Group #41005803 — subgroups

Subgroups — #41005803

Order 26973.

248 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 52 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	#22582553	C₃	1	1	yes	maximal, characteristic
2	#22582553	C₃	1	2	yes	maximal
3	C₃²⋊C₉×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
4	#22582458	C₃	1	4	yes	maximal, complemented
5	#22582457	C₃	1	2	yes	maximal, complemented
6	#22582456	C₃	1	2	yes	maximal, complemented
7	C₃₇×#6861	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
9	C₃×#11758605	—	3	2	no	—
10	C₃×#11758605	—	3	1	no	—
11	C₃×C₉×C₃₇⋊C₃	—	3	1	no	—
12	C₃²⋊#66524	C₃²	1	2	yes	—
13	C₃²⋊#66524	C₃²	1	4	yes	—
14	C₃²⋊#66524	C₃²	1	2	yes	—
15	#15688502	—	3	2	no	—
16	#15688502	—	3	2	no	—
17	#15688502	—	3	2	no	—
18	#15688502	—	3	2	no	—
19	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
20	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
21	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
22	C₃³×C₃₇	C₃³	1	1	yes	characteristic
23	C₃²×C₃₇⋊C₃	—	3	3	no	—
24	C₃²×C₃₇⋊C₃	—	9	2	no	—
25	C₃²×C₃₇⋊C₃	—	9	1	no	—
26	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	2	yes	—
27	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
28	C₃×C₉×C₃₇	—	3	1	no	—
29	C₃×C₉×C₃₇	—	3	1	no	—
30	C₃×C₉×C₃₇	—	3	2	no	—
31	C₃×#66524	—	3	2	no	—
32	C₃×#66524	—	3	2	no	—
33	C₃×#66524	—	3	4	no	—
34	#11758605	—	9	6	no	—
35	#11758605	—	9	3	no	—
36	C₉×C₃₇⋊C₃	—	9	3	no	—
37	#10624643	—	37	1	no	maximal, Sylow
38	C₃²×C₃₇	—	3	2	no	—
39	C₃²×C₃₇	—	3	1	no	—
40	C₃²×C₃₇	—	3	1	no	—
41	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
42	C₃×C₃₇⋊C₃	—	3	2	no	—
43	C₃×C₃₇⋊C₃	—	3	1	no	—
44	C₃×C₃₇⋊C₃	—	3	4	no	—
45	C₃×C₃₇⋊C₃	—	3	2	no	—
46	C₃×C₃₇⋊C₃	—	3	2	no	—
47	C₃×C₃₇⋊C₃	—	9	9	no	—
48	C₃×C₃₇⋊C₃	#731	1	3	yes	—
49	C₉×C₃₇	—	9	1	no	—
50	C₉×C₃₇	—	9	1	no	—
51	C₉×C₃₇	—	9	2	no	—
52	#66524	—	9	2	no	—
53	#66524	—	9	2	no	—
54	#66524	—	9	4	no	—
55	#6891	—	37	2	no	—
56	#6891	—	37	1	no	—
57	C₃×C₃²⋊C₉	—	37	1	no	—
58	#6861	—	37	2	no	—
59	#6861	—	37	2	no	—
60	#6861	—	37	4	no	—
61	#6861	C₃₇⋊C₃	1	1	yes	characteristic, complemented
62	C₃×C₃₇	—	9	1	no	—
63	C₃×C₃₇	C₃×#731	1	1	yes	characteristic
64	C₃×C₃₇	#6911	1	1	yes	characteristic
65	C₃×C₃₇	#6912	1	2	yes	—
66	C₃₇⋊C₃	—	3	3	no	—
67	C₃₇⋊C₃	—	9	2	no	—
68	C₃⁴	—	37	1	no	—
69	C₃×3₋¹⁺²	—	111	1	no	—
70	C₃×3₋¹⁺²	—	111	2	no	—
71	C₃²×C₉	—	111	1	no	—
72	C₃²⋊C₉	—	37	2	no	—
73	C₃²⋊C₉	—	37	4	no	—
74	C₃²⋊C₉	—	37	2	no	—
75	C₃²⋊C₉	—	111	2	no	—
76	C₃²⋊C₉	—	111	2	no	—
77	C₃²⋊C₉	—	111	2	no	—
78	C₃²⋊C₉	—	111	2	no	—
79	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
80	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
81	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	—
82	C₃₇	#10624643	1	1	yes	characteristic, complemented, Sylow
83	C₃³	—	37	1	no	—
84	C₃³	—	37	2	no	—
85	C₃³	—	333	1	no	—
86	C₃³	—	333	2	no	—
87	C₃³	—	111	3	no	—
88	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
89	3₋¹⁺²	—	333	3	no	—
90	3₋¹⁺²	—	333	6	no	—
91	C₃×C₉	—	3	1	no	—
92	C₃×C₉	—	3	1	no	—
93	C₃×C₉	—	3	2	no	—
94	C₃×C₉	—	333	3	no	—
95	C₃×C₉	—	111	2	no	—
96	C₃×C₉	—	111	2	no	—
97	C₃×C₉	—	111	4	no	—
98	C₃²	—	3	1	no	—
99	C₃²	—	3	1	no	—
100	C₃²	—	3	2	no	—
101	C₃²	—	37	3	no	—
102	C₃²	—	333	9	no	—
103	C₃²	—	111	2	no	—
104	C₃²	—	111	2	no	—
105	C₃²	—	111	4	no	—
106	C₃²	—	111	1	no	—
107	C₃²	—	111	2	no	—
108	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
109	C₉	—	9	1	no	—
110	C₉	—	9	1	no	—
111	C₉	—	9	2	no	—
112	C₉	—	333	2	no	—
113	C₉	—	333	2	no	—
114	C₉	—	333	4	no	—
115	C₃	—	9	1	no	—
116	C₃	—	333	2	no	—
117	C₃	—	111	3	no	—
118	C₃	#731×C₃₇⋊C₃	1	1	yes	characteristic
119	C₃	#22582645	1	1	yes	characteristic
120	C₃	#22582649	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22582553

