Group #18018469 — subgroups

Subgroups — C₁₉×#12510 #18018469

Order 4864.

262 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 110 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×#2241	C₂	1	4	yes	maximal
2	C₁₉×#1713	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1713	C₂	1	1	yes	maximal, characteristic
4	C₈×C₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#500	—	2	1	no	—
6	C₁₉×#500	—	2	1	no	—
7	C₁₉×#500	—	2	1	no	—
8	C₁₉×#500	C₂²	1	2	yes	—
9	C₁₉×#494	—	2	1	no	—
10	C₁₉×#494	C₂²	1	2	yes	—
11	C₁₉×#358	—	2	8	no	—
12	C₄×C₁₆×C₁₉	C₂²	1	2	yes	—
13	C₈²×C₁₉	C₂²	1	1	yes	characteristic
14	C₂×Q₁₆×C₁₉	—	2	4	no	—
15	C₂×Q₁₆×C₁₉	—	4	2	no	—
16	C₁₉×C₄⋊Q₈	—	2	2	no	—
17	C₁₉×C₄⋊Q₈	—	2	2	no	—
18	C₂×C₁₆×C₁₉	D₄	1	4	yes	—
19	C₁₉×#107	—	2	4	no	—
20	C₁₉×#107	—	4	2	no	—
21	C₁₉×#107	—	4	2	no	—
22	C₁₉×#107	—	4	2	no	—
23	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
24	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
25	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
26	C₂×Q₈×C₁₉	—	4	2	no	—
27	C₂×Q₈×C₁₉	—	4	2	no	—
28	Q₁₆×C₁₉	—	4	8	no	—
29	Q₁₆×C₁₉	—	8	4	no	—
30	C₂×C₈×C₁₉	D₈	1	2	yes	—
31	C₂×C₈×C₁₉	D₈	1	2	yes	—
32	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
33	C₄⋊C₄×C₁₉	—	4	2	no	—
34	C₄⋊C₄×C₁₉	—	4	2	no	—
35	C₄⋊C₄×C₁₉	—	4	2	no	—
36	C₄⋊C₄×C₁₉	—	4	2	no	—
37	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
38	C₁₆×C₁₉	—	2	4	no	—
39	#12510	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	Q₈×C₁₉	—	8	4	no	—
41	Q₈×C₁₉	—	8	4	no	—
42	C₂×C₄×C₁₉	—	8	2	no	—
43	C₂×C₄×C₁₉	—	8	2	no	—
44	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
45	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
46	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
47	C₈×C₁₉	SD₃₂	1	4	yes	—
48	C₈×C₁₉	Q₃₂	1	4	yes	—
49	C₈×C₁₉	#135	1	4	yes	—
50	#2241	C₂×C₁₉	1	4	yes	—
51	#1713	C₂×C₁₉	1	1	yes	characteristic
52	#1713	C₂×C₁₉	1	1	yes	characteristic
53	C₈×C₁₆	C₂×C₁₉	1	1	yes	characteristic
54	C₂²×C₁₉	#493	1	1	yes	characteristic
55	C₄×C₁₉	—	16	2	no	—
56	C₄×C₁₉	—	16	2	no	—
57	C₄×C₁₉	#495	1	2	yes	—
58	C₄×C₁₉	C₂×SD₃₂	1	2	yes	—
59	C₄×C₁₉	C₂×Q₃₂	1	2	yes	—
60	#500	—	2	1	no	—
61	#500	—	2	1	no	—
62	#500	—	2	1	no	—
63	#500	C₂²×C₁₉	1	2	yes	—
64	#494	—	2	1	no	—
65	#494	C₂²×C₁₉	1	2	yes	—
66	#358	—	2	8	no	—
67	C₄×C₁₆	C₂²×C₁₉	1	2	yes	—
68	C₈²	C₂²×C₁₉	1	1	yes	characteristic
69	C₂×C₁₉	#1709	1	1	yes	characteristic
70	C₂×C₁₉	#2247	1	1	yes	characteristic
71	C₂×C₁₉	#2248	1	1	yes	characteristic
72	C₂×Q₁₆	—	2	4	no	—
73	C₂×Q₁₆	—	4	2	no	—
74	C₄⋊Q₈	—	2	2	no	—
75	C₄⋊Q₈	—	2	2	no	—
76	C₂×C₁₆	D₄×C₁₉	1	4	yes	—
77	#107	—	2	4	no	—
78	#107	—	4	2	no	—
79	#107	—	4	2	no	—
80	#107	—	4	2	no	—
81	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
82	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
83	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
84	C₁₉	#12510	1	1	yes	characteristic, direct product factor, Sylow
85	C₂×Q₈	—	4	2	no	—
86	C₂×Q₈	—	4	2	no	—
87	Q₁₆	—	4	8	no	—
88	Q₁₆	—	8	4	no	—
89	C₂×C₈	D₈×C₁₉	1	2	yes	—
90	C₂×C₈	D₈×C₁₉	1	2	yes	—
91	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	4	2	no	—
95	C₄⋊C₄	—	4	2	no	—
96	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
97	C₁₆	—	2	4	no	—
98	Q₈	—	8	4	no	—
99	Q₈	—	8	4	no	—
100	C₂×C₄	—	8	2	no	—
101	C₂×C₄	—	8	2	no	—
102	C₂×C₄	C₁₉×#127	1	1	yes	characteristic
103	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
104	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
105	C₈	C₁₉×SD₃₂	1	4	yes	—
106	C₈	C₁₉×Q₃₂	1	4	yes	—
107	C₈	C₁₉×#135	1	4	yes	—
108	C₂²	C₁₉×#493	1	1	yes	characteristic
109	C₄	—	16	2	no	—
110	C₄	—	16	2	no	—
111	C₄	C₁₉×#495	1	2	yes	—
112	C₄	C₂×C₁₉×SD₃₂	1	2	yes	—
113	C₄	C₂×C₁₉×Q₃₂	1	2	yes	—
114	C₂	C₁₉×#1709	1	1	yes	characteristic
115	C₂	C₁₉×#2247	1	1	yes	characteristic
116	C₂	C₁₉×#2248	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#2241

