Group #41008058 — subgroups

Subgroups — #41008058

Order 26973.

596 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 250 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	#22582593	C₃	1	18	yes	maximal, complemented
2	#22582585	C₃	1	9	yes	maximal, complemented
3	#22582580	C₃	1	9	yes	maximal, complemented
4	#22582554	C₃	1	2	yes	maximal
5	#22582554	C₃	1	1	yes	maximal, characteristic
6	C₃₇×#6891	C₃	1	1	yes	maximal, characteristic
7	C₃×#11758608	C₃²	1	3	yes	—
8	C₃×#11758608	C₃²	1	6	yes	—
9	#15688506	C₃²	1	9	yes	complemented
10	#15688505	C₃²	1	9	yes	complemented
11	#15688504	C₃²	1	54	yes	complemented
12	C₃²⋊#66524	C₃²	1	6	yes	—
13	C₃²⋊#66524	C₃²	1	6	yes	—
14	C₃²⋊#66524	C₃²	1	3	yes	—
15	C₃²⋊#66524	C₃²	1	6	yes	—
16	C₃²⋊#66524	C₃²	1	6	yes	—
17	C₉×#66524	C₃²	1	9	yes	complemented
18	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
19	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
20	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
21	C₃₇×C₃²⋊C₉	C₃²	1	6	yes	—
22	C₃³×C₃₇	—	9	2	no	—
23	C₃³×C₃₇	—	9	1	no	—
24	C₃³×C₃₇	—	9	1	no	—
25	C₃³×C₃₇	C₃³	1	1	yes	characteristic
26	C₃³×C₃₇	C₃³	1	2	yes	—
27	C₃³×C₃₇	C₃³	1	1	yes	characteristic
28	3₋¹⁺²×C₃₇	—	9	3	no	—
29	C₃×C₉×C₃₇	C₃³	1	9	yes	—
30	#11758608	—	9	3	no	—
31	#11758608	—	9	6	no	—
32	C₃×#66524	C₃³	1	18	yes	—
33	C₃×#66524	C₃³	1	9	yes	—
34	#10624911	—	37	1	no	maximal, Sylow
35	C₃²×C₃₇	—	3	2	no	—
36	C₃²×C₃₇	—	3	1	no	—
37	C₃²×C₃₇	—	3	1	no	—
38	C₃²×C₃₇	—	3	2	no	—
39	C₃²×C₃₇	—	3	1	no	—
40	C₃²×C₃₇	—	3	2	no	—
41	C₃²×C₃₇	—	3	2	no	—
42	C₃²×C₃₇	—	3	2	no	—
43	C₃²×C₃₇	—	3	1	no	—
44	C₃²×C₃₇	—	3	2	no	—
45	C₃²×C₃₇	—	9	9	no	—
46	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
47	C₉×C₃₇	—	3	9	no	—
48	#66524	—	3	18	no	—
49	#66524	—	3	9	no	—
50	#6899	—	37	18	no	—
51	#6897	—	37	9	no	—
52	#6896	—	37	9	no	—
53	#6891	—	37	2	no	—
54	#6891	—	37	1	no	—
55	#6891	C₃₇⋊C₃	1	1	yes	characteristic
56	C₃×C₃₇	—	9	1	no	—
57	C₃×C₃₇	—	9	2	no	—
58	C₃×C₃₇	—	9	1	no	—
59	C₃×C₃₇	3₋¹⁺⁴	1	2	yes	—
60	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
61	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
62	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
63	C₃×3₋¹⁺²	—	37	3	no	—
64	C₃×3₋¹⁺²	—	37	6	no	—
65	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	3	yes	—
66	C₉⋊C₉	—	37	54	no	—
67	C₉⋊C₉	—	37	9	no	—
68	C₉⋊C₉	—	37	9	no	—
69	C₃²⋊C₉	—	37	6	no	—
70	C₃²⋊C₉	—	37	6	no	—
71	C₃²⋊C₉	—	37	3	no	—
72	C₃²⋊C₉	—	37	6	no	—
73	C₃²⋊C₉	—	37	6	no	—
74	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
75	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	6	yes	—
76	C₉²	—	37	9	no	—
77	C₃₇	#10624911	1	1	yes	characteristic, complemented, Sylow
78	C₃³	—	9	2	no	—
79	C₃³	—	9	1	no	—
80	C₃³	—	9	1	no	—
81	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
82	C₃³	C₃²×C₃₇⋊C₃	1	2	yes	—
83	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
84	3₋¹⁺²	—	9	3	no	—
85	3₋¹⁺²	—	333	3	no	—
86	3₋¹⁺²	—	333	6	no	—
87	C₃×C₉	—	37	18	no	—
88	C₃×C₉	—	37	9	no	—
89	C₃×C₉	C₃²×C₃₇⋊C₃	1	9	yes	—
90	C₃²	—	3	2	no	—
91	C₃²	—	3	1	no	—
92	C₃²	—	3	2	no	—
93	C₃²	—	3	1	no	—
94	C₃²	—	3	2	no	—
95	C₃²	—	3	2	no	—
96	C₃²	—	3	2	no	—
97	C₃²	—	3	1	no	—
98	C₃²	—	3	1	no	—
99	C₃²	—	3	2	no	—
100	C₃²	—	9	9	no	—
101	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
102	C₉	—	3	9	no	—
103	C₉	—	111	18	no	—
104	C₉	—	111	9	no	—
105	C₃	—	9	1	no	—
106	C₃	—	9	2	no	—
107	C₃	—	9	1	no	—
108	C₃	#22582679	1	2	yes	—
109	C₃	#22582679	1	1	yes	characteristic
110	C₃	#22582680	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22582593

