Group #14517218 — subgroups

Subgroups — #14517218

Order 1944.

202 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 49 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	#10619916	C₃	1	3	yes	maximal, complemented
2	C₃×C₉⋊SL(2,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
3	Q₈×#733	C₃	1	1	yes	maximal, characteristic, complemented
4	#10619583	C₃	1	3	yes	maximal, complemented
5	#10619582	C₃	1	3	yes	maximal, complemented
6	#10619576	C₃	1	1	yes	maximal, characteristic, complemented
7	#10619575	C₃	1	1	yes	maximal, characteristic, complemented
8	C₂×#6913	—	4	1	no	maximal
9	C₄×#733	—	3	1	no	—
10	#6913	—	4	1	no	Sylow
11	C₃²×SL(2,𝔽₃)	C₃²	1	1	yes	characteristic
12	Q₈×3₋¹⁺²	C₃²	1	3	yes	—
13	C₃×Q₈×C₉	C₃²	1	1	yes	characteristic
14	Q₈⋊H(3,𝔽₃)	C₃²	1	3	yes	—
15	#6204	C₃²	1	3	yes	—
16	C₃×Q₈⋊C₉	C₃²	1	1	yes	characteristic
17	C₃×Q₈⋊C₉	C₃²	1	1	yes	characteristic
18	C₉⋊SL(2,𝔽₃)	—	3	3	no	—
19	C₉⋊SL(2,𝔽₃)	—	3	6	no	—
20	C₉×SL(2,𝔽₃)	—	3	3	no	—
21	C₂×#737	—	4	3	no	—
22	C₂×C₃×3₋¹⁺²	—	4	1	no	—
23	C₂×#733	—	4	1	no	—
24	C₂×#733	A₄	1	1	yes	characteristic
25	C₂×#732	—	4	1	no	—
26	C₂×#731	—	4	3	no	—
27	C₂×AΣL(1,𝔽₂₇)	—	4	3	no	—
28	C₄×3₋¹⁺²	—	3	3	no	—
29	C₃×C₄×C₉	—	3	1	no	—
30	#737	—	4	3	no	—
31	C₃×3₋¹⁺²	—	4	1	no	—
32	#733	—	4	1	no	—
33	#733	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
34	#732	—	4	1	no	—
35	#731	—	4	3	no	—
36	AΣL(1,𝔽₂₇)	—	4	3	no	—
37	C₃²×Q₈	C₃³	1	1	yes	characteristic
38	C₃×SL(2,𝔽₃)	—	3	3	no	—
39	C₃×SL(2,𝔽₃)	—	3	1	no	—
40	C₃×SL(2,𝔽₃)	—	9	1	no	—
41	Q₈×C₉	—	3	3	no	—
42	Q₈×C₉	—	3	1	no	—
43	Q₈⋊C₉	—	3	3	no	—
44	Q₈⋊C₉	—	3	1	no	—
45	Q₈⋊C₉	H(3,𝔽₃)	1	3	yes	—
46	C₂×C₃³	—	4	1	no	—
47	C₂×3₋¹⁺²	—	4	3	no	—
48	C₂×3₋¹⁺²	—	12	3	no	—
49	C₂×3₋¹⁺²	—	12	6	no	—
50	C₂×3₋¹⁺²	C₃×A₄	1	3	yes	—
51	C₂×H(3,𝔽₃)	—	4	3	no	—
52	C₂×C₃×C₉	—	4	1	no	—
53	C₂×C₃×C₉	—	4	1	no	—
54	C₂×C₃×C₉	—	12	3	no	—
55	C₂×C₃×C₉	C₃×A₄	1	1	yes	characteristic
56	C₃²×C₄	—	3	1	no	—
57	C₄×C₉	—	9	1	no	—
58	C₄×C₉	—	9	3	no	—
59	C₃³	—	4	1	no	—
60	3₋¹⁺²	—	4	3	no	—
61	3₋¹⁺²	—	12	3	no	—
62	3₋¹⁺²	—	12	6	no	—
63	3₋¹⁺²	C₃×SL(2,𝔽₃)	1	3	yes	—
64	H(3,𝔽₃)	—	4	3	no	—
65	C₃×C₉	—	4	1	no	—
66	C₃×C₉	—	4	1	no	—
67	C₃×C₉	—	12	3	no	—
68	C₃×C₉	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
69	C₃×Q₈	—	3	1	no	—
70	C₃×Q₈	C₃×H(3,𝔽₃)	1	1	yes	characteristic
71	SL(2,𝔽₃)	—	9	3	no	—
72	SL(2,𝔽₃)	—	9	1	no	—
73	C₂×C₃²	—	12	3	no	—
74	C₂×C₃²	—	12	1	no	—
75	C₂×C₃²	—	36	1	no	—
76	C₂×C₃²	C₃²×A₄	1	1	yes	characteristic
77	C₂×C₉	—	3	3	no	—
78	C₂×C₉	—	3	1	no	—
79	C₂×C₉	—	4	3	no	—
80	C₂×C₉	—	12	1	no	—
81	C₂×C₉	—	12	3	no	—
82	C₃×C₄	—	3	1	no	—
83	C₃×C₄	—	9	1	no	—
84	C₃²	—	12	3	no	—
85	C₃²	—	12	1	no	—
86	C₃²	—	36	1	no	—
87	C₃²	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
88	C₉	—	3	3	no	—
89	C₉	—	3	1	no	—
90	C₉	—	4	3	no	—
91	C₉	—	12	1	no	—
92	C₉	—	12	3	no	—
93	Q₈	#6913	1	1	yes	characteristic, complemented, Sylow
94	C₂×C₃	—	3	1	no	—
95	C₂×C₃	—	36	1	no	—
96	C₂×C₃	—	36	3	no	—
97	C₂×C₃	A₄×H(3,𝔽₃)	1	1	yes	characteristic
98	C₄	—	3	1	no	—
99	C₃	—	3	1	no	—
100	C₃	—	36	3	no	—
101	C₃	—	36	1	no	—
102	C₃	SL(2,𝔽₃)×H(3,𝔽₃)	1	1	yes	characteristic
103	C₂	#11757811	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10619916

