Group #18069164 — subgroups

Subgroups — #18069164

Order 4864.

196 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 59 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	#14561575	C₂	1	2	yes	maximal
2	C₁₉×#1984	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊(C₁₉⋊C₁₆)	—	2	2	no	—
4	C₂×C₄×Q₈×C₁₉	C₄	1	1	yes	characteristic
5	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₈×C₁₉	C₄	1	1	yes	characteristic
7	C₁₉×#336	—	2	2	no	—
8	C₂²×Q₈×C₁₉	C₈	1	1	yes	characteristic
9	C₂²×C₈×C₁₉	—	2	1	no	—
10	C₂²×C₈×C₁₉	D₄	1	2	yes	—
11	C₄×Q₈×C₁₉	—	4	1	no	—
12	C₄×Q₈×C₁₉	—	4	1	no	—
13	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
14	C₂×C₄⋊C₄×C₁₉	C₈	1	1	yes	characteristic
15	C₂×C₄²×C₁₉	—	2	1	no	—
16	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
17	C₁₉×C₄⋊C₈	—	4	1	no	—
18	C₄×C₈×C₁₉	—	4	1	no	—
19	C₂×C₁₉⋊C₁₆	—	4	2	no	—
20	C₂×Q₈×C₁₉	—	2	1	no	—
21	C₂×Q₈×C₁₉	—	4	1	no	—
22	C₂×Q₈×C₁₉	—	4	1	no	—
23	C₂×Q₈×C₁₉	C₁₆	1	2	yes	complemented
24	C₂²×C₄×C₁₉	—	2	1	no	—
25	C₂²×C₄×C₁₉	—	2	1	no	—
26	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₉	C₂×C₈	1	1	yes	characteristic
28	C₂²×C₄×C₁₉	M₁₆	1	1	yes	characteristic
29	C₂×C₈×C₁₉	—	2	2	no	—
30	C₂×C₈×C₁₉	—	2	2	no	—
31	C₂×C₈×C₁₉	—	4	1	no	—
32	C₂×C₈×C₁₉	—	4	1	no	—
33	C₂×C₈×C₁₉	—	4	2	no	—
34	C₄⋊C₄×C₁₉	—	4	1	no	—
35	C₄⋊C₄×C₁₉	—	4	2	no	—
36	C₄²×C₁₉	—	2	1	no	—
37	C₄²×C₁₉	—	2	1	no	—
38	C₄²×C₁₉	—	4	1	no	—
39	C₄²×C₁₉	—	4	1	no	—
40	C₁₉⋊C₁₆	—	8	2	no	—
41	#7057	—	19	1	no	maximal, Sylow
42	C₂³×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
43	Q₈×C₁₉	—	4	2	no	—
44	Q₈×C₁₉	—	4	2	no	—
45	C₂×C₄×C₁₉	—	2	1	no	—
46	C₂×C₄×C₁₉	—	2	1	no	—
47	C₂×C₄×C₁₉	—	2	1	no	—
48	C₂×C₄×C₁₉	—	2	1	no	—
49	C₂×C₄×C₁₉	—	2	1	no	—
50	C₂×C₄×C₁₉	—	2	2	no	—
51	C₂×C₄×C₁₉	—	4	1	no	—
52	C₂×C₄×C₁₉	—	4	1	no	—
53	C₂×C₄×C₁₉	—	4	1	no	—
54	C₂×C₄×C₁₉	—	4	2	no	—
55	C₂×C₄×C₁₉	—	4	1	no	—
56	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
57	C₂×C₄×C₁₉	#100	1	1	yes	characteristic
58	C₂×C₄×C₁₉	C₂×C₁₆	1	1	yes	characteristic
59	C₂×C₄×C₁₉	M₃₂	1	1	yes	characteristic
60	C₈×C₁₉	—	4	2	no	—
61	C₈×C₁₉	—	4	2	no	—
62	#1984	D₁₉	1	1	yes	characteristic
63	#1322	—	19	2	no	—
64	C₂²×C₁₉	—	2	1	no	—
65	C₂²×C₁₉	—	2	1	no	—
66	C₂²×C₁₉	#323	1	1	yes	characteristic
67	C₂²×C₁₉	C₂²⋊C₁₆	1	1	yes	characteristic
68	C₂²×C₁₉	#349	1	1	yes	characteristic
69	C₄×C₁₉	—	2	1	no	—
70	C₄×C₁₉	—	2	1	no	—
