Group #13090285 — subgroups

Subgroups — C₂×#10624799 #13090285

Order 1458.

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

normal subgroups only

This group has 222 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	#10624799	C₂	1	1	yes	maximal, characteristic, direct product factor, Sylow
2	C₂×C₃×#733	C₃	1	1	yes	maximal, characteristic
3	C₂×#6894	C₃	1	2	yes	maximal
4	C₂×C₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
5	C₂×C₃×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic
6	C₂×3₋¹⁺²⋊C₉	C₃	1	4	yes	maximal
7	C₂×3₋¹⁺²⋊C₉	C₃	1	2	yes	maximal
8	C₂×#6868	C₃	1	2	yes	maximal
9	C₃×#733	C₂×C₃	1	1	yes	characteristic
10	#6894	C₂×C₃	1	2	yes	—
11	C₉×3₋¹⁺²	C₂×C₃	1	1	yes	characteristic
12	C₃×C₉⋊C₉	C₂×C₃	1	1	yes	characteristic
13	3₋¹⁺²⋊C₉	C₂×C₃	1	4	yes	—
14	3₋¹⁺²⋊C₉	C₂×C₃	1	2	yes	—
15	#6868	C₂×C₃	1	2	yes	—
16	C₂×C₃×3₋¹⁺²	C₃²	1	2	yes	—
17	C₂×C₃×3₋¹⁺²	C₃²	1	1	yes	characteristic
18	C₂×C₃²×C₉	C₃²	1	1	yes	characteristic
19	C₂×C₃²×C₉	C₃²	1	2	yes	—
20	C₂×C₃²×C₉	C₃²	1	1	yes	characteristic
21	C₂×#733	—	3	3	no	—
22	C₂×C₉⋊C₉	—	3	2	no	—
23	C₂×C₉⋊C₉	—	3	1	no	—
24	C₂×C₉⋊C₉	—	3	2	no	—
25	C₂×C₉⋊C₉	—	3	4	no	—
26	C₂×C₉⋊C₉	—	3	2	no	—
27	C₂×C₃²⋊C₉	C₃²	1	2	yes	—
28	C₂×C₃²⋊C₉	C₃²	1	4	yes	—
29	C₂×C₉²	—	3	1	no	—
30	C₃×3₋¹⁺²	C₂×C₃²	1	2	yes	—
31	C₃×3₋¹⁺²	C₂×C₃²	1	1	yes	characteristic
32	C₃²×C₉	C₂×C₃²	1	1	yes	characteristic
33	C₃²×C₉	C₂×C₃²	1	1	yes	characteristic
34	C₃²×C₉	C₂×C₃²	1	2	yes	—
35	#733	—	3	3	no	—
36	C₉⋊C₉	—	3	2	no	—
37	C₉⋊C₉	—	3	1	no	—
38	C₉⋊C₉	—	3	2	no	—
39	C₉⋊C₉	—	3	4	no	—
40	C₉⋊C₉	—	3	2	no	—
41	C₃²⋊C₉	C₂×C₃²	1	2	yes	—
42	C₃²⋊C₉	C₂×C₃²	1	4	yes	—
43	C₉²	—	3	1	no	—
44	C₂×C₃³	C₃³	1	1	yes	characteristic
45	C₂×3₋¹⁺²	—	3	2	no	—
46	C₂×3₋¹⁺²	—	3	2	no	—
47	C₂×3₋¹⁺²	—	3	4	no	—
48	C₂×3₋¹⁺²	3₋¹⁺²	1	3	yes	—
