Group #44534574 — subgroups

Subgroups — #44534574

Order 31347.

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	#23782905	C₃	1	2	yes	maximal, complemented
2	#23782845	C₃	1	1	yes	maximal, characteristic, complemented
3	#23782766	C₃	1	2	yes	maximal, complemented
4	#23782764	C₃	1	1	yes	maximal, characteristic, complemented
5	#23782760	C₃	1	2	yes	maximal, complemented
6	#23782758	C₃	1	1	yes	maximal, characteristic, complemented
7	#23782752	C₃	1	1	yes	maximal, characteristic, complemented
8	#23782749	C₃	1	2	yes	maximal, complemented
9	C₄₃×#6912	C₃	1	1	yes	maximal, characteristic, complemented
10	#16812525	—	3	2	no	—
11	C₃×#11923958	—	3	1	no	—
12	#16812515	C₃²	1	2	yes	—
13	#16812515	C₃²	1	2	yes	—
14	#16812513	C₃²	1	2	yes	—
15	#16812510	—	3	2	no	—
16	#16812507	—	3	2	no	—
17	#16812500	C₃²	1	1	yes	characteristic
18	#16812499	C₃²	1	1	yes	characteristic
19	(C₄₃⋊C₉)⋊C₉	—	3	2	no	—
20	C₉×C₄₃⋊C₉	C₃²	1	1	yes	characteristic
21	C₃×3₋¹⁺²×C₄₃	—	3	1	no	—
22	C₃×3₋¹⁺²×C₄₃	C₃²	1	1	yes	characteristic
23	C₄₃×#733	—	3	2	no	—
24	C₄₃×#733	C₃²	1	2	yes	—
25	C₄₃×#733	C₃²	1	1	yes	characteristic
26	#11923958	—	3	2	no	—
27	#11923958	—	3	2	no	—
28	#11923958	—	9	3	no	—
29	#11923957	—	9	2	no	—
30	#11923956	—	9	2	no	—
31	C₃²⋊(C₄₃⋊C₃)	—	3	2	no	—
32	C₃×C₄₃⋊C₉	—	3	1	no	—
33	C₃×C₄₃⋊C₉	—	3	1	no	—
34	C₃×C₄₃⋊C₉	—	3	1	no	—
35	C₉×C₄₃⋊C₃	—	9	2	no	—
36	C₃³×C₄₃	H(3,𝔽₃)	1	1	yes	characteristic
37	3₋¹⁺²×C₄₃	—	3	2	no	—
38	3₋¹⁺²×C₄₃	—	3	1	no	—
39	3₋¹⁺²×C₄₃	—	9	1	no	—
40	3₋¹⁺²×C₄₃	—	9	2	no	—
41	C₃×C₉×C₄₃	—	9	1	no	—
42	C₃×C₉×C₄₃	H(3,𝔽₃)	1	2	yes	—
43	C₃×C₉×C₄₃	C₃³	1	1	yes	characteristic
44	#10624852	—	43	1	no	maximal, Sylow
45	C₃²×C₄₃	—	3	1	no	—
46	C₃²×C₄₃	—	9	1	no	—
47	C₃²×C₄₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
48	C₃×C₄₃⋊C₃	—	9	2	no	—
49	C₉×C₄₃	—	3	2	no	—
50	C₉×C₄₃	—	3	1	no	—
51	C₉×C₄₃	—	9	2	no	—
52	C₉×C₄₃	—	9	1	no	—
53	C₄₃⋊C₉	—	3	3	no	—
54	C₄₃⋊C₉	—	9	1	no	—
55	C₄₃⋊C₉	—	9	1	no	—
56	C₄₃⋊C₉	—	9	2	no	—
57	C₄₃⋊C₉	—	9	2	no	—
58	#6913	—	43	2	no	—
59	#6912	C₄₃⋊C₃	1	1	yes	characteristic, complemented
60	#6900	—	43	1	no	—
61	#6883	—	43	2	no	—
62	#6882	—	43	2	no	—
63	#6882	—	43	1	no	—
64	#6881	—	43	1	no	—
65	#6880	—	43	1	no	—
66	#6878	—	43	2	no	—
67	C₃×C₄₃	—	3	1	no	—
68	C₃×C₄₃	—	9	1	no	—
69	C₃×C₄₃	#6911	1	1	yes	characteristic
70	C₄₃⋊C₃	—	27	2	no	—
71	#737	—	129	2	no	—
72	C₃×3₋¹⁺²	—	3	1	no	—
73	C₃×3₋¹⁺²	—	129	1	no	—
74	C₃×3₋¹⁺²	C₃×C₄₃⋊C₃	1	1	yes	characteristic
75	#733	—	3	2	no	—
76	#733	—	43	2	no	—
77	#733	—	43	2	no	—
78	#733	C₃×C₄₃⋊C₃	1	2	yes	—
79	#733	C₃×C₄₃⋊C₃	1	1	yes	characteristic
80	#732	—	43	2	no	—
81	#731	—	129	2	no	—
82	AΣL(1,𝔽₂₇)	—	129	2	no	—
83	C₉⋊C₉	—	43	1	no	—
84	C₉⋊C₉	—	43	1	no	—
85	C₉⋊C₉	—	129	2	no	—
86	C₉²	—	43	1	no	—
87	C₄₃	#10624852	1	1	yes	characteristic, complemented, Sylow
88	C₃³	C₃²⋊(C₄₃⋊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	—	129	2	no	—
94	3₋¹⁺²	—	129	2	no	—
95	3₋¹⁺²	—	387	2	no	—
96	3₋¹⁺²	—	387	2	no	—
97	3₋¹⁺²	—	387	3	no	—
98	H(3,𝔽₃)	—	129	2	no	—
99	C₃×C₉	—	9	1	no	—
100	C₃×C₉	—	129	1	no	—
101	C₃×C₉	—	129	1	no	—
102	C₃×C₉	—	129	1	no	—
103	C₃×C₉	—	387	2	no	—
104	C₃×C₉	C₃²⋊(C₄₃⋊C₃)	1	2	yes	—
105	C₃×C₉	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
106	C₃²	—	3	1	no	—
107	C₃²	—	9	1	no	—
108	C₃²	—	387	2	no	—
109	C₃²	C₃×C₃²⋊(C₄₃⋊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	—	129	3	no	—
115	C₉	—	387	1	no	—
116	C₉	—	387	1	no	—
117	C₉	—	387	2	no	—
118	C₉	—	387	2	no	—
119	C₃	—	3	1	no	—
120	C₃	—	9	1	no	—
121	C₃	—	1161	2	no	—
122	C₃	#23782891	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23782905

