Group #41008606 — subgroups

Subgroups — #41008606

Order 26973.

220 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 96 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×(C₃₇⋊C₃)⋊C₂₇	C₃	1	1	yes	maximal, characteristic
2	#22582706	C₃	1	2	yes	maximal, complemented
3	#22582705	C₃	1	2	yes	maximal, complemented
4	#22582704	C₃	1	1	yes	maximal, characteristic, complemented
5	#22582703	C₃	1	1	yes	maximal, characteristic, complemented
6	#22582690	C₃	1	2	yes	maximal
7	C₉×#11758603	C₃	1	1	yes	maximal, characteristic, complemented
8	C₉×#11758603	C₃	1	2	yes	maximal, complemented
9	#22582543	C₃	1	1	yes	maximal, characteristic
10	(C₃₇⋊C₃)⋊C₂₇	—	3	3	no	—
11	C₃×#11758603	C₃²	1	2	yes	—
12	C₃×#11758603	C₃²	1	2	yes	—
13	C₃×#11758603	C₃²	1	2	yes	—
14	C₃×#11758603	C₃²	1	1	yes	characteristic
15	C₃×#11758603	C₃²	1	1	yes	characteristic
16	C₃×#11758603	C₃²	1	1	yes	characteristic
17	C₃×#11758605	C₉	1	1	yes	characteristic
18	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
19	#15688506	C₉	1	1	yes	characteristic
20	#15688504	C₉	1	2	yes	—
21	#15688504	C₉	1	2	yes	—
22	#15688502	C₉	1	2	yes	—
23	C₉×#66524	C₃²	1	1	yes	characteristic
24	C₉×#66524	C₃²	1	2	yes	—
25	C₃₇×C₉⋊C₉	C₉	1	1	yes	characteristic
26	C₃²×C₃₇⋊C₃	C₃×C₉	1	1	yes	characteristic
27	C₃×C₉×C₃₇	C₃×C₉	1	2	yes	—
28	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
29	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
30	C₃×#66524	C₃×C₉	1	2	yes	—
31	C₃×#66524	C₃×C₉	1	2	yes	—
32	C₃×#66524	C₃×C₉	1	1	yes	characteristic
33	C₃×#66524	C₃×C₉	1	2	yes	—
34	C₃×#66524	C₃³	1	1	yes	characteristic
35	#11758605	C₂₇	1	9	yes	complemented
36	C₉×C₃₇⋊C₃	—	3	1	no	—
37	C₉×C₃₇⋊C₃	—	3	2	no	—
38	#11758603	—	3	1	no	—
39	#11758603	—	3	1	no	—
40	#11758603	—	3	2	no	—
41	#11758603	—	3	2	no	—
42	#11758603	—	3	2	no	—
43	#11758603	3₋¹⁺²	1	3	yes	complemented
44	C₂₇×3₋¹⁺²	—	37	1	no	maximal, Sylow
45	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
46	C₃×C₃₇⋊C₃	—	3	1	no	—
47	C₃×C₃₇⋊C₃	—	3	2	no	—
