Group #41008612 — subgroups

Subgroups — #41008612

Order 26973.

192 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 78 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×#15688541	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	C₃²⋊#11758603	C₃	1	2	yes	maximal
7	#22582688	C₃	1	2	yes	maximal, complemented
8	C₉×#11758603	C₃	1	1	yes	maximal, characteristic, complemented
9	3₋¹⁺²×#66524	C₃	1	1	yes	maximal, characteristic
10	#15688541	—	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	C₃²×#66524	C₃²	1	1	yes	characteristic
18	#15688506	C₉	1	2	yes	—
19	#15688506	C₉	1	1	yes	characteristic
20	#15688505	C₉	1	2	yes	—
21	#15688505	C₉	1	1	yes	characteristic
22	C₃²⋊#66524	C₉	1	2	yes	—
23	C₉×#66524	C₃²	1	1	yes	characteristic
24	C₉×#66524	C₃²	1	2	yes	—
25	C₃×3₋¹⁺²×C₃₇	C₉	1	1	yes	characteristic
26	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
27	3₋¹⁺²×C₃₇	—	3	3	no	—
28	C₃×C₉×C₃₇	C₃×C₉	1	2	yes	—
29	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
30	C₃×#66524	—	3	1	no	—
31	C₃×#66524	—	3	2	no	—
32	C₃×#66524	C₃×C₉	1	2	yes	—
33	C₃×#66524	C₃×C₉	1	2	yes	—
34	C₃×#66524	C₃×C₉	1	2	yes	—
35	C₃×#66524	C₃×C₉	1	1	yes	characteristic
36	C₃×#66524	C₃×C₉	1	1	yes	characteristic
37	C₃×#66524	C₃³	1	1	yes	characteristic
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	#10624728	—	37	1	no	maximal, Sylow
45	C₃²×C₃₇	—	3	1	no	—
46	C₃²×C₃₇	—	3	1	no	—
47	C₃²×C₃₇	—	3	2	no	—
48	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
49	C₉×C₃₇	—	3	2	no	—
50	C₉×C₃₇	C₂₇⋊C₃	1	3	yes	complemented
51	#66524	—	3	1	no	—
52	#66524	—	3	1	no	—
53	#66524	—	3	2	no	—
54	#66524	—	3	2	no	—
55	#66524	—	3	2	no	—
56	#66524	C₃×3₋¹⁺²	1	1	yes	characteristic
57	#66524	#737	1	2	yes	—
58	C₃×C₂₇⋊C₃	—	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	#6864	—	37	2	no	—
66	C₉×C₂₇	—	37	1	no	—
67	C₃×C₃₇	—	9	1	no	—
68	C₃×C₃₇	C₉×3₋¹⁺²	1	1	yes	characteristic
69	C₃×C₃₇	C₃×C₂₇⋊C₃	1	1	yes	characteristic
70	C₃×C₃₇	#6903	1	2	yes	—
71	C₃×3₋¹⁺²	#66526	1	1	yes	characteristic
72	C₃²×C₉	—	37	1	no	—
73	C₂₇⋊C₃	—	111	3	no	—
74	C₃×C₂₇	—	37	1	no	—
75	C₃×C₂₇	—	37	2	no	—
76	C₃×C₂₇	—	37	2	no	—
77	C₃×C₂₇	—	37	2	no	—
78	C₃×C₂₇	—	37	1	no	—
79	C₃×C₂₇	—	37	1	no	—
80	C₉⋊C₉	—	37	2	no	—
81	C₉⋊C₉	—	37	1	no	—
82	C₉⋊C₉	—	37	2	no	—
83	C₉⋊C₉	—	37	1	no	—
84	C₃²⋊C₉	—	37	2	no	—
85	C₉²	—	37	1	no	—
86	C₉²	—	37	2	no	—
87	C₃₇	#10624728	1	1	yes	characteristic, complemented, Sylow
88	C₃³	C₃×#66526	1	1	yes	characteristic
89	3₋¹⁺²	—	3	3	no	—
90	C₃×C₉	—	37	1	no	—
91	C₃×C₉	—	37	2	no	—
92	C₃×C₉	—	37	2	no	—
93	C₃×C₉	—	37	2	no	—
94	C₃×C₉	—	37	1	no	—
95	C₃×C₉	—	37	1	no	—
96	C₃×C₉	—	111	1	no	—
97	C₃×C₉	—	111	2	no	—
98	C₃×C₉	C₃×#66526	1	2	yes	—
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	—	3	1	no	—
107	C₃²	—	3	1	no	—
108	C₃²	—	3	2	no	—
109	C₃²	C₃²×#66526	1	1	yes	characteristic
110	C₉	—	3	2	no	—
111	C₉	—	37	1	no	—
112	C₉	—	37	2	no	—
113	C₉	—	111	1	no	—
114	C₉	—	111	1	no	—
115	C₉	—	111	2	no	—
116	C₉	—	111	2	no	—
117	C₉	—	111	2	no	—
118	C₉	#15688541	1	3	yes	complemented
119	C₃	—	9	1	no	—
120	C₃	3₋¹⁺²×#66526	1	1	yes	characteristic
121	C₃	C₃×#15688541	1	1	yes	characteristic
122	C₃	#22582732	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688541

