Group #16814267 — subgroups

Subgroups — C₅×#10624758 #16814267

Order 3645.

402 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 92 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×C₅×#737	C₃	1	1	yes	maximal, characteristic
2	C₅×#6912	C₃	1	6	yes	maximal, complemented
3	C₅×#6910	C₃	1	3	yes	maximal
4	C₃×C₅×#733	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#733	C₃	1	2	yes	maximal, complemented
6	#10624758	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₅×#737	—	3	6	no	—
8	C₅×#737	—	3	3	no	—
9	C₃×C₅×3₋¹⁺²	—	3	2	no	—
10	C₃×C₅×3₋¹⁺²	C₃²	1	3	yes	—
11	C₃×C₅×3₋¹⁺²	C₃²	1	6	yes	—
12	C₃×C₅×3₋¹⁺²	C₃²	1	2	yes	—
13	C₃×C₅×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
14	C₃²×C₅×C₉	—	3	1	no	—
15	C₃²×C₅×C₉	C₃²	1	1	yes	characteristic
16	C₅×#733	—	3	18	no	—
17	C₅×#733	—	3	1	no	—
18	C₅×#733	—	3	2	no	—
19	C₅×#733	—	3	2	no	—
20	C₅×#733	—	3	4	no	—
21	C₅×#731	—	3	9	no	—
22	C₃×#737	C₃×C₅	1	1	yes	characteristic
23	#6912	C₃×C₅	1	6	yes	complemented
24	#6910	C₃×C₅	1	3	yes	—
25	C₃×#733	C₃×C₅	1	1	yes	characteristic, complemented
26	C₃×#733	C₃×C₅	1	2	yes	complemented
27	C₃³×C₅	—	3	1	no	—
28	C₃³×C₅	C₃³	1	1	yes	characteristic
29	C₅×3₋¹⁺²	—	3	18	no	—
30	C₅×3₋¹⁺²	—	3	2	no	—
31	C₅×3₋¹⁺²	—	3	2	no	—
32	C₅×3₋¹⁺²	—	3	6	no	—
33	C₅×3₋¹⁺²	—	3	6	no	—
34	C₅×3₋¹⁺²	—	3	6	no	—
35	C₅×3₋¹⁺²	—	3	3	no	—
36	C₅×3₋¹⁺²	—	3	6	no	—
37	C₅×3₋¹⁺²	H(3,𝔽₃)	1	6	yes	—
38	C₅×H(3,𝔽₃)	—	3	2	no	—
39	C₅×H(3,𝔽₃)	H(3,𝔽₃)	1	3	yes	—
40	C₃×C₅×C₉	—	3	9	no	—
41	C₃×C₅×C₉	—	3	2	no	—
42	C₃×C₅×C₉	—	3	2	no	—
43	C₃×C₅×C₉	—	3	1	no	—
44	C₃×C₅×C₉	—	3	3	no	—
45	C₃×C₅×C₉	—	3	6	no	—
46	C₃×C₅×C₉	H(3,𝔽₃)	1	1	yes	characteristic
47	C₃×C₅×C₉	H(3,𝔽₃)	1	2	yes	—
48	#737	—	3	6	no	—
49	#737	—	3	3	no	—
50	C₃×3₋¹⁺²	—	3	2	no	—
51	C₃×3₋¹⁺²	C₃²×C₅	1	3	yes	—
52	C₃×3₋¹⁺²	C₃²×C₅	1	6	yes	—
53	C₃×3₋¹⁺²	C₃²×C₅	1	2	yes	—
54	C₃×H(3,𝔽₃)	C₃²×C₅	1	1	yes	characteristic
55	C₃²×C₉	—	3	1	no	—
56	C₃²×C₉	C₃²×C₅	1	1	yes	characteristic
57	#733	—	3	18	no	—
58	#733	—	3	1	no	—
59	#733	—	3	2	no	—
60	#733	—	3	2	no	—
61	#733	—	3	4	no	—
62	#731	—	3	9	no	—
63	C₃²×C₅	—	3	3	no	—
64	C₃²×C₅	—	9	1	no	—
65	C₃²×C₅	—	9	3	no	—
66	C₃²×C₅	C₃×H(3,𝔽₃)	1	2	yes	—
67	C₃²×C₅	C₃×H(3,𝔽₃)	1	1	yes	characteristic
68	C₃²×C₅	C₃×H(3,𝔽₃)	1	1	yes	characteristic
69	C₅×C₉	—	3	1	no	—
70	C₅×C₉	—	3	6	no	—
71	C₅×C₉	—	3	2	no	—
72	C₅×C₉	—	9	3	no	—
73	C₅×C₉	—	9	6	no	—
74	C₃³	—	3	1	no	—
75	C₃³	C₃³×C₅	1	1	yes	characteristic
76	3₋¹⁺²	—	3	18	no	—
77	3₋¹⁺²	—	3	2	no	—
78	3₋¹⁺²	—	3	2	no	—
79	3₋¹⁺²	—	3	6	no	—
80	3₋¹⁺²	—	3	6	no	—
81	3₋¹⁺²	—	3	6	no	—
82	3₋¹⁺²	—	3	3	no	—
83	3₋¹⁺²	—	3	6	no	—
84	3₋¹⁺²	C₅×H(3,𝔽₃)	1	6	yes	—
85	H(3,𝔽₃)	—	3	2	no	—
86	H(3,𝔽₃)	C₅×H(3,𝔽₃)	1	3	yes	—
87	C₃×C₉	—	3	9	no	—
88	C₃×C₉	—	3	2	no	—
89	C₃×C₉	—	3	2	no	—
90	C₃×C₉	—	3	1	no	—
91	C₃×C₉	—	3	3	no	—
92	C₃×C₉	—	3	6	no	—
93	C₃×C₉	C₅×H(3,𝔽₃)	1	1	yes	characteristic
94	C₃×C₉	C₅×H(3,𝔽₃)	1	2	yes	—
95	C₃×C₅	—	3	1	no	—
96	C₃×C₅	—	3	2	no	—
97	C₃×C₅	—	3	1	no	—
98	C₃×C₅	—	9	3	no	—
99	C₃×C₅	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
100	C₃²	—	3	3	no	—
101	C₃²	—	9	1	no	—
102	C₃²	—	9	3	no	—
103	C₃²	C₃×C₅×H(3,𝔽₃)	1	2	yes	—
104	C₃²	C₃×C₅×H(3,𝔽₃)	1	1	yes	characteristic
105	C₃²	C₃×C₅×H(3,𝔽₃)	1	1	yes	characteristic
106	C₉	—	3	1	no	—
107	C₉	—	3	6	no	—
108	C₉	—	3	2	no	—
109	C₉	—	9	3	no	—
110	C₉	—	9	6	no	—
111	C₅	#10624758	1	1	yes	characteristic, direct product factor, Sylow
112	C₃	—	3	1	no	—
113	C₃	—	3	2	no	—
114	C₃	—	3	1	no	—
115	C₃	—	9	3	no	—
116	C₃	C₅×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₅×#737

