Group #36382176 — subgroups

Subgroups — #36382176

Order 22599.

188 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 46 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	#21419419	C₃	1	2	yes	maximal, complemented
2	#21419359	C₃	1	1	yes	maximal, characteristic, complemented
3	#21419280	C₃	1	2	yes	maximal, complemented
4	#21419278	C₃	1	1	yes	maximal, characteristic, complemented
5	#21419274	C₃	1	2	yes	maximal, complemented
6	#21419272	C₃	1	1	yes	maximal, characteristic, complemented
7	#21419266	C₃	1	1	yes	maximal, characteristic, complemented
8	#21419263	C₃	1	2	yes	maximal, complemented
9	C₃₁×#6912	C₃	1	1	yes	maximal, characteristic, complemented
10	#14578912	—	3	2	no	—
11	C₃×#11719689	—	3	1	no	—
12	#14578902	C₃²	1	2	yes	—
13	#14578902	C₃²	1	2	yes	—
14	#14578900	C₃²	1	2	yes	—
15	#14578897	—	3	2	no	—
16	#14578894	—	3	2	no	—
17	#14578887	C₃²	1	1	yes	characteristic
18	#14578886	C₃²	1	1	yes	characteristic
19	(C₃₁⋊C₉)⋊C₉	—	3	2	no	—
20	C₉×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
21	C₃×3₋¹⁺²×C₃₁	—	3	1	no	—
22	C₃×3₋¹⁺²×C₃₁	C₃²	1	1	yes	characteristic
23	C₃₁×#733	—	3	2	no	—
24	C₃₁×#733	C₃²	1	1	yes	characteristic
25	C₃₁×#733	C₃²	1	2	yes	—
26	C₃³×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
27	3₋¹⁺²×C₃₁	—	3	1	no	—
28	3₋¹⁺²×C₃₁	—	3	2	no	—
29	3₋¹⁺²×C₃₁	—	9	1	no	—
30	3₋¹⁺²×C₃₁	—	9	2	no	—
31	C₃×C₉×C₃₁	—	9	1	no	—
32	C₃×C₉×C₃₁	H(3,𝔽₃)	1	2	yes	—
33	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
34	C₃²⋊(C₃₁⋊C₃)	—	3	2	no	—
35	#11719689	—	3	2	no	—
36	#11719689	—	3	2	no	—
37	#11719689	—	9	3	no	—
38	C₃×C₃₁⋊C₉	—	3	1	no	—
39	C₃×C₃₁⋊C₉	—	3	1	no	—
40	C₃×C₃₁⋊C₉	—	3	1	no	—
41	#11719687	—	9	2	no	—
42	#11719686	—	9	2	no	—
43	C₉×C₃₁⋊C₃	—	9	2	no	—
44	#10624852	—	31	1	no	maximal, Sylow
45	C₃²×C₃₁	—	3	1	no	—
46	C₃²×C₃₁	—	9	1	no	—
47	C₃²×C₃₁	C₃×H(3,𝔽₃)	1	1	yes	characteristic
48	C₃×C₃₁⋊C₃	—	9	2	no	—
49	C₉×C₃₁	—	3	2	no	—
50	C₉×C₃₁	—	3	1	no	—
51	C₉×C₃₁	—	9	1	no	—
52	C₉×C₃₁	—	9	2	no	—
53	C₃₁⋊C₉	—	3	3	no	—
54	C₃₁⋊C₉	—	9	1	no	—
55	C₃₁⋊C₉	—	9	1	no	—
56	C₃₁⋊C₉	—	9	2	no	—
57	C₃₁⋊C₉	—	9	2	no	—
58	#6913	—	31	2	no	—
59	#6912	C₃₁⋊C₃	1	1	yes	characteristic, complemented
60	#6900	—	31	1	no	—
61	#6883	—	31	2	no	—
62	#6882	—	31	2	no	—
63	#6882	—	31	1	no	—
64	#6881	—	31	1	no	—
65	#6880	—	31	1	no	—
66	#6878	—	31	2	no	—
67	C₃×C₃₁	—	3	1	no	—
68	C₃×C₃₁	—	9	1	no	—
69	C₃×C₃₁	#6911	1	1	yes	characteristic
70	C₃₁⋊C₃	—	27	2	no	—
71	#737	—	93	2	no	—
72	C₃×3₋¹⁺²	—	3	1	no	—
73	C₃×3₋¹⁺²	—	93	1	no	—
74	C₃×3₋¹⁺²	C₃×C₃₁⋊C₃	1	1	yes	characteristic
75	#733	—	3	2	no	—
76	#733	—	31	2	no	—
77	#733	—	31	2	no	—
78	#733	C₃×C₃₁⋊C₃	1	2	yes	—
79	#733	C₃×C₃₁⋊C₃	1	1	yes	characteristic
80	#732	—	31	2	no	—
81	#731	—	93	2	no	—
82	AΣL(1,𝔽₂₇)	—	93	2	no	—
83	C₉⋊C₉	—	31	1	no	—
84	C₉⋊C₉	—	31	1	no	—
85	C₉⋊C₉	—	93	2	no	—
86	C₉²	—	31	1	no	—
87	C₃₁	#10624852	1	1	yes	characteristic, complemented, Sylow
88	C₃³	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
89	3₋¹⁺²	—	3	2	no	—
90	3₋¹⁺²	—	3	1	no	—
91	3₋¹⁺²	—	9	1	no	—
92	3₋¹⁺²	—	9	2	no	—
93	3₋¹⁺²	—	279	2	no	—
94	3₋¹⁺²	—	279	2	no	—
95	3₋¹⁺²	—	279	3	no	—
96	3₋¹⁺²	—	93	2	no	—
97	3₋¹⁺²	—	93	2	no	—
98	H(3,𝔽₃)	—	93	2	no	—
99	C₃×C₉	—	9	1	no	—
100	C₃×C₉	—	279	2	no	—
101	C₃×C₉	—	93	1	no	—
102	C₃×C₉	—	93	1	no	—
103	C₃×C₉	—	93	1	no	—
104	C₃×C₉	C₃²⋊(C₃₁⋊C₃)	1	2	yes	—
105	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
106	C₃²	—	3	1	no	—
107	C₃²	—	9	1	no	—
108	C₃²	—	279	2	no	—
109	C₃²	C₃×C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
110	C₉	—	3	2	no	—
111	C₉	—	3	1	no	—
112	C₉	—	9	2	no	—
113	C₉	—	9	1	no	—
114	C₉	—	279	1	no	—
115	C₉	—	279	1	no	—
116	C₉	—	279	2	no	—
117	C₉	—	279	2	no	—
118	C₉	—	93	3	no	—
119	C₃	—	3	1	no	—
120	C₃	—	9	1	no	—
121	C₃	—	837	2	no	—
122	C₃	#21419405	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21419419

