	—
102	C₉	—	111	2	no	—
103	C₉	—	111	4	no	—
104	C₉	#15688510	1	3	yes	complemented
105	C₉	3₋¹⁺²×C₃₇⋊C₃	1	1	yes	characteristic
106	C₉	#15688532	1	2	yes	—
107	C₃	—	333	1	no	—
108	C₃	#22582543	1	1	yes	characteristic
109	C₃	C₃×#15688510	1	1	yes	characteristic
110	C₃	#22582614	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688510

C₃

yes

maximal, characteristic

C₉×#11758605

C₃

yes

maximal, characteristic

#22582490

C₃

yes

maximal, characteristic, complemented

#22582488

C₃

yes

maximal, complemented

#22582488

C₃

yes

maximal, complemented

#22582465

C₃

yes

maximal

#22582463

C₃

yes

maximal, complemented

C₉×#11758601

C₃

yes

maximal, characteristic, complemented

C₃₇×C₉⋊C₂₇

C₃

yes

maximal, characteristic, complemented

C₃×#11758605

C₉

yes

characteristic

C₃×C₉×C₃₇⋊C₃

C₃²

yes

characteristic

#15688510

—

C₃×#11758601

C₃²

yes

characteristic

C₃×#11758601

C₃²

yes

—

C₃×#11758601

C₃²

yes

—

C₃×#11758601

C₃²

yes

characteristic

#15688505

C₉

yes

—

#15688504

C₉

yes

—

#15688502

C₉

yes

—

C₉×#66524

C₃²

yes

—

C₃×C₂₇×C₃₇

C₃²

yes

characteristic

C₃×C₂₇×C₃₇

C₃²

yes

—

C₉²×C₃₇

C₃²

yes

characteristic

C₃²×C₃₇⋊C₃

C₃×C₉

yes

characteristic

C₃×C₉×C₃₇

C₃×C₉

yes

—

C₃×C₉×C₃₇

C₃×C₉

yes

characteristic

C₃×C₉×C₃₇

C₃³

yes

characteristic

C₃×#66524

C₃×C₉

yes

—

C₃×#66524

C₃×C₉

yes

—

C₃×#66524

C₃×C₉

yes

—

#11758605

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₂₇×C₃₇

—

C₂₇×C₃₇

—

—

—

—

3₋¹⁺²

yes

complemented

#10624728

—

maximal, Sylow

C₃²×C₃₇

C₃²×C₉

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

C₂₇⋊C₃

yes

complemented

C₉×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₉×C₃₇

#737

yes

—

#66524

—

#66524

—

#66524

—

C₃×C₂₇⋊C₃

—

C₉×3₋¹⁺²

—

C₉⋊C₂₇

—

C₉⋊C₂₇

—

C₉⋊C₂₇

—

C₉⋊C₂₇

C₃₇⋊C₃

yes

characteristic, complemented

C₃²⋊C₂₇

—

#6864

—

C₉×C₂₇

—

C₃×C₃₇

C₉×3₋¹⁺²

yes

characteristic

C₃×C₃₇

C₃×C₂₇⋊C₃

yes

characteristic

C₃×C₃₇

#6903

yes

—

C₃₇⋊C₃

—

C₃×3₋¹⁺²

—

C₃²×C₉

—

C₂₇⋊C₃

—

111

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

C₃×C₃₇⋊C₃

yes

—

C₃×C₂₇

C₃×C₃₇⋊C₃

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₉²

—

C₉²

C₃×C₃₇⋊C₃

yes

characteristic

C₃₇

#10624728

yes

characteristic, complemented, Sylow

C₃³

—

3₋¹⁺²

—

111

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

C₉×C₃₇⋊C₃

yes

—

C₃×C₉

C₉×C₃₇⋊C₃

yes

characteristic

C₃×C₉

C₃²×C₃₇⋊C₃

yes

characteristic

C₂₇

—

C₂₇

—

C₂₇

—

C₂₇

—

111

—

C₂₇

—

111

—

C₂₇

—

111

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

C₃×C₉×C₃₇⋊C₃

yes

characteristic

100

C₉

—

101

C₉

—

111

—

102

C₉

—

111

—

103

C₉

—

111

—

104

C₉

#15688510

yes

complemented

105

C₉

3₋¹⁺²×C₃₇⋊C₃

yes

characteristic

106

C₉

#15688532

yes

—

107

C₃

—

333

—

108

C₃

#22582543

yes

characteristic

109

C₃

C₃×#15688510

yes

characteristic

110

C₃

#22582614

yes

—

	subgroup	G/H
center, Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
commutator, socle	C₃²×C₃₇	C₃²×C₉
Fitting	C₃₇×C₉⋊C₂₇	C₃
prime radical	C₃²×C₃₇⋊C₃	C₃×C₉
cosocle	C₃×C₉×C₃₇	C₃³
hypercenter	C₉⋊C₂₇	C₃₇⋊C₃
hypocenter	C₃₇	#10624728
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₉ ◁ C₉⋊C₂₇

subgroup

G/H

center, Frattini

C₃×C₉

C₃²×C₃₇⋊C₃

commutator, socle

C₃²×C₃₇

C₃²×C₉

Fitting

C₃₇×C₉⋊C₂₇

C₃

prime radical

C₃²×C₃₇⋊C₃

C₃×C₉

cosocle

C₃×C₉×C₃₇

C₃³

hypercenter

C₉⋊C₂₇

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃₇

C₁ ◁ C₃×C₉ ◁ C₉⋊C₂₇

N	H ≅ G/N	type
3. #22582490	C₃	semidirect
4. #22582488	C₃	semidirect
5. #22582488	C₃	semidirect
7. #22582463	C₃	semidirect
8. C₉×#11758601	C₃	semidirect
9. C₃₇×C₉⋊C₂₇	C₃	semidirect
39. #11758601	3₋¹⁺²	semidirect
46. C₉×C₃₇	C₂₇⋊C₃	semidirect
57. C₉⋊C₂₇	C₃₇⋊C₃	semidirect
79. C₃₇	#10624728	semidirect
104. C₉	#15688510	semidirect

H ≅ G/N

type

3. #22582490

C₃

semidirect

4. #22582488

C₃

semidirect

5. #22582488

C₃

semidirect

7. #22582463

C₃

semidirect

8. C₉×#11758601

C₃

semidirect

9. C₃₇×C₉⋊C₂₇

C₃

semidirect

39. #11758601

3₋¹⁺²

semidirect

46. C₉×C₃₇

C₂₇⋊C₃

semidirect

57. C₉⋊C₂₇

C₃₇⋊C₃

semidirect

79. C₃₇

#10624728

semidirect

104. C₉

#15688510

semidirect

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

subgroup	index
C₃×#15688510	3
C₉×#11758605	3
#22582490	3
#22582488	3
#22582465	3
#22582463	3
C₉×#11758601	3
C₃₇×C₉⋊C₂₇	3
#10624728	37

subgroup

index

C₃₇×C₉⋊C₂₇

Subgroups — #41006465

Conjugacy classes of subgroups

Complements

Maximal subgroups