Group #18019702 — subgroups

Subgroups — C₁₉×#13743 #18019702

Order 4864.

126 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 58 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×#2262	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×Q₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×Q₁₂₈	C₂	1	2	yes	maximal, complemented
4	C₁₉×SD₁₂₈	C₂	1	2	yes	maximal, complemented
5	C₁₉×M₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#508	—	2	1	no	—
7	C₂×C₁₉×Q₃₂	—	2	1	no	—
8	C₁₉×Q₆₄	—	2	1	no	—
9	C₁₉×Q₆₄	C₂²	1	2	yes	—
10	C₁₉×Q₆₄	C₂²	1	1	yes	characteristic
11	C₁₉×SD₆₄	—	2	1	no	—
12	C₁₉×D₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₁₉×C₃₂	C₂²	1	1	yes	characteristic
14	C₁₉×C₆₄	C₂²	1	2	yes	—
15	C₁₉×#135	—	4	1	no	—
16	C₂×Q₁₆×C₁₉	—	4	1	no	—
17	C₁₉×Q₃₂	—	2	1	no	—
18	C₁₉×Q₃₂	—	2	2	no	—
19	C₁₉×Q₃₂	—	4	1	no	—
20	C₁₉×SD₃₂	—	4	1	no	—
21	C₁₉×D₁₆	—	2	1	no	—
22	C₂×C₁₆×C₁₉	D₄	1	1	yes	characteristic
23	C₁₉×C₃₂	D₄	1	1	yes	characteristic
24	C₁₉×C₃₂	C₂³	1	1	yes	characteristic
25	#41×C₁₉	—	8	1	no	—
26	C₂×Q₈×C₁₉	—	8	1	no	—
27	Q₁₆×C₁₉	—	4	2	no	—
28	Q₁₆×C₁₉	—	4	1	no	—
29	Q₁₆×C₁₉	—	8	1	no	—
30	SD₁₆×C₁₉	—	8	1	no	—
31	D₈×C₁₉	—	4	1	no	—
32	C₂×C₈×C₁₉	D₈	1	1	yes	characteristic
33	C₁₆×C₁₉	D₈	1	1	yes	characteristic
34	C₁₆×C₁₉	C₂×D₄	1	1	yes	characteristic
35	#13743	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	Q₈×C₁₉	—	8	1	no	—
37	Q₈×C₁₉	—	8	2	no	—
38	Q₈×C₁₉	—	16	1	no	—
39	D₄×C₁₉	—	8	1	no	—
40	D₄×C₁₉	—	16	1	no	—
41	C₂×C₄×C₁₉	—	16	1	no	—
42	C₂×C₄×C₁₉	—	16	1	no	—
43	C₂×C₄×C₁₉	D₁₆	1	1	yes	characteristic
44	C₈×C₁₉	D₁₆	1	1	yes	characteristic
45	C₈×C₁₉	C₂×D₈	1	1	yes	characteristic
46	#2262	C₂×C₁₉	1	1	yes	characteristic
47	C₂×Q₆₄	C₂×C₁₉	1	1	yes	characteristic, complemented
