Group #14517373 — subgroups

Subgroups — Dic₆×#733 #14517373

Order 1944.

216 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 72 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×C₄×#733	C₂	1	1	yes	maximal, characteristic
2	Dic₃×#733	C₂	1	2	yes	maximal
3	Dic₆×3₋¹⁺²	C₃	1	3	yes	maximal
4	C₃×C₉×Dic₆	C₃	1	1	yes	maximal, characteristic
5	Q₈×#733	—	3	1	no	maximal
6	C₂×C₃×#733	C₂²	1	1	yes	characteristic
7	C₃×C₄×3₋¹⁺²	C₂×C₃	1	3	yes	—
8	C₃²×C₄×C₉	C₂×C₃	1	1	yes	characteristic
9	Dic₃×3₋¹⁺²	C₂×C₃	1	6	yes	—
10	C₃×C₉×Dic₃	C₂×C₃	1	2	yes	—
11	C₄×#733	—	2	3	no	—
12	C₄×#733	—	2	1	no	—
13	C₄×#733	—	3	2	no	—
14	C₄×#733	D₃	1	1	yes	characteristic
15	C₃×#733	Q₈	1	1	yes	characteristic, complemented, Sylow
16	C₃²×Dic₆	C₃²	1	1	yes	characteristic
17	Q₈×3₋¹⁺²	—	3	3	no	—
18	C₃×Q₈×C₉	—	3	1	no	—
19	C₉×Dic₆	—	3	1	no	—
20	C₉×Dic₆	—	3	3	no	—
21	C₂×C₃×3₋¹⁺²	C₂²×C₃	1	3	yes	—
22	C₂×C₃²×C₉	C₂²×C₃	1	1	yes	characteristic
23	C₂×#733	—	2	3	no	—
24	C₂×#733	—	2	1	no	—
25	C₂×#733	C₂×D₃	1	1	yes	characteristic
26	C₃³×C₄	C₂×C₃²	1	1	yes	characteristic
27	C₃²×Dic₃	C₂×C₃²	1	2	yes	—
28	C₄×3₋¹⁺²	—	2	3	no	—
29	C₄×3₋¹⁺²	—	3	6	no	—
30	C₄×3₋¹⁺²	—	6	3	no	—
31	C₄×3₋¹⁺²	C₃×D₃	1	3	yes	—
32	C₃×C₄×C₉	—	2	1	no	—
33	C₃×C₄×C₉	—	3	1	no	—
34	C₃×C₄×C₉	—	3	3	no	—
35	C₃×C₄×C₉	—	3	2	no	—
36	C₃×C₄×C₉	—	6	1	no	—
37	C₃×C₄×C₉	C₃×D₃	1	1	yes	characteristic
38	C₉×Dic₃	—	3	2	no	—
39	C₉×Dic₃	—	3	6	no	—
40	C₃×3₋¹⁺²	C₃×Q₈	1	3	yes	—
41	C₃²×C₉	C₃×Q₈	1	1	yes	characteristic
42	#733	—	2	3	no	—
43	#733	—	2	1	no	—
44	#733	Dic₆	1	1	yes	characteristic, direct product factor
45	C₃²×Q₈	—	3	1	no	—
46	C₃×Dic₆	—	3	1	no	—
47	C₃×Dic₆	H(3,𝔽₃)	1	1	yes	characteristic
48	Q₈×C₉	—	9	1	no	—
49	Q₈×C₉	—	9	3	no	—
50	C₂×C₃³	C₂²×C₃²	1	1	yes	characteristic
51	C₂×3₋¹⁺²	—	2	3	no	—
52	C₂×3₋¹⁺²	—	6	3	no	—
53	C₂×3₋¹⁺²	C₂×C₃×D₃	1	3	yes	—
54	C₂×C₃×C₉	—	2	1	no	—
55	C₂×C₃×C₉	—	3	1	no	—
56	C₂×C₃×C₉	—	3	3	no	—
57	C₂×C₃×C₉	—	6	1	no	—
58	C₂×C₃×C₉	C₂×C₃×D₃	1	1	yes	characteristic
59	C₃²×C₄	—	2	1	no	—
60	C₃²×C₄	—	3	1	no	—
61	C₃²×C₄	—	3	2	no	—
62	C₃²×C₄	—	6	1	no	—
63	C₃²×C₄	C₂×H(3,𝔽₃)	1	1	yes	characteristic
64	C₃²×C₄	C₃²×D₃	1	1	yes	characteristic
65	C₃×Dic₃	—	3	2	no	—
66	C₃×Dic₃	C₂×H(3,𝔽₃)	1	2	yes	—
67	C₄×C₉	—	3	1	no	—
68	C₄×C₉	—	3	3	no	—
69	C₄×C₉	—	6	1	no	—
70	C₄×C₉	—	6	3	no	—
71	C₄×C₉	—	9	2	no	—
72	C₄×C₉	—	9	6	no	—
73	C₃³	C₃²×Q₈	1	1	yes	characteristic
74	3₋¹⁺²	—	2	3	no	—
75	3₋¹⁺²	—	6	3	no	—
76	3₋¹⁺²	C₃×Dic₆	1	3	yes	—
77	C₃×C₉	—	2	1	no	—
78	C₃×C₉	—	3	1	no	—
79	C₃×C₉	—	3	3	no	—
80	C₃×C₉	—	6	1	no	—
81	C₃×C₉	C₃×Dic₆	1	1	yes	characteristic
82	C₃×Q₈	—	3	1	no	—
83	C₃×Q₈	—	9	1	no	—
84	Dic₆	#733	1	1	yes	characteristic, direct product factor
85	C₂×C₃²	—	2	1	no	—
86	C₂×C₃²	—	3	1	no	—
87	C₂×C₃²	—	6	1	no	—
88	C₂×C₃²	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
89	C₂×C₃²	C₂×C₃²×D₃	1	1	yes	characteristic
90	C₂×C₉	—	3	1	no	—
91	C₂×C₉	—	3	3	no	—
92	C₂×C₉	—	6	1	no	—
93	C₂×C₉	—	6	3	no	—
94	C₃×C₄	—	2	1	no	—
95	C₃×C₄	—	3	2	no	—
96	C₃×C₄	—	3	1	no	—
97	C₃×C₄	—	6	1	no	—
98	C₃×C₄	—	9	2	no	—
99	C₃×C₄	C₂×#733	1	1	yes	characteristic
100	C₃×C₄	D₃×H(3,𝔽₃)	1	1	yes	characteristic
101	Dic₃	C₂×#733	1	2	yes	—
102	C₃²	—	2	1	no	—
103	C₃²	—	3	1	no	—
104	C₃²	—	6	1	no	—
105	C₃²	Q₈×H(3,𝔽₃)	1	1	yes	characteristic
106	C₃²	C₃²×Dic₆	1	1	yes	characteristic
107	C₉	—	3	1	no	—
108	C₉	—	3	3	no	—
109	C₉	—	6	1	no	—
110	C₉	—	6	3	no	—
111	Q₈	—	3	1	no	Sylow
112	C₂×C₃	—	2	1	no	—
113	C₂×C₃	—	3	1	no	—
114	C₂×C₃	—	6	1	no	—
115	C₂×C₃	C₂²×#733	1	1	yes	characteristic
116	C₂×C₃	C₂×D₃×H(3,𝔽₃)	1	1	yes	characteristic
117	C₄	—	3	2	no	—
118	C₄	D₃×#733	1	1	yes	characteristic
119	C₃	—	2	1	no	—
120	C₃	—	3	1	no	—
121	C₃	—	6	1	no	—
122	C₃	Q₈×#733	1	1	yes	characteristic, complemented
123	C₃	Dic₆×H(3,𝔽₃)	1	1	yes	characteristic
124	C₂	C₂×D₃×#733	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₄×#733