C₃

yes

maximal, complemented

#21419359

C₃

yes

maximal, characteristic, complemented

#21419280

C₃

yes

maximal, complemented

#21419278

C₃

yes

maximal, characteristic, complemented

#21419274

C₃

yes

maximal, complemented

#21419272

C₃

yes

maximal, characteristic, complemented

#21419266

C₃

yes

maximal, characteristic, complemented

#21419263

C₃

yes

maximal, complemented

C₃₁×#6912

C₃

yes

maximal, characteristic, complemented

#14578912

—

C₃×#11719689

—

#14578902

C₃²

yes

—

#14578902

C₃²

yes

—

#14578900

C₃²

yes

—

#14578897

—

#14578894

—

#14578887

C₃²

yes

characteristic

#14578886

C₃²

yes

characteristic

(C₃₁⋊C₉)⋊C₉

—

C₉×C₃₁⋊C₉

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₁

—

C₃×3₋¹⁺²×C₃₁

C₃²

yes

characteristic

C₃₁×#733

—

C₃₁×#733

C₃²

yes

characteristic

C₃₁×#733

C₃²

yes

—

C₃³×C₃₁

H(3,𝔽₃)

yes

characteristic

3₋¹⁺²×C₃₁

—

3₋¹⁺²×C₃₁

—

3₋¹⁺²×C₃₁

—

3₋¹⁺²×C₃₁

—

C₃×C₉×C₃₁

—

C₃×C₉×C₃₁

H(3,𝔽₃)

yes

—

C₃×C₉×C₃₁

C₃³

yes

characteristic

C₃²⋊(C₃₁⋊C₃)