71	C₄×C₁₉	—	2	1	no	—
72	C₄×C₁₉	—	4	2	no	—
73	C₄×C₁₉	—	4	1	no	—
74	C₄×C₁₉	—	8	1	no	—
75	C₄×C₁₉	#354	1	1	yes	characteristic
76	C₄×C₁₉	#356	1	1	yes	characteristic
77	C₂×C₄×Q₈	Dic₁₉	1	1	yes	characteristic
78	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
79	C₂×C₄×C₈	Dic₁₉	1	1	yes	characteristic
80	C₂²⋊C₁₆	—	38	2	no	—
81	#336	—	2	2	no	—
82	C₂×C₁₉	—	2	1	no	—
83	C₂×C₁₉	—	2	1	no	—
84	C₂×C₁₉	#1314	1	1	yes	characteristic
85	C₂×C₁₉	#1319	1	1	yes	characteristic
86	C₂×C₁₉	#1321	1	1	yes	characteristic
87	C₂²×Q₈	C₁₉⋊C₈	1	1	yes	characteristic
88	C₂²×C₈	—	2	1	no	—
89	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
90	C₄×Q₈	—	4	1	no	—
91	C₄×Q₈	—	4	1	no	—
92	C₂×C₄⋊C₄	—	2	1	no	—
93	C₂×C₄⋊C₄	C₁₉⋊C₈	1	1	yes	characteristic
94	C₂×C₄²	—	2	1	no	—
95	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
96	C₂×C₁₆	—	76	2	no	—
97	C₄⋊C₈	—	4	1	no	—
98	C₄×C₈	—	4	1	no	—
99	C₁₉	#7057	1	1	yes	characteristic, complemented, Sylow
100	C₂×Q₈	—	2	1	no	—
101	C₂×Q₈	—	4	1	no	—
102	C₂×Q₈	—	4	1	no	—
103	C₂×Q₈	C₁₉⋊C₁₆	1	2	yes	complemented
104	C₂²×C₄	—	2	1	no	—
105	C₂²×C₄	—	2	1	no	—
106	C₂²×C₄	C₂×C₁₉⋊C₈	1	1	yes	characteristic
107	C₂²×C₄	#64520	1	1	yes	characteristic
108	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	4	1	no	—
112	C₂×C₈	—	4	1	no	—
113	C₂×C₈	—	4	2	no	—
114	C₄⋊C₄	—	4	1	no	—
115	C₄⋊C₄	—	4	2	no	—
116	C₄²	—	2	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	—	4	1	no	—
119	C₄²	—	4	1	no	—
120	C₁₆	—	152	2	no	—
121	C₂³	C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
122	Q₈	—	4	2	no	—
123	Q₈	—	4	2	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	C₂×C₁₉⋊C₁₆	1	1	yes	characteristic
136	C₂×C₄	#10597269	1	1	yes	characteristic
137	C₂×C₄	#10597290	1	1	yes	characteristic
138	C₂×C₄	#10597291	1	1	yes	characteristic
139	C₈	—	4	2	no	—
140	C₈	—	4	2	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	#11926268	1	1	yes	characteristic
144	C₂²	C₂²⋊(C₁₉⋊C₁₆)	1	1	yes	characteristic
145	C₂²	#11926326	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	4	2	no	—
150	C₄	—	4	1	no	—
151	C₄	—	8	1	no	—
152	C₄	#11926339	1	1	yes	characteristic
153	C₄	#11926343	1	1	yes	characteristic
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	#14561559	1	1	yes	characteristic
157	C₂	#14561569	1	1	yes	characteristic
158	C₂	#14561573	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14561575

