Group #18018472 — subgroups

Subgroups — C₁₉×#12513 #18018472

Order 4864.

290 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 94 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	C₁₉×#2240	C₂	1	4	yes	maximal, complemented
2	C₁₉×#1712	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1710	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1311	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#500	C₂²	1	2	yes	—
6	C₁₉×#493	—	2	2	no	—
7	C₁₉×#493	C₂²	1	2	yes	—
8	C₁₉×#487	—	2	2	no	—
9	C₁₉×#357	—	2	8	no	—
10	C₄×C₁₆×C₁₉	C₂²	1	2	yes	—
11	C₈²×C₁₉	C₂²	1	1	yes	characteristic
12	C₂×D₈×C₁₉	—	2	4	no	—
13	C₂×D₈×C₁₉	—	4	4	no	—
14	C₁₉×C₄⋊Q₈	—	2	2	no	—
15	C₁₉×#127	—	2	2	no	—
16	C₂×C₁₆×C₁₉	D₄	1	4	yes	—
17	C₁₉×#107	—	2	4	no	—
18	C₁₉×#103	—	4	4	no	—
19	C₄×C₈×C₁₉	D₄	1	2	yes	—
20	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
21	C₂×Q₈×C₁₉	—	4	2	no	—
22	C₂×D₄×C₁₉	—	4	2	no	—
23	C₂×D₄×C₁₉	—	4	4	no	—
24	D₈×C₁₉	—	4	8	no	—
25	D₈×C₁₉	—	8	8	no	—
26	C₂×C₈×C₁₉	D₈	1	4	yes	—
27	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
28	C₄⋊C₄×C₁₉	—	4	2	no	—
29	C₄⋊C₄×C₁₉	—	4	2	no	—
30	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
31	C₁₆×C₁₉	—	2	4	no	—
32	#12513	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂³×C₁₉	—	8	2	no	—
34	Q₈×C₁₉	—	8	4	no	—
35	D₄×C₁₉	—	8	4	no	—
36	D₄×C₁₉	—	8	8	no	—
37	C₂×C₄×C₁₉	—	8	2	no	—
38	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
39	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
40	C₈×C₁₉	—	2	4	no	—
41	C₈×C₁₉	#135	1	4	yes	—
42	#2240	C₂×C₁₉	1	4	yes	complemented
43	#1712	C₂×C₁₉	1	1	yes	characteristic
44	#1710	C₂×C₁₉	1	1	yes	characteristic, complemented
45	#1311	C₂×C₁₉	1	1	yes	characteristic, complemented
46	C₂²×C₁₉	—	16	2	no	—
47	C₂²×C₁₉	—	16	4	no	—
48	C₂²×C₁₉	#493	1	1	yes	characteristic
49	C₄×C₁₉	—	16	2	no	—
50	C₄×C₁₉	#495	1	2	yes	—
51	C₄×C₁₉	#509	1	4	yes	—
52	#500	C₂²×C₁₉	1	2	yes	—
53	#493	—	2	2	no	—
54	#493	C₂²×C₁₉	1	2	yes	—
55	#487	—	2	2	no	—
56	#357	—	2	8	no	—
57	C₄×C₁₆	C₂²×C₁₉	1	2	yes	—
58	C₈²	C₂²×C₁₉	1	1	yes	characteristic
59	C₂×C₁₉	—	32	2	no	—
60	C₂×C₁₉	#1709	1	1	yes	characteristic
61	C₂×C₁₉	#2250	1	2	yes	—
62	C₂×D₈	—	2	4	no	—
63	C₂×D₈	—	4	4	no	—
64	C₄⋊Q₈	—	2	2	no	—
65	#127	—	2	2	no	—
66	C₂×C₁₆	D₄×C₁₉	1	4	yes	—
67	#107	—	2	4	no	—
68	#103	—	4	4	no	—
69	C₄×C₈	D₄×C₁₉	1	2	yes	—
70	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
71	C₁₉	#12513	1	1	yes	characteristic, direct product factor, Sylow
72	C₂×Q₈	—	4	2	no	—
73	C₂×D₄	—	4	2	no	—
74	C₂×D₄	—	4	4	no	—
75	D₈	—	4	8	no	—
76	D₈	—	8	8	no	—
77	C₂×C₈	D₈×C₁₉	1	4	yes	—
78	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
79	C₄⋊C₄	—	4	2	no	—
80	C₄⋊C₄	—	4	2	no	—
81	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
82	C₁₆	—	2	4	no	—
83	C₂³	—	8	2	no	—
84	Q₈	—	8	4	no	—
85	D₄	—	8	4	no	—
86	D₄	—	8	8	no	—
87	C₂×C₄	—	8	2	no	—
88	C₂×C₄	C₁₉×#127	1	1	yes	characteristic
89	C₂×C₄	C₂×D₈×C₁₉	1	2	yes	—
90	C₈	—	2	4	no	—
91	C₈	C₁₉×#135	1	4	yes	—
92	C₂²	—	16	2	no	—
93	C₂²	—	16	4	no	—
94	C₂²	C₁₉×#493	1	1	yes	characteristic
95	C₄	—	16	2	no	—
96	C₄	C₁₉×#495	1	2	yes	—
97	C₄	C₁₉×#509	1	4	yes	—
98	C₂	—	32	2	no	—
99	C₂	C₁₉×#1709	1	1	yes	characteristic
100	C₂	C₁₉×#2250	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#2240

