Group #18069970 — subgroups

Subgroups — #18069970

Order 4864.

170 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 39 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	#14563854	C₂	1	1	yes	maximal, characteristic
2	#14561561	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1321	C₂	1	1	yes	maximal, characteristic
4	#11927311	C₄	1	1	yes	characteristic
5	#11926592	—	2	1	no	—
6	#11926528	C₄	1	1	yes	characteristic
7	C₂²⋊(C₁₉⋊C₁₆)	—	2	1	no	—
8	#11926309	—	2	1	no	—
9	C₁₉×#408	C₂²	1	1	yes	characteristic
10	C₁₉×#349	—	2	1	no	—
11	C₂×#41×C₁₉	C₂×C₄	1	1	yes	characteristic
12	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
14	#10597394	—	4	2	no	—
15	C₂×#64520	—	2	1	no	—
16	C₂×#64520	—	2	1	no	—
17	C₂×#64520	—	4	1	no	—
18	C₁₉×M₃₂	—	4	1	no	—
19	C₁₉×C₂²⋊C₈	—	4	1	no	—
20	#10597293	—	2	1	no	—
21	#10597290	C₈	1	2	yes	—
22	#10597279	—	4	2	no	—
23	C₂×C₁₉⋊C₁₆	—	4	1	no	—
24	#41×C₁₉	—	4	1	no	—
25	#41×C₁₉	—	4	1	no	—
26	C₂×Q₈×C₁₉	M₁₆	1	1	yes	characteristic
27	C₂×D₄×C₁₉	—	2	1	no	—
28	C₂×D₄×C₁₉	C₂×C₈	1	1	yes	characteristic
29	C₂²×C₄×C₁₉	—	2	1	no	—
30	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
31	M₁₆×C₁₉	—	4	1	no	—
32	C₂×C₈×C₁₉	—	2	1	no	—
33	C₂×C₈×C₁₉	—	2	1	no	—
34	C₂×C₈×C₁₉	—	4	1	no	—
35	#64520	—	2	2	no	—
36	#64520	—	2	2	no	—
37	#64520	—	4	1	no	—
38	#64520	—	4	2	no	—
39	#64520	—	4	1	no	—
40	C₂×C₁₉⋊C₈	—	4	1	no	—
41	C₂×C₁₉⋊C₈	—	4	1	no	—
42	C₂×C₁₉⋊C₈	—	4	2	no	—
43	C₁₆×C₁₉	—	8	1	no	—
44	C₁₉⋊C₁₆	—	8	1	no	—
45	#7377	—	19	1	no	maximal, Sylow
46	C₂³×C₁₉	—	2	1	no	—
47	C₂³×C₁₉	#101	1	1	yes	characteristic
48	Q₈×C₁₉	—	4	1	no	—
49	D₄×C₁₉	—	4	1	no	—
50	D₄×C₁₉	—	4	2	no	—
51	C₂×C₄×C₁₉	—	2	1	no	—
52	C₂×C₄×C₁₉	—	2	1	no	—
53	C₂×C₄×C₁₉	—	2	1	no	—
54	C₂×C₄×C₁₉	—	4	1	no	—
55	C₂×C₄×C₁₉	—	4	1	no	—
56	C₂×C₄×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
58	C₈×C₁₉	—	4	1	no	—
59	C₈×C₁₉	—	4	1	no	—
60	C₁₉⋊C₈	—	4	2	no	—
61	C₁₉⋊C₈	—	4	2	no	—
62	#1858	—	19	1	no	—
63	#1321	D₁₉	1	1	yes	characteristic
64	#1315	—	19	1	no	—
65	C₂²×C₁₉	—	2	1	no	—
66	C₂²×C₁₉	—	4	1	no	—
67	C₂²×C₁₉	—	4	1	no	—
68	C₂²×C₁₉	—	8	1	no	—
69	C₂²×C₁₉	#324	1	1	yes	characteristic
70	C₄×C₁₉	—	2	1	no	—
71	C₄×C₁₉	—	4	1	no	—
72	C₄×C₁₉	AΣL(1,𝔽₁₆)	1	1	yes	characteristic
73	C₄×C₁₉	#352	1	1	yes	characteristic
74	#568	—	19	1	no	—
75	#430	—	38	1	no	—
76	#413	—	19	1	no	—
77	#408	C₂×D₁₉	1	1	yes	characteristic
78	#349	—	2	1	no	—
79	C₂²⋊C₁₆	—	38	1	no	—
80	#341	—	38	1	no	—
81	C₂×C₁₉	—	2	1	no	—
82	C₂×C₁₉	—	4	1	no	—
83	C₂×C₁₉	—	8	1	no	—
84	C₂×C₁₉	#1326	1	1	yes	characteristic
85	C₂×#41	C₄×D₁₉	1	1	yes	characteristic
86	#131	—	76	2	no	—
87	C₂×M₁₆	—	38	1	no	—
88	C₂×M₁₆	—	38	1	no	—
89	C₂×M₁₆	—	76	1	no	—
90	C₂×M₁₆	C₂²⋊D₁₉	1	1	yes	characteristic
91	C₂²×C₈	D₇₆	1	1	yes	characteristic
92	M₃₂	—	4	1	no	—
93	C₂×C₁₆	—	76	1	no	—
94	#108	—	76	2	no	—
95	#101	—	38	1	no	—
96	#100	—	19	2	no	—
97	C₂²⋊C₈	—	4	1	no	—
98	C₁₉	#7377	1	1	yes	characteristic, complemented, Sylow
99	#41	—	4	1	no	—
100	#41	—	4	1	no	—
101	C₂×Q₈	#64515	1	1	yes	characteristic
102	C₂×D₄	—	2	1	no	—
103	C₂×D₄	C₈×D₁₉	1	1	yes	characteristic
104	C₂²×C₄	—	2	1	no	—
105	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
106	M₁₆	—	4	1	no	—
107	M₁₆	—	38	2	no	—
108	M₁₆	—	38	2	no	—
109	M₁₆	—	76	1	no	—
110	M₁₆	—	76	2	no	—
111	M₁₆	—	76	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	4	1	no	—
115	C₂×C₈	—	76	1	no	—
116	C₂×C₈	—	76	1	no	—
117	C₂×C₈	—	76	2	no	—
118	C₁₆	—	8	1	no	—
119	C₁₆	—	152	1	no	—
120	C₂³	—	2	1	no	—
121	C₂³	#10597281	1	1	yes	characteristic
122	Q₈	—	4	1	no	—
123	D₄	—	4	1	no	—
124	D₄	—	4	2	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	#10597263	1	1	yes	characteristic
131	C₂×C₄	(C₂×D₁₉)⋊C₈	1	1	yes	characteristic
132	C₈	—	4	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	76	2	no	—
135	C₈	—	76	2	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	4	1	no	—
138	C₂²	—	4	1	no	—
139	C₂²	—	8	1	no	—
140	C₂²	#11926269	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	4	1	no	—
143	C₄	#11926330	1	1	yes	characteristic
144	C₄	#11926335	1	1	yes	characteristic
145	C₂	—	2	1	no	—
146	C₂	—	4	1	no	—
147	C₂	—	8	1	no	—
148	C₂	#14561582	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14563854

