Group #14269630 — subgroups

Subgroups — C₁₉×#878 #14269630

Order 1824.

148 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 70 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	C₂²×Dic₃×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₂²⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#217	C₂	1	1	yes	maximal, characteristic
5	C₁₉×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₉×C₂²⋊Q₈	—	3	1	no	maximal, Hall
9	C₂³×C₃×C₁₉	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄×C₁₉	C₂²	1	1	yes	characteristic
12	C₂×Dic₃×C₁₉	—	2	2	no	—
13	C₂×Dic₃×C₁₉	C₂²	1	1	yes	characteristic
14	C₂×Dic₃×C₁₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₃×C₁₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₃×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×Dic₆	—	2	2	no	—
18	C₂×Q₈×C₁₉	—	3	1	no	—
19	C₂²×C₄×C₁₉	—	3	1	no	—
20	C₄⋊C₄×C₁₉	—	3	1	no	—
21	C₄⋊C₄×C₁₉	—	3	1	no	—
22	C₄⋊C₄×C₁₉	—	3	1	no	—
23	C₂²⋊C₄×C₁₉	—	3	1	no	—
24	C₂²⋊C₄×C₁₉	D₃	1	1	yes	characteristic
25	C₂²×C₃×C₁₉	—	2	1	no	—
26	C₂²×C₃×C₁₉	—	2	1	no	—
27	C₂²×C₃×C₁₉	Q₈	1	2	yes	complemented
28	C₂²×C₃×C₁₉	C₂³	1	1	yes	characteristic
29	C₃×C₄×C₁₉	—	2	1	no	—
30	C₃×C₄×C₁₉	—	2	1	no	—
31	Dic₃×C₁₉	—	2	1	no	—
32	Dic₃×C₁₉	—	2	1	no	—
33	Dic₃×C₁₉	—	2	1	no	—
34	Dic₃×C₁₉	D₄	1	2	yes	—
35	C₂³×C₁₉	C₂×D₃	1	1	yes	characteristic
36	Q₈×C₁₉	—	6	2	no	—
37	C₂×C₄×C₁₉	—	3	1	no	—
38	C₂×C₄×C₁₉	—	3	1	no	—
39	C₂×C₄×C₁₉	—	3	1	no	—
40	C₂×C₄×C₁₉	—	3	1	no	—
41	C₂×C₄×C₁₉	—	6	2	no	—
42	C₂×C₄×C₁₉	C₂×D₃	1	1	yes	characteristic
43	C₂×C₄×C₁₉	C₂×D₃	1	1	yes	characteristic
44	C₂×C₃×C₁₉	—	2	2	no	—
45	C₂×C₃×C₁₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₃×C₁₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₃×C₁₉	#41	1	1	yes	characteristic
48	#878	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
49	C₂²×C₁₉	—	2	1	no	—
50	C₂²×C₁₉	—	2	1	no	—
51	C₂²×C₁₉	Dic₆	1	2	yes	complemented
52	C₂²×C₁₉	C₂²×D₃	1	1	yes	characteristic
53	C₄×C₁₉	—	2	1	no	—
54	C₄×C₁₉	—	2	1	no	—
55	C₄×C₁₉	—	3	2	no	—
56	C₄×C₁₉	—	6	1	no	—
57	C₄×C₁₉	—	6	1	no	—
58	C₄×C₁₉	—	6	1	no	—
59	C₃×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic, complemented, Hall
60	C₂²×Dic₃	C₂×C₁₉	1	1	yes	characteristic