C₃

yes

maximal, complemented

#22582585

C₃

yes

maximal, complemented

#22582580

C₃

yes

maximal, complemented

#22582554

C₃

yes

maximal

#22582554

C₃

yes

maximal, characteristic

C₃₇×#6891

C₃

yes

maximal, characteristic

C₃×#11758608

C₃²

yes

—

C₃×#11758608

C₃²

yes

—

#15688506

C₃²

yes

complemented

#15688505

C₃²

yes

complemented

#15688504

C₃²

yes

complemented

C₃²⋊#66524

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

C₉×#66524

C₃²

yes

complemented

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃³×C₃₇

C₃³

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

3₋¹⁺²×C₃₇

—

C₃×C₉×C₃₇

C₃³

yes

—

#11758608

—

#11758608

—

C₃×#66524

C₃³

yes

—

C₃×#66524

C₃³

yes

—

#10624911

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃⁴

yes

characteristic

C₉×C₃₇

—

—

—

—

—

—

—

—

C₃₇⋊C₃

yes

characteristic

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

3₋¹⁺⁴

yes

—

C₃×C₃₇

3₋¹⁺⁴

yes

characteristic

C₃×C₃₇

3₋¹⁺⁴

yes

characteristic

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₉²

—

C₃₇

#10624911

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

C₃³

C₃²×C₃₇⋊C₃

yes

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

333

—

3₋¹⁺²

—

333

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃²×C₃₇⋊C₃

yes

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

C₃³×C₃₇⋊C₃

yes

characteristic

102

C₉

—

103

C₉

—

111

—

104

C₉

—

111

—

105

C₃

—

106

C₃

—

107

C₃

—

108

C₃

#22582679

yes

—

109

C₃

#22582679

yes

characteristic

110

C₃

#22582680

yes

characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting	C₃₇×#6891	C₃
prime radical	C₃⁴×C₃₇	C₃²
hypercenter	#6891	C₃₇⋊C₃
hypocenter	C₃₇	#10624911
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ #6891

subgroup

G/H

center, Frattini

C₃²

C₃³×C₃₇⋊C₃

commutator, socle, cosocle

C₃²×C₃₇

C₃⁴

Fitting

C₃₇×#6891

C₃

prime radical

C₃⁴×C₃₇

C₃²

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃₇

C₁ ◁ C₃² ◁ #6891

N	H ≅ G/N	type
1. #22582593	C₃	semidirect
2. #22582585	C₃	semidirect
3. #22582580	C₃	semidirect
9. #15688506	C₃²	semidirect
10. #15688505	C₃²	semidirect
11. #15688504	C₃²	semidirect
17. C₉×#66524	C₃²	semidirect
77. C₃₇	#10624911	semidirect

H ≅ G/N

type

1. #22582593

C₃

semidirect

2. #22582585

C₃

semidirect

3. #22582580

C₃

semidirect

9. #15688506

C₃²

semidirect

10. #15688505

C₃²

semidirect

11. #15688504

C₃²

semidirect

17. C₉×#66524

C₃²

semidirect

77. C₃₇

#10624911

semidirect

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

subgroup	index
#22582593	3
#22582585	3
#22582580	3
#22582554	3
C₃₇×#6891	3
#10624911	37

subgroup

index

C₃₇×#6891

Subgroups — #41008058

Conjugacy classes of subgroups

Complements

Maximal subgroups