C₃

yes

maximal, characteristic

C₅×#6912

C₃

yes

maximal, complemented

C₅×#6910

C₃

yes

maximal

C₃×C₅×#733

C₃

yes

maximal, characteristic, complemented

C₃×C₅×#733

C₃

yes

maximal, complemented

#10624758

C₅

yes

maximal, characteristic, direct product factor, Sylow

C₅×#737

—

C₅×#737

—

C₃×C₅×3₋¹⁺²

—

C₃×C₅×3₋¹⁺²

C₃²

yes

—

C₃×C₅×3₋¹⁺²

C₃²

yes

—

C₃×C₅×3₋¹⁺²

C₃²

yes

—

C₃×C₅×H(3,𝔽₃)

C₃²

yes

characteristic

C₃²×C₅×C₉

—

C₃²×C₅×C₉

C₃²

yes

—

—

—

—

—

—

C₃×C₅

yes

characteristic

#6912

C₃×C₅

yes

complemented

#6910

C₃×C₅

yes

—

C₃×#733

C₃×C₅

yes

characteristic, complemented

C₃×#733

C₃×C₅

yes

complemented

C₃³×C₅

—

C₃³×C₅

C₃³

yes

characteristic

C₅×3₋¹⁺²

—

C₅×3₋¹⁺²

—

C₅×3₋¹⁺²

—

C₅×3₋¹⁺²

—

C₅×3₋¹⁺²

—

C₅×3₋¹⁺²

—

C₅×3₋¹⁺²

—

C₅×3₋¹⁺²

—

C₅×3₋¹⁺²

H(3,𝔽₃)

yes

—

C₅×H(3,𝔽₃)