61	C₂×Dic₆	C₂×C₁₉	1	1	yes	characteristic, complemented
62	C₃×C₂²⋊C₄	C₂×C₁₉	1	1	yes	characteristic
63	#217	C₂×C₁₉	1	1	yes	characteristic
64	C₄⋊Dic₃	C₂×C₁₉	1	1	yes	characteristic, complemented
65	Dic₃⋊C₄	C₂×C₁₉	1	1	yes	characteristic, complemented
66	Dic₃⋊C₄	C₂×C₁₉	1	1	yes	characteristic, complemented
67	C₂×C₁₉	—	2	2	no	—
68	C₂×C₁₉	C₂×Dic₆	1	1	yes	characteristic
69	C₂×C₁₉	D₃×D₄	1	1	yes	characteristic
70	C₂×C₁₉	#237	1	1	yes	characteristic
71	C₂²⋊Q₈	—	3	1	no	Sylow
72	C₂³×C₃	C₂²×C₁₉	1	1	yes	characteristic
73	C₂×C₃×C₄	C₂²×C₁₉	1	1	yes	characteristic
74	C₂×C₃×C₄	C₂²×C₁₉	1	1	yes	characteristic
75	C₂×Dic₃	—	2	2	no	—
76	C₂×Dic₃	C₂²×C₁₉	1	1	yes	characteristic
77	C₂×Dic₃	C₂²×C₁₉	1	1	yes	characteristic
78	C₂×Dic₃	C₂²×C₁₉	1	1	yes	characteristic
79	C₂×Dic₃	C₂²×C₁₉	1	1	yes	characteristic
80	Dic₆	—	2	2	no	—
81	C₁₉	#878	1	1	yes	characteristic, direct product factor, Sylow
82	C₂×Q₈	—	3	1	no	—
83	C₂²×C₄	—	3	1	no	—
84	C₄⋊C₄	—	3	1	no	—
85	C₄⋊C₄	—	3	1	no	—
86	C₄⋊C₄	—	3	1	no	—
87	C₂²⋊C₄	—	3	1	no	—
88	C₂²⋊C₄	D₃×C₁₉	1	1	yes	characteristic
89	C₂²×C₃	—	2	1	no	—
90	C₂²×C₃	—	2	1	no	—
91	C₂²×C₃	Q₈×C₁₉	1	2	yes	complemented
92	C₂²×C₃	C₂³×C₁₉	1	1	yes	characteristic
93	C₃×C₄	—	2	1	no	—
94	C₃×C₄	—	2	1	no	—
95	Dic₃	—	2	1	no	—
96	Dic₃	—	2	1	no	—
97	Dic₃	—	2	1	no	—
98	Dic₃	D₄×C₁₉	1	2	yes	—
99	C₂³	C₂×D₃×C₁₉	1	1	yes	characteristic
100	Q₈	—	6	2	no	—
101	C₂×C₄	—	3	1	no	—
102	C₂×C₄	—	3	1	no	—
103	C₂×C₄	—	3	1	no	—
104	C₂×C₄	—	3	1	no	—
105	C₂×C₄	—	6	2	no	—
106	C₂×C₄	C₂×D₃×C₁₉	1	1	yes	characteristic
107	C₂×C₄	C₂×D₃×C₁₉	1	1	yes	characteristic
108	C₂×C₃	—	2	2	no	—
109	C₂×C₃	C₂×D₄×C₁₉	1	1	yes	characteristic
110	C₂×C₃	C₂×Q₈×C₁₉	1	1	yes	characteristic
111	C₂×C₃	#41×C₁₉	1	1	yes	characteristic
112	C₂²	—	2	1	no	—
113	C₂²	—	2	1	no	—
114	C₂²	C₁₉×Dic₆	1	2	yes	complemented
115	C₂²	C₂²×D₃×C₁₉	1	1	yes	characteristic
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	3	2	no	—
119	C₄	—	6	1	no	—
120	C₄	—	6	1	no	—
121	C₄	—	6	1	no	—
122	C₃	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
123	C₂	—	2	2	no	—
124	C₂	C₂×C₁₉×Dic₆	1	1	yes	characteristic
125	C₂	D₃×D₄×C₁₉	1	1	yes	characteristic
126	C₂	C₁₉×#237	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₃×C₁₉