48	C₃×C₃₇⋊C₃	C₃×C₂₇	1	3	yes	—
49	C₉×C₃₇	—	3	1	no	—
50	C₉×C₃₇	—	3	2	no	—
51	C₉×C₃₇	C₃×C₂₇	1	3	yes	complemented
52	#66524	—	3	1	no	—
53	#66524	—	3	2	no	—
54	#66524	—	3	2	no	—
55	#66524	C₃×C₂₇	1	6	yes	complemented
56	#66524	C₃×3₋¹⁺²	1	1	yes	characteristic
57	#66524	#737	1	2	yes	—
58	C₃²×C₂₇	—	37	1	no	—
59	C₉×3₋¹⁺²	—	37	1	no	—
60	C₉⋊C₂₇	—	37	1	no	—
61	C₉⋊C₂₇	—	37	1	no	—
62	C₉⋊C₂₇	—	37	2	no	—
63	C₉⋊C₂₇	—	37	2	no	—
64	C₃²⋊C₂₇	—	37	2	no	—
65	C₉×C₂₇	—	37	2	no	—
66	C₉×C₂₇	—	37	1	no	—
67	C₃×C₃₇	C₉×3₋¹⁺²	1	1	yes	characteristic
68	C₃×C₃₇	C₃²×C₂₇	1	1	yes	characteristic
69	C₃×C₃₇	#6903	1	2	yes	—
70	C₃₇⋊C₃	—	3	3	no	—
71	C₃×3₋¹⁺²	—	37	1	no	—
72	C₃²×C₉	—	37	1	no	—
73	C₃×C₂₇	—	37	1	no	—
74	C₃×C₂₇	—	37	2	no	—
75	C₃×C₂₇	—	37	2	no	—
76	C₃×C₂₇	—	37	2	no	—
77	C₃×C₂₇	—	37	1	no	—
78	C₃×C₂₇	—	37	1	no	—
79	C₃×C₂₇	—	111	3	no	—
80	C₉⋊C₉	—	37	2	no	—
81	C₉⋊C₉	—	37	2	no	—
82	C₉⋊C₉	—	37	1	no	—
83	C₉⋊C₉	#66526	1	1	yes	characteristic
84	C₃²⋊C₉	—	37	2	no	—
85	C₉²	—	37	2	no	—
86	C₉²	—	37	1	no	—
87	C₃₇	C₂₇×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
88	C₃³	—	37	1	no	—
89	3₋¹⁺²	—	37	9	no	—
90	C₃×C₉	—	37	2	no	—
91	C₃×C₉	—	37	1	no	—
92	C₃×C₉	—	37	2	no	—
93	C₃×C₉	—	37	2	no	—
94	C₃×C₉	—	37	1	no	—
95	C₃×C₉	—	111	1	no	—
96	C₃×C₉	—	111	2	no	—
97	C₃×C₉	C₃×#66526	1	2	yes	—
98	C₃×C₉	C₃×#66526	1	1	yes	characteristic
99	C₃×C₉	C₃×#66526	1	1	yes	characteristic
100	C₂₇	—	37	3	no	—
101	C₂₇	—	111	2	no	—
102	C₂₇	—	111	2	no	—
103	C₂₇	—	111	2	no	—
104	C₂₇	—	111	1	no	—
105	C₂₇	—	111	1	no	—
106	C₃²	—	37	3	no	—
107	C₃²	—	111	1	no	—
108	C₃²	—	111	2	no	—
109	C₃²	C₃²×#66526	1	1	yes	characteristic
110	C₉	—	3	1	no	—
111	C₉	—	3	2	no	—
112	C₉	—	37	1	no	—
113	C₉	—	37	2	no	—
114	C₉	—	37	6	no	—
115	C₉	—	111	1	no	—
116	C₉	—	111	2	no	—
117	C₉	—	111	2	no	—
118	C₉	(C₃₇⋊C₃)⋊C₂₇	1	3	yes	complemented
119	C₃	—	111	3	no	—
120	C₃	3₋¹⁺²×#66526	1	1	yes	characteristic
121	C₃	C₃×(C₃₇⋊C₃)⋊C₂₇	1	1	yes	characteristic
122	C₃	#22582732	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×(C₃₇⋊C₃)⋊C₂₇

