Group #18069388 — subgroups

Subgroups — #18069388

Order 4864.

222 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 119 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	#14565434	C₂	1	2	yes	maximal
2	C₂×C₁₉×#322	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₁₉⋊C₁₆)	—	2	2	no	—
4	C₂²⋊(C₁₉⋊C₁₆)	—	2	2	no	—
5	C₄×C₁₉⋊C₁₆	C₄	1	4	yes	—
6	C₂×C₄×C₈×C₁₉	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₉	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
9	C₁₉×#322	—	2	2	no	—
10	C₂²×C₈×C₁₉	—	2	1	no	—
11	C₂²×C₈×C₁₉	C₈	1	2	yes	—
12	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₉	Q₈	1	1	yes	characteristic
14	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
15	C₄×C₈×C₁₉	—	2	1	no	—
16	C₄×C₈×C₁₉	—	2	1	no	—
17	C₄×C₈×C₁₉	—	2	1	no	—
18	C₄×C₈×C₁₉	C₈	1	2	yes	—
19	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
20	C₄×C₈×C₁₉	D₄	1	2	yes	—
21	C₂×C₁₉⋊C₁₆	—	2	4	no	—
22	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₉	C₂×C₈	1	1	yes	characteristic
24	C₂²×C₄×C₁₉	M₁₆	1	1	yes	characteristic
25	C₂×C₈×C₁₉	—	2	2	no	—
26	C₂×C₈×C₁₉	—	2	2	no	—
27	C₂×C₈×C₁₉	—	2	2	no	—
28	C₂×C₈×C₁₉	—	4	1	no	—
29	C₂×C₈×C₁₉	—	4	1	no	—
30	C₂×C₈×C₁₉	—	4	2	no	—
31	C₂×C₈×C₁₉	C₁₆	1	4	yes	complemented
32	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
33	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
34	C₄²×C₁₉	C₄²	1	1	yes	characteristic
35	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
36	C₄²×C₁₉	C₂×C₈	1	1	yes	characteristic
37	C₄²×C₁₉	M₁₆	1	1	yes	characteristic
38	C₁₉⋊C₁₆	—	4	4	no	—
39	#7145	—	19	1	no	maximal, Sylow
40	C₂³×C₁₉	C₄⋊C₈	1	1	yes	characteristic
41	C₂×C₄×C₁₉	—	2	1	no	—
42	C₂×C₄×C₁₉	—	2	1	no	—
43	C₂×C₄×C₁₉	—	2	2	no	—
44	C₂×C₄×C₁₉	#95	1	1	yes	characteristic
45	C₂×C₄×C₁₉	C₄×C₈	1	1	yes	characteristic
46	C₂×C₄×C₁₉	#97	1	1	yes	characteristic
47	C₂×C₄×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
48	C₂×C₄×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
49	C₂×C₄×C₁₉	C₄⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₁₉	C₂×C₁₆	1	2	yes	—
51	C₂×C₄×C₁₉	M₃₂	1	2	yes	—
52	C₈×C₁₉	—	2	2	no	—
53	C₈×C₁₉	—	2	2	no	—
54	C₈×C₁₉	—	2	4	no	—
55	C₈×C₁₉	—	4	2	no	—
56	#2162	—	19	2	no	—
57	C₂×#322	D₁₉	1	1	yes	characteristic
58	C₂²×C₁₉	—	2	1	no	—
59	C₂²×C₁₉	—	2	1	no	—
60	C₂²×C₁₉	#336	1	1	yes	characteristic
61	C₂²×C₁₉	C₄⋊C₁₆	1	2	yes	—
62	C₄×C₁₉	—	2	1	no	—
63	C₄×C₁₉	—	2	1	no	—
64	C₄×C₁₉	H(3,ℤ/4)	1	1	yes	characteristic
65	C₄×C₁₉	#338	1	1	yes	characteristic
66	C₄×C₁₉	C₄×C₁₆	1	1	yes	characteristic
