Group #13090103 — subgroups

Subgroups — C₂×#10624617 #13090103

Order 1458.

260 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 86 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	#10624617	C₂	1	1	yes	maximal, characteristic, direct product factor, Sylow
2	C₂×#6891	C₃	1	1	yes	maximal, characteristic
3	C₂×#6891	C₃	1	1	yes	maximal, characteristic
4	C₂×C₃×C₃²⋊C₉	C₃	1	2	yes	maximal
5	C₂×#6861	C₃	1	9	yes	maximal, complemented
6	#6891	C₂×C₃	1	1	yes	characteristic
7	#6891	C₂×C₃	1	1	yes	characteristic
8	C₃×C₃²⋊C₉	C₂×C₃	1	2	yes	—
9	#6861	C₂×C₃	1	9	yes	complemented
10	C₂×C₃⁴	C₃²	1	1	yes	characteristic
11	C₂×C₃×3₋¹⁺²	—	3	1	no	—
12	C₂×C₃×3₋¹⁺²	—	3	1	no	—
13	C₂×C₃²×C₉	—	3	2	no	—
14	C₂×C₃²⋊C₉	—	3	2	no	—
15	C₂×C₃²⋊C₉	—	3	2	no	—
16	C₂×C₃²⋊C₉	—	3	2	no	—
17	C₂×C₃²⋊C₉	—	3	2	no	—
18	C₂×C₃²⋊C₉	C₃²	1	6	yes	—
19	C₂×C₃²⋊C₉	C₃²	1	3	yes	—
20	C₂×C₃²⋊C₉	C₃²	1	3	yes	—
21	C₃⁴	C₂×C₃²	1	1	yes	characteristic
22	C₃×3₋¹⁺²	—	3	1	no	—
23	C₃×3₋¹⁺²	—	3	1	no	—
24	C₃²×C₉	—	3	2	no	—
25	C₃²⋊C₉	—	3	2	no	—
26	C₃²⋊C₉	—	3	2	no	—
27	C₃²⋊C₉	—	3	2	no	—
28	C₃²⋊C₉	—	3	2	no	—
29	C₃²⋊C₉	C₂×C₃²	1	6	yes	—
30	C₃²⋊C₉	C₂×C₃²	1	3	yes	—
31	C₃²⋊C₉	C₂×C₃²	1	3	yes	—
32	C₂×C₃³	—	3	6	no	—
33	C₂×C₃³	—	9	1	no	—
34	C₂×C₃³	—	9	1	no	—
35	C₂×C₃³	H(3,𝔽₃)	1	1	yes	characteristic
36	C₂×C₃³	H(3,𝔽₃)	1	2	yes	—
37	C₂×C₃³	C₃³	1	1	yes	characteristic
38	C₂×3₋¹⁺²	—	9	3	no	—
39	C₂×3₋¹⁺²	—	9	3	no	—
40	C₂×C₃×C₉	—	3	6	no	—
41	C₂×C₃×C₉	—	3	3	no	—
42	C₂×C₃×C₉	—	3	3	no	—
43	C₂×C₃×C₉	—	9	6	no	—
44	C₃³	—	3	6	no	—
45	C₃³	—	9	1	no	—
46	C₃³	—	9	1	no	—
47	C₃³	C₂×H(3,𝔽₃)	1	1	yes	characteristic
48	C₃³	C₂×H(3,𝔽₃)	1	2	yes	—
49	C₃³	C₂×C₃³	1	1	yes	characteristic
50	3₋¹⁺²	—	9	3	no	—
51	3₋¹⁺²	—	9	3	no	—
52	C₃×C₉	—	3	6	no	—
53	C₃×C₉	—	3	3	no	—
54	C₃×C₉	—	3	3	no	—
55	C₃×C₉	—	9	6	no	—
56	C₂×C₃²	—	3	2	no	—
57	C₂×C₃²	—	3	2	no	—
58	C₂×C₃²	—	3	2	no	—
59	C₂×C₃²	—	3	2	no	—
60	C₂×C₃²	—	3	2	no	—
61	C₂×C₃²	—	3	1	no	—
62	C₂×C₃²	—	3	1	no	—
63	C₂×C₃²	—	3	1	no	—
64	C₂×C₃²	—	3	1	no	—
65	C₂×C₃²	—	9	9	no	—
66	C₂×C₃²	#733	1	6	yes	—
67	C₂×C₃²	C₃×H(3,𝔽₃)	1	1	yes	characteristic
68	C₂×C₉	—	9	3	no	—
69	C₂×C₉	—	9	6	no	—
70	C₂×C₉	—	9	3	no	—
71	C₃²	—	3	2	no	—
72	C₃²	—	3	2	no	—
73	C₃²	—	3	2	no	—
74	C₃²	—	3	2	no	—
75	C₃²	—	3	2	no	—
76	C₃²	—	3	1	no	—
77	C₃²	—	3	1	no	—
78	C₃²	—	3	1	no	—
79	C₃²	—	3	1	no	—
80	C₃²	—	9	9	no	—
81	C₃²	C₂×#733	1	6	yes	—
82	C₃²	C₂×C₃×H(3,𝔽₃)	1	1	yes	characteristic
83	C₉	—	9	3	no	—
84	C₉	—	9	6	no	—
85	C₉	—	9	3	no	—
86	C₂×C₃	—	3	6	no	—
87	C₂×C₃	—	9	1	no	—
88	C₂×C₃	—	9	1	no	—
89	C₂×C₃	C₃×#733	1	2	yes	—
90	C₂×C₃	#6910	1	1	yes	characteristic
91	C₂×C₃	#6911	1	1	yes	characteristic
92	C₃	—	3	6	no	—
93	C₃	—	9	1	no	—
94	C₃	—	9	1	no	—
95	C₃	C₂×C₃×#733	1	2	yes	—
96	C₃	C₂×#6910	1	1	yes	characteristic
97	C₃	C₂×#6911	1	1	yes	characteristic
98	C₂	#10624617	1	1	yes	characteristic, direct product factor, Sylow