C₃

yes

maximal, characteristic

#22582706

C₃

yes

maximal, complemented

#22582705

C₃

yes

maximal, complemented

#22582704

C₃

yes

maximal, characteristic, complemented

#22582703

C₃

yes

maximal, characteristic, complemented

#22582690

C₃

yes

maximal

C₉×#11758603

C₃

yes

maximal, characteristic, complemented

C₉×#11758603

C₃

yes

maximal, complemented

#22582543

C₃

yes

maximal, characteristic

(C₃₇⋊C₃)⋊C₂₇

—

C₃×#11758603

C₃²

yes

—

C₃×#11758603

C₃²

yes

—

C₃×#11758603

C₃²

yes

—

C₃×#11758603

C₃²

yes

characteristic

C₃×#11758603

C₃²

yes

characteristic

C₃×#11758603

C₃²

yes

characteristic

C₃×#11758605

C₉

yes

characteristic

C₃×C₉×C₃₇⋊C₃

C₃²

yes

characteristic

#15688506

C₉

yes

characteristic

#15688504

C₉

yes

—

#15688504

C₉

yes

—

#15688502

C₉

yes

—

C₉×#66524

C₃²

yes

characteristic

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×C₉×C₃₇

C₃×C₉

yes

characteristic

C₃×#66524

C₃×C₉

yes

—

C₃×#66524

C₃×C₉

yes

—

C₃×#66524

C₃×C₉

yes

characteristic

C₃×#66524

C₃×C₉

yes

—

C₃×#66524

C₃³

yes

characteristic

#11758605

C₂₇

yes

complemented

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

—

—

—

—

—

3₋¹⁺²

yes

complemented

C₂₇×3₋¹⁺²

—

maximal, Sylow

C₃²×C₃₇

C₃²×C₉

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

C₃×C₂₇

yes

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

C₃×C₂₇

yes

—

—

—

C₃×C₂₇

yes

complemented

#66524

C₃×3₋¹⁺²

yes

characteristic

#66524

#737

yes

—

C₃²×C₂₇

—

C₉×3₋¹⁺²

—

C₉⋊C₂₇

—

C₉⋊C₂₇

—

C₉⋊C₂₇

—

C₉⋊C₂₇

—

C₃²⋊C₂₇

—

C₉×C₂₇

—

C₉×C₂₇

—

C₃×C₃₇

C₉×3₋¹⁺²

yes

characteristic

C₃×C₃₇

C₃²×C₂₇

yes

characteristic

C₃×C₃₇

#6903

yes

—

C₃₇⋊C₃

—

C₃×3₋¹⁺²

—

C₃²×C₉

—

C₃×C₂₇

—

C₃×C₂₇

—

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

—

C₉²

—

C₃₇

C₂₇×3₋¹⁺²

yes

characteristic, complemented, Sylow

C₃³

—

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

C₃×#66526

yes

—

C₃×C₉

yes

C₃×C₉

yes

100

C₂₇

—

101

C₂₇

—

111

—

102

C₂₇

—

111

—

103

C₂₇

—

111

—

104

C₂₇

—

111

—

105

C₂₇

—

111

—

106

C₃²

—

107

C₃²

—

111

—

108

C₃²

—

111

—

109

C₃²

C₃²×#66526

yes

characteristic

110

C₉

—

111

C₉

—

112

C₉

—

113

C₉

—

114

C₉

—

115

C₉

—

111

—

116

C₉

—

111

—

117

C₉

—

111

—

118

C₉

(C₃₇⋊C₃)⋊C₂₇

yes

complemented

119

C₃

—

111

—

120

C₃

3₋¹⁺²×#66526

yes

characteristic

121

C₃

C₃×(C₃₇⋊C₃)⋊C₂₇

yes

characteristic

122

C₃

#22582732

yes

—

	subgroup	G/H
center, Frattini	C₃²	C₃²×#66526
commutator	C₃×C₃₇	C₃²×C₂₇
Fitting	C₃₇×C₉⋊C₉	C₉
socle	C₃²×C₃₇	C₃²×C₉
prime radical	C₃²×C₃₇⋊C₃	C₃×C₉
cosocle	C₃×#66524	C₃³
hypercenter	C₉⋊C₉	#66526
hypocenter	C₃₇	C₂₇×3₋¹⁺²
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₉⋊C₉

subgroup

G/H

center, Frattini

C₃²

C₃²×#66526

commutator

C₃×C₃₇

C₃²×C₂₇

Fitting

C₃₇×C₉⋊C₉

C₉

socle

C₃²×C₃₇

C₃²×C₉

prime radical

C₃²×C₃₇⋊C₃

C₃×C₉

C₃³

C₉⋊C₉

C₃₇

C₂₇×3₋¹⁺²

LCS

G ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₉⋊C₉

N	H ≅ G/N	type
2. #22582706	C₃	semidirect
3. #22582705	C₃	semidirect
4. #22582704	C₃	semidirect
5. #22582703	C₃	semidirect
7. C₉×#11758603	C₃	semidirect
8. C₉×#11758603	C₃	semidirect
35. #11758605	C₂₇	semidirect
43. #11758603	3₋¹⁺²	semidirect
51. C₉×C₃₇	C₃×C₂₇	semidirect
55. #66524	C₃×C₂₇	semidirect
87. C₃₇	C₂₇×3₋¹⁺²	semidirect
118. C₉	(C₃₇⋊C₃)⋊C₂₇	semidirect

H ≅ G/N

type

2. #22582706

C₃

semidirect

3. #22582705

C₃

semidirect

4. #22582704

C₃

semidirect

5. #22582703

C₃

semidirect

7. C₉×#11758603

C₃

semidirect

8. C₉×#11758603

C₃

semidirect

35. #11758605

C₂₇

semidirect

43. #11758603

3₋¹⁺²

semidirect

51. C₉×C₃₇

C₃×C₂₇

semidirect

55. #66524

C₃×C₂₇

semidirect

87. C₃₇

C₂₇×3₋¹⁺²

semidirect

118. C₉

(C₃₇⋊C₃)⋊C₂₇

semidirect

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

subgroup	index
C₃×(C₃₇⋊C₃)⋊C₂₇	3
#22582706	3
#22582705	3
#22582704	3
#22582703	3
#22582690	3
C₉×#11758603	3
#22582543	3
C₂₇×3₋¹⁺²	37

subgroup

index

C₃×(C₃₇⋊C₃)⋊C₂₇

C₂₇×3₋¹⁺²

Subgroups — #41008606

Conjugacy classes of subgroups

Complements

Maximal subgroups