67	C₄×C₁₉	#346	1	1	yes	characteristic
68	C₄×C₁₉	#347	1	2	yes	—
69	C₄×C₁₉	C₂²⋊C₁₆	1	2	yes	—
70	C₂×C₄×C₈	Dic₁₉	1	1	yes	characteristic
71	C₂×C₄×C₈	Dic₁₉	1	1	yes	characteristic
72	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
73	C₄⋊C₁₆	—	38	2	no	—
74	C₂²⋊C₁₆	—	38	2	no	—
75	C₄×C₁₆	—	19	4	no	—
76	#322	—	2	2	no	—
77	C₂×C₁₉	—	2	2	no	—
78	C₂×C₁₉	#1275	1	1	yes	characteristic
79	C₂×C₁₉	#1374	1	1	yes	characteristic
80	C₂×C₁₉	#1375	1	1	yes	characteristic
81	C₂²×C₈	—	2	1	no	—
82	C₂²×C₈	C₁₉⋊C₈	1	2	yes	—
83	C₂²×C₈	Dic₃₈	1	1	yes	characteristic
84	C₂²×C₈	D₇₆	1	1	yes	characteristic
85	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
86	C₂×C₁₆	—	38	4	no	—
87	C₄×C₈	—	2	1	no	—
88	C₄×C₈	—	2	1	no	—
89	C₄×C₈	—	2	1	no	—
90	C₄×C₈	C₁₉⋊C₈	1	2	yes	—
91	C₄×C₈	C₄×D₁₉	1	2	yes	—
92	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
93	C₁₉	#7145	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₄	C₂×C₁₉⋊C₈	1	1	yes	characteristic
95	C₂²×C₄	#64520	1	1	yes	characteristic
96	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	4	1	no	—
101	C₂×C₈	—	4	1	no	—
102	C₂×C₈	—	4	2	no	—
103	C₂×C₈	C₁₉⋊C₁₆	1	4	yes	complemented
104	C₂×C₈	Dic₁₉⋊C₄	1	2	yes	—
105	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
106	C₄²	C₂×C₁₉⋊C₈	1	1	yes	characteristic
107	C₄²	#64520	1	1	yes	characteristic
108	C₄²	C₄×Dic₁₉	1	1	yes	characteristic
109	C₄²	C₂²⋊Dic₁₉	1	1	yes	characteristic
110	C₁₆	—	76	4	no	—
111	C₂³	C₄⋊(C₁₉⋊C₈)	1	1	yes	characteristic
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	C₄×C₁₉⋊C₈	1	1	yes	characteristic
116	C₂×C₄	#10597260	1	1	yes	characteristic
117	C₂×C₄	C₄⋊(C₁₉⋊C₈)	1	1	yes	characteristic
118	C₂×C₄	C₂×C₁₉⋊C₁₆	1	2	yes	—
119	C₂×C₄	#10597269	1	2	yes	—
120	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
121	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
122	C₂×C₄	C₁₉⋊#95	1	1	yes	characteristic
123	C₈	—	2	2	no	—
124	C₈	—	2	2	no	—
125	C₈	—	2	4	no	—
126	C₈	—	4	2	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	#11926299	1	1	yes	characteristic
130	C₂²	C₄⋊(C₁₉⋊C₁₆)	1	2	yes	—
131	C₄	—	2	1	no	—
132	C₄	—	2	1	no	—
133	C₄	#11926301	1	1	yes	characteristic
134	C₄	C₁₉⋊#338	1	1	yes	characteristic
135	C₄	C₄×C₁₉⋊C₁₆	1	1	yes	characteristic
136	C₄	#11926320	1	1	yes	characteristic
137	C₄	#11926322	1	2	yes	—
138	C₄	C₂²⋊(C₁₉⋊C₁₆)	1	2	yes	—
139	C₂	—	2	2	no	—
140	C₂	#14561466	1	1	yes	characteristic
141	C₂	#14561713	1	1	yes	characteristic
142	C₂	#14561715	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14565434