C₃

yes

maximal, complemented

C₃×C₉⋊SL(2,𝔽₃)

C₃

yes

maximal, characteristic, complemented

Q₈×#733

C₃

yes

maximal, characteristic, complemented

#10619583

C₃

yes

maximal, complemented

#10619582

C₃

yes

maximal, complemented

#10619576

C₃

yes

maximal, characteristic, complemented

#10619575

C₃

yes

maximal, characteristic, complemented

—

—

—

C₃²×SL(2,𝔽₃)

C₃²

yes

characteristic

Q₈×3₋¹⁺²

C₃²

yes

—

C₃×Q₈×C₉

C₃²

yes

characteristic

Q₈⋊H(3,𝔽₃)

C₃²

yes

—

#6204

C₃²

yes

—

C₃×Q₈⋊C₉

C₃²

yes

characteristic

C₃×Q₈⋊C₉

C₃²

yes

characteristic

C₉⋊SL(2,𝔽₃)

—

C₉⋊SL(2,𝔽₃)

—

C₉×SL(2,𝔽₃)

—

C₂×#737

—

C₂×C₃×3₋¹⁺²

—

C₂×#733

—

C₂×#733

A₄

yes

characteristic

C₂×#732

—

C₂×#731

—

C₂×AΣL(1,𝔽₂₇)

—

C₄×3₋¹⁺²

—

C₃×C₄×C₉

—

#737

—

C₃×3₋¹⁺²

—

#733

—

#733

SL(2,𝔽₃)

yes

characteristic, complemented

#732

—

#731

—

AΣL(1,𝔽₂₇)

—

C₃²×Q₈

C₃³

yes

characteristic

C₃×SL(2,𝔽₃)

—

C₃×SL(2,𝔽₃)

—

C₃×SL(2,𝔽₃)

—

Q₈×C₉

—

Q₈×C₉

—

Q₈⋊C₉

—

Q₈⋊C₉

—

Q₈⋊C₉

H(3,𝔽₃)

yes

—

C₂×C₃³

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

C₃×A₄

yes

—

C₂×H(3,𝔽₃)

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

C₃×A₄

yes

characteristic

C₃²×C₄

—

C₄×C₉

—

C₄×C₉

—

C₃³

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₃×SL(2,𝔽₃)

yes

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃×SL(2,𝔽₃)

yes

characteristic

C₃×Q₈

—

C₃×Q₈

C₃×H(3,𝔽₃)

yes

characteristic

SL(2,𝔽₃)

—

SL(2,𝔽₃)

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₃²×A₄

yes

characteristic

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₃×C₄

—

C₃×C₄

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃²×SL(2,𝔽₃)

yes

characteristic

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

Q₈

#6913

yes

characteristic, complemented, Sylow

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

A₄×H(3,𝔽₃)

yes

characteristic

C₄

—

C₃

—

100

C₃

—

101

C₃

—

102

C₃

SL(2,𝔽₃)×H(3,𝔽₃)

yes

characteristic

103

C₂

#11757811

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₃	A₄×H(3,𝔽₃)
commutator, cosocle	C₃²×Q₈	C₃³
Frattini	C₂×C₃²	C₃²×A₄
Fitting	Q₈×#733	C₃
hypercenter	C₂×#733	A₄
hypocenter	Q₈	#6913
LCS	G ▷ C₃²×Q₈ ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃² ◁ C₂×#733

subgroup

G/H

center, socle

C₂×C₃

A₄×H(3,𝔽₃)

commutator, cosocle

C₃²×Q₈

C₃³

Frattini

C₂×C₃²

C₃²×A₄

C₃

A₄

Q₈

G ▷ C₃²×Q₈ ▷ C₃×Q₈ ▷ Q₈

UCS

C₁ ◁ C₂×C₃ ◁ C₂×C₃² ◁ C₂×#733

N	H ≅ G/N	type
1. #10619916	C₃	semidirect
2. C₃×C₉⋊SL(2,𝔽₃)	C₃	semidirect
3. Q₈×#733	C₃	semidirect
4. #10619583	C₃	semidirect
5. #10619582	C₃	semidirect
6. #10619576	C₃	semidirect
7. #10619575	C₃	semidirect
33. #733	SL(2,𝔽₃)	semidirect
93. Q₈	#6913	semidirect

H ≅ G/N

type

1. #10619916

C₃

semidirect

2. C₃×C₉⋊SL(2,𝔽₃)

C₃

semidirect

3. Q₈×#733

C₃

semidirect

4. #10619583

C₃

semidirect

5. #10619582

C₃

semidirect

6. #10619576

C₃

semidirect

7. #10619575

C₃

semidirect

33. #733

SL(2,𝔽₃)

semidirect

93. Q₈

#6913

semidirect

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

subgroup	index
#10619916	3
C₃×C₉⋊SL(2,𝔽₃)	3
Q₈×#733	3
#10619583	3
#10619582	3
#10619576	3
#10619575	3
C₂×#6913	4

subgroup

index

C₃×C₉⋊SL(2,𝔽₃)

Subgroups — #14517218

Conjugacy classes of subgroups

Complements

Maximal subgroups