C₂

yes

maximal, characteristic

#14561561

C₂

yes

maximal, characteristic

C₁₉×#1321

C₂

yes

maximal, characteristic

#11927311

C₄

yes

characteristic

#11926592

—

#11926528

C₄

yes

characteristic

C₂²⋊(C₁₉⋊C₁₆)

—

#11926309

—

C₁₉×#408

C₂²

yes

characteristic

C₁₉×#349

—

C₂×#41×C₁₉

C₂×C₄

yes

characteristic

C₂×M₁₆×C₁₉

D₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

—

—

—

—

C₁₉×M₃₂

—

C₁₉×C₂²⋊C₈

—

#10597293

—

#10597290

C₈

yes

—

#10597279

—

C₂×C₁₉⋊C₁₆

—

#41×C₁₉

—

#41×C₁₉

—

C₂×Q₈×C₁₉

M₁₆

yes

characteristic

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

—

—

—

—

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₁₆×C₁₉

—

C₁₉⋊C₁₆

—

#7377

—

maximal, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

#101

yes

characteristic

Q₈×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₉

#99

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#1858

—

#1321

D₁₉

yes

characteristic

#1315

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#324

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

AΣL(1,𝔽₁₆)

yes

C₄×C₁₉

yes

—

—

—

C₂×D₁₉

yes

characteristic

#349

—

C₂²⋊C₁₆

—

#341

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

#1326

yes

characteristic

C₂×#41

C₄×D₁₉

yes

characteristic

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂²⋊D₁₉

yes

characteristic

C₂²×C₈

D₇₆

yes

characteristic

M₃₂

—

C₂×C₁₆

—

#108

—

#101

—

#100

—

C₂²⋊C₈

—

C₁₉

#7377

yes

characteristic, complemented, Sylow

#41

—

100

#41

—

101

C₂×Q₈

#64515

yes

characteristic

102

C₂×D₄

—

103

C₂×D₄

C₈×D₁₉

yes

characteristic

104

C₂²×C₄

—

105

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

111

M₁₆

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₁₆

—

119

C₁₆

—

152

—

120

C₂³

—

121

C₂³

#10597281

yes

characteristic

122

Q₈

—

123

D₄

—

124

D₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

#10597263

yes

characteristic

131

C₂×C₄

(C₂×D₁₉)⋊C₈

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

#11926269

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

#11926330

yes

characteristic

144

C₄

#11926335

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

#14561582

yes

characteristic

	subgroup	G/H
center	C₄	#11926330
commutator	C₂×D₄×C₁₉	C₂×C₈
Frattini	#408	C₂×D₁₉
Fitting	C₁₉×#1321	C₂
socle	C₂×C₁₉	#1326
prime radical	C₂×#41×C₁₉	C₂×C₄
cosocle	C₁₉×#408	C₂²
hypercenter	#1321	D₁₉
hypocenter	C₁₉	#7377
LCS	G ▷ C₂×D₄×C₁₉ ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #408 ◁ #1321

subgroup

G/H

center

C₄

#11926330

commutator

C₂×D₄×C₁₉

C₂×C₈

Frattini

#408

C₂×D₁₉

Fitting

C₁₉×#1321

C₂

socle

C₂×C₁₉

#1326

prime radical

C₂×#41×C₁₉

C₂×C₄

cosocle

C₁₉×#408

C₂²

D₁₉

C₁₉

G ▷ C₂×D₄×C₁₉ ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #408 ◁ #1321

N	H ≅ G/N	type
98. C₁₉	#7377	semidirect

H ≅ G/N

type

98. C₁₉

#7377

semidirect

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

subgroup	index
#14563854	2
#14561561	2
C₁₉×#1321	2
#7377	19

subgroup

index

#14563854

#14561561

C₁₉×#1321

#7377

Subgroups — #18069970

Conjugacy classes of subgroups

Complements

Maximal subgroups