C₃

yes

maximal, complemented

#23782845

C₃

yes

maximal, characteristic, complemented

#23782766

C₃

yes

maximal, complemented

#23782764

C₃

yes

maximal, characteristic, complemented

#23782760

C₃

yes

maximal, complemented

#23782758

C₃

yes

maximal, characteristic, complemented

#23782752

C₃

yes

maximal, characteristic, complemented

#23782749

C₃

yes

maximal, complemented

C₄₃×#6912

C₃

yes

maximal, characteristic, complemented

#16812525

—

C₃×#11923958

—

#16812515

C₃²

yes

—

#16812515

C₃²

yes

—

#16812513

C₃²

yes

—

#16812510

—

#16812507

—

#16812500

C₃²

yes

characteristic

#16812499

C₃²

yes

characteristic

(C₄₃⋊C₉)⋊C₉

—

C₉×C₄₃⋊C₉

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₄₃

—

C₃×3₋¹⁺²×C₄₃

C₃²

yes

characteristic

C₄₃×#733

—

C₄₃×#733

C₃²

yes

—

C₄₃×#733

C₃²

yes

—

—

—

—

—

C₃²⋊(C₄₃⋊C₃)

—

C₃×C₄₃⋊C₉

—

C₃×C₄₃⋊C₉

—

C₃×C₄₃⋊C₉

—

C₉×C₄₃⋊C₃

—

C₃³×C₄₃

H(3,𝔽₃)

yes

characteristic

3₋¹⁺²×C₄₃

—

3₋¹⁺²×C₄₃

—

3₋¹⁺²×C₄₃

—

3₋¹⁺²×C₄₃

—

C₃×C₉×C₄₃

—

C₃×C₉×C₄₃

H(3,𝔽₃)

