Group #18019668 — subgroups

Subgroups — C₁₉×#13709 #18019668

Order 4864.

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

normal subgroups only

This group has 154 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	C₁₉×#2253	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₉×#2178	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1426	C₂	1	2	yes	maximal, complemented
4	C₁₉×#1422	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1418	C₂	1	2	yes	maximal
6	C₁₉×#508	C₂²	1	1	yes	characteristic
7	C₁₉×#499	—	2	1	no	—
8	C₁₉×#443	—	2	1	no	—
9	C₁₉×M₆₄	C₂²	1	2	yes	—
10	C₁₉×#368	C₂²	1	1	yes	characteristic
11	C₁₉×#367	C₂²	1	2	yes	—
12	C₁₉×#366	—	2	1	no	—
13	C₁₉×#359	—	2	1	no	—
14	C₄×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
15	C₁₉×#135	—	2	1	no	—
16	C₁₉×#131	—	4	1	no	—
17	C₁₉×#125	—	4	1	no	—
18	C₁₉×Q₃₂	Q₈	1	1	yes	characteristic
19	C₁₉×SD₃₂	—	2	1	no	—
20	C₁₉×D₁₆	Q₈	1	1	yes	characteristic
21	C₂×C₁₆×C₁₉	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₁₉	C₂³	1	1	yes	characteristic
23	C₁₉×#108	—	2	1	no	—
24	C₁₉×#107	—	2	2	no	—
25	C₁₉×#106	—	4	1	no	—
26	C₁₉×#104	—	4	1	no	—
27	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
28	C₁₉×C₃₂	—	2	2	no	—
29	#41×C₁₉	—	4	1	no	—
30	Q₁₆×C₁₉	—	2	1	no	—
31	SD₁₆×C₁₉	—	4	1	no	—
32	D₈×C₁₉	—	2	1	no	—
33	M₁₆×C₁₉	—	4	1	no	—
34	M₁₆×C₁₉	—	8	1	no	—
35	C₂×C₈×C₁₉	—	8	1	no	—
36	C₂×C₈×C₁₉	D₈	1	1	yes	characteristic
37	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
38	C₄⋊C₄×C₁₉	—	4	2	no	—
39	C₄⋊C₄×C₁₉	—	8	2	no	—
40	C₄²×C₁₉	D₈	1	1	yes	characteristic
41	C₁₆×C₁₉	—	2	1	no	—
42	C₁₆×C₁₉	C₂×Q₈	1	1	yes	characteristic
43	C₁₆×C₁₉	#41	1	1	yes	characteristic
44	#13709	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	Q₈×C₁₉	—	4	1	no	—
46	D₄×C₁₉	—	4	1	no	—
47	D₄×C₁₉	—	8	1	no	—
48	C₂×C₄×C₁₉	—	2	1	no	—
49	C₂×C₄×C₁₉	—	8	1	no	—
50	C₂×C₄×C₁₉	—	8	2	no	—
51	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
52	C₈×C₁₉	—	8	1	no	—
53	C₈×C₁₉	SD₃₂	1	1	yes	characteristic
54	C₈×C₁₉	SD₃₂	1	1	yes	characteristic
55	C₈×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
56	C₈×C₁₉	#137	1	1	yes	characteristic
57	#2253	C₂×C₁₉	1	1	yes	characteristic, complemented
58	#2178	C₂×C₁₉	1	1	yes	characteristic