C₂

yes

maximal

C₁₉×#1713

C₂

yes

maximal, characteristic

C₁₉×#1713

C₂

yes

maximal, characteristic

C₈×C₁₆×C₁₉

C₂

yes

maximal, characteristic

C₁₉×#500

—

C₁₉×#500

—

C₁₉×#500

—

C₁₉×#500

C₂²

yes

—

C₁₉×#494

—

C₁₉×#494

C₂²

yes

—

C₁₉×#358

—

C₄×C₁₆×C₁₉

C₂²

yes

—

C₈²×C₁₉

C₂²

yes

characteristic

C₂×Q₁₆×C₁₉

—

C₂×Q₁₆×C₁₉

—

C₁₉×C₄⋊Q₈

—

C₁₉×C₄⋊Q₈

—

C₂×C₁₆×C₁₉

D₄

yes

—

C₁₉×#107

—

C₁₉×#107

—

C₁₉×#107

—

C₁₉×#107

—

C₄×C₈×C₁₉

D₄

yes

characteristic

C₄×C₈×C₁₉

D₄

yes

characteristic

C₄×C₈×C₁₉

C₂³

yes

characteristic

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

Q₁₆×C₁₉

—

Q₁₆×C₁₉

—

C₂×C₈×C₁₉

D₈

yes

—

C₂×C₈×C₁₉

D₈

yes

—

C₂×C₈×C₁₉

C₂×D₄

yes

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₁₆×C₁₉

—

#12510

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

Q₈×C₁₉

—

Q₈×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#127

yes

characteristic

C₂×C₄×C₁₉

C₂×D₈

yes

characteristic

C₂×C₄×C₁₉

C₂×D₈

yes

characteristic

C₈×C₁₉

SD₃₂

yes

—

C₈×C₁₉

Q₃₂

yes

—

C₈×C₁₉

#135

yes

—

#2241

C₂×C₁₉

yes

—

#1713

C₂×C₁₉

yes

characteristic

#1713

C₂×C₁₉

yes

characteristic

C₈×C₁₆

C₂×C₁₉

yes

characteristic

C₂²×C₁₉

#493

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#495

yes

—

C₄×C₁₉

C₂×SD₃₂

yes

—

C₄×C₁₉

C₂×Q₃₂

yes

—

—

—

—

C₂²×C₁₉

yes

—

#494

—

#494

C₂²×C₁₉

yes

—

#358

—

C₄×C₁₆

C₂²×C₁₉

yes

—

C₈²

C₂²×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

D₄×C₁₉

yes

—

—

—

—

—

C₄×C₈

D₄×C₁₉

yes

characteristic

C₄×C₈

D₄×C₁₉

yes

characteristic

C₄×C₈

C₂³×C₁₉

yes

characteristic

C₁₉

#12510

yes

characteristic, direct product factor, Sylow

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

D₈×C₁₉

yes

—

C₂×C₈

D₈×C₁₉

yes

—

C₂×C₈

C₂×D₄×C₁₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₄×C₁₉

yes

characteristic

C₁₆

—

Q₈

—

Q₈

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

C₁₉×#127

yes

characteristic

103

C₂×C₄

C₂×D₈×C₁₉

yes

characteristic

104

C₂×C₄

C₂×D₈×C₁₉

yes

characteristic

105

C₈

C₁₉×SD₃₂

yes

—

106

C₈

C₁₉×Q₃₂

yes

—

107

C₈

C₁₉×#135

yes

—

108

C₂²

C₁₉×#493

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

C₁₉×#495

yes

—

112

C₄

C₂×C₁₉×SD₃₂

yes

—

113

C₄

C₂×C₁₉×Q₃₂

yes

—

114

C₂

C₁₉×#1709

yes

characteristic

115

C₂

C₁₉×#2247

yes

characteristic

116

C₂

C₁₉×#2248

yes

characteristic

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

subgroup

G/H

center, socle, prime radical

C₂²×C₁₉

#493

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₁₉

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₉ ◁ C₄²×C₁₉ ◁ C₈²×C₁₉ ◁ G

N	H ≅ G/N	type
39. #12510	C₁₉	direct (N × H)
84. C₁₉	#12510	direct (N × H)

H ≅ G/N

type

39. #12510

C₁₉

direct (N × H)

84. C₁₉

#12510

direct (N × H)

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

subgroup	index
C₁₉×#2241	2
C₁₉×#1713	2
C₈×C₁₆×C₁₉	2
#12510	19

subgroup

index

C₁₉×#2241

C₁₉×#1713

C₈×C₁₆×C₁₉

#12510

Subgroups — C₁₉×#12510 #18018469

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#12510 #18018469

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12510 #18018469