subgroup

G/H

class size

multiplicity

normal?

properties

#10624617

C₂

yes

maximal, characteristic, direct product factor, Sylow

C₂×#6891

C₃

yes

maximal, characteristic

C₂×#6891

C₃

yes

maximal, characteristic

C₂×C₃×C₃²⋊C₉

C₃

yes

maximal

C₂×#6861

C₃

yes

maximal, complemented

#6891

C₂×C₃

yes

characteristic

#6891

C₂×C₃

yes

characteristic

C₃×C₃²⋊C₉

C₂×C₃

yes

—

#6861

C₂×C₃

yes

complemented

C₂×C₃⁴

C₃²

yes

characteristic

C₂×C₃×3₋¹⁺²

—

C₂×C₃×3₋¹⁺²

—

C₂×C₃²×C₉

—

C₂×C₃²⋊C₉

—

C₂×C₃²⋊C₉

—

C₂×C₃²⋊C₉

—

C₂×C₃²⋊C₉

—

C₂×C₃²⋊C₉

C₃²

yes

—

C₂×C₃²⋊C₉

C₃²

yes

—

C₂×C₃²⋊C₉

C₃²

yes

—

C₃⁴

C₂×C₃²

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₂×C₃²

yes

—

C₃²⋊C₉

C₂×C₃²

yes

—

C₃²⋊C₉

C₂×C₃²

yes

—

C₂×C₃³

—

C₂×C₃³

—

C₂×C₃³

—

C₂×C₃³

H(3,𝔽₃)

yes

characteristic

C₂×C₃³

H(3,𝔽₃)

yes

—

C₂×C₃³

C₃³

yes

characteristic

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₂×H(3,𝔽₃)

yes

characteristic

C₃³

C₂×H(3,𝔽₃)

yes

—

C₃³

C₂×C₃³

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

#733

yes

—

C₂×C₃²

C₃×H(3,𝔽₃)

yes

characteristic

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₂×#733

yes

—

C₃²

C₂×C₃×H(3,𝔽₃)

yes

characteristic

C₉

—

C₉

—

C₉

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

C₃×#733

yes

—

C₂×C₃

yes

C₂×C₃

yes

C₃

—

C₃

—

C₃

—

C₃

C₂×C₃×#733

yes

—

C₃

yes

C₃

yes

C₂

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×C₃⁴	C₃²
LCS	G ▷ C₃³ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃⁴ ◁ G

subgroup

G/H

center, socle

C₂×C₃²

C₃×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₂×C₃³

prime radical

C₂×C₃⁴

C₃²

LCS

G ▷ C₃³ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₂×C₃² ◁ C₂×C₃⁴ ◁ G

N	H ≅ G/N	type
1. #10624617	C₂	direct (N × H)
5. C₂×#6861	C₃	semidirect
9. #6861	C₂×C₃	semidirect
98. C₂	#10624617	direct (N × H)

H ≅ G/N

type

1. #10624617

C₂

direct (N × H)

5. C₂×#6861

C₃

semidirect

9. #6861

C₂×C₃

semidirect

98. C₂

#10624617

direct (N × H)

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

subgroup	index
#10624617	2
C₂×#6891	3
C₂×C₃×C₃²⋊C₉	3
C₂×#6861	3

subgroup

index

#10624617

C₂×#6891

C₂×C₃×C₃²⋊C₉

C₂×#6861

Subgroups — C₂×#10624617 #13090103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#10624617 #13090103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10624617 #13090103