C₂

yes

maximal, characteristic

Dic₃×#733

C₂

yes

maximal

Dic₆×3₋¹⁺²

C₃

yes

maximal

C₃×C₉×Dic₆

C₃

yes

maximal, characteristic

Q₈×#733

—

maximal

C₂×C₃×#733

C₂²

yes

characteristic

C₃×C₄×3₋¹⁺²

C₂×C₃

yes

—

C₃²×C₄×C₉

C₂×C₃

yes

characteristic

Dic₃×3₋¹⁺²

C₂×C₃

yes

—

C₃×C₉×Dic₃

C₂×C₃

yes

—

—

—

—

D₃

yes

characteristic

C₃×#733

Q₈

yes

characteristic, complemented, Sylow

C₃²×Dic₆

C₃²

yes

characteristic

Q₈×3₋¹⁺²

—

C₃×Q₈×C₉

—

C₉×Dic₆

—

C₉×Dic₆

—

C₂×C₃×3₋¹⁺²

C₂²×C₃

yes

—

C₂×C₃²×C₉

C₂²×C₃

yes

—

—

C₂×D₃

yes

characteristic

C₃³×C₄

C₂×C₃²

yes

characteristic

C₃²×Dic₃

C₂×C₃²

yes

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

C₃×D₃

yes

—

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

C₃×D₃

yes

characteristic

C₉×Dic₃

—

C₉×Dic₃

—

C₃×3₋¹⁺²

C₃×Q₈

yes

—

C₃²×C₉

C₃×Q₈

yes

—

—

Dic₆

yes

characteristic, direct product factor

C₃²×Q₈

—

C₃×Dic₆

—

C₃×Dic₆

H(3,𝔽₃)