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

C₃²⋊#11758603

C₃

yes

maximal

#22582688

C₃

yes

maximal, complemented

C₉×#11758603

C₃

yes

maximal, characteristic, complemented

3₋¹⁺²×#66524

C₃

yes

maximal, characteristic

#15688541

—

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

C₃²×#66524

C₃²

yes

characteristic

#15688506

C₉

yes

—

#15688506

C₉

yes

characteristic

#15688505

C₉

yes

—

#15688505

C₉

yes

characteristic

C₃²⋊#66524

C₉

yes

—

C₉×#66524

C₃²

yes

characteristic

C₉×#66524

C₃²

yes

—

C₃×3₋¹⁺²×C₃₇

C₉

yes

characteristic

C₃³×C₃₇

C₃×C₉

yes

characteristic

3₋¹⁺²×C₃₇

—

C₃×C₉×C₃₇

C₃×C₉

yes

—

C₃×C₉×C₃₇

C₃×C₉

yes

—

—

C₃×C₉

yes

—

C₃×#66524

C₃×C₉

yes

—

C₃×#66524

C₃×C₉

yes

—

C₃×#66524

C₃×C₉

yes

characteristic

C₃×#66524

C₃×C₉

yes

characteristic

C₃×#66524

C₃³

yes

—

—

—

—

—

3₋¹⁺²

yes

complemented

#10624728

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃²×C₉

yes

characteristic

C₉×C₃₇

—

C₉×C₃₇

C₂₇⋊C₃

yes

—

—

—

—

—

C₃×3₋¹⁺²

yes

characteristic

#66524

#737

yes

—

C₃×C₂₇⋊C₃

—

C₉×3₋¹⁺²

—

C₉⋊C₂₇

—

C₉⋊C₂₇

—

C₉⋊C₂₇

—

C₉⋊C₂₇

—

C₃²⋊C₂₇

—

#6864

—

C₉×C₂₇

—

C₃×C₃₇

—

C₃×C₃₇

C₉×3₋¹⁺²

yes

characteristic

C₃×C₃₇

C₃×C₂₇⋊C₃

yes

characteristic

C₃×C₃₇

#6903

yes

—

C₃×3₋¹⁺²

#66526

yes

characteristic

C₃²×C₉

—

C₂₇⋊C₃

—

111

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₉²

—

C₉²

—

C₃₇

#10624728

yes

characteristic, complemented, Sylow

C₃³

C₃×#66526

yes

characteristic

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

C₃×#66526

yes

—

C₃×C₉

C₃×#66526

yes

characteristic

100

C₂₇

—

101

C₂₇

—

111

—

102

C₂₇

—

111

—

103

C₂₇

—

111

—

104

C₂₇

—

111

—

105

C₂₇

—

111

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

C₃²×#66526

yes

characteristic

110

C₉

—

111

C₉

—

112

C₉

—

113

C₉

—

111

—

114

C₉

—

111

—

115

C₉

—

111

—

116

C₉

—

111

—

117

C₉

—

111

—

118

C₉

#15688541

yes

complemented

119

C₃

—

120

C₃

3₋¹⁺²×#66526

yes

characteristic

121

C₃

C₃×#15688541

yes

characteristic

122

C₃

#22582732

yes

—

	subgroup	G/H
center, Frattini	C₃²	C₃²×#66526
commutator, socle	C₃²×C₃₇	C₃²×C₉
Fitting	C₃×3₋¹⁺²×C₃₇	C₉
prime radical	C₃³×C₃₇	C₃×C₉
cosocle	C₃×#66524	C₃³
hypercenter	C₃×3₋¹⁺²	#66526
hypocenter	C₃₇	#10624728
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃×3₋¹⁺²

subgroup

G/H

center, Frattini

C₃²

C₃²×#66526

commutator, socle

C₃²×C₃₇

C₃²×C₉

Fitting

C₃×3₋¹⁺²×C₃₇

C₉

prime radical

C₃³×C₃₇

C₃×C₉

C₃³

C₃×3₋¹⁺²

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃₇

C₁ ◁ C₃² ◁ C₃×3₋¹⁺²

N	H ≅ G/N	type
2. #22582706	C₃	semidirect
3. #22582705	C₃	semidirect
4. #22582704	C₃	semidirect
5. #22582703	C₃	semidirect
7. #22582688	C₃	semidirect
8. C₉×#11758603	C₃	semidirect
43. #11758603	3₋¹⁺²	semidirect
50. C₉×C₃₇	C₂₇⋊C₃	semidirect
87. C₃₇	#10624728	semidirect
118. C₉	#15688541	semidirect

H ≅ G/N

type

2. #22582706

C₃

semidirect

3. #22582705

C₃

semidirect

4. #22582704

C₃

semidirect

5. #22582703

C₃

semidirect

7. #22582688

C₃

semidirect

8. C₉×#11758603

C₃

semidirect

43. #11758603

3₋¹⁺²

semidirect

50. C₉×C₃₇

C₂₇⋊C₃

semidirect

87. C₃₇

#10624728

semidirect

118. C₉

#15688541

semidirect

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

subgroup	index
C₃×#15688541	3
#22582706	3
#22582705	3
#22582704	3
#22582703	3
C₃²⋊#11758603	3
#22582688	3
C₉×#11758603	3
3₋¹⁺²×#66524	3
#10624728	37

subgroup

index

3₋¹⁺²×#66524

Subgroups — #41008612

Conjugacy classes of subgroups

Complements

Maximal subgroups