—

#11719689

—

#11719689

—

#11719689

—

C₃×C₃₁⋊C₉

—

C₃×C₃₁⋊C₉

—

C₃×C₃₁⋊C₉

—

#11719687

—

#11719686

—

C₉×C₃₁⋊C₃

—

#10624852

—

maximal, Sylow

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₃₁⋊C₃

—

C₉×C₃₁

—

C₉×C₃₁

—

C₉×C₃₁

—

C₉×C₃₁

—

C₃₁⋊C₉

—

C₃₁⋊C₉

—

C₃₁⋊C₉

—

C₃₁⋊C₉

—

C₃₁⋊C₉

—

#6913

—

#6912

C₃₁⋊C₃

yes

characteristic, complemented

—

—

—

—

—

—

—

C₃×C₃₁

—

C₃×C₃₁

—

C₃×C₃₁

#6911

yes

characteristic

C₃₁⋊C₃

—

#737

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₃₁⋊C₃

yes

—

—

—

C₃×C₃₁⋊C₃

yes

—

#733

C₃×C₃₁⋊C₃

yes

characteristic

#732

—

#731

—

AΣL(1,𝔽₂₇)

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉²

—

C₃₁

#10624852

yes

characteristic, complemented, Sylow

C₃³

C₃²⋊(C₃₁⋊C₃)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

279

—

3₋¹⁺²

—

279

—

3₋¹⁺²

—

279

—

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

C₃×C₉

—

100

C₃×C₉

—

279

—

101

C₃×C₉

—

102

C₃×C₉

—

103

C₃×C₉

—

104

C₃×C₉

C₃²⋊(C₃₁⋊C₃)

yes

—

105

C₃×C₉

C₃²×C₃₁⋊C₃

yes

characteristic

106

C₃²

—

107

C₃²

—

108

C₃²

—

279

—

109

C₃²

C₃×C₃²⋊(C₃₁⋊C₃)

yes

characteristic

110

C₉

—

111

C₉

—

112

C₉

—

113

C₉

—

114

C₉

—

279

—

115

C₉

—

279

—

116

C₉

—

279

—

117

C₉

—

279

—

118

C₉

—

119

C₃

—

120

C₃

—

121

C₃

—

837

—

122

C₃

#21419405

yes

characteristic

	subgroup	G/H
center	C₃	#21419405
commutator, cosocle	C₃×C₉×C₃₁	C₃³
Frattini	C₃×C₉	C₃²×C₃₁⋊C₃
Fitting	C₃₁×#6912	C₃
socle	C₃×C₃₁	#6911
hypercenter	#6912	C₃₁⋊C₃
hypocenter	C₃₁	#10624852
LCS	G ▷ C₃×C₉×C₃₁ ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×3₋¹⁺² ◁ #6912

subgroup

G/H

center

C₃

#21419405

commutator, cosocle

C₃×C₉×C₃₁

C₃³

Frattini

C₃×C₉

C₃²×C₃₁⋊C₃

Fitting

C₃₁×#6912

C₃

C₃×C₃₁

C₃₁⋊C₃

C₃₁

G ▷ C₃×C₉×C₃₁ ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₃ ◁ C₃² ◁ C₃×3₋¹⁺² ◁ #6912

N	H ≅ G/N	type
1. #21419419	C₃	semidirect
2. #21419359	C₃	semidirect
3. #21419280	C₃	semidirect
4. #21419278	C₃	semidirect
5. #21419274	C₃	semidirect
6. #21419272	C₃	semidirect
7. #21419266	C₃	semidirect
8. #21419263	C₃	semidirect
9. C₃₁×#6912	C₃	semidirect
59. #6912	C₃₁⋊C₃	semidirect
87. C₃₁	#10624852	semidirect

H ≅ G/N

type

1. #21419419

C₃

semidirect

2. #21419359

C₃

semidirect

3. #21419280

C₃

semidirect

4. #21419278

C₃

semidirect

5. #21419274

C₃

semidirect

6. #21419272

C₃

semidirect

7. #21419266

C₃

semidirect

8. #21419263

C₃

semidirect

9. C₃₁×#6912

C₃

semidirect

59. #6912

C₃₁⋊C₃

semidirect

87. C₃₁

#10624852

semidirect

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

subgroup	index
#21419419	3
#21419359	3
#21419280	3
#21419278	3
#21419274	3
#21419272	3
#21419266	3
#21419263	3
C₃₁×#6912	3
#10624852	31

subgroup

index

C₃₁×#6912

Subgroups — #36382176

Conjugacy classes of subgroups

Complements

Maximal subgroups