C₂

yes

maximal, characteristic

C₂×C₁₉×Dic₆

C₂

yes

maximal, characteristic, complemented

C₃×C₂²⋊C₄×C₁₉

C₂

yes

maximal, characteristic

C₁₉×#217

C₂

yes

maximal, characteristic

C₁₉×C₄⋊Dic₃

C₂

yes

maximal, characteristic, complemented

C₁₉×Dic₃⋊C₄

C₂

yes

maximal, characteristic, complemented

C₁₉×Dic₃⋊C₄

C₂

yes

maximal, characteristic, complemented

C₁₉×C₂²⋊Q₈

—

maximal, Hall

C₂³×C₃×C₁₉

C₂²

yes

characteristic

C₂×C₃×C₄×C₁₉

C₂²

yes

characteristic

C₂×C₃×C₄×C₁₉

C₂²

yes

characteristic

C₂×Dic₃×C₁₉

—

C₂×Dic₃×C₁₉

C₂²

yes

characteristic

C₂×Dic₃×C₁₉

C₂²

yes

characteristic

C₂×Dic₃×C₁₉

C₂²

yes

characteristic

C₂×Dic₃×C₁₉

C₂²

yes

characteristic

C₁₉×Dic₆

—

C₂×Q₈×C₁₉

—

C₂²×C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

D₃

yes

characteristic

C₂²×C₃×C₁₉

—

C₂²×C₃×C₁₉

—

C₂²×C₃×C₁₉

Q₈

yes

complemented

C₂²×C₃×C₁₉

C₂³

yes

characteristic

C₃×C₄×C₁₉

—

C₃×C₄×C₁₉

—

Dic₃×C₁₉

—

Dic₃×C₁₉

—

Dic₃×C₁₉

—

Dic₃×C₁₉

D₄

yes

—

C₂³×C₁₉

C₂×D₃

yes

characteristic

Q₈×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂×D₃

yes

characteristic

C₂×C₄×C₁₉

C₂×D₃

yes

characteristic

C₂×C₃×C₁₉

—

C₂×C₃×C₁₉

C₂×D₄

yes

characteristic

C₂×C₃×C₁₉

C₂×Q₈

yes

C₂×C₃×C₁₉

yes

C₁₉

yes

maximal, characteristic, direct product factor, Hall

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

Dic₆

yes

complemented

C₂²×C₁₉

C₂²×D₃

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₃×C₁₉

C₂²⋊Q₈

yes

characteristic, complemented, Hall

C₂²×Dic₃

C₂×C₁₉

yes

characteristic

C₂×Dic₆

C₂×C₁₉

yes

characteristic, complemented

C₃×C₂²⋊C₄

C₂×C₁₉

yes

characteristic

#217

C₂×C₁₉

yes

characteristic

C₄⋊Dic₃

C₂×C₁₉

yes

characteristic, complemented

Dic₃⋊C₄

C₂×C₁₉

yes

characteristic, complemented

Dic₃⋊C₄

C₂×C₁₉

yes

characteristic, complemented

C₂×C₁₉

—

C₂×C₁₉

C₂×Dic₆

yes

characteristic

C₂×C₁₉

D₃×D₄

yes

characteristic

C₂×C₁₉

#237

yes

characteristic

C₂²⋊Q₈

—

Sylow

C₂³×C₃

C₂²×C₁₉

yes

characteristic

C₂×C₃×C₄

C₂²×C₁₉

yes

characteristic

C₂×C₃×C₄

C₂²×C₁₉

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

C₂²×C₁₉

yes

characteristic

C₂×Dic₃

C₂²×C₁₉

yes

characteristic

C₂×Dic₃

C₂²×C₁₉

yes

characteristic

C₂×Dic₃

C₂²×C₁₉

yes

characteristic

Dic₆

—

C₁₉

#878

yes

characteristic, direct product factor, Sylow

C₂×Q₈

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

D₃×C₁₉

yes

characteristic

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

Q₈×C₁₉

yes

complemented

C₂²×C₃

C₂³×C₁₉

yes

characteristic

C₃×C₄

—

C₃×C₄

—

Dic₃

—

Dic₃

—

Dic₃

—

Dic₃

D₄×C₁₉

yes

—

C₂³

C₂×D₃×C₁₉

yes

characteristic

100

Q₈