59	#1426	C₂×C₁₉	1	2	yes	complemented
60	#1422	C₂×C₁₉	1	1	yes	characteristic, complemented
61	#1418	C₂×C₁₉	1	2	yes	—
62	C₂²×C₁₉	—	8	1	no	—
63	C₂²×C₁₉	#509	1	1	yes	characteristic
64	C₄×C₁₉	—	4	1	no	—
65	C₄×C₁₉	—	8	1	no	—
66	C₄×C₁₉	—	16	2	no	—
67	C₄×C₁₉	#474	1	1	yes	characteristic
68	C₄×C₁₉	C₂×SD₃₂	1	1	yes	characteristic
69	#508	C₂²×C₁₉	1	1	yes	characteristic
70	#499	—	2	1	no	—
71	#443	—	2	1	no	—
72	M₆₄	C₂²×C₁₉	1	2	yes	—
73	#368	C₂²×C₁₉	1	1	yes	characteristic
74	#367	C₂²×C₁₉	1	2	yes	—
75	#366	—	2	1	no	—
76	#359	—	2	1	no	—
77	C₄×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
78	C₂×C₁₉	—	2	1	no	—
79	C₂×C₁₉	—	16	1	no	—
80	C₂×C₁₉	#2226	1	1	yes	characteristic
81	#135	—	2	1	no	—
82	#131	—	4	1	no	—
83	#125	—	4	1	no	—
84	Q₃₂	Q₈×C₁₉	1	1	yes	characteristic
85	SD₃₂	—	2	1	no	—
86	D₁₆	Q₈×C₁₉	1	1	yes	characteristic
87	C₂×C₁₆	D₄×C₁₉	1	1	yes	characteristic
88	C₂×C₁₆	C₂³×C₁₉	1	1	yes	characteristic
89	#108	—	2	1	no	—
90	#107	—	2	2	no	—
91	#106	—	4	1	no	—
92	#104	—	4	1	no	—
93	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
94	C₃₂	—	2	2	no	—
95	C₁₉	#13709	1	1	yes	characteristic, direct product factor, Sylow
96	#41	—	4	1	no	—
97	Q₁₆	—	2	1	no	—
98	SD₁₆	—	4	1	no	—
99	D₈	—	2	1	no	—
100	M₁₆	—	4	1	no	—
101	M₁₆	—	8	1	no	—
102	C₂×C₈	—	8	1	no	—
103	C₂×C₈	D₈×C₁₉	1	1	yes	characteristic
104	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	8	2	no	—
107	C₄²	D₈×C₁₉	1	1	yes	characteristic
108	C₁₆	—	2	1	no	—
109	C₁₆	C₂×Q₈×C₁₉	1	1	yes	characteristic
110	C₁₆	#41×C₁₉	1	1	yes	characteristic
111	Q₈	—	4	1	no	—
112	D₄	—	4	1	no	—
113	D₄	—	8	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	8	1	no	—
116	C₂×C₄	—	8	2	no	—
117	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
118	C₈	—	8	1	no	—
119	C₈	C₁₉×SD₃₂	1	1	yes	characteristic
120	C₈	C₁₉×SD₃₂	1	1	yes	characteristic
121	C₈	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
122	C₈	C₁₉×#137	1	1	yes	characteristic
123	C₂²	—	8	1	no	—
124	C₂²	C₁₉×#509	1	1	yes	characteristic
125	C₄	—	4	1	no	—
126	C₄	—	8	1	no	—
127	C₄	—	16	2	no	—
128	C₄	C₁₉×#474	1	1	yes	characteristic
129	C₄	C₂×C₁₉×SD₃₂	1	1	yes	characteristic
130	C₂	—	2	1	no	—
131	C₂	—	16	1	no	—
132	C₂	C₁₉×#2226	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#2253