yes

—

C₃×C₉×C₄₃

C₃³

yes

characteristic

#10624852

—

maximal, Sylow

C₃²×C₄₃

—

C₃²×C₄₃

—

C₃²×C₄₃

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₄₃⋊C₃

—

C₉×C₄₃

—

C₉×C₄₃

—

C₉×C₄₃

—

C₉×C₄₃

—

C₄₃⋊C₉

—

C₄₃⋊C₉

—

C₄₃⋊C₉

—

C₄₃⋊C₉

—

C₄₃⋊C₉

—

#6913

—

#6912

C₄₃⋊C₃

yes

characteristic, complemented

—

—

—

—

—

—

—

C₃×C₄₃

—

C₃×C₄₃

—

C₃×C₄₃

#6911

yes

characteristic

C₄₃⋊C₃

—

#737

—

129

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

129

—

C₃×3₋¹⁺²

C₃×C₄₃⋊C₃

yes

—

—

—

C₃×C₄₃⋊C₃

yes

—

#733

C₃×C₄₃⋊C₃

yes

characteristic

#732

—

#731

—

129

—

AΣL(1,𝔽₂₇)

—

129

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

129

—

C₉²

—

C₄₃

#10624852

yes

characteristic, complemented, Sylow

C₃³

C₃²⋊(C₄₃⋊C₃)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

129

—

3₋¹⁺²

—

129

—

3₋¹⁺²

—

387

—

3₋¹⁺²

—

387

—

3₋¹⁺²

—

387

—

H(3,𝔽₃)

—

129

—

C₃×C₉

—

100

C₃×C₉

—

129

—

101

C₃×C₉

—

129

—

102

C₃×C₉

—

129

—

103

C₃×C₉

—

387

—

104

C₃×C₉

C₃²⋊(C₄₃⋊C₃)

yes

—

105

C₃×C₉

C₃²×C₄₃⋊C₃

yes

characteristic

106

C₃²

—

107

C₃²

—

108

C₃²

—

387

—

109

C₃²

C₃×C₃²⋊(C₄₃⋊C₃)

yes

characteristic

110

C₉

—

111

C₉

—

112

C₉

—

113

C₉

—

114

C₉

—

129

—

115

C₉

—

387

—

116

C₉

—

387

—

117

C₉

—

387

—

118

C₉

—

387

—

119

C₃

—

120

C₃

—

121

C₃

—

1161

—

122

C₃

#23782891

yes

characteristic

	subgroup	G/H
center	C₃	#23782891
commutator, cosocle	C₃×C₉×C₄₃	C₃³
Frattini	C₃×C₉	C₃²×C₄₃⋊C₃
Fitting	C₄₃×#6912	C₃
socle	C₃×C₄₃	#6911
hypercenter	#6912	C₄₃⋊C₃
hypocenter	C₄₃	#10624852
LCS	G ▷ C₃×C₉×C₄₃ ▷ C₃²×C₄₃ ▷ C₃×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×3₋¹⁺² ◁ #6912

subgroup

G/H

center

C₃

#23782891

commutator, cosocle

C₃×C₉×C₄₃

C₃³

Frattini

C₃×C₉

C₃²×C₄₃⋊C₃

Fitting

C₄₃×#6912

C₃

C₃×C₄₃

C₄₃⋊C₃

C₄₃

G ▷ C₃×C₉×C₄₃ ▷ C₃²×C₄₃ ▷ C₃×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₃ ◁ C₃² ◁ C₃×3₋¹⁺² ◁ #6912

N	H ≅ G/N	type
1. #23782905	C₃	semidirect
2. #23782845	C₃	semidirect
3. #23782766	C₃	semidirect
4. #23782764	C₃	semidirect
5. #23782760	C₃	semidirect
6. #23782758	C₃	semidirect
7. #23782752	C₃	semidirect
8. #23782749	C₃	semidirect
9. C₄₃×#6912	C₃	semidirect
59. #6912	C₄₃⋊C₃	semidirect
87. C₄₃	#10624852	semidirect

H ≅ G/N

type

1. #23782905

C₃

semidirect

2. #23782845

C₃

semidirect

3. #23782766

C₃

semidirect

4. #23782764

C₃

semidirect

5. #23782760

C₃

semidirect

6. #23782758

C₃

semidirect

7. #23782752

C₃

semidirect

8. #23782749

C₃

semidirect

9. C₄₃×#6912

C₃

semidirect

59. #6912

C₄₃⋊C₃

semidirect

87. C₄₃

#10624852

semidirect

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

subgroup	index
#23782905	3
#23782845	3
#23782766	3
#23782764	3
#23782760	3
#23782758	3
#23782752	3
#23782749	3
C₄₃×#6912	3
#10624852	43

subgroup

index

C₄₃×#6912

Subgroups — #44534574

Conjugacy classes of subgroups

Complements

Maximal subgroups