—

C₅×H(3,𝔽₃)

H(3,𝔽₃)

yes

—

C₃×C₅×C₉

—

C₃×C₅×C₉

—

C₃×C₅×C₉

—

C₃×C₅×C₉

—

C₃×C₅×C₉

—

C₃×C₅×C₉

—

C₃×C₅×C₉

H(3,𝔽₃)

yes

characteristic

C₃×C₅×C₉

H(3,𝔽₃)

yes

—

#737

—

#737

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃²×C₅

yes

—

C₃×3₋¹⁺²

C₃²×C₅

yes

—

C₃×3₋¹⁺²

C₃²×C₅

yes

—

C₃×H(3,𝔽₃)

C₃²×C₅

yes

characteristic

C₃²×C₉

—

C₃²×C₉

C₃²×C₅

yes

—

—

—

—

—

—

C₃²×C₅

—

C₃²×C₅

—

C₃²×C₅

—

C₃²×C₅

C₃×H(3,𝔽₃)

yes

—

C₃²×C₅

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₅

C₃×H(3,𝔽₃)

yes

characteristic

C₅×C₉

—

C₅×C₉

—

C₅×C₉

—

C₅×C₉

—

C₅×C₉

—

C₃³

—

C₃³

C₃³×C₅

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₅×H(3,𝔽₃)

yes

—

H(3,𝔽₃)

—

H(3,𝔽₃)

C₅×H(3,𝔽₃)

yes

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₅×H(3,𝔽₃)

yes

characteristic

C₃×C₉

C₅×H(3,𝔽₃)

yes

—

C₃×C₅

—

C₃×C₅

—

C₃×C₅

—

C₃×C₅

—

C₃×C₅

C₃²⋊H(3,𝔽₃)

yes

characteristic

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

C₃×C₅×H(3,𝔽₃)

yes

—

104

C₃²

C₃×C₅×H(3,𝔽₃)

yes

characteristic

105

C₃²

C₃×C₅×H(3,𝔽₃)

yes

characteristic

106

C₉

—

107

C₉

—

108

C₉

—

109

C₉

—

110

C₉

—

111

C₅

#10624758

yes

characteristic, direct product factor, Sylow

112

C₃

—

113

C₃

—

114

C₃

—

115

C₃

—

116

C₃

C₅×C₃²⋊H(3,𝔽₃)

yes

characteristic

	subgroup	G/H
center, socle	C₃×C₅	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₅
prime radical	C₃×C₅×H(3,𝔽₃)	C₃²
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₅ ◁ C₃³×C₅ ◁ G

subgroup

G/H

center, socle

C₃×C₅

C₃²⋊H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₃³×C₅

prime radical

C₃×C₅×H(3,𝔽₃)

C₃²

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃×C₅ ◁ C₃³×C₅ ◁ G

N	H ≅ G/N	type
2. C₅×#6912	C₃	semidirect
4. C₃×C₅×#733	C₃	semidirect
5. C₃×C₅×#733	C₃	semidirect
6. #10624758	C₅	direct (N × H)
23. #6912	C₃×C₅	semidirect
25. C₃×#733	C₃×C₅	semidirect
26. C₃×#733	C₃×C₅	semidirect
111. C₅	#10624758	direct (N × H)

H ≅ G/N

type

2. C₅×#6912

C₃

semidirect

4. C₃×C₅×#733

C₃

semidirect

5. C₃×C₅×#733

C₃

semidirect

6. #10624758

C₅

direct (N × H)

23. #6912

C₃×C₅

semidirect

25. C₃×#733

C₃×C₅

semidirect

26. C₃×#733

C₃×C₅

semidirect

111. C₅

#10624758

direct (N × H)

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

subgroup	index
C₃×C₅×#737	3
C₅×#6912	3
C₅×#6910	3
C₃×C₅×#733	3
#10624758	5

subgroup

index

C₃×C₅×#737

C₅×#6912

C₅×#6910

C₃×C₅×#733

#10624758

Subgroups — C₅×#10624758 #16814267

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#10624758 #16814267

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10624758 #16814267