Group #14517728 — subgroups

Subgroups — #14517728

Order 1944.

801 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 801 conjugacy classes of subgroups — too many to draw. Only its 114 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	C₄×C₃²⋊H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	#11757642	C₂	1	2	yes	maximal
3	C₃²×#634	—	3	1	no	maximal
4	C₃×#6229	—	3	12	no	maximal
5	C₂×C₃²⋊H(3,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₃⁴×C₄	D₃	1	1	yes	characteristic
7	C₃²×C₃⋊Dic₃	—	3	2	no	—
8	C₃×C₄×H(3,𝔽₃)	D₃	1	12	yes	—
9	C₃×#1084	—	3	24	no	—
10	C₃²⋊H(3,𝔽₃)	Q₈	1	1	yes	characteristic, complemented, Sylow
11	C₃×#634	—	3	4	no	—
12	C₃²×Dic₆	—	9	4	no	—
13	C₃²×Dic₆	—	9	9	no	—
14	#6229	—	3	12	no	—
15	C₂×C₃⁴	C₂×D₃	1	1	yes	characteristic
16	C₂×C₃×H(3,𝔽₃)	C₂×D₃	1	12	yes	—
17	C₃³×C₄	—	3	4	no	—
18	C₃³×C₄	—	6	4	no	—
19	C₃³×C₄	C₃⋊D₃	1	4	yes	—
20	C₃³×C₄	C₃⋊D₃	1	9	yes	—
21	C₃×C₃⋊Dic₃	—	3	8	no	—
22	C₃²×Dic₃	—	9	18	no	—
23	C₃²×Dic₃	—	9	8	no	—
24	C₄×H(3,𝔽₃)	—	3	12	no	—
25	C₄×H(3,𝔽₃)	—	6	12	no	—
26	#1084	—	3	24	no	—
27	C₃⁴	Dic₆	1	1	yes	characteristic, complemented
28	C₃×H(3,𝔽₃)	Dic₆	1	12	yes	complemented
29	C₃²×Q₈	—	27	1	no	—
30	#634	—	3	1	no	—
31	C₃×Dic₆	—	9	36	no	—
32	C₃×Dic₆	—	9	12	no	—
33	C₃×Dic₆	—	9	4	no	—
34	C₂×C₃³	—	3	4	no	—
35	C₂×C₃³	—	6	4	no	—
36	C₂×C₃³	C₂×C₃⋊D₃	1	4	yes	—
37	C₂×C₃³	C₂×C₃⋊D₃	1	9	yes	—
38	C₂×H(3,𝔽₃)	—	3	12	no	—
39	C₂×H(3,𝔽₃)	—	6	12	no	—
40	C₃²×C₄	—	2	4	no	—
41	C₃²×C₄	—	3	1	no	—
42	C₃²×C₄	—	3	12	no	—
43	C₃²×C₄	—	3	36	no	—
44	C₃²×C₄	—	6	6	no	—
45	C₃²×C₄	—	6	4	no	—
46	C₃²×C₄	—	6	3	no	—
47	C₃²×C₄	—	27	2	no	—
48	C₃²×C₄	#272	1	4	yes	—
49	C₃²×C₄	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
50	C₃⋊Dic₃	—	3	2	no	—
51	C₃×Dic₃	—	9	72	no	—
52	C₃×Dic₃	—	9	24	no	—
53	C₃×Dic₃	—	9	8	no	—
54	C₃³	—	3	4	no	—
55	C₃³	—	6	4	no	—
56	C₃³	#634	1	4	yes	—
57	C₃³	#634	1	9	yes	complemented
58	H(3,𝔽₃)	—	3	12	no	—
59	H(3,𝔽₃)	—	6	12	no	—
60	C₃×Q₈	—	27	4	no	—
61	Dic₆	—	9	4	no	—
62	Dic₆	—	9	9	no	—
63	C₂×C₃²	—	2	4	no	—
64	C₂×C₃²	—	3	1	no	—
65	C₂×C₃²	—	3	12	no	—
66	C₂×C₃²	—	3	36	no	—
67	C₂×C₃²	—	6	6	no	—
68	C₂×C₃²	—	6	4	no	—
69	C₂×C₃²	—	6	3	no	—
70	C₂×C₃²	C₂×#272	1	4	yes	—
71	C₂×C₃²	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
72	C₃×C₄	—	3	4	no	—
73	C₃×C₄	—	6	4	no	—
74	C₃×C₄	—	9	9	no	—
75	C₃×C₄	—	27	8	no	—
76	C₃×C₄	C₃⋊#272	1	4	yes	—
77	Dic₃	—	9	8	no	—
78	Dic₃	—	9	18	no	—
79	C₃²	—	2	4	no	—
80	C₃²	—	3	1	no	—
81	C₃²	—	3	12	no	—
82	C₃²	—	3	36	no	—
83	C₃²	—	6	6	no	—
84	C₃²	—	6	4	no	—
85	C₃²	—	6	3	no	—
86	C₃²	#6229	1	4	yes	complemented
87	C₃²	#6308	1	1	yes	characteristic
88	Q₈	—	27	1	no	Sylow
89	C₂×C₃	—	3	4	no	—
90	C₂×C₃	—	6	4	no	—
91	C₂×C₃	—	9	9	no	—
92	C₂×C₃	C₂×C₃⋊#272	1	4	yes	—
93	C₄	—	27	2	no	—
94	C₄	#92338	1	1	yes	characteristic
95	C₃	—	3	4	no	—
96	C₃	—	6	4	no	—
97	C₃	—	9	9	no	—
98	C₃	#10619988	1	4	yes	—
99	C₂	C₂×#92338	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₃²⋊H(3,𝔽₃)

