Group #14517259 — subgroups

Subgroups — #14517259

Order 1944.

227 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 227 conjugacy classes of subgroups — too many to draw. Only its 55 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×Q₈×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
2	#10619912	C₃	1	3	yes	maximal, complemented
3	#10619585	C₃	1	3	yes	maximal, complemented
4	#10619576	C₃	1	2	yes	maximal, complemented
5	#10619575	C₃	1	1	yes	maximal, characteristic, complemented
6	#10619574	C₃	1	3	yes	maximal, complemented
7	C₂×#6913	—	4	1	no	maximal
8	C₃×C₄×3₋¹⁺²	—	3	1	no	—
9	#6913	—	4	1	no	Sylow
10	C₃³×Q₈	C₃²	1	1	yes	characteristic
11	Q₈×3₋¹⁺²	—	3	3	no	—
12	C₃×Q₈×C₉	C₃²	1	1	yes	characteristic
13	C₃×Q₈×C₉	C₃²	1	1	yes	characteristic
14	C₃×Q₈×C₉	C₃²	1	1	yes	characteristic
15	Q₈⋊H(3,𝔽₃)	C₃²	1	3	yes	—
16	#6204	C₃²	1	6	yes	—
17	C₉⋊SL(2,𝔽₃)	—	3	6	no	—
18	C₉×SL(2,𝔽₃)	—	3	3	no	—
19	C₂×#737	—	4	3	no	—
20	C₂×C₃×3₋¹⁺²	A₄	1	1	yes	characteristic
21	C₂×#733	—	4	2	no	—
22	C₂×#732	—	4	1	no	—
23	C₂×#731	—	4	3	no	—
24	C₂×AΣL(1,𝔽₂₇)	—	4	3	no	—
25	C₃³×C₄	—	3	1	no	—
26	C₄×3₋¹⁺²	—	3	9	no	—
27	C₃×C₄×C₉	—	3	1	no	—
28	C₃×C₄×C₉	—	3	1	no	—
29	C₃×C₄×C₉	—	3	1	no	—
30	#737	—	4	3	no	—
31	C₃×3₋¹⁺²	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
32	#733	—	4	2	no	—
33	#732	—	4	1	no	—
34	#731	—	4	3	no	—
35	AΣL(1,𝔽₂₇)	—	4	3	no	—
36	C₃²×Q₈	—	3	1	no	—
37	C₃²×Q₈	—	9	1	no	—
38	C₃²×Q₈	C₃³	1	1	yes	characteristic
39	C₃×SL(2,𝔽₃)	—	3	3	no	—
40	Q₈×C₉	—	3	1	no	—
41	Q₈×C₉	—	3	1	no	—
42	Q₈×C₉	H(3,𝔽₃)	1	3	yes	—
43	Q₈⋊C₉	—	3	6	no	—
44	C₂×C₃³	C₃×A₄	1	1	yes	characteristic
45	C₂×3₋¹⁺²	—	3	3	no	—
46	C₂×3₋¹⁺²	—	4	6	no	—
47	C₂×3₋¹⁺²	—	12	6	no	—
48	C₂×H(3,𝔽₃)	—	4	3	no	—
49	C₂×C₃×C₉	—	12	3	no	—
50	C₂×C₃×C₉	C₃×A₄	1	1	yes	characteristic
51	C₂×C₃×C₉	C₃×A₄	1	1	yes	characteristic
52	C₂×C₃×C₉	C₃×A₄	1	1	yes	characteristic
53	C₃²×C₄	—	3	1	no	—
54	C₃²×C₄	—	3	3	no	—
55	C₃²×C₄	—	9	3	no	—
56	C₄×C₉	—	3	3	no	—
57	C₄×C₉	—	3	3	no	—
58	C₄×C₉	—	3	3	no	—
59	C₃³	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
60	3₋¹⁺²	—	3	3	no	—
61	3₋¹⁺²	—	4	6	no	—
62	3₋¹⁺²	—	12	6	no	—
63	H(3,𝔽₃)	—	4	3	no	—
64	C₃×C₉	—	12	3	no	—
65	C₃×C₉	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
66	C₃×C₉	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
67	C₃×C₉	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
68	C₃×Q₈	—	3	1	no	—
69	C₃×Q₈	—	9	1	no	—
70	C₃×Q₈	C₃×H(3,𝔽₃)	1	1	yes	characteristic
71	SL(2,𝔽₃)	—	9	3	no	—
72	C₂×C₃²	—	3	1	no	—
73	C₂×C₃²	—	9	1	no	—
74	C₂×C₃²	—	12	3	no	—
75	C₂×C₃²	C₃²×A₄	1	1	yes	characteristic
76	C₂×C₉	—	3	1	no	—
77	C₂×C₉	—	3	1	no	—
78	C₂×C₉	—	12	6	no	—
79	C₂×C₉	C₂²⋊H(3,𝔽₃)	1	3	yes	—
80	C₃×C₄	—	3	3	no	—
81	C₃×C₄	—	3	1	no	—
82	C₃×C₄	—	9	3	no	—
83	C₃²	—	3	1	no	—
84	C₃²	—	9	1	no	—
85	C₃²	—	12	3	no	—
86	C₃²	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
87	C₉	—	3	1	no	—
88	C₉	—	3	1	no	—
89	C₉	—	12	6	no	—
90	C₉	Q₈⋊H(3,𝔽₃)	1	3	yes	—
91	Q₈	#6913	1	1	yes	characteristic, complemented, Sylow
92	C₂×C₃	—	3	1	no	—
93	C₂×C₃	—	9	1	no	—
94	C₂×C₃	—	36	3	no	—
95	C₂×C₃	C₃×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
96	C₄	—	3	1	no	—
97	C₃	—	3	1	no	—
98	C₃	—	9	1	no	—
99	C₃	—	36	3	no	—
100	C₃	C₃×Q₈⋊H(3,𝔽₃)	1	1	yes	characteristic
101	C₂	#11757853	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×Q₈×3₋¹⁺²