C₂

yes

maximal

C₂×C₁₉×#322

C₂

yes

maximal, characteristic

C₄⋊(C₁₉⋊C₁₆)

—

C₂²⋊(C₁₉⋊C₁₆)

—

C₄×C₁₉⋊C₁₆

C₄

yes

—

C₂×C₄×C₈×C₁₉

C₄

yes

characteristic

C₂×C₄×C₈×C₁₉

C₄

yes

characteristic

C₂×C₄×C₈×C₁₉

C₂²

yes

characteristic

C₁₉×#322

—

C₂²×C₈×C₁₉

—

C₂²×C₈×C₁₉

C₈

yes

—

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂²×C₈×C₁₉

Q₈

yes

characteristic

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₄×C₈×C₁₉

—

C₄×C₈×C₁₉

—

C₄×C₈×C₁₉

—

C₄×C₈×C₁₉

C₈

yes

—

C₄×C₈×C₁₉

C₂×C₄

yes

—

C₄×C₈×C₁₉

D₄

yes

—

C₂×C₁₉⋊C₁₆

—

C₂²×C₄×C₁₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₉

M₁₆

yes

characteristic

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₁₆

yes

complemented

C₂×C₈×C₁₉

C₂²⋊C₄

yes

—

C₂×C₈×C₁₉

C₄⋊C₄

yes

—

C₄²×C₁₉

C₄²

yes

characteristic

C₄²×C₁₉

C₂²⋊C₄

yes

characteristic

C₄²×C₁₉

C₂×C₈

yes

characteristic

C₄²×C₁₉

M₁₆

yes

characteristic

C₁₉⋊C₁₆

—

#7145

—

maximal, Sylow

C₂³×C₁₉

C₄⋊C₈

yes

characteristic

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#95

yes

characteristic

C₂×C₄×C₁₉

C₄×C₈

yes

characteristic

C₂×C₄×C₁₉

#97

yes

characteristic

C₂×C₄×C₁₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₉

C₄⋊C₈

yes

characteristic

C₂×C₄×C₁₉

C₂×C₁₆

yes

—

C₂×C₄×C₁₉

M₃₂

yes

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

#2162

—

C₂×#322

D₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#336

yes

characteristic

C₂²×C₁₉

C₄⋊C₁₆

yes

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

C₄×C₁₆

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

—

C₄×C₁₉

C₂²⋊C₁₆

yes

—

C₂×C₄×C₈

Dic₁₉

yes

characteristic

C₂×C₄×C₈

Dic₁₉

yes

characteristic

C₂×C₄×C₈

C₂×D₁₉

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

#322

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂²×C₈

—

C₂²×C₈

C₁₉⋊C₈

yes

—

C₂²×C₈

Dic₃₈

yes

characteristic

C₂²×C₈

D₇₆

yes

characteristic

C₂×C₄²

C₂×Dic₁₉

yes

characteristic

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₁₉⋊C₈

yes

—

C₄×C₈

C₄×D₁₉

yes

—

C₄×C₈

C₂²⋊D₁₉

yes

—

C₁₉

#7145

yes

characteristic, complemented, Sylow

C₂²×C₄

C₂×C₁₉⋊C₈

yes

characteristic

C₂²×C₄

#64520

yes

characteristic

C₂²×C₄

C₄⋊Dic₁₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

C₁₉⋊C₁₆

yes

complemented

104

C₂×C₈

Dic₁₉⋊C₄

yes

—

105

C₂×C₈

(C₂×D₁₉)⋊C₄

yes

—

106

C₄²

C₂×C₁₉⋊C₈

yes

characteristic

107

C₄²

#64520

yes

characteristic

108

C₄²

C₄×Dic₁₉

yes

characteristic

109

C₄²

C₂²⋊Dic₁₉

yes

characteristic

110

C₁₆

—

111

C₂³

C₄⋊(C₁₉⋊C₈)

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

C₄×C₁₉⋊C₈

yes

characteristic

116

C₂×C₄

#10597260

yes

characteristic

117

C₂×C₄

C₄⋊(C₁₉⋊C₈)

yes

characteristic

118

C₂×C₄

C₂×C₁₉⋊C₁₆

yes

—

119

C₂×C₄

#10597269

yes

—

120

C₂×C₄

C₂²⋊(C₁₉⋊C₈)

yes

characteristic

121

C₂×C₄

C₂²⋊(C₁₉⋊C₈)

yes

characteristic

122

C₂×C₄

C₁₉⋊#95

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₂²

—

128

C₂²

—

129

C₂²

#11926299

yes

characteristic

130

C₂²

C₄⋊(C₁₉⋊C₁₆)

yes

—

131

C₄

—

132

C₄

—

133

C₄

#11926301

yes

characteristic

134

C₄

C₁₉⋊#338

yes

characteristic

135

C₄

C₄×C₁₉⋊C₁₆

yes

characteristic

136

C₄

#11926320

yes

characteristic

137

C₄

#11926322

yes

—

138

C₄

C₂²⋊(C₁₉⋊C₁₆)

yes

—

139

C₂

—

140

C₂

#14561466

yes

characteristic

141

C₂

#14561713

yes

characteristic

142

C₂

#14561715

yes

characteristic

	subgroup	G/H
center	C₄²	C₄×Dic₁₉
commutator	C₄×C₁₉	C₄×C₁₆
Frattini	C₂×C₄×C₈	C₂×D₁₉
Fitting	C₂×C₁₉×#322	C₂
socle	C₂²×C₁₉	#336
prime radical	C₂³×C₁₉	C₄⋊C₈
cosocle	C₂×C₄×C₈×C₁₉	C₂²
hypercenter	C₂×#322	D₁₉
hypocenter	C₁₉	#7145
LCS	G ▷ C₄×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₄² ◁ C₂×#322

subgroup

G/H

center

C₄²

C₄×Dic₁₉

commutator

C₄×C₁₉

C₄×C₁₆

Frattini

C₂×C₄×C₈

C₂×D₁₉

Fitting

C₂×C₁₉×#322

C₂

socle

C₂²×C₁₉

#336

prime radical

C₂³×C₁₉

C₄⋊C₈

cosocle

C₂×C₄×C₈×C₁₉

C₂²

D₁₉

C₁₉

G ▷ C₄×C₁₉ ▷ C₁₉

C₁ ◁ C₄² ◁ C₂×#322

N	H ≅ G/N	type
31. C₂×C₈×C₁₉	C₁₆	semidirect
93. C₁₉	#7145	semidirect
103. C₂×C₈	C₁₉⋊C₁₆	semidirect

H ≅ G/N

type

31. C₂×C₈×C₁₉

C₁₆

semidirect

93. C₁₉

#7145

semidirect

103. C₂×C₈

C₁₉⋊C₁₆

semidirect

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

subgroup	index
#14565434	2
C₂×C₁₉×#322	2
#7145	19

subgroup

index

#14565434

C₂×C₁₉×#322

#7145

Subgroups — #18069388

Conjugacy classes of subgroups

Complements

Maximal subgroups