yes

characteristic

Q₈×C₉

—

Q₈×C₉

—

C₂×C₃³

C₂²×C₃²

yes

characteristic

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

C₂×C₃×D₃

yes

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

C₂×C₃×D₃

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

C₂×H(3,𝔽₃)

yes

characteristic

C₃²×C₄

C₃²×D₃

yes

characteristic

C₃×Dic₃

—

C₃×Dic₃

C₂×H(3,𝔽₃)

yes

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₃³

C₃²×Q₈

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₃×Dic₆

yes

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃×Dic₆

yes

characteristic

C₃×Q₈

—

C₃×Q₈

—

Dic₆

#733

yes

characteristic, direct product factor

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₂²×H(3,𝔽₃)

yes

characteristic

C₂×C₃²

C₂×C₃²×D₃

yes

characteristic

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₂×#733

yes

characteristic

100

C₃×C₄

D₃×H(3,𝔽₃)

yes

characteristic

101

Dic₃

C₂×#733

yes

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

Q₈×H(3,𝔽₃)

yes

characteristic

106

C₃²

C₃²×Dic₆

yes

characteristic

107

C₉

—

108

C₉

—

109

C₉

—

110

C₉

—

111

Q₈

—

Sylow

112

C₂×C₃

—

113

C₂×C₃

—

114

C₂×C₃

—

115

C₂×C₃

C₂²×#733

yes

characteristic

116

C₂×C₃

C₂×D₃×H(3,𝔽₃)

yes

characteristic

117

C₄

—

118

C₄

D₃×#733

yes

characteristic

119

C₃

—

120

C₃

—

121

C₃

—

122

C₃

Q₈×#733

yes

characteristic, complemented

123

C₃

Dic₆×H(3,𝔽₃)

yes

characteristic

124

C₂

C₂×D₃×#733

yes

characteristic

	subgroup	G/H
center	C₂×C₃	C₂×D₃×H(3,𝔽₃)
commutator, prime radical, cosocle	C₂×C₃³	C₂²×C₃²
Frattini	C₂×C₃²	C₂×C₃²×D₃
Fitting	C₃×C₄×#733	C₂
socle	C₂×C₃²	C₂²×H(3,𝔽₃)
hypercenter	C₄×#733	D₃
hypocenter	C₃	Q₈×#733
LCS	G ▷ C₂×C₃³ ▷ C₃² ▷ C₃
UCS	C₁ ◁ C₂×C₃ ◁ C₃²×C₄ ◁ C₄×#733

subgroup

G/H

center

C₂×C₃

C₂×D₃×H(3,𝔽₃)

commutator, prime radical, cosocle

C₂×C₃³

C₂²×C₃²

Frattini

C₂×C₃²

C₂×C₃²×D₃

Fitting

C₃×C₄×#733

C₂

socle

C₂×C₃²

C₂²×H(3,𝔽₃)

D₃

C₃

G ▷ C₂×C₃³ ▷ C₃² ▷ C₃

UCS

C₁ ◁ C₂×C₃ ◁ C₃²×C₄ ◁ C₄×#733

N	H ≅ G/N	type
15. C₃×#733	Q₈	semidirect
44. #733	Dic₆	direct (N × H)
84. Dic₆	#733	direct (N × H)
122. C₃	Q₈×#733	semidirect

H ≅ G/N

type

15. C₃×#733

Q₈

semidirect

44. #733

Dic₆

direct (N × H)

84. Dic₆

#733

direct (N × H)

122. C₃

Q₈×#733

semidirect

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

subgroup	index
C₃×C₄×#733	2
Dic₃×#733	2
Dic₆×3₋¹⁺²	3
C₃×C₉×Dic₆	3
Q₈×#733	3

subgroup

index

C₃×C₄×#733

Dic₃×#733

Dic₆×3₋¹⁺²

C₃×C₉×Dic₆

Q₈×#733

Subgroups — Dic₆×#733 #14517373

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic6×#733 #14517373

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₆×#733 #14517373