Group #14517245 — subgroups

Subgroups — #14517245

Order 1944.

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

normal subgroups only

This group has 260 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×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	C₃×#6204	C₃	1	2	yes	maximal, complemented
3	C₃×#6204	C₃	1	1	yes	maximal, characteristic, complemented
4	#10619578	C₃	1	9	yes	maximal, complemented
5	C₂×#6910	—	4	1	no	maximal
6	C₃×C₄×H(3,𝔽₃)	—	3	1	no	—
7	#6910	—	4	1	no	Sylow
8	C₃³×Q₈	—	3	1	no	—
9	C₃³×Q₈	C₃²	1	1	yes	characteristic
10	Q₈×H(3,𝔽₃)	—	3	2	no	—
11	Q₈×H(3,𝔽₃)	C₃²	1	3	yes	—
12	#6204	—	3	2	no	—
13	#6204	—	3	2	no	—
14	#6204	—	3	2	no	—
15	#6204	—	3	1	no	—
16	#6204	C₃²	1	6	yes	complemented
17	C₃×Q₈⋊C₉	—	3	2	no	—
18	C₃×Q₈⋊C₉	C₃²	1	3	yes	complemented
19	C₂×C₃×3₋¹⁺²	—	4	1	no	—
20	C₂×C₃×3₋¹⁺²	—	4	2	no	—
21	C₂×C₃×H(3,𝔽₃)	A₄	1	1	yes	characteristic
22	C₂×#731	—	4	9	no	—
23	C₃³×C₄	—	3	3	no	—
24	C₃³×C₄	—	3	1	no	—
25	C₄×H(3,𝔽₃)	—	3	6	no	—
26	C₄×H(3,𝔽₃)	—	3	3	no	—
27	C₃×3₋¹⁺²	—	4	1	no	—
28	C₃×3₋¹⁺²	—	4	2	no	—
29	C₃×H(3,𝔽₃)	SL(2,𝔽₃)	1	1	yes	characteristic
30	#731	—	4	9	no	—
31	C₃²×Q₈	—	3	3	no	—
32	C₃²×Q₈	—	9	1	no	—
33	C₃²×Q₈	—	9	3	no	—
34	C₃²×Q₈	H(3,𝔽₃)	1	2	yes	—
35	C₃²×Q₈	H(3,𝔽₃)	1	1	yes	characteristic
36	C₃²×Q₈	C₃³	1	1	yes	characteristic
37	Q₈⋊C₉	—	3	6	no	—
38	Q₈⋊C₉	—	3	3	no	—
39	C₂×C₃³	—	3	1	no	—
40	C₂×C₃³	C₃×A₄	1	1	yes	characteristic
41	C₂×3₋¹⁺²	—	4	6	no	—
42	C₂×3₋¹⁺²	—	12	2	no	—
43	C₂×3₋¹⁺²	—	12	1	no	—
44	C₂×3₋¹⁺²	—	12	2	no	—
45	C₂×3₋¹⁺²	—	12	2	no	—
46	C₂×H(3,𝔽₃)	—	3	2	no	—
47	C₂×H(3,𝔽₃)	C₃×A₄	1	3	yes	—
48	C₂×C₃×C₉	—	4	3	no	—
49	C₂×C₃×C₉	—	12	2	no	—
50	C₃²×C₄	—	3	1	no	—
51	C₃²×C₄	—	3	9	no	—
52	C₃²×C₄	—	3	2	no	—
53	C₃²×C₄	—	3	1	no	—
54	C₃²×C₄	—	9	3	no	—
55	C₃²×C₄	—	9	9	no	—
56	C₃³	—	3	1	no	—
57	C₃³	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
58	3₋¹⁺²	—	4	6	no	—
59	3₋¹⁺²	—	12	2	no	—
60	3₋¹⁺²	—	12	1	no	—
61	3₋¹⁺²	—	12	2	no	—
62	3₋¹⁺²	—	12	2	no	—
63	H(3,𝔽₃)	—	3	2	no	—
64	H(3,𝔽₃)	C₃×SL(2,𝔽₃)	1	3	yes	—
65	C₃×C₉	—	4	3	no	—
66	C₃×C₉	—	12	2	no	—
67	C₃×Q₈	—	3	1	no	—
68	C₃×Q₈	—	3	2	no	—
69	C₃×Q₈	—	3	1	no	—
70	C₃×Q₈	—	9	3	no	—
71	C₃×Q₈	C₃×H(3,𝔽₃)	1	1	yes	characteristic
72	C₂×C₃²	—	3	3	no	—
73	C₂×C₃²	—	9	1	no	—
74	C₂×C₃²	—	9	3	no	—
75	C₂×C₃²	C₂²⋊H(3,𝔽₃)	1	2	yes	—
76	C₂×C₃²	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
77	C₂×C₃²	C₃²×A₄	1	1	yes	characteristic
78	C₂×C₉	—	12	3	no	—
79	C₂×C₉	—	12	6	no	—
80	C₃×C₄	—	3	3	no	—
81	C₃×C₄	—	3	1	no	—
82	C₃×C₄	—	9	2	no	—
83	C₃×C₄	—	9	1	no	—
84	C₃×C₄	—	9	9	no	—
85	C₃²	—	3	3	no	—
86	C₃²	—	9	1	no	—
87	C₃²	—	9	3	no	—
88	C₃²	Q₈⋊H(3,𝔽₃)	1	2	yes	—
89	C₃²	Q₈⋊H(3,𝔽₃)	1	1	yes	characteristic
90	C₃²	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
91	C₉	—	12	3	no	—
92	C₉	—	12	6	no	—
93	Q₈	#6910	1	1	yes	characteristic, complemented, Sylow
94	C₂×C₃	—	3	1	no	—
95	C₂×C₃	—	3	2	no	—
96	C₂×C₃	—	3	1	no	—
97	C₂×C₃	—	9	3	no	—
98	C₂×C₃	C₃×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
99	C₄	—	3	1	no	—
100	C₃	—	3	1	no	—
101	C₃	—	3	2	no	—
102	C₃	—	3	1	no	—
103	C₃	—	9	3	no	—
104	C₃	C₃×Q₈⋊H(3,𝔽₃)	1	1	yes	characteristic
105	C₂	#11757838	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×Q₈×H(3,𝔽₃)