C₃

yes

maximal, characteristic

#22582553

C₃

yes

maximal

C₃²⋊C₉×C₃₇⋊C₃

C₃

yes

maximal, characteristic

#22582458

C₃

yes

maximal, complemented

#22582457

C₃

yes

maximal, complemented

#22582456

C₃

yes

maximal, complemented

C₃₇×#6861

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

C₃²

yes

characteristic

C₃×#11758605

—

C₃×#11758605

—

C₃×C₉×C₃₇⋊C₃

—

C₃²⋊#66524

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

—

—

—

—

C₃₇×C₃²⋊C₉

C₃²

yes

characteristic

C₃₇×C₃²⋊C₉

C₃²

yes

characteristic

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

H(3,𝔽₃)

yes

—

C₃²×C₃₇⋊C₃

H(3,𝔽₃)

yes

characteristic

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

—

—

—

—

—

C₉×C₃₇⋊C₃

—

#10624643

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

#731

yes

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

—

—

—

—

—

C₃×C₃²⋊C₉

—

—

—

—

C₃₇⋊C₃

yes

characteristic, complemented

C₃×C₃₇

—

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

—

C₃₇⋊C₃

—

C₃₇⋊C₃

—

C₃⁴

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃²×C₉

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₃₇

#10624643

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

333

—

C₃³

—

333

—

C₃³

—

111

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

333

—

3₋¹⁺²

—

333

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

333

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

333

—

103

C₃²

—

111

—

104

C₃²

—

111

—

105

C₃²

—

111

—

106

C₃²

—

111

—

107

C₃²

—

111

—

108

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

109

C₉

—

110

C₉

—

111

C₉

—

112

C₉

—

333

—

113

C₉

—

333

—

114

C₉

—

333

—

115

C₃

—

116

C₃

—

333

—

117

C₃

—

111

—

118

C₃

#731×C₃₇⋊C₃

yes

characteristic

119

C₃

#22582645

yes

characteristic

120

C₃

#22582649

yes

—

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6861	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	#6861	C₃₇⋊C₃
hypocenter	C₃₇	#10624643
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ #6861

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6861

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃³×C₃₇⋊C₃

C₃²

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ #6861

N	H ≅ G/N	type
4. #22582458	C₃	semidirect
5. #22582457	C₃	semidirect
6. #22582456	C₃	semidirect
7. C₃₇×#6861	C₃	semidirect
61. #6861	C₃₇⋊C₃	semidirect
82. C₃₇	#10624643	semidirect

H ≅ G/N

type

4. #22582458

C₃

semidirect

5. #22582457

C₃

semidirect

6. #22582456

C₃

semidirect

7. C₃₇×#6861

C₃

semidirect

61. #6861

C₃₇⋊C₃

semidirect

82. C₃₇

#10624643

semidirect

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

subgroup	index
#22582553	3
C₃²⋊C₉×C₃₇⋊C₃	3
#22582458	3
#22582457	3
#22582456	3
C₃₇×#6861	3
#10624643	37

subgroup

index

C₃²⋊C₉×C₃₇⋊C₃

C₃₇×#6861

Subgroups — #41005803

Conjugacy classes of subgroups

Complements

Maximal subgroups