48	Q₁₂₈	C₂×C₁₉	1	2	yes	complemented
49	SD₁₂₈	C₂×C₁₉	1	2	yes	complemented
50	M₁₂₈	C₂×C₁₉	1	1	yes	characteristic, complemented
51	C₂²×C₁₉	—	16	1	no	—
52	C₂²×C₁₉	D₃₂	1	1	yes	characteristic
53	C₄×C₁₉	—	16	1	no	—
54	C₄×C₁₉	—	16	2	no	—
55	C₄×C₁₉	D₃₂	1	1	yes	characteristic
56	C₄×C₁₉	C₂×D₁₆	1	1	yes	characteristic
57	#508	—	2	1	no	—
58	C₂×Q₃₂	—	2	1	no	—
59	Q₆₄	—	2	1	no	—
60	Q₆₄	C₂²×C₁₉	1	2	yes	—
61	Q₆₄	C₂²×C₁₉	1	1	yes	characteristic
62	SD₆₄	—	2	1	no	—
63	D₃₂	C₂²×C₁₉	1	1	yes	characteristic
64	C₂×C₃₂	C₂²×C₁₉	1	1	yes	characteristic
65	C₆₄	C₂²×C₁₉	1	2	yes	—
66	C₂×C₁₉	—	2	1	no	—
67	C₂×C₁₉	—	32	1	no	—
68	C₂×C₁₉	C₂×D₃₂	1	1	yes	characteristic
69	#135	—	4	1	no	—
70	C₂×Q₁₆	—	4	1	no	—
71	Q₃₂	—	2	1	no	—
72	Q₃₂	—	2	2	no	—
73	Q₃₂	—	4	1	no	—
74	SD₃₂	—	4	1	no	—
75	D₁₆	—	2	1	no	—
76	C₂×C₁₆	D₄×C₁₉	1	1	yes	characteristic
77	C₃₂	D₄×C₁₉	1	1	yes	characteristic
78	C₃₂	C₂³×C₁₉	1	1	yes	characteristic
79	C₁₉	#13743	1	1	yes	characteristic, direct product factor, Sylow
80	#41	—	8	1	no	—
81	C₂×Q₈	—	8	1	no	—
82	Q₁₆	—	4	1	no	—
83	Q₁₆	—	4	2	no	—
84	Q₁₆	—	8	1	no	—
85	SD₁₆	—	8	1	no	—
86	D₈	—	4	1	no	—
87	C₂×C₈	D₈×C₁₉	1	1	yes	characteristic
88	C₁₆	D₈×C₁₉	1	1	yes	characteristic
89	C₁₆	C₂×D₄×C₁₉	1	1	yes	characteristic
90	Q₈	—	8	1	no	—
91	Q₈	—	8	2	no	—
92	Q₈	—	16	1	no	—
93	D₄	—	8	1	no	—
94	D₄	—	16	1	no	—
95	C₂×C₄	—	16	1	no	—
96	C₂×C₄	—	16	1	no	—
97	C₂×C₄	C₁₉×D₁₆	1	1	yes	characteristic
98	C₈	C₁₉×D₁₆	1	1	yes	characteristic
99	C₈	C₂×D₈×C₁₉	1	1	yes	characteristic
100	C₂²	—	16	1	no	—
101	C₂²	C₁₉×D₃₂	1	1	yes	characteristic
102	C₄	—	16	1	no	—
103	C₄	—	16	2	no	—
104	C₄	C₁₉×D₃₂	1	1	yes	characteristic
105	C₄	C₂×C₁₉×D₁₆	1	1	yes	characteristic
106	C₂	—	2	1	no	—
107	C₂	—	32	1	no	—
108	C₂	C₂×C₁₉×D₃₂	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#2262