C₂

yes

maximal, characteristic

#11757642

C₂

yes

maximal

C₃²×#634

—

maximal

C₃×#6229

—

maximal

C₂×C₃²⋊H(3,𝔽₃)

C₂²

yes

characteristic

C₃⁴×C₄

D₃

yes

characteristic

C₃²×C₃⋊Dic₃

—

C₃×C₄×H(3,𝔽₃)

D₃

yes

—

C₃×#1084

—

C₃²⋊H(3,𝔽₃)

Q₈

yes

characteristic, complemented, Sylow

C₃×#634

—

C₃²×Dic₆

—

C₃²×Dic₆

—

#6229

—

C₂×C₃⁴

C₂×D₃

yes

characteristic

C₂×C₃×H(3,𝔽₃)

C₂×D₃

yes

—

C₃³×C₄

—

C₃³×C₄

—

C₃³×C₄

C₃⋊D₃

yes

—

C₃³×C₄

C₃⋊D₃

yes

—

C₃×C₃⋊Dic₃

—

C₃²×Dic₃

—

C₃²×Dic₃

—

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

—

#1084

—

C₃⁴

Dic₆

yes

characteristic, complemented

C₃×H(3,𝔽₃)

Dic₆

yes

complemented

C₃²×Q₈

—

#634

—

C₃×Dic₆

—

C₃×Dic₆

—

C₃×Dic₆

—

C₂×C₃³

—

C₂×C₃³

—

C₂×C₃³

C₂×C₃⋊D₃

yes

—

C₂×C₃³

C₂×C₃⋊D₃

yes

—

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

#272

yes

—

C₃²×C₄

C₃⋊(C₃⋊D₃)

yes

characteristic

C₃⋊Dic₃

—

C₃×Dic₃

—

C₃×Dic₃

—

C₃×Dic₃

—

C₃³

—

C₃³

—

C₃³

#634

yes

—

C₃³

yes

—

—

C₃×Q₈

—

Dic₆

—

Dic₆

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₂×#272

yes

—

C₂×C₃²

C₂×C₃⋊(C₃⋊D₃)

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₃⋊#272

yes

—

Dic₃

—

Dic₃

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

yes

C₃²

yes

Q₈

—

Sylow

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

C₂×C₃⋊#272

yes

—

C₄

—

C₄

#92338

yes

characteristic

C₃

—

C₃

—

C₃

—

C₃

#10619988

yes

—

C₂

C₂×#92338

yes

characteristic

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

subgroup

G/H

center, Frattini, socle

C₂×C₃²

C₂×C₃⋊(C₃⋊D₃)

commutator, prime radical, cosocle

C₂×C₃²⋊H(3,𝔽₃)

C₂²

Fitting

C₄×C₃²⋊H(3,𝔽₃)

C₂

hypercenter

C₃²×C₄

C₃⋊(C₃⋊D₃)

hypocenter

C₃²⋊H(3,𝔽₃)

Q₈

LCS

G ▷ C₂×C₃²⋊H(3,𝔽₃) ▷ C₃²⋊H(3,𝔽₃)

UCS

C₁ ◁ C₂×C₃² ◁ C₃²×C₄

N	H ≅ G/N	type
10. C₃²⋊H(3,𝔽₃)	Q₈	semidirect
27. C₃⁴	Dic₆	semidirect
28. C₃×H(3,𝔽₃)	Dic₆	semidirect
57. C₃³	#634	semidirect
86. C₃²	#6229	semidirect

H ≅ G/N

type

10. C₃²⋊H(3,𝔽₃)

Q₈

semidirect

27. C₃⁴

Dic₆

semidirect

28. C₃×H(3,𝔽₃)

Dic₆

semidirect

57. C₃³

#634

semidirect

86. C₃²

#6229

semidirect

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

subgroup	index
C₄×C₃²⋊H(3,𝔽₃)	2
#11757642	2
C₃²×#634	3
C₃×#6229	3

subgroup

index

C₄×C₃²⋊H(3,𝔽₃)

#11757642

C₃²×#634

C₃×#6229

Subgroups — #14517728

Conjugacy classes of subgroups

Complements

Maximal subgroups