C₂

yes

maximal

C₁₉×#1984

C₂

yes

maximal, characteristic

C₂²⋊(C₁₉⋊C₁₆)

—

C₂×C₄×Q₈×C₁₉

C₄

yes

characteristic

C₂×C₁₉×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₈×C₁₉

C₄

yes

characteristic

C₁₉×#336

—

C₂²×Q₈×C₁₉

C₈

yes

characteristic

C₂²×C₈×C₁₉

—

C₂²×C₈×C₁₉

D₄

yes

—

C₄×Q₈×C₁₉

—

C₄×Q₈×C₁₉

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

C₈

yes

characteristic

C₂×C₄²×C₁₉

—

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₁₉×C₄⋊C₈

—

C₄×C₈×C₁₉

—

C₂×C₁₉⋊C₁₆

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

C₁₆

yes

complemented

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊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

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₁₉⋊C₁₆

—

#7057

—

maximal, Sylow

C₂³×C₁₉

C₂²⋊C₈

yes

characteristic

Q₈×C₁₉

—

Q₈×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₁₉

M₃₂

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

—

#1984

D₁₉

yes

characteristic

#1322

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#323

yes

characteristic

C₂²×C₁₉

C₂²⋊C₁₆

yes

characteristic

C₂²×C₁₉

#349

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₂×C₄×Q₈

Dic₁₉

yes

characteristic

C₂×C₄⋊C₈

C₂×D₁₉

yes

characteristic

C₂×C₄×C₈

Dic₁₉

yes

characteristic

C₂²⋊C₁₆

—

#336

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂²×Q₈

C₁₉⋊C₈

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₁₉

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₁₉⋊C₈

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×Dic₁₉

yes

characteristic

C₂×C₁₆

—

C₄⋊C₈

—

C₄×C₈

—

C₁₉

#7057

yes

characteristic, complemented, Sylow

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

C₁₉⋊C₁₆

yes

complemented

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

C₂×C₁₉⋊C₈

yes

characteristic

107

C₂²×C₄

#64520

yes

characteristic

108

C₂²×C₄

C₂²⋊Dic₁₉

yes

characteristic

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

120

C₁₆

—

152

—

121

C₂³

C₂²⋊(C₁₉⋊C₈)

yes

characteristic

122

Q₈

—

123

Q₈

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₂×C₁₉⋊C₁₆

yes

characteristic

136

C₂×C₄

#10597269

yes

characteristic

137

C₂×C₄

#10597290

yes

characteristic

138

C₂×C₄

#10597291

yes

characteristic

139

C₈

—

140

C₈

—

141

C₂²

—

142

C₂²

—

143

C₂²

#11926268

yes

characteristic

144

C₂²

C₂²⋊(C₁₉⋊C₁₆)

yes

characteristic

145

C₂²

#11926326

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

#11926339

yes

characteristic

153

C₄

#11926343

yes

characteristic

154

C₂

—

155

C₂

—

156

C₂

#14561559

yes

characteristic

157

C₂

#14561569

yes

characteristic

158

C₂

#14561573

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10597291
commutator	C₂×C₄×C₁₉	C₂×C₁₆
Frattini	C₂×C₄⋊C₈	C₂×D₁₉
Fitting	C₁₉×#1984	C₂
socle	C₂²×C₁₉	#323
prime radical	C₂³×C₁₉	C₂²⋊C₈
cosocle	C₂×C₁₉×C₄⋊C₈	C₂²
hypercenter	#1984	D₁₉
hypocenter	C₁₉	#7057
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ #1984

subgroup

G/H

center

C₂×C₄

#10597291

commutator

C₂×C₄×C₁₉

C₂×C₁₆

Frattini

C₂×C₄⋊C₈

C₂×D₁₉

Fitting

C₁₉×#1984

C₂

socle

C₂²×C₁₉

#323

prime radical

C₂³×C₁₉

C₂²⋊C₈

cosocle

C₂×C₁₉×C₄⋊C₈

C₂²

D₁₉

C₁₉

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

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ #1984

N	H ≅ G/N	type
23. C₂×Q₈×C₁₉	C₁₆	semidirect
99. C₁₉	#7057	semidirect
103. C₂×Q₈	C₁₉⋊C₁₆	semidirect

H ≅ G/N

type

23. C₂×Q₈×C₁₉

C₁₆

semidirect

99. C₁₉

#7057

semidirect

103. C₂×Q₈

C₁₉⋊C₁₆

semidirect

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

subgroup	index
#14561575	2
C₁₉×#1984	2
#7057	19

subgroup

index

#14561575

C₁₉×#1984

#7057

Subgroups — #18069164

Conjugacy classes of subgroups

Complements

Maximal subgroups