49	C₂×C₃×C₉	—	3	1	no	—
50	C₂×C₃×C₉	—	3	3	no	—
51	C₂×C₃×C₉	—	3	2	no	—
52	C₂×C₃×C₉	—	3	4	no	—
53	C₂×C₃×C₉	—	3	2	no	—
54	C₂×C₃×C₉	—	3	1	no	—
55	C₂×C₃×C₉	—	3	2	no	—
56	C₂×C₃×C₉	—	3	1	no	—
57	C₂×C₃×C₉	—	3	2	no	—
58	C₂×C₃×C₉	—	9	2	no	—
59	C₂×C₃×C₉	H(3,𝔽₃)	1	2	yes	—
60	C₂×C₃×C₉	H(3,𝔽₃)	1	1	yes	characteristic
61	C₃³	C₂×C₃³	1	1	yes	characteristic
62	3₋¹⁺²	—	3	2	no	—
63	3₋¹⁺²	—	3	2	no	—
64	3₋¹⁺²	—	3	4	no	—
65	3₋¹⁺²	C₂×3₋¹⁺²	1	3	yes	—
66	C₃×C₉	—	3	1	no	—
67	C₃×C₉	—	3	3	no	—
68	C₃×C₉	—	3	2	no	—
69	C₃×C₉	—	3	4	no	—
70	C₃×C₉	—	3	1	no	—
71	C₃×C₉	—	3	2	no	—
72	C₃×C₉	—	3	2	no	—
73	C₃×C₉	—	3	1	no	—
74	C₃×C₉	—	3	2	no	—
75	C₃×C₉	—	9	2	no	—
76	C₃×C₉	C₂×H(3,𝔽₃)	1	2	yes	—
77	C₃×C₉	C₂×H(3,𝔽₃)	1	1	yes	characteristic
78	C₂×C₃²	—	3	2	no	—
79	C₂×C₃²	—	3	1	no	—
80	C₂×C₃²	C₃×H(3,𝔽₃)	1	1	yes	characteristic
81	C₂×C₃²	C₃×3₋¹⁺²	1	1	yes	characteristic
82	C₂×C₃²	#737	1	2	yes	—
83	C₂×C₉	—	3	3	no	—
84	C₂×C₉	—	3	2	no	—
85	C₂×C₉	—	9	2	no	—
86	C₂×C₉	—	9	4	no	—
87	C₂×C₉	—	9	2	no	—
88	C₂×C₉	—	9	1	no	—
89	C₂×C₉	—	9	2	no	—
90	C₂×C₉	#733	1	3	yes	complemented
91	C₃²	—	3	1	no	—
92	C₃²	—	3	2	no	—
93	C₃²	C₂×C₃×H(3,𝔽₃)	1	1	yes	characteristic
94	C₃²	C₂×C₃×3₋¹⁺²	1	1	yes	characteristic
95	C₃²	C₂×#737	1	2	yes	—
96	C₉	—	3	3	no	—
97	C₉	—	3	2	no	—
98	C₉	—	9	1	no	—
99	C₉	—	9	2	no	—
100	C₉	—	9	2	no	—
101	C₉	—	9	4	no	—
102	C₉	—	9	2	no	—
103	C₉	C₂×#733	1	3	yes	complemented
104	C₂×C₃	—	3	1	no	—
105	C₂×C₃	—	3	2	no	—
106	C₂×C₃	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
107	C₂×C₃	C₃×#733	1	1	yes	characteristic
108	C₂×C₃	#6913	1	2	yes	—
109	C₃	—	3	1	no	—
110	C₃	—	3	2	no	—
111	C₃	C₂×C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
112	C₃	C₂×C₃×#733	1	1	yes	characteristic
113	C₃	C₂×#6913	1	2	yes	—
114	C₂	#10624799	1	1	yes	characteristic, direct product factor, Sylow