C₂

yes

maximal, characteristic, complemented

C₁₉×#2178

C₂

yes

maximal, characteristic

C₁₉×#1426

C₂

yes

maximal, complemented

C₁₉×#1422

C₂

yes

maximal, characteristic, complemented

C₁₉×#1418

C₂

yes

maximal

C₁₉×#508

C₂²

yes

characteristic

C₁₉×#499

—

C₁₉×#443

—

C₁₉×M₆₄

C₂²

yes

—

C₁₉×#368

C₂²

yes

characteristic

C₁₉×#367

C₂²

yes

—

C₁₉×#366

—

C₁₉×#359

—

C₄×C₁₆×C₁₉

C₂²

yes

characteristic

C₁₉×#135

—

C₁₉×#131

—

C₁₉×#125

—

C₁₉×Q₃₂

Q₈

yes

characteristic

C₁₉×SD₃₂

—

C₁₉×D₁₆

Q₈

yes

characteristic

C₂×C₁₆×C₁₉

D₄

yes

characteristic

C₂×C₁₆×C₁₉

C₂³

yes

characteristic

C₁₉×#108

—

C₁₉×#107

—

C₁₉×#106

—

C₁₉×#104

—

C₄×C₈×C₁₉

D₄

yes

characteristic

C₁₉×C₃₂

—

#41×C₁₉

—

Q₁₆×C₁₉

—

SD₁₆×C₁₉

—

D₈×C₁₉

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

D₈

yes

characteristic

C₂×C₈×C₁₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

D₈

yes

characteristic

C₁₆×C₁₉

—

C₁₆×C₁₉

C₂×Q₈

yes

C₁₆×C₁₉

yes

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

Q₈×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂×D₈

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

SD₃₂

yes

characteristic

C₈×C₁₉

SD₃₂

yes

characteristic

C₈×C₁₉

C₂²⋊Q₈

yes

C₈×C₁₉

yes

C₂×C₁₉

yes

characteristic, complemented

#2178

C₂×C₁₉

yes

characteristic

#1426

C₂×C₁₉

yes

complemented

#1422

C₂×C₁₉

yes

characteristic, complemented

#1418

C₂×C₁₉

yes

—

C₂²×C₁₉

—

C₂²×C₁₉

#509

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#474

yes

characteristic

C₄×C₁₉

C₂×SD₃₂

yes

characteristic

#508

C₂²×C₁₉

yes

characteristic

#499

—

#443

—

M₆₄

C₂²×C₁₉

yes

—

#368

C₂²×C₁₉

yes

characteristic

#367

C₂²×C₁₉

yes

—

#366

—

#359

—

C₄×C₁₆

C₂²×C₁₉

yes

characteristic

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

yes

—

—

—

Q₃₂

Q₈×C₁₉

yes

characteristic

SD₃₂

—

D₁₆

Q₈×C₁₉

yes

characteristic

C₂×C₁₆

D₄×C₁₉

yes

characteristic

C₂×C₁₆

C₂³×C₁₉

yes

—

—

—

—

C₄×C₈

D₄×C₁₉

yes

characteristic

C₃₂

—

C₁₉

#13709

yes

characteristic, direct product factor, Sylow

#41

—

Q₁₆

—

SD₁₆

—

D₈

—

100

M₁₆

—

101

M₁₆

—

102

C₂×C₈

—

103

C₂×C₈

D₈×C₁₉

yes

characteristic

104

C₂×C₈

C₂×D₄×C₁₉

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄²

D₈×C₁₉

yes

characteristic

108

C₁₆

—

109

C₁₆

C₂×Q₈×C₁₉

yes

characteristic

110

C₁₆

#41×C₁₉

yes

characteristic

111

Q₈

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂×D₈×C₁₉

yes

characteristic

118

C₈

—

119

C₈

C₁₉×SD₃₂

yes

characteristic

120

C₈

C₁₉×SD₃₂

yes

characteristic

121

C₈

C₁₉×C₂²⋊Q₈

yes

characteristic

122

C₈

C₁₉×#137

yes

characteristic

123

C₂²

—

124

C₂²

C₁₉×#509

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

C₁₉×#474

yes

characteristic

129

C₄

C₂×C₁₉×SD₃₂

yes

characteristic

130

C₂

—

131

C₂

—

132

C₂

C₁₉×#2226

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₁₉	#2226
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₁₉
prime radical	C₁₉×#508	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₄×C₈×C₁₉ ◁ C₄×C₁₆×C₁₉ ◁ G

subgroup

G/H

center, socle

C₂×C₁₉

#2226

commutator, Frattini, cosocle

C₂×C₁₆

C₂³×C₁₉

prime radical

C₁₉×#508

C₂²

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

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

N	H ≅ G/N	type
1. C₁₉×#2253	C₂	semidirect
3. C₁₉×#1426	C₂	semidirect
4. C₁₉×#1422	C₂	semidirect
44. #13709	C₁₉	direct (N × H)
57. #2253	C₂×C₁₉	semidirect
59. #1426	C₂×C₁₉	semidirect
60. #1422	C₂×C₁₉	semidirect
95. C₁₉	#13709	direct (N × H)

H ≅ G/N

type

1. C₁₉×#2253

C₂

semidirect

3. C₁₉×#1426

C₂

semidirect

4. C₁₉×#1422

C₂

semidirect

44. #13709

C₁₉

direct (N × H)

57. #2253

C₂×C₁₉

semidirect

59. #1426

C₂×C₁₉

semidirect

60. #1422

C₂×C₁₉

semidirect

95. C₁₉

#13709

direct (N × H)

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

subgroup	index
C₁₉×#2253	2
C₁₉×#2178	2
C₁₉×#1426	2
C₁₉×#1422	2
C₁₉×#1418	2
#13709	19

subgroup

index

C₁₉×#2253

C₁₉×#2178

C₁₉×#1426

C₁₉×#1422

C₁₉×#1418

#13709

Subgroups — C₁₉×#13709 #18019668

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#13709 #18019668

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13709 #18019668