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂×D₃×C₁₉

yes

characteristic

107

C₂×C₄

C₂×D₃×C₁₉

yes

characteristic

108

C₂×C₃

—

109

C₂×C₃

C₂×D₄×C₁₉

yes

characteristic

110

C₂×C₃

C₂×Q₈×C₁₉

yes

characteristic

111

C₂×C₃

#41×C₁₉

yes

characteristic

112

C₂²

—

113

C₂²

—

114

C₂²

C₁₉×Dic₆

yes

complemented

115

C₂²

C₂²×D₃×C₁₉

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₃

C₁₉×C₂²⋊Q₈

yes

characteristic, complemented, Sylow

123

C₂

—

124

C₂

C₂×C₁₉×Dic₆

yes

characteristic

125

C₂

D₃×D₄×C₁₉

yes

characteristic

126

C₂

C₁₉×#237

yes

characteristic

	subgroup	G/H
center	C₂²×C₁₉	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂³×C₁₉
Frattini	C₂²	C₂²×D₃×C₁₉
Fitting	C₃×C₂²⋊C₄×C₁₉	C₂
socle	C₂²×C₃×C₁₉	C₂³
prime radical	C₂³×C₃×C₁₉	C₂²
hypercenter	C₂²⋊C₄×C₁₉	D₃
hypocenter	C₃	C₁₉×C₂²⋊Q₈
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²⋊C₄×C₁₉

subgroup

G/H

center

C₂²×C₁₉

C₂²×D₃

commutator, cosocle

C₂²×C₃

C₂³×C₁₉

Frattini

C₂²

C₂²×D₃×C₁₉

Fitting

C₃×C₂²⋊C₄×C₁₉

C₂

socle

C₂²×C₃×C₁₉

C₂³

prime radical

C₂³×C₃×C₁₉

C₂²

hypercenter

C₂²⋊C₄×C₁₉

D₃

hypocenter

C₃

C₁₉×C₂²⋊Q₈

LCS

G ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂²×C₁₉ ◁ C₂²⋊C₄×C₁₉

N	H ≅ G/N	type
2. C₂×C₁₉×Dic₆	C₂	semidirect
5. C₁₉×C₄⋊Dic₃	C₂	semidirect
6. C₁₉×Dic₃⋊C₄	C₂	semidirect
7. C₁₉×Dic₃⋊C₄	C₂	semidirect
27. C₂²×C₃×C₁₉	Q₈	semidirect
48. #878	C₁₉	direct (N × H)
51. C₂²×C₁₉	Dic₆	semidirect
59. C₃×C₁₉	C₂²⋊Q₈	semidirect
61. C₂×Dic₆	C₂×C₁₉	semidirect
64. C₄⋊Dic₃	C₂×C₁₉	semidirect
65. Dic₃⋊C₄	C₂×C₁₉	semidirect
66. Dic₃⋊C₄	C₂×C₁₉	semidirect
81. C₁₉	#878	direct (N × H)
91. C₂²×C₃	Q₈×C₁₉	semidirect
114. C₂²	C₁₉×Dic₆	semidirect
122. C₃	C₁₉×C₂²⋊Q₈	semidirect

H ≅ G/N

type

2. C₂×C₁₉×Dic₆

C₂

semidirect

5. C₁₉×C₄⋊Dic₃

C₂

semidirect

6. C₁₉×Dic₃⋊C₄

C₂

semidirect

7. C₁₉×Dic₃⋊C₄

C₂

semidirect

27. C₂²×C₃×C₁₉

Q₈

semidirect

48. #878

C₁₉

direct (N × H)

51. C₂²×C₁₉

Dic₆

semidirect

59. C₃×C₁₉

C₂²⋊Q₈

semidirect

61. C₂×Dic₆

C₂×C₁₉

semidirect

64. C₄⋊Dic₃

C₂×C₁₉

semidirect

65. Dic₃⋊C₄

C₂×C₁₉

semidirect

66. Dic₃⋊C₄

C₂×C₁₉

semidirect

81. C₁₉

#878

direct (N × H)

91. C₂²×C₃

Q₈×C₁₉

semidirect

114. C₂²

C₁₉×Dic₆

semidirect

122. C₃

C₁₉×C₂²⋊Q₈

semidirect

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

subgroup	index
C₂²×Dic₃×C₁₉	2
C₂×C₁₉×Dic₆	2
C₃×C₂²⋊C₄×C₁₉	2
C₁₉×#217	2
C₁₉×C₄⋊Dic₃	2
C₁₉×Dic₃⋊C₄	2
C₁₉×C₂²⋊Q₈	3
#878	19

subgroup

index

C₂²×Dic₃×C₁₉

C₂×C₁₉×Dic₆

C₃×C₂²⋊C₄×C₁₉

C₁₉×#217

C₁₉×C₄⋊Dic₃

C₁₉×Dic₃⋊C₄

C₁₉×C₂²⋊Q₈

#878

Subgroups — C₁₉×#878 #14269630

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#878 #14269630

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#878 #14269630