C₂

yes

maximal, characteristic

C₂×C₁₉×Q₆₄

C₂

yes

maximal, characteristic, complemented

C₁₉×Q₁₂₈

C₂

yes

maximal, complemented

C₁₉×SD₁₂₈

C₂

yes

maximal, complemented

C₁₉×M₁₂₈

C₂

yes

maximal, characteristic, complemented

C₁₉×#508

—

C₂×C₁₉×Q₃₂

—

C₁₉×Q₆₄

—

C₁₉×Q₆₄

C₂²

yes

—

C₁₉×Q₆₄

C₂²

yes

characteristic

C₁₉×SD₆₄

—

C₁₉×D₃₂

C₂²

yes

characteristic

C₂×C₁₉×C₃₂

C₂²

yes

characteristic

C₁₉×C₆₄

C₂²

yes

—

C₁₉×#135

—

C₂×Q₁₆×C₁₉

—

C₁₉×Q₃₂

—

C₁₉×Q₃₂

—

C₁₉×Q₃₂

—

C₁₉×SD₃₂

—

C₁₉×D₁₆

—

C₂×C₁₆×C₁₉

D₄

yes

characteristic

C₁₉×C₃₂

D₄

yes

characteristic

C₁₉×C₃₂

C₂³

yes

characteristic

#41×C₁₉

—

C₂×Q₈×C₁₉

—

Q₁₆×C₁₉

—

Q₁₆×C₁₉

—

Q₁₆×C₁₉

—

SD₁₆×C₁₉

—

D₈×C₁₉

—

C₂×C₈×C₁₉

D₈

yes

characteristic

C₁₆×C₁₉

D₈

yes

characteristic

C₁₆×C₁₉

C₂×D₄

yes

characteristic

#13743

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

Q₈×C₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

D₁₆

yes

characteristic

C₈×C₁₉

D₁₆

yes

characteristic

C₈×C₁₉

C₂×D₈

yes

characteristic

#2262

C₂×C₁₉

yes

characteristic

C₂×Q₆₄

C₂×C₁₉

yes

characteristic, complemented

Q₁₂₈

C₂×C₁₉

yes

complemented

SD₁₂₈

C₂×C₁₉

yes

complemented

M₁₂₈

C₂×C₁₉

yes

characteristic, complemented

C₂²×C₁₉

—

C₂²×C₁₉

D₃₂

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

D₃₂

yes

characteristic

C₄×C₁₉

C₂×D₁₆

yes

characteristic

#508

—

C₂×Q₃₂

—

Q₆₄

—

Q₆₄

C₂²×C₁₉

yes

—

Q₆₄

C₂²×C₁₉

yes

characteristic

SD₆₄

—

D₃₂

C₂²×C₁₉

yes

characteristic

C₂×C₃₂

C₂²×C₁₉

yes

characteristic

C₆₄

C₂²×C₁₉

yes

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

C₂×D₃₂

yes

characteristic

#135

—

C₂×Q₁₆

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

SD₃₂

—

D₁₆

—

C₂×C₁₆

D₄×C₁₉

yes

characteristic

C₃₂

D₄×C₁₉

yes

characteristic

C₃₂

C₂³×C₁₉

yes

characteristic

C₁₉

#13743

yes

characteristic, direct product factor, Sylow

#41

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

SD₁₆

—

D₈

—

C₂×C₈

D₈×C₁₉

yes

characteristic

C₁₆

D₈×C₁₉

yes

characteristic

C₁₆

C₂×D₄×C₁₉

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₁₉×D₁₆

yes

characteristic

C₈

C₁₉×D₁₆

yes

characteristic

C₈

C₂×D₈×C₁₉

yes

characteristic

100

C₂²

—

101

C₂²

C₁₉×D₃₂

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

C₁₉×D₃₂

yes

characteristic

105

C₄

C₂×C₁₉×D₁₆

yes

characteristic

106

C₂

—

107

C₂

—

108

C₂

C₂×C₁₉×D₃₂

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₁₉	C₂×D₃₂
commutator, Frattini, cosocle	C₃₂	C₂³×C₁₉
prime radical	C₁₉×#2262	C₂
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂×C₈×C₁₉ ◁ C₂×C₁₆×C₁₉ ◁ C₂×C₁₉×C₃₂ ◁ G

subgroup

G/H

center, socle

C₂×C₁₉

C₂×D₃₂

commutator, Frattini, cosocle

C₃₂

C₂³×C₁₉

prime radical

C₁₉×#2262

C₂

LCS

G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂×C₈×C₁₉ ◁ C₂×C₁₆×C₁₉ ◁ C₂×C₁₉×C₃₂ ◁ G

N	H ≅ G/N	type
2. C₂×C₁₉×Q₆₄	C₂	semidirect
3. C₁₉×Q₁₂₈	C₂	semidirect
4. C₁₉×SD₁₂₈	C₂	semidirect
5. C₁₉×M₁₂₈	C₂	semidirect
35. #13743	C₁₉	direct (N × H)
47. C₂×Q₆₄	C₂×C₁₉	semidirect
48. Q₁₂₈	C₂×C₁₉	semidirect
49. SD₁₂₈	C₂×C₁₉	semidirect
50. M₁₂₈	C₂×C₁₉	semidirect
79. C₁₉	#13743	direct (N × H)

H ≅ G/N

type

2. C₂×C₁₉×Q₆₄

C₂

semidirect

3. C₁₉×Q₁₂₈

C₂

semidirect

4. C₁₉×SD₁₂₈

C₂

semidirect

5. C₁₉×M₁₂₈

C₂

semidirect

35. #13743

C₁₉

direct (N × H)

47. C₂×Q₆₄

C₂×C₁₉

semidirect

48. Q₁₂₈

C₂×C₁₉

semidirect

49. SD₁₂₈

C₂×C₁₉

semidirect

50. M₁₂₈

C₂×C₁₉

semidirect

79. C₁₉

#13743

direct (N × H)

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

subgroup	index
C₁₉×#2262	2
C₂×C₁₉×Q₆₄	2
C₁₉×Q₁₂₈	2
C₁₉×SD₁₂₈	2
C₁₉×M₁₂₈	2
#13743	19

subgroup

index

C₁₉×#2262

C₂×C₁₉×Q₆₄

C₁₉×Q₁₂₈

C₁₉×SD₁₂₈

C₁₉×M₁₂₈

#13743

Subgroups — C₁₉×#13743 #18019702

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#13743 #18019702

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13743 #18019702