C₃

yes

maximal, characteristic

C₃×#6204

C₃

yes

maximal, complemented

C₃×#6204

C₃

yes

maximal, characteristic, complemented

#10619578

C₃

yes

maximal, complemented

C₂×#6910

—

maximal

C₃×C₄×H(3,𝔽₃)

—

#6910

—

Sylow

C₃³×Q₈

—

C₃³×Q₈

C₃²

yes

characteristic

Q₈×H(3,𝔽₃)

—

Q₈×H(3,𝔽₃)

C₃²

yes

—

—

—

—

—

C₃²

yes

complemented

C₃×Q₈⋊C₉

—

C₃×Q₈⋊C₉

C₃²

yes

complemented

C₂×C₃×3₋¹⁺²

—

C₂×C₃×3₋¹⁺²

—

C₂×C₃×H(3,𝔽₃)

A₄

yes

characteristic

C₂×#731

—

C₃³×C₄

—

C₃³×C₄

—

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

SL(2,𝔽₃)

yes

characteristic

#731

—

C₃²×Q₈

—

C₃²×Q₈

—

C₃²×Q₈

—

C₃²×Q₈

H(3,𝔽₃)

yes

—

C₃²×Q₈

H(3,𝔽₃)

yes

characteristic

C₃²×Q₈

C₃³

yes

characteristic

Q₈⋊C₉

—

Q₈⋊C₉

—

C₂×C₃³

—

C₂×C₃³

C₃×A₄

yes

characteristic

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

C₃×A₄

yes

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃³

—

C₃³

C₃×SL(2,𝔽₃)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

H(3,𝔽₃)

C₃×SL(2,𝔽₃)

yes

—

C₃×C₉

—

C₃×C₉

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

C₃×H(3,𝔽₃)

yes

characteristic

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₂²⋊H(3,𝔽₃)

yes

—

C₂×C₃²

C₂²⋊H(3,𝔽₃)

yes

characteristic

C₂×C₃²

C₃²×A₄

yes

characteristic

C₂×C₉

—

C₂×C₉

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃²

—

C₃²

—

C₃²

—

C₃²

Q₈⋊H(3,𝔽₃)

yes

—

C₃²

Q₈⋊H(3,𝔽₃)

yes

characteristic

C₃²

C₃²×SL(2,𝔽₃)

yes

characteristic

C₉

—

C₉

—

Q₈

#6910

yes

characteristic, complemented, Sylow

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

C₃×C₂²⋊H(3,𝔽₃)

yes

characteristic

C₄

—

100

C₃

—

101

C₃

—

102

C₃

—

103

C₃

—

104

C₃

C₃×Q₈⋊H(3,𝔽₃)

yes

characteristic

105

C₂

#11757838

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×H(3,𝔽₃)	C₃
prime radical, hypercenter	C₂×C₃×H(3,𝔽₃)	A₄
hypocenter	Q₈	#6910
LCS	G ▷ C₃²×Q₈ ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃³ ◁ C₂×C₃×H(3,𝔽₃)

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×H(3,𝔽₃)

C₃

prime radical, hypercenter

C₂×C₃×H(3,𝔽₃)

A₄

hypocenter

Q₈

#6910

LCS

G ▷ C₃²×Q₈ ▷ C₃×Q₈ ▷ Q₈

UCS

C₁ ◁ C₂×C₃ ◁ C₂×C₃³ ◁ C₂×C₃×H(3,𝔽₃)

N	H ≅ G/N	type
2. C₃×#6204	C₃	semidirect
3. C₃×#6204	C₃	semidirect
4. #10619578	C₃	semidirect
16. #6204	C₃²	semidirect
18. C₃×Q₈⋊C₉	C₃²	semidirect
93. Q₈	#6910	semidirect

H ≅ G/N

type

2. C₃×#6204

C₃

semidirect

3. C₃×#6204

C₃

semidirect

4. #10619578

C₃

semidirect

16. #6204

C₃²

semidirect

18. C₃×Q₈⋊C₉

C₃²

semidirect

93. Q₈

#6910

semidirect

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

subgroup	index
C₃×Q₈×H(3,𝔽₃)	3
C₃×#6204	3
#10619578	3
C₂×#6910	4

subgroup

index

C₃×Q₈×H(3,𝔽₃)

C₃×#6204

#10619578

C₂×#6910

Subgroups — #14517245

Conjugacy classes of subgroups

Complements

Maximal subgroups