C₂

yes

maximal, complemented

C₁₉×#1712

C₂

yes

maximal, characteristic

C₁₉×#1710

C₂

yes

maximal, characteristic, complemented

C₁₉×#1311

C₂

yes

maximal, characteristic, complemented

C₁₉×#500

C₂²

yes

—

C₁₉×#493

—

C₁₉×#493

C₂²

yes

—

C₁₉×#487

—

C₁₉×#357

—

C₄×C₁₆×C₁₉

C₂²

yes

—

C₈²×C₁₉

C₂²

yes

characteristic

C₂×D₈×C₁₉

—

C₂×D₈×C₁₉

—

C₁₉×C₄⋊Q₈

—

C₁₉×#127

—

C₂×C₁₆×C₁₉

D₄

yes

—

C₁₉×#107

—

C₁₉×#103

—

C₄×C₈×C₁₉

D₄

yes

—

C₄×C₈×C₁₉

C₂³

yes

characteristic

C₂×Q₈×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

D₈×C₁₉

—

D₈×C₁₉

—

C₂×C₈×C₁₉

D₈

yes

—

C₂×C₈×C₁₉

C₂×D₄

yes

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₁₆×C₁₉

—

#12513

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

—

Q₈×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#127

yes

characteristic

C₂×C₄×C₁₉

C₂×D₈

yes

—

C₈×C₁₉

—

C₈×C₁₉

#135

yes

—

#2240

C₂×C₁₉

yes

complemented

#1712

C₂×C₁₉

yes

characteristic

#1710

C₂×C₁₉

yes

characteristic, complemented

#1311

C₂×C₁₉

yes

characteristic, complemented

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#493

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

#495

yes

—

C₄×C₁₉

#509

yes

—

#500

C₂²×C₁₉

yes

—

#493

—

#493

C₂²×C₁₉

yes

—

#487

—

#357

—

C₄×C₁₆

C₂²×C₁₉

yes

—

C₈²

C₂²×C₁₉

yes

characteristic

C₂×C₁₉

—

C₂×C₁₉

#1709

yes

characteristic

C₂×C₁₉

#2250

yes

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

D₄×C₁₉

yes

—

#107

—

#103

—

C₄×C₈

D₄×C₁₉

yes

—

C₄×C₈

C₂³×C₁₉

yes

characteristic

C₁₉

#12513

yes

characteristic, direct product factor, Sylow

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

C₂×C₈

D₈×C₁₉

yes

—

C₂×C₈

C₂×D₄×C₁₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₄×C₁₉

yes

characteristic

C₁₆

—

C₂³

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

C₁₉×#127

yes

characteristic

C₂×C₄

C₂×D₈×C₁₉

yes

—

C₈

—

C₈

C₁₉×#135

yes

—

C₂²

—

C₂²

—

C₂²

C₁₉×#493

yes

characteristic

C₄

—

C₄

C₁₉×#495

yes

—

C₄

C₁₉×#509

yes

—

C₂

—

C₂

C₁₉×#1709

yes

characteristic

100

C₂

C₁₉×#2250

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₉	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₉
prime radical	C₁₉×#1712	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₄²×C₁₉ ◁ C₈²×C₁₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₉

#493

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₁₉

prime radical

C₁₉×#1712

C₂

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₉ ◁ C₄²×C₁₉ ◁ C₈²×C₁₉ ◁ G

N	H ≅ G/N	type
1. C₁₉×#2240	C₂	semidirect
3. C₁₉×#1710	C₂	semidirect
4. C₁₉×#1311	C₂	semidirect
32. #12513	C₁₉	direct (N × H)
42. #2240	C₂×C₁₉	semidirect
44. #1710	C₂×C₁₉	semidirect
45. #1311	C₂×C₁₉	semidirect
71. C₁₉	#12513	direct (N × H)

H ≅ G/N

type

1. C₁₉×#2240

C₂

semidirect

3. C₁₉×#1710

C₂

semidirect

4. C₁₉×#1311

C₂

semidirect

32. #12513

C₁₉

direct (N × H)

42. #2240

C₂×C₁₉

semidirect

44. #1710

C₂×C₁₉

semidirect

45. #1311

C₂×C₁₉

semidirect

71. C₁₉

#12513

direct (N × H)

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

subgroup	index
C₁₉×#2240	2
C₁₉×#1712	2
C₁₉×#1710	2
C₁₉×#1311	2
#12513	19

subgroup

index

C₁₉×#2240

C₁₉×#1712

C₁₉×#1710

C₁₉×#1311

#12513

Subgroups — C₁₉×#12513 #18018472

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#12513 #18018472

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12513 #18018472