C₃

yes

maximal, characteristic, complemented

#10619912

C₃

yes

maximal, complemented

#10619585

C₃

yes

maximal, complemented

#10619576

C₃

yes

maximal, complemented

#10619575

C₃

yes

maximal, characteristic, complemented

#10619574

C₃

yes

maximal, complemented

C₂×#6913

—

maximal

C₃×C₄×3₋¹⁺²

—

#6913

—

Sylow

C₃³×Q₈

C₃²

yes

characteristic

Q₈×3₋¹⁺²

—

C₃×Q₈×C₉

C₃²

yes

characteristic

C₃×Q₈×C₉

C₃²

yes

characteristic

C₃×Q₈×C₉

C₃²

yes

characteristic

Q₈⋊H(3,𝔽₃)

C₃²

yes

—

#6204

C₃²

yes

—

C₉⋊SL(2,𝔽₃)

—

C₉×SL(2,𝔽₃)

—

C₂×#737

—

C₂×C₃×3₋¹⁺²

A₄

yes

—

—

—

C₂×AΣL(1,𝔽₂₇)

—

C₃³×C₄

—

C₄×3₋¹⁺²

—

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

—

#737

—

C₃×3₋¹⁺²

SL(2,𝔽₃)

yes

characteristic, complemented

#733

—

#732

—

#731

—

AΣL(1,𝔽₂₇)

—

C₃²×Q₈

—

C₃²×Q₈

—

C₃²×Q₈

C₃³

yes

characteristic

C₃×SL(2,𝔽₃)

—

Q₈×C₉

—

Q₈×C₉

—

Q₈×C₉

H(3,𝔽₃)

yes

—

Q₈⋊C₉

—

C₂×C₃³

C₃×A₄

yes

characteristic

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×H(3,𝔽₃)

—

C₂×C₃×C₉

—

C₂×C₃×C₉

C₃×A₄

yes

characteristic

C₂×C₃×C₉

C₃×A₄

yes

characteristic

C₂×C₃×C₉

C₃×A₄

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₃³

C₃×SL(2,𝔽₃)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

C₃×SL(2,𝔽₃)

yes

characteristic

C₃×C₉

C₃×SL(2,𝔽₃)

yes

characteristic

C₃×C₉

C₃×SL(2,𝔽₃)

yes

characteristic

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

C₃×H(3,𝔽₃)

yes

characteristic

SL(2,𝔽₃)

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₃²×A₄

yes

characteristic

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

C₂²⋊H(3,𝔽₃)

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃²×SL(2,𝔽₃)

yes

characteristic

C₉

—

C₉

—

C₉

—

C₉

Q₈⋊H(3,𝔽₃)

yes

—

Q₈

#6913

yes

characteristic, complemented, Sylow

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

C₃×C₂²⋊H(3,𝔽₃)

yes

characteristic

C₄

—

C₃

—

C₃

—

C₃

—

100

C₃

C₃×Q₈⋊H(3,𝔽₃)

yes

characteristic

101

C₂

#11757853

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₃	C₃×C₂²⋊H(3,𝔽₃)
commutator, cosocle	C₃²×Q₈	C₃³
Frattini	C₂×C₃²	C₃²×A₄
Fitting	C₃×Q₈×3₋¹⁺²	C₃
hypercenter	C₂×C₃×3₋¹⁺²	A₄
hypocenter	Q₈	#6913
LCS	G ▷ C₃²×Q₈ ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₉ ◁ C₂×C₃×3₋¹⁺²

subgroup

G/H

center, socle

C₂×C₃

C₃×C₂²⋊H(3,𝔽₃)

commutator, cosocle

C₃²×Q₈

C₃³

Frattini

C₂×C₃²

C₃²×A₄

Fitting

C₃×Q₈×3₋¹⁺²

C₃

hypercenter

C₂×C₃×3₋¹⁺²

A₄

hypocenter

Q₈

#6913

LCS

G ▷ C₃²×Q₈ ▷ C₃×Q₈ ▷ Q₈

UCS

C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₉ ◁ C₂×C₃×3₋¹⁺²

N	H ≅ G/N	type
1. C₃×Q₈×3₋¹⁺²	C₃	semidirect
2. #10619912	C₃	semidirect
3. #10619585	C₃	semidirect
4. #10619576	C₃	semidirect
5. #10619575	C₃	semidirect
6. #10619574	C₃	semidirect
31. C₃×3₋¹⁺²	SL(2,𝔽₃)	semidirect
91. Q₈	#6913	semidirect

H ≅ G/N

type

1. C₃×Q₈×3₋¹⁺²

C₃

semidirect

2. #10619912

C₃

semidirect

3. #10619585

C₃

semidirect

4. #10619576

C₃

semidirect

5. #10619575

C₃

semidirect

6. #10619574

C₃

semidirect

31. C₃×3₋¹⁺²

SL(2,𝔽₃)

semidirect

91. Q₈

#6913

semidirect

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

subgroup	index
C₃×Q₈×3₋¹⁺²	3
#10619912	3
#10619585	3
#10619576	3
#10619575	3
#10619574	3
C₂×#6913	4

subgroup

index

C₃×Q₈×3₋¹⁺²

Subgroups — #14517259

Conjugacy classes of subgroups

Complements

Maximal subgroups