subgroup

G/H

class size

multiplicity

normal?

properties

#10624799

C₂

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₃×#733

C₃

yes

maximal, characteristic

C₂×#6894

C₃

yes

maximal

C₂×C₉×3₋¹⁺²

C₃

yes

maximal, characteristic

C₂×C₃×C₉⋊C₉

C₃

yes

maximal, characteristic

C₂×3₋¹⁺²⋊C₉

C₃

yes

maximal

C₂×3₋¹⁺²⋊C₉

C₃

yes

maximal

C₂×#6868

C₃

yes

maximal

C₃×#733

C₂×C₃

yes

characteristic

#6894

C₂×C₃

yes

—

C₉×3₋¹⁺²

C₂×C₃

yes

characteristic

C₃×C₉⋊C₉

C₂×C₃

yes

characteristic

3₋¹⁺²⋊C₉

C₂×C₃

yes

—

3₋¹⁺²⋊C₉

C₂×C₃

yes

—

#6868

C₂×C₃

yes

—

C₂×C₃×3₋¹⁺²

C₃²

yes

—

C₂×C₃×3₋¹⁺²

C₃²

yes

characteristic

C₂×C₃²×C₉

C₃²

yes

characteristic

C₂×C₃²×C₉

C₃²

yes

—

C₂×C₃²×C₉

C₃²

yes

characteristic

C₂×#733

—

C₂×C₉⋊C₉

—

C₂×C₉⋊C₉

—

C₂×C₉⋊C₉

—

C₂×C₉⋊C₉

—

C₂×C₉⋊C₉

—

C₂×C₃²⋊C₉

C₃²

yes

—

C₂×C₃²⋊C₉

C₃²

yes

—

C₂×C₉²

—

C₃×3₋¹⁺²

C₂×C₃²

yes

—

C₃×3₋¹⁺²

C₂×C₃²

yes

characteristic

C₃²×C₉

C₂×C₃²

yes

characteristic

C₃²×C₉

C₂×C₃²

yes

characteristic

C₃²×C₉

C₂×C₃²

yes

—

#733

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

C₂×C₃²

yes

—

C₃²⋊C₉

C₂×C₃²

yes

—

C₉²

—

C₂×C₃³

C₃³

yes

characteristic

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

3₋¹⁺²

yes

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

H(3,𝔽₃)

yes

—

C₂×C₃×C₉

H(3,𝔽₃)

yes

characteristic

C₃³

C₂×C₃³

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₂×3₋¹⁺²

yes

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₂×H(3,𝔽₃)

yes

—

C₃×C₉

C₂×H(3,𝔽₃)

yes

characteristic

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₃×H(3,𝔽₃)

yes

characteristic

C₂×C₃²

C₃×3₋¹⁺²

yes

characteristic

C₂×C₃²

#737

yes

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

#733

yes

complemented

C₃²

—

C₃²

—

C₃²

C₂×C₃×H(3,𝔽₃)

yes

characteristic

C₃²

C₂×C₃×3₋¹⁺²

yes

characteristic

C₃²

C₂×#737

yes

—

C₉

—

C₉

—

C₉

—

C₉

—

100

C₉

—

101

C₉

—

102

C₉

—

103

C₉

C₂×#733

yes

complemented

104

C₂×C₃

—

105

C₂×C₃

—

106

C₂×C₃

C₉⋊H(3,𝔽₃)

yes

characteristic

107

C₂×C₃

C₃×#733

yes

characteristic

108

C₂×C₃

#6913

yes

—

109

C₃

—

110

C₃

—

111

C₃

C₂×C₉⋊H(3,𝔽₃)

yes

characteristic

112

C₃

C₂×C₃×#733

yes

characteristic

113

C₃

C₂×#6913

yes

—

114

C₂

#10624799

yes

characteristic, direct product factor, Sylow

	subgroup	G/H
center, socle	C₂×C₃²	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₂×C₃³
prime radical	C₂×C₃³	C₃³
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃²×C₉ ◁ G

subgroup

G/H

center, socle

C₂×C₃²

C₃×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₂×C₃³

prime radical

C₂×C₃³

C₃³

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₂×C₃² ◁ C₂×C₃²×C₉ ◁ G

N	H ≅ G/N	type
1. #10624799	C₂	direct (N × H)
90. C₂×C₉	#733	semidirect
103. C₉	C₂×#733	semidirect
114. C₂	#10624799	direct (N × H)

H ≅ G/N

type

1. #10624799

C₂

direct (N × H)

90. C₂×C₉

#733

semidirect

103. C₉

C₂×#733

semidirect

114. C₂

#10624799

direct (N × H)

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

subgroup	index
#10624799	2
C₂×C₃×#733	3
C₂×#6894	3
C₂×C₉×3₋¹⁺²	3
C₂×C₃×C₉⋊C₉	3
C₂×3₋¹⁺²⋊C₉	3
C₂×#6868	3

subgroup

index

#10624799

C₂×C₃×#733

C₂×#6894

C₂×C₉×3₋¹⁺²

C₂×C₃×C₉⋊C₉

C₂×3₋¹⁺²⋊C₉

C₂×#6868

Subgroups — C₂×#10624799 #13090285